diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzppwr" "b/data_all_eng_slimpj/shuffled/split2/finalzzppwr" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzppwr" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\\label{sec:Introduction}\nSleepiness is a state identified by reduced alertness that varies according to a circadian rhythm, i.\\,e., with the time of the day~\\cite{Dement.1982,Killgore.2010}. Its detection is important for safety applications, as it has been shown, for example, that sleepiness impacts driving performance, even more so than fatigue~\\cite{Philip.2005}. \n\\cite{lyznicki1998sleepiness}\nMost systems that aim to detect a sleepy driver rely on signals derived from interaction with the vehicle, such as abnormal steering behaviour, failures in lane keeping or irregular use of the pedals~\\cite{Mashko.2015}. Research dealing with automatic sleepiness recognition has investigated methods to derive the state based on different bio-signals. Performing visual analysis of a subject's face, e.\\,g., measuring blinking, can serve in assessing sleepiness but may be negatively affected by changing environmental parameters, such as illumination~\\cite{F.Friedrichs.2010}. While electroencephalography (EEG) has also been shown to be a robust approach to the problem~\\cite{Balandong.2018}, it is far more intrusive and can only be achieved with special equipment and professional setup. In contrast to this, performing acoustic analysis of speech is non-obtrusive and does not need as much sensor application and calibration effort~\\cite{Krajewski09-ASD}. \n\nIn order to define a suitable target for the automatic analysis of sleepiness, the state can be described by the Karolinska Sleepiness Scale (KSS) in terms of ratings ranging from 1 to 9, sometimes additionally extended by an extra category 10 for extreme sleepiness. A binary classification of sleepy speech has been previously performed as part of the INTERSPEECH 2011 Speaker State Challenge~\\cite{Schuller11-TI2}. The subchallenge of INTERSPEECH 2019's \\mbox{\\textsc{ComParE }} challenge deals with the detection of continuous sleepiness, and the sleepiness of a speaker is assessed as a regression problem~\\cite{Schuller19-TI2}. In the challenge baseline, the problem was approached in a number of ways, including a traditional acoustic feature extraction pipeline, Bag-of-Audio-Words, and a deep recurrent autoencoder framework. Here, the unsupervised sequence to sequence model \\mbox{\\textsc{auDeep }} achieved the strongest results. \n\nDeep learning models, such as \\mbox{\\textsc{auDeep }}, that process raw or low level inputs yield state-of-the-art results for a wide range of machine learning problems~\\cite{young2018recent, amodei2016deep, he2016deep}. Many of these approaches consider inputs as a whole, treating every part with the same importance. Often, however, some pieces of the input contain more information pertinent to solving the task at hand than others. A popular approach that takes this notion into consideration can be found with attention mechanisms, such as the one introduced by Bahdanau~et\\,al.~\\cite{bahdanau} for machine translation. Compared to regular sequence to sequence autoencoders where all information is compressed into the last hidden state of the encoder, the dynamic context vector in the attention model retains information about all hidden states of the encoder and their alignment to the current decoding step. Since their introduction, attention mechanisms have also been adapted to speech recognition~\\cite{chorowski2015attention, bahdanau2016end}, visual image captioning~\\cite{xu2015show} or question answering~\\cite{anderson2018bottom}, and speech emotion classification~\\cite{huang2017deep}. \n\nMotivated both by the effectiveness of recurrent autoencoders for acoustic analysis of sleepiness from speech as well as by the improvements to sequence to sequence models achieved with attention mechanisms, we evaluate the impacts of combining the two approaches for the detection of continuous sleepiness on the respective 2019 INTERSPEECH \\mbox{\\textsc{ComParE }} sub-challenge.\n\n\n\\begin{figure*}[ht!]\n \\centering\n \\includegraphics[width=.85\\linewidth]{figs\/sys-overview-s2s.pdf}\n \\caption{An overview of our framework for unsupervised representation learning with sequence to sequence recurrent autoencoders. Except for the SVR training, the approach is fully unsupervised. A detailed description of the procedure is given in~\\Cref{sec:approach}.}\n \\label{fig:sys-overview-s2s}\n\\end{figure*}\n\\section{Dataset}\\label{sec:dataset}\nWe use a subset of the SLEEP Corpus that was employed in the 2019 edition of the INTERSPEECH Computational Paralinguistics Challenge (\\mbox{\\textsc{ComParE }})~\\cite{Schuller19-TI2}. The corpus contains speech recordings of 915 individuals (364 females, 551 males) at varying levels of sleepiness. The audio files have a sample rate of 16\\,kHz with a 16\\,bit quantisation. Each file is annotated with a KSS-score, ranging from 1 to 9, with 9 denoting extreme sleepiness. The labels were derived by averaging self-report with the scores of two external observers. The dataset is split into three partitions with 5\\,564, 5\\,328, and 5\\,570 samples. \n\n\n\\section{Approach}\\label{sec:approach}\nA high-level overview of our proposed approach is depicted in~\\Cref{fig:sys-overview-s2s}. First, Mel-spectrograms are extracted from audio signals. Recurrent autoencoders (AEs), both with and without attention mechanism, are then trained on the Mel-spectrograms. Afterwards, the learnt representations are obtained from the AEs. We then fuse the representations and classify them.\n\n\n\n\n\\subsection{Spectrogram Extraction}\\label{ssec:spectrogram-extraction}\nFirst, the Mel-spectrograms of audio recordings are extracted using periodic Hamming windows with width $w$ and overlap $0.5w$. From these, a given number of log-scaled Mel-frequency\nbands are then computed. Finally, we normalise the Mel-spectra to have values in [-1; 1], since the outputs of the recurrent sequence to sequence autoencoders are constrained to this interval.\n\n\\subsection{Autoencoder Architecture without Attention}\\label{ssec:ae-wo-att}\nFor this architecture, we utilise \\textsc{auDeep}\\footnote{\\url{https:\/\/github.com\/auDeep\/auDeep}}~\\cite{Amiriparian17-STS,Freitag18-AUL}, our recurrent sequence to sequence autoencoder. For the representation learning with our framework, we can adjust a range of autoencoder parameters, including the direction (e.\\,g., uni- or bidirectional) of the encoder and decoder RNNs, types of RNN cells, e.\\,g., gated recurrent units (GRUs), or long short-term memory (LSTM) cells, and the number of hidden layers and units. To use LSTM-RNNs in the decoder, the LSTM cell is modified to work with a context vector similar to the GRUs in the encoder-decoder model proposed by Cho~et\\,al.~\\cite{seq2seq_cho}. The weight matrices $C_i, C_f, C_o$, and $C_z$ are added to the input $z_t$, input gate $i_t$, forget gate $f_t$, and output gate $o_t$ to enable an LSTM \ncell \nto work with the context vector:\n\\begin{equation}\n\\begin{aligned}\n z_t &= \\tanh \\left(W_zx_t+R_z y_{t-1} + C_z c + b_z \\right) \\\\\ni_t &= \\sigma \\left(W_ix_t + R_iy_{t-1} + C_i c + p_i \\odot c_{t-1} + b_i\\right)\\\\\nf_t &= \\sigma \\left(W_fx_t + R_fy_{t-1} + C_f c + p_f \\odot c_{t-1} + b_f\\right)\\\\\no_t &= \\sigma \\left(W_ox_t + R_oy_{t-1} + C_o c + p_o \\odot c_t + b_o\\right).\n\\end{aligned}\n\\end{equation}\n\nFor each input sequence, the initial hidden state vector of the encoder is zero-padded. The last concatenated hidden state vector of the encoder $h_T^{e} = \\left[\\overrightarrow{h}_{T} \\overleftarrow{h}_{T}\\right]^T$ is then passed through a fully connected layer with $\\tanh$ activation which has the same number of units as the decoder RNN. The output of this layer represents the context vector and is used as the first hidden state vector of the decoder $h^{d}_0$. During the feature extraction, the context vector also represents the feature vector.\nThe outputs of the decoder are passed through a fully connected projection layer with $\\tanh$ activation at each time step in order to map the decoder output dimensionality to the target dimensionality. The weights of this output projection are shared across time steps.\nFor the network training the teacher forcing algorithm~\\cite{lamb2016professor} is applied. Following this method, instead of feeding the decoder with the predicted output at time step $t-1$ ($\\hat{y}_{t-1}$), the expected decoder output at time step $t-1$ ($y_{t-1}$) is fed as an input to the decoder.\nThis means that the decoder input is the same original spectrogram, only shifted by one step in time. Instead of the first step, the zero vector is inserted and the frequency vector at the last step is removed. \nDuring the training, the autoencoder learns to reconstruct the reversed input spectrogram~\\cite{Amiriparian17-STS, seq2seq_sutskever}. Mean squared error (MSE) is used as the loss function to compare the reversed source spectrogram with the concatenated spectrogram obtained from the projection layer. \n\n\n\\begin{figure*}[tph!]\n\\centering\n\\includegraphics[width=.90\\linewidth]{figs\/Figure2_cropped.pdf}\n\\caption{Schematic structure of our attention-based autoencoder with stacked encoder and decoder RNNs. Feature vectors 1 and 2 are extracted from the activations of the fully connected layer of the encoder RNN and the last hidden state of the decoder RNN, respectively. A detailed account of the proposed architecture is provided in~\\Cref{ssec:ae-w-att}.} \\label{fig:att-model}\n\\end{figure*}\n\n\\subsection{Autoencoder Architecture With Attention}\\label{ssec:ae-w-att}\n\nIn the second model, we add an attention mechanism to the autoencoder architecture. Here, encoder and decoder have almost the same structure as the baseline autoencoder. The problem of the sequence to sequence model is that the encoder must map all essential information of the input sequence to a fixed-length vector. This may not be enough to represent a long input sequence. To circumvent this, Bahdanau et\\,al.~\\cite{bahdanau} have introduced an attention mechanism that includes dynamic computation of the context vector. At each time step of the decoder, the attention mechanism enables to choose the hidden state vectors of the encoder that contain the most significant information to generate the context vector $c_t$ which is used to generate the output $y_t$. This computation is based on all hidden state vectors of the encoder and the last hidden state vectors of the decoder. The context vector $c_i$ is the linear combination of the hidden state vectors of the encoder $h^{e}_j$:\n\\begin{gather}\n\tc_i = \\sum_{j=1}^{T_x} \\alpha_{ij} h_j^{e}. \n\\end{gather}\nHere, the weights $\\alpha_{ij}$ are numbers between 0 and 1 and define which hidden states $h^{e}_j$ have the biggest influence on $y_i$. $\\alpha_{ij}$ is calculated with the softmax-normalised inner activation of the alignment model $e_{ij}$:\n\\begin{gather}\n\t\\alpha_{ij}=\\frac{\\exp({e_{ij}})}{\\sum_{k=1}^{T_x} \\exp({e_{ik}})}. \n\\end{gather}\nThe alignment model $e_{ij} = a(h^e_{i-1}, h^d_j)$ is a small feedforward neural network trained together with the sequence to sequence model using backpropagation and is defined as\n\\begin{gather}\n v^\\top_{a_i}\\tanh \\left( W_a h^{d}_{i-1} + U_a h^e_j \\right),\n\\end{gather}\nwhere $v_a\\in\\mathbb{R}^l$ is a weight vector, and $W_a\\in\\mathbb{R}^{l\\times n}$ and $U_a\\in\\mathbb{R}^{l\\times 2n}$ are weight matrices, and $l$ is the number of units in the alignment model.\n\n\n\nThe difference to the autoencoder described in \\Cref{ssec:ae-wo-att} is that the context vector is calculated dynamically using the alignment model and is not solely used as a feature vector. If we compute the feature vector from the last hidden state vector of the encoder, the attention mechanism will have less influence on the feature extraction. The feature vector, however, should contain information of all context vectors and, therefore, is extracted after their creation. For this reason, an additional set of feature vectors is extracted from the hidden state vector of the last decoder layer at the last time-step. The attention mechanism in the sequence to sequence model is visualised in~\\Cref{fig:att-model}.\n\n\\section{Experimental Settings}\\label{sec:experimental-settings}\nFor all experiments, we train autoencoder models on the sleepiness spectrograms and then extract features for the three partitions. These feature vectors are used to train a linear support vector regressor (SVR) for which we optimise the complexity parameter on a logarithmic scale of $10^{-5}$ to 1. The complexity parameter is chosen based on the Spearman's correlation coefficient ($\\rho$) achieved on the development partition.\n\nOur proposed autoencoder approaches contain a large amount of adjustable hyperparameters (cf.~\\Cref{ssec:ae-w-att,ssec:ae-wo-att}), which prohibits an exhaustive exploration of the parameter space. For this reason, we choose suitable values for the hyperparameters in multiple stages, using the results of our initial experiments to bootstrap the process. In preliminary experiments, we test different configurations for the spectrograms that are used as input for the autoencoders. We arrived at using Mel-spectrograms with 160 and \\{128, 256\\} Mel-bands extracted from the audio samples for our autoencoders with and without attention, respectively. \nFor the fast Fourier transform (FFT), we apply Hamming windows of 40\\,ms width and 20\\,ms overlap. We further experimented with the architecture of our recurrent autoencoder models. Here, we tested one and two layer variants for both encoder and decoder using either GRU or LSTM cells with 128, 256, or 512 hidden units. Moreover, bidirectional and unidirectional encoders were compared. For the attention autoencoder, models with two-layer bidirectional encoders and two-layer unidirectional decoders worked best with 512 hidden units. For the results presented herein, we therefore settled on this architecture with the choice of either GRU or LSTM layers (cf.~\\Cref{tab:attention-results}). Furthermore, we introduce an additional RNN layer in the decoder that serves to produce hidden states for the attention mechanism. The architecture of those models is visualised in~\\Cref{fig:att-model}. We evaluate features extracted from both the last hidden state of the encoder and decoder. All attention models are trained using a batch size of 256 with the Adam optimiser~\\cite{adam} and the learning rate set to $10^{-4}$ for a maximum of 40 epochs. The model checkpoints at 20, 25, 30, 35, and 40 epochs further serve as feature extractors.\nFor the \\mbox{\\textsc{auDeep }} experiments (cf.~\\Cref{ssec:ae-wo-att}), we found the best configuration with 2 hidden layers each with 256 hidden units. We then optimise the direction of the encoder and decoder and adjust the RNN cell type. Additionally, we filter some of the background noise in the recordings by clipping amplitudes below \\{-40, -50, -60, -70\\}\\,dB thresholds, and fuse them together resulting in five different feature vectors for each data partition. In~\\cite{Amiriparian18-DUR,Amiriparian18-AFO}, we have shown the effectiveness of our amplitude clipping approach for various audio processing tasks.\n\n\\section{Results and Discussion}\n\\label{sec:results}\nAll results obtained with our autoencoders and their early fusion are shown in~\\Cref{tab:attention-results,tab:audeep-results,tab:fusion-results}. In the attention model (cf.~\\Cref{tab:attention-results}), features from the fully connected layer of the encoder RNN (fc$_{enc}$) generalise better when GRUs are applied ($\\rho_{devel} = .250$, $\\rho_{test} = .314$), and the features from the last hidden state of the decoder RNN (state$_{dec}$) perform better with LSTM cells ($\\rho_{devel} = .324$, $\\rho_{test} = .311$). \nFrom both autoencoder approaches, the recurrent model without attention shows the best performance on the test partition ($\\rho_{test} = .359$), whilst the attention model achieves the highest results on the development partition ($\\rho_{devel} = .324$). The result \nimplies \npossible overfitting of the attention model on the development data. This issue is not strongly present in our model without attention, and we hypothesise that this is mainly because of the filtering of some of the background noise found in the audio data by clipping amplitudes below a certain threshold. In~\\Cref{tab:audeep-results}, we provide the highest achieved results with various thresholds and hyperparameter combinations. Furthermore, we fuse the best performing attention feature set on the development set ($\\rho_{devel} = .324$, $\\rho_{test} = .311$) with all non-attention (\\textsc{auDeep}) features to analyse the complementarity of the learnt representations. The results in~\\Cref{tab:fusion-results} demonstrate an improvement of all results after early fusion. The highest improvement on the test partition after fusion is achieved when the best attention feature set is combined with the fourth \\mbox{\\textsc{auDeep }} feature with GRUs, \nand the \nunidirectional encoder and bidirectional encoder trained on Mel-spectrograms with 256 Mel-band (FFT window width of 80\\,ms and overlap of 40\\,ms) and -60\\,dB amplitude clipping (cf.~\\Cref{tab:fusion-results}). It is worth mentioning that the dimensionality of the attention features are either $1\/2$ or $1\/8$ of the \\mbox{\\textsc{auDeep }} features, leading to a faster classifier training. Moreover, the training process with attention autoencoders can be performed faster, as the encoder RNN is relieved from encoding all information in the whole input sequence of the Mel-spectrograms into a fixed-length vector~\\cite{Amiriparian19-DRL}.\nWe further compare our best performing approaches with \nthe \nbest challenge baselines~\\cite{Schuller19-TI2}, the winner of the challenge who combined Fisher vectors with baseline features~\\cite{Gosztolya.2019}, and the runner-up who utilised a fusion of convolutional neural networks (CNNs) and RNNs~\\cite{yeh2019using} (cf.~\\Cref{tab:compare_results}). \n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Performance comparison of the features obtained from the fully connected layer of the encoder (fc$_{enc}$) and the last hidden state of the decoder (state$_{dec}$) in our attention autoencoder. id: feature identifier, Dim.: feature dimensionality.}\n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{rrlrrrrr}\n \\toprule\n \\multicolumn{4}{c}{Parameters} & \\multicolumn{2}{c}{fc$_{enc}$} & \\multicolumn{2}{c}{state$_{dec}$} \\\\\n \\midrule\n id & Epoch & Cell & Dim. & $\\rho_{devel} $ & $\\rho_{test}$ & $\\rho_{devel} $ & $\\rho_{test}$ \\\\\n \\midrule\n 1 & 20 & GRU & 512 & .262 & .308 & .276 & .294 \\\\\n 2 & 25 & GRU & 512 & .258 & .308 & .278 & .298 \\\\\n 3 & 30 & GRU & 512 & .250 & .314 & .267 & .292 \\\\\n 4 & 35 & GRU & 512 & .260 & .312 & .266 & .298 \\\\\n 5 & 40 & GRU & 512 & .253 & .307 & .265 & .288 \\\\\n \\midrule\n 6 & 20 & LSTM & 512 & .293 & .294 & .318 & .303 \\\\\n 7 & 25 & LSTM & 512 & .303 & .276 & \\textbf{.324} & \\textbf{.311} \\\\\n 8 & 30 & LSTM & 512 & .298 & .263 & .322 & .305 \\\\\n 9 & 35 & LSTM & 512 & .303 & .285 & .289 & .304 \\\\\n 10 & 40 & LSTM & 512 & .307 & .302 & .288 & .299 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:attention-results}\n\\end{table}\n\n\\begin{table}[tp]\n \\centering\n \\caption{Results obtained from our autoencoder without attention. id: feature identifier, w: width of the Hamming window, Mel: number of Mel-bands, Clip: clipped amplitudes below a certain threshold to filter some noise from audio, Dim.: feature dimensionality, Dir.: direction of the encoder-decoder.}\n \n \n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{rrrrrrrrr}\n \\toprule\n \\multicolumn{8}{c}{Autoencoders without attention} \\\\\n \\midrule\n id & w & Mel & Clip & Cell & Dim. & Dir. & {$\\rho_{devel}$} & {$\\rho_{test}$} \\\\\n \\midrule\n 1 & 0.08 & 256 & fused & GRU & 4\\,096 & uni-bi & .286 & .338 \\\\\n 2 & 0.08 & 256 & -70\\,dB & GRU & 1\\,024 & bi-uni & \\textbf{.283} & \\textbf{.359}\\\\\n 3 & 0.06 & 256 & -70\\,dB & GRU & 1\\,024 & bi-uni & .281 & .357 \\\\\n 4 & 0.08 & 256 & -60\\,dB & GRU & 1\\,024 & uni-bi & .278 & .331 \\\\\n 5 & 0.04 & 256 & -70\\,dB & GRU & 1\\,024 & uni-bi & .278 & .340 \\\\\n 6 & 0.06 & 256 & fused & GRU & 4\\,096 & bi-uni & .277 & .346 \\\\\n 7 & 0.06 & 256 & -70\\,dB & GRU & 1\\,024 & uni-bi & .277 & .348 \\\\\n 8 & 0.06 & 128 & -70\\,dB & LSTM & 1\\,024 & uni-bi & .277 & .317 \\\\\n 9 & 0.08 & 128 & -60\\,dB & GRU & 1\\,024 & bi-uni & .275 & .324 \\\\\n 10 & 0.04 & 256 & -60\\,dB & GRU & 1\\,024 & uni-bi & .275 & .336 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:audeep-results}\n\\end{table}{}\n\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:conclusions}\nIn~\\Cref{ssec:ae-wo-att}, we introduced a novel attention mechanism for recurrent sequence to sequence autoencoders to process audio signals in an unsupervised manner\\footnote{Our audio-based attention framework (\\textsc{auttention}), and all codes to reproduce the attention results are provided here: \\mbox{\\url{https:\/\/github.com\/auttention\/SleepyAttention}}}. We have demonstrated the suitability of two fully unsupervised representation learning techniques for continuous sleepiness recognition, and their superior performance to expert-designed, hand-crafted features (cf.~\\Cref{tab:compare_results}). Furthermore, we conducted a feature fusion strategy and demonstrated the complementarity of the learnt representations from both autoencoder architectures (cf.~\\Cref{sec:results}). In future work, we will be evaluating our systems over a wide range of audio recognition tasks. We also want to utilise dimensionality reduction techniques~\\cite{Amiriparian17-FSI} to cope with the high-dimensionality of our autoencoder features. Finally, we want to combine our methods with CNN-based representation learning systems, such as deep convolutional generative adversarial networks~\\cite{radford2015unsupervised}.\n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Results of our early fusion experiments with the best attention result (id$_{att}$ = 7) and all results provided in~\\Cref{tab:audeep-results}. id$_{att}$ and id$_{audeep}$: identifiers for the attention feature and \\mbox{\\textsc{auDeep }} features which are fused. $C_{SVR}$: Complexity of the SVR which is optimised on the development partition after fusion.}\n \\begin{tabular}{rr|rrr}\n \\toprule\n \\multicolumn{5}{c}{Early fusion} \\\\\n \\midrule\n id$_{att}$ &\n id$_{audeep}$ &\n $C_{SVR}$ &\n {$\\rho_{devel}$} &\n {$\\rho_{test}$} \\\\\n \\midrule\n 7 & 1 & $10^{-3}$ & .315 & .359 \\\\\n 7 & 2 & $10^{-2}$ & .336 & .360 \\\\\n 7 & 3 & $10^{-2}$ & .334 & .365 \\\\\n 7 & 4 & $10^{-1}$ & \\textbf{.320} & \\textbf{.367} \\\\\n 7 & 5 & $10^{-2}$ & .333 & .349 \\\\\n 7 & 6 & $10^{-3}$ & .319 & .363 \\\\\n 7 & 7 & $10^{-2}$ & .326 & .361 \\\\\n 7 & 8 & $10^{-2}$ & .333 & .341 \\\\\n 7 & 9 & $10^{-2}$ & .340 & .351 \\\\\n 7 & 10 & $10^{-2}$ & .339 & .357 \\\\\n \\bottomrule\n \\end{tabular}\n \\label{tab:fusion-results}\n\\end{table}{}\n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Comparison of our best performing models with best performing challenge baselines and the challenge winner. $Dim.$: feature dimensionality of each system.}\n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{lrrr}\n \\toprule\n System & Dim. & $\\rho_{dev}$ & $\\rho_{test}$ \\\\\n \\midrule\n \\multicolumn{4}{l}{\\textbf{Challenge Winner}~\\cite{Gosztolya.2019}} \\\\\n \\mbox{\\textsc{ComParE }} + BoAW + Fisher vectors & -- & -- & \\textbf{.383} \\\\\n \\multicolumn{4}{l}{\\textbf{Runner-up}~\\cite{yeh2019using}} \\\\\n CNNs and BLSTMs with attention & -- & .373 & .369 \\\\ \n \n \\midrule\n \\multicolumn{4}{l}{\\textbf{Best Challenge Baselines}~\\cite{Schuller19-TI2}} \\\\\n \\mbox{\\textsc{ComParE }} & 6\\,373 & .251 & .314 \\\\\n Bag-of-Audio-Words & 500 & .250 & .304 \\\\\n Autoencoders & 1\\,024 & .243 & .325 \\\\\n Late fusion of best & -- & -- & .343 \\\\\n \\midrule\n \\multicolumn{4}{l}{\\textbf{Best of Our Proposed Approaches}} \\\\\n With attention & 512 & .324 & .311 \\\\\n Without attention & 4\\,096 & .286 & .338 \\\\\n Early fusion & 4\\,608 & \\textbf{.320} & \\textbf{.367} \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:compare_results}\n\\end{table}{}\n\\section{Acknowledgements}\nThis research was partially supported by BMW AG and Deutsche Forschungsgemeinschaft (DFG) under grant agreement No. 421613952 (ParaStiChaD).\n\\clearpage\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{CONCLUSIONS}\n\nFirst results from simulations using the non-perturbatively\\ improved Wilson action\nand currents in the quenched approximation show that lattice artefacts\nare drastically reduced. Unlike the case of unimproved actions, the\nresults for the vector meson mass are practically independent of~$a$\nat $\\beta=6.0$ and~6.2, so that residual~$a^2$ effects are around\n$2\\%$ at $a\\simeq0.1\\,{\\rm fm}$. \nA real test of the scaling behaviour is still lacking and will only\nbecome possible when more and different values of the lattice spacing\nare studied.\n\nThe analysis of data for the pseudoscalar decay constant has shown\nthat a non-perturbative determination of $b_{\\rm A}$ is required for quark\nmasses above $m_{\\rm strange}$. The improvement coefficient $c_{\\rm A}$ has\na large influence on the scaling behaviour, which motivates further\ninvestigation. On the whole, it appears that non-perturbative\\ improvement leads to\nbetter agreement between the continuum result for $f_K$ and the\nexperimental value.\n\nThe systematic nature of non-perturbative\\ improvement makes it easily applicable\nto other situations: results for $c_{\\rm sw}$ computed for two flavours of\ndynamical quarks have been reported\\,\\cite{Karl_Rainer_lat97}.\nFurthermore, the formalism has been applied to quenched QCD with an\nimproved gauge action\\,\\cite{KlasEd_lat97}.\n\nI thank P. Rowland for help with the UKQCD data and also M. Guagnelli,\nT. Mendes and G. Schierholz for communicating their data prior to\npublication. I wish to thank M. L\\\"uscher and R. Sommer for many\nhelpful discussions.\n\n\\section{DECAY CONSTANTS}\n\nFrom the definitions of the renormalised and improved currents,\neq.\\,(\\ref{EQ:currents}), one sees that in order to obtain O($a$)\nimproved matrix elements of axial and vector currents one requires\nknowledge of the improvement coefficients $c_{\\rm A},b_{\\rm A},c_{\\rm V},b_{\\rm V}$ and the\nrenormalisation factors $Z_{\\rm A}$ and $Z_{\\rm V}$. In this section we will focus\non the pseudoscalar decay constant $(f_{\\rm PS})_{\\rm R}$, which we write as\n\\begin{equation}\n (f_{\\rm PS})_{\\rm R} = Z_{\\rm A}(1+b_{\\rm A} am_{\\rm q})\\big\\{f_{\\rm PS}^{(0)}+c_{\\rm A} g_{\\rm PS}^{(0)}\\big\\}, \n\\end{equation}\nwhere the unrenormalised matrix elements are parametrised as\n\\begin{eqnarray}\n \\mpsf_{\\rm PS}^{(0)} & = & \\langle0|A_4(0)|{\\rm PS}\\rangle \\\\\n g_{\\rm PS}^{(0)} & = & \\langle0|P(0)|{\\rm PS}\\rangle.\n\\end{eqnarray}\nA major advantage of non-perturbative\\ improvement is that the renormalisation\nfactor $Z_{\\rm A}$ has been determined with an accuracy of around\n1\\%. At $\\beta=6.0$ and 6.2 one finds\\,\\cite{paperIV}\n\\begin{equation}\n Z_{\\rm A}=\\left\\{\\begin{array}{lr}\n\t0.791(9), & \\quad\\beta=6.0 \\\\\n\t0.807(8), & \\quad\\beta=6.2 \n\t \\end{array}\n \\right.\n\\end{equation}\nNote that the error on $Z_{\\rm A}$ must be combined with the statistical\nerror on $f_{\\rm PS}^{(0)}+c_{\\rm A} g_{\\rm PS}^{(0)}$.\n\nThe improvement coefficient $c_{\\rm A}$ has also been determined\nnon-perturbatively\\,\\cite{paperIII}:\n\\begin{equation}\n c_{\\rm A}=\\left\\{\\begin{array}{lr}\n -0.083, & \\quad\\beta=6.0 \\\\\n -0.037, & \\quad\\beta=6.2\n \\end{array}\n \\right.\n\\end{equation}\nThe effect of $c_{\\rm A}$ on the decay constant can be studied by comparing\n$f_{\\rm PS}^{(0)}$ with $f_{\\rm PS}^{(0)}+c_{\\rm A}g_{\\rm PS}^{(0)}$. Here the contribution from $c_{\\rm A}g_{\\rm PS}^{(0)}$\nleads to a decrease in the pseudoscalar decay constant of $\\sim4\\%$ at\n$\\beta=6.2$ and even $\\sim15\\%$ at $\\beta=6.0$. This effect is\nparticularly pronounced for $\\beta<6.2$.\n\nThe improvement coefficient $b_{\\rm A}$ is relevant for the determination of\nthe kaon decay constant $f_K$. However, unlike $b_{\\rm V}$, the coefficient\n$b_{\\rm A}$ has so far not been determined non-perturbatively. In one-loop perturbation\ntheory one finds\\,\\cite{paperV}\n\\begin{equation}\n b_{\\rm A}=1+0.1522g_0^2+O(g_0^4).\n\\end{equation}\nIn order to study the influence of $b_{\\rm A}$, one can evaluate $(f_{\\rm PS})_{\\rm R}$\naround the strange quark mass for different choices of $b_{\\rm A}$. Here we\ncompare\n\\begin{itemize}\n\\item $b_{\\rm A}=0$\n\\item $b_{\\rm A} = 1 + 0.1522g_0^2$\n\\item $b_{\\rm A}=b_{\\rm V}$\n\\end{itemize}\nThe seemingly {\\it ad hoc\\\/} choice of $b_{\\rm A}=b_{\\rm V}$ (here we use the non-perturbative\\\ndetermination of $b_{\\rm V}$) is motivated by the observation that the\none-loop coefficients for $b_{\\rm A}$ and $b_{\\rm V}$ are approximately\nequal\\,\\cite{paperV}. By applying different choices of $b_{\\rm A}$ to the\nanalysis of the UKQCD data around $m_K$, one finds that $b_{\\rm A}$ leads to\nan increase in $(f_{\\rm PS})_{\\rm R}$ of at most $\\sim2\\%$ at $\\beta=6.2$ and\n$\\sim3\\%$ at $\\beta=6.0$, which is fairly small. Since the choice\n$b_{\\rm A}=b_{\\rm V}$ gives essentially the same mass behaviour compared\nto choosing $b_{\\rm A} = 1 + 0.1522g_0^2$, one concludes that perturbative\nestimates of $b_{\\rm A}$ are quite acceptable for quark masses up to and\naround $m_{\\rm strange}$.\n\nWe now analyse the scaling behaviour of $(f_{\\rm PS})_{\\rm R}$. In\nFig.\\,\\ref{Fig:fpsvsmps2} we plot $(f_{\\rm PS})_{\\rm R} r_0$ versus $(m_{\\rm PS} r_0)^2$,\nwhere $r_0$ is the hadronic radius defined in\\,\\cite{rainer_r0}. Data\nfor $r_0\/a$ were taken from\\,\\cite{how_r0}. If lattice effects are\nsmall the data in Fig.\\,\\ref{Fig:fpsvsmps2} should lie on a universal\ncurve. The results for $f_Kr_0$ computed at $\\beta=6.0$ and~6.2 show a\nslight dependence on the lattice spacing. In order to study residual\nlattice artefacts, we employ a similar procedure as in the case of the\nvector mass and extrapolate $(f_{\\rm PS})_{\\rm R} r_0$ to $(m_K\nr_0)^2$. Fig.\\,\\ref{Fig:fK} shows the resulting values of $f_Kr_0$ as\na function of $(a\/r_0)^2$. There is good agreement between the data\nfrom all three collaborations, and in principle their results could be\ncombined. Furthermore, it appears that a linear extrapolation in $a^2$\nyields a continuum result which is compatible with the experimentally\nobserved value (although there is {\\it a priori\\\/} no reason why the\nquenched approximation should reproduce the measured result). In\ncontrast, the authors of\\,\\cite{GF11_decay} have found that, for the\nunimproved action, the continuum value of $f_K$ is significantly lower\nthan the experimental result.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{fpi_r0.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{The pseudoscalar decay constant as a function of $m_{\\rm PS}^2$ in\nunits of $r_0$ for $\\beta=6.0$ (open symbols) and $\\beta=6.2$ (full\nsymbols). Asterisks denote the experimental values for $f_\\pi$ and\n$f_K$ using $r_0=0.5\\,{\\rm fm}$.} \n\\label{Fig:fpsvsmps2}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{fK_r0_vs_r0_notad.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{$f_Kr_0$ plotted versus $(a\/r_0)^2$. The asterisk denotes the\nexperimental result.}\n\\label{Fig:fK}\n\\end{figure}\n\nFrom the slope in $(a\/r_0)^2$ one estimates that residual lattice\nartefacts in $f_K r_0$ amount to ca. 10\\% at\n$a\\simeq0.1\\,{\\rm fm}$. Compared to the previously discussed case of the\nvector meson, this is a fairly large correction. Given the substantial\ncontribution of $c_{\\rm A}g_{\\rm PS}^{(0)}$ to $(f_{\\rm PS})_{\\rm R}$ at $\\beta=6.0$, one can ask\nwhether $c_{\\rm A}$ has a large effect on the scaling behaviour. If one\nformulates a similar improvement condition for $c_{\\rm A}$ as in\neq.\\,(\\ref{EQ:phenimp}), for instance\n\\begin{equation}\n f_K r_0 = {\\rm const.}\n\\end{equation}\none can study how well it is satisfied for different choices of\n$c_{\\rm A}$. Using the UKQCD data it turns out that the above condition\n``favours'' smaller values of $c_{\\rm A}$ at the lower end of the\n$\\beta$-range. However, at this stage one should not jump to\nconclusions before a more thorough investigation of improvement\nconditions has been performed.\n\n\\section{INTRODUCTION}\n\nOne of the most important systematic effects in lattice simulations of\nQCD is the finiteness of the lattice spacing~$a$. It is well known\nthat physical observables computed using the Wilson action are subject\nto corrections of order~$a$, which can be rather large. In order to\nobtain reliable results in the continuum limit it is desirable to\nreduce lattice artefacts, either through the Symanzik improvement\nprogramme\\,\\cite{SymanzikI,SymanzikII}, or by employing a\nrenormalisation group approach\\,\\cite{HasNied94,Has_lat97}. Recently\nthe ALPHA Collaboration has carried out the Symanzik improvement\nprogramme to leading order through a non-perturbative determination of\nthe O($a$) improved fermion action and isospin\ncurrents\\,\\cite{Alpha_lat96,paperIII,paperIV,marco_lat97}. This\napproach should lead to the complete removal of lattice artefacts of\norder~$a$ in spectral quantities and matrix elements of local\ncurrents, so that the remaining cutoff effects are of order~$a^2$. The\nnon-perturbatively\\ improved action has already been employed in a number of\nsimulations in the quenched\napproximation\\,\\cite{QCDSF_97,par_lat97,mendes_lat97}. Here, we\nassess the impact of non-perturbative\\ improvement on the calculation of the mass\nspectrum and decay constants in the light hadron sector, analyse the\nscaling behaviour and estimate the size of residual lattice artefacts.\n\nThe general expression of the O($a$) improved fermion action\nreads\\,\\cite{SW85} \n\\begin{eqnarray}\nS_F^I[U,\\overline{\\psi},\\psi] & = & S_F^W[U,\\overline{\\psi},\\psi] \\nonumber\\\\\n & & \\hspace{-1.5cm}+\\,c_{\\rm sw}{\\textstyle\\frac{ia}{4}}\\sum_{x,\\mu,\\nu}\n \\overline{\\psi}(x)\\sigma_{\\mu\\nu}F_{\\mu\\nu}(x)\\psi(x),\n\\label{EQ:currents}\n\\end{eqnarray}\nwhere $S_F^W$ is the (unimproved) Wilson fermion action and $c_{\\rm sw}$ is\nan improvement coefficient. The improved and renormalised axial and\nvector currents are defined as\\,\\cite{paperI}\n\\begin{eqnarray}\n(A_{\\rm R})_\\mu^a\\hspace{-0.2cm} & = &\\hspace{-0.2cm}\n Z_{\\rm A}(1+b_{\\rm A} am_{\\rm q})\\big\\{A_\\mu^a+c_{\\rm A} a\\partial_\\mu\nP^a\\big\\}, \\\\\n(V_{\\rm R})_\\mu^a\\hspace{-0.2cm} & = &\\hspace{-0.2cm}\n Z_{\\rm V}(1+b_{\\rm V} am_{\\rm q})\\big\\{V_\\mu^a+c_{\\rm V} a\\partial_\\nu\nT_{\\mu\\nu}^a\\big\\},\n\\end{eqnarray}\nwhere $Z_{\\rm A}, Z_{\\rm V}$ are the renormalisation factors of the respective\ncurrents, and $b_{\\rm A},c_{\\rm A},b_{\\rm V}$ and $c_{\\rm V}$ are further improvement\ncoefficients. The unimproved currents and densities are defined as\n\\begin{eqnarray}\nA_\\mu^a & = & \\overline{\\psi}\\gamma_\\mu\\gamma_5\\frac{\\tau^a}{2}\\psi,\\quad\n~V_\\mu^a = \\overline{\\psi}\\gamma_\\mu\\frac{\\tau^a}{2}\\psi,\\nonumber\\\\\nP^a & = & \\overline{\\psi}\\gamma_5\\frac{\\tau^a}{2}\\psi,\\qquad\nT_{\\mu\\nu}^a = i\\overline{\\psi}\\sigma_{\\mu\\nu}\\frac{\\tau^a}{2}\\psi,\n\\end{eqnarray}\nwhere $\\tau^a$ are the Pauli matrices acting in flavour space.\nThe normalisations $Z_{\\rm A},Z_{\\rm V}$ and improvement coefficients $c_{\\rm sw}, c_{\\rm A},\nc_{\\rm V}, b_{\\rm V}$ have been determined non-perturbatively\\ for bare couplings $g_0$ in the\nrange $0\\leq g_0\\leq1$\\,\\cite{paperIII,paperIV,marco_lat97}. Results\nand proposals for non-perturbative determinations of $b_{\\rm A}$ at\n$g_0\\simeq1$ have been\nreported\\,\\cite{MRSSTT97,giulia_lat97,ssharpe_lat97}. Furthermore, all\nof the above improvement coefficients have been calculated in\nperturbation theory to one-loop order\\,\\cite{paperII,paperV}.\n\nThe parameters of the simulations discussed here are listed in\nTable\\,\\ref{Tab:simpar}. In addition to the non-perturbative\\ value of $c_{\\rm sw}$, the\nQCDSF collaboration have also used unimproved Wilson fermions for a\ndirect comparison\\,\\cite{QCDSF_97}. UKQCD have also used the tadpole\nimproved value of $c_{\\rm sw}$ on the same set of configurations and at\n$\\beta=5.7$\\,\\cite{UKQCD_lat96,tadpole_comp}, where a non-perturbatively\\\ndetermined value of $c_{\\rm sw}$ is not available. Table\\,\\ref{Tab:simpar}\nshows that all collaborations have used the same two values of\n$\\beta$. This implies that one cannot test as yet whether non-perturbative\\\nimprovement indeed leads to a scaling behaviour of physical\nobservables which is consistent with O($a^2$) corrections. However, if\none {\\it assumes\\\/} this to be the leading scaling behaviour after\nimprovement, one can still estimate the size of residual lattice\nartefacts at a given value of~$a$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\caption{Simulations using non-perturbative\\\timprovement. The number of exceptional\nconfigurations discarded from the ensemble is shown in brackets.}\n\\begin{tabular}{lccr}\n\\hline\nCollab.\t& $\\beta$ & $L^3\\cdot T$ & Statistics \\\\\n\\hline\nQCDSF\\,\\cite{QCDSF_97}\n\t& 6.0\t & $16^3\\cdot48$ & $\\sim1000$~~~\t\\\\\n\t& \t & $24^3\\cdot48$ & $\\sim200$~~~\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & $\\sim300$~~~\t\\\\\n\\hline\nUKQCD\\,\\cite{par_lat97}\n\t& 6.0 & $16^3\\cdot48$ & 497(3)\t\\\\\n\t&\t & $32^3\\cdot64$ & 70(1)\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & 251~~~\t\\\\\n\\hline\nAPETOV\\,\\cite{mendes_lat97}\n\t& 6.0\t & $16^3\\cdot48$ & 50(1)\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & 50~~~~\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{Tab:simpar} \n\\vspace{-0.8cm}\n\\end{table}\n\n\n\n\n\n\n\\section{SPECTRAL QUANTITIES}\n\nIn Fig.\\,\\ref{Fig:edin} we show the Edinburgh plot. In addition to the\nnon-perturbatively\\ improved data, we also display the tadpole improved results by\nUKQCD at $\\beta=5.7$ and the data by GF11 using the unimproved action\nat $\\beta=5.7, 5.93, 6.17$\\,\\cite{GF11_spec}. One observes that\nimprovement yields consistently lower values for $m_N\/m_\\rho$ compared\nto the unimproved action. The most dramatic effect is observed at\n$\\beta=5.7$ when one compares the unimproved results (full triangles)\nto the tadpole improved ones (full circles). In fact, it seems that\nthe mass behaviour obtained using tadpole improvement at $\\beta=5.7$\nis indistinguishable from the non-perturbatively\\ improved action. However, as we\nshall see later, the residual lattice artefacts in the tadpole\nimproved data at $\\beta=5.7$ are still large. Thus, the Edinburgh plot\ndisguises rather than exposes lattice artefacts and should therefore\nnot be used to draw conclusions about the scaling behaviour.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{edin2.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{Edinburgh plot.}\n\\label{Fig:edin}\n\\end{figure}\n\nIt is a well-known fact that lattice simulations using unimproved\nactions fail to reproduce the experimentally observed behaviour of the\nvector-pseudoscalar mass splitting, i.e. $m_{\\rm V}^2-m_{\\rm PS}^2\\sim {\\rm\nconst}$, which holds up to the mass of the charm quark. This is\nusually ascribed to the influence of lattice artefacts in the\ncomputation of these splittings. Data for $m_{\\rm V}^2-m_{\\rm PS}^2$ by\nUKQCD\\,\\cite{par_lat97} and QCDSF\\,\\cite{QCDSF_97} using the\nnon-perturbatively\\ improved action show that the splittings are close to the\nexperimental values for the $(\\rho,\\pi)$ and $(K^*,K)$ systems (see\nFig.\\,\\ref{Fig:splitting} for a plot of the UKQCD data). Furthermore,\nby comparing the results at $\\beta=6.0$ and~6.2 one observes that the\ndependence on the lattice spacing is small. However, for quark masses\nabove $m_{\\rm strange}$ the two collaborations see a slight downward\ntrend in the data, so that one can expect that the $D^*-D$ splitting\nis still not reproduced correctly.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mass_splitting.eps}{90mm}\n\\vspace{-4.3cm}\n\\caption{UKQCD data for the vector-pseudo-scalar mass splitting\nin units of $m_K^2$. Asterisks denote the experimentally observed\nvalues.} \n\\vspace{-0.3cm}\n\\label{Fig:splitting}\n\\end{figure}\n\nWe now discuss the chiral limit and the critical value of the hopping\nparamter, $\\kappa_{\\rm c}$. Usually $\\kappa_{\\rm c}$ is defined at the point where the\npseudoscalar mass vanishes, $m_{\\rm PS}=0$. In accordance with the quark\nmass behaviour of $m_{\\rm PS}^2$ implied by the PCAC relation, one can\ndetermine $\\kappa_{\\rm c}$ from a linear fit to\n\\begin{equation}\n (am_{\\rm PS})^2 = a^2B\\frac{1}{2}\\left(\\frac{1}{\\kappa}-\\frac{1}{\\kappa_{\\rm c}}\n \\right) = a^2B\\,m_{\\rm q}.\n\\label{EQ:mps2}\n\\end{equation}\nBoth QCDSF and UKQCD have reported that this linear ansatz in\n$1\/\\kappa$ results in poor fits with large correlated $\\chi^2\/{\\rm\ndof}$ and have therefore resorted to using model functions which also\ncontain quadratic terms in $1\/\\kappa$. However, in order to be\nconsistent with O($a$) improvement, the quark mass $m_{\\rm q}$ in\neq.\\,(\\ref{EQ:mps2}) should be replaced by\n\\begin{equation}\n\\widetilde{m}_{\\rm q}=m_{\\rm q}(1+b_{\\rm m} am_{\\rm q}),\n\\end{equation}\nwhere $b_{\\rm m}=-\\frac{1}{2}-0.0962g_0^2$ in one-loop perturbation\ntheory\\,\\cite{paperV}. This modification of the fitting ansatz in the\ndetermination of $\\kappa_{\\rm c}$ has so far not been used.\n\nAnother method defines $\\kappa_{\\rm c}$ at the point where the quark mass defined\nthrough the PCAC relation vanishes, i.e. $m_{\\rm PCAC}=0$. Here it is\nimportant to realise that \neven after non-perturbative\\ improvement\nchiral symmetry is only approximately restored, so that\n\\begin{eqnarray}\n &\\partial_\\mu &\\hspace{-0.3cm} \\big\\{A_\\mu(x)+c_{\\rm A} a\\partial_\\mu P(x)\\big\\}\n\t\\nonumber\\\\\n & &\\hspace{1.0cm} =\\hspace{0.1cm} 2m_{\\rm PCAC}P(x) + {\\rm O}(a^2).\n\\end{eqnarray}\nTherefore, the values of $\\kappa_{\\rm c}$ determined by requiring either\n$m_{\\rm PS}^2=0$ or $m_{\\rm PCAC}=0$ will differ by terms of order\n$a^2$. The compilation of results in Table\\,\\ref{Tab:kappac} shows\nthat in the range of $\\beta$ under study the difference in $\\kappa_{\\rm c}$ using\neither method is of the order of $10^{-4}$ and thus statistically\nsignificant.\n\n\\begin{table}[tb]\n\\begin{center}\n\\caption{Values for $\\kappa_{\\rm c}$ determined at $m_{\\rm PS}^2=0$ and at $m_{\\rm\nPCAC}=0$.} \n\\begin{tabular}{cr@{.}lr@{.}lr@{.}l}\n\\hline\n\\hline\n\t& \\multicolumn{6}{c}{$m_{\\rm PS}^2=0$} \\\\\n\\cline{2-7}\n$\\beta$ & \\multicolumn{2}{c}{UKQCD} & \\multicolumn{2}{c}{QCDSF}\n & \\multicolumn{2}{c}{APETOV} \\\\\n\\hline\n6.0 & 0&135335${}^{+20}_{-17}$ & 0&13531(1)\n\t& \\multicolumn{2}{c}{--} \\\\[0.1cm]\n6.2 & 0&135895${}^{+14}_{-55}$ & 0&13589(2) & 0&135861(19) \\\\[0.05cm]\n\\hline\n\\hline\n\t& \\multicolumn{6}{c}{$m_{\\rm PCAC}=0$} \\\\\n\\cline{2-7}\n$\\beta$ & \\multicolumn{3}{c}{ALPHA} & \\multicolumn{3}{c}{APETOV} \\\\[0.05cm]\n\\hline\n6.0\t& \\multicolumn{3}{c}{0.135196(14)}\n\t& \\multicolumn{3}{c}{--}\t\\\\[0.1cm]\n6.2\t& \\multicolumn{3}{c}{0.135795(13)}\n\t& \\multicolumn{3}{c}{0.135802(6)}\t\\\\[0.05cm]\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{Tab:kappac}\n\\vspace{-0.8cm}\n\\end{table}\n\nWe now analyse the scaling behaviour of the mass of the vector meson\nby comparing the approach to the continuum limit for unimproved and\nimproved actions. To this end, we note that the continuum limit should\nbe studied for constant physical volume, so that finite-volume effects\ndo not distort the scaling behaviour. Since chiral extrapolations are\npoorly understood, we will use the available lattice data for the\nvector and pseudoscalar masses $am_{\\rm V}, am_{\\rm PS}$ as well as the string\ntension $a\\sqrt{\\sigma}$ in order to interpolate the dimensionless ratio\n$m_{\\rm V}\/\\sqrt{\\sigma}$ to $m_{\\rm PS}\/\\sqrt{\\sigma}=1.125$. For $\\sqrt{\\sigma}=440\\,{\\rm MeV}$ this implies\n$m_{\\rm PS}=m_K=495\\,{\\rm MeV}$. If $m_{\\rm PS}^2$ and $m_{\\rm V}$ are linear functions of\nthe quark masses then one expects that $m_{\\rm V}\\simeq m_{K^*}=892\\,{\\rm MeV}$\nor $m_{\\rm V}\/\\sqrt{\\sigma}\\simeq2.027$. In Fig.\\,\\ref{Fig:mKstar} we have plotted\nthe interpolated data for $m_{\\rm V}\/\\sqrt{\\sigma}$ versus $a\\sqrt{\\sigma}$ for\nunimproved\\,\\cite{GF11_spec} and improved\nactions\\,\\cite{par_lat97,UKQCD_lat96,tadpole_comp}. For the unimproved\naction one observes that $m_{\\rm V}\\simeq m_{K^*}$ is satisfied, but only\nafter the extrapolation to the continuum limit. In contrast, the data\nobtained using the non-perturbatively\\ or tadpole improved actions at $\\beta=6.0,\n6.2$ show very little dependence on~$a$ and are rather close to\n$m_{\\rm V}\\simeq m_{K^*}$. This, however, can no longer be claimed for the\ntadpole improved data at $\\beta=5.7$, where instead one observes large\nresidual lattice artefacts. From the slope of the linear fit to\n$m_{\\rm V}\/\\sqrt{\\sigma}$ for the unimproved data, we infer the size of residual\nlattice artefacts at $a\\simeq0.1\\,{\\rm fm}$ to be $(12\\pm1)\\%$. For the\nnon-perturbatively\\ improved data we estimate the leading corrections of O($a^2$)\nto be only $\\simeq2\\%$.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mK.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{Scaling behaviour of the vector mass at $m_{\\rm PS}=m_K$. The\ndashed line indicates the extrapolated result using the GF11 data.}\n\\label{Fig:mKstar}\n\\end{figure}\n\nWe can now turn the tables and ask how well various prescriptions to\nfix $c_{\\rm sw}$ satisfy the ``phenomenological'' improvement condition\n\\begin{equation}\n{m_{\\rm V}}\/{\\sqrt{\\sigma}}={\\rm const}\n\\label{EQ:phenimp}\n\\end{equation}\nBesides the data obtained using tadpole and non-perturbative\\ estimates for $c_{\\rm sw}$,\none can use further spectrum data obtained using\n$c_{\\rm sw}=0$\\,\\cite{GF11_spec} and $c_{\\rm sw}=1$\\,\\cite{strange}. In\nFig.\\,\\ref{Fig:mK_vs_csw} we plot $m_{\\rm V}\/\\sqrt{\\sigma}$ at $m_{\\rm PS}=m_K$ versus\n$c_{\\rm sw}$. From the plot one infers that the condition (\\ref{EQ:phenimp})\nis satisfied within statistical errors if\n\\begin{equation}\n c_{\\rm sw}\\,\\,\\raisebox{-.6ex}{$\\stackrel{\\textstyle{>}}{\\sim}$}\\,\\, c_{\\rm sw}^{\\rm tadpole}\\equiv\n u_0^{-3},\\qquad\\beta\\,\\,\\raisebox{-.6ex}{$\\stackrel{\\textstyle{>}}{\\sim}$}\\,\\,6.0,\n\\label{EQ:cswtad}\n\\end{equation}\nwhere $u_0$ denotes the average link variable. Thus, with the present\nstatistical accuracy, already the tadpole improved estimate of $c_{\\rm sw}$\nleads to a large reduction of O($a$) effects, provided the lattice\nspacing is not too large. Fig.\\,\\ref{Fig:mK_vs_csw} also shows that in\norder to satisfy (\\ref{EQ:phenimp}) at $\\beta=5.7$ one would have to\ngo to much larger values of $c_{\\rm sw}$ than those implied by the tadpole\nprescription. This, however, appears impossible due to the appearance\nof exceptional configurations\\,\\cite{paperIII}.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mK_vs_csw.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{$m_{\\rm V}\/\\sqrt{\\sigma}$ as a function of $c_{\\rm sw}$. The sets of\npoints from right to left have been obtained using non-perturbative, tadpole,\ntree-level and no improvement. Data at $\\beta=5.7$ are shown only for\nunimproved and tadpole improved actions. The dashed line shows the\nextrapolated GF11 data.}\n\\label{Fig:mK_vs_csw}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe characterization of clock performance by means of average measures such as Allan variance, Hadamard variance, Theo, and modified forms of such variances, is widely applied within the time and frequency community as well as by most clock and oscillator fabricators. Such variances are measured by comparing the times $t_k$ on a device under test, with the times at regular intervals $k\\tau_0$ on a perfect reference, or at least on a better reference. Imperfections in performance of the clock under test are studied by analyzing noise in the time deviation sequence $x_k=t_k-k\\tau_0$, or the fractional frequency difference during the sampling interval $\\tau=s\\tau_0$:\n\\be\n\\Delta_{k,s}^{(1)}=(x_{k+s}-x_k)\/(s\\tau_0).\n\\ee\nThe frequency spectrum of fractional frequency differences can usually be adequately characterized by linear superposition of a small set of types of power law noise. The frequency spectrum of the fractional frequency differences of a particular noise type is given by a one-sided spectral density \n\\be\\label{onesided}\nS_y(f)=h_{\\alpha}f^{\\alpha},\\quad f>0.\n\\ee\n(The units of $S_y(f)$ are Hz$^{-1}$.) For the common power-law noise types, $\\alpha$ varies in integral steps from +2 down to -2 corresponding respectively to white phase modulation, flicker phase modulation, white frequency modulation, flicker frequency modulation, and random walk of frequency. \n\n\tSimulation of clock noise can be extremely useful in testing software algorithms that use various windowing functions and Fourier transform algorithms to extract spectral density and stability information from measured time deviations, and especially in predicting the probability for observing a particular value of some clock stability variance. This paper develops a simple simulation method for a time difference sequence that guarantees the spectral density will have some chosen average power law dependence. Expressions for the common variances and their modified forms are derived here that agree with expressions found in the literature, with some exceptions. This approach also leads to predictions of probabilities for observing a variance of a particular type at particular values of the sampling time. A broad class of probability functions naturally arises. These only rarely correspond to chi-squared distributions.\n\n\tThis paper is organized as follows. Sect. 2 introduces the basic simulation method, and Sect 3 applies the method to the overlapping Allan variance. Sect. 4 shows how diagonalization of the averaged squared second-difference operator, applied to the simulated time series, leads to expressions for the probability of observing a value of the variance for some chosen value of the sampling or averaging time. Expressions for the mean squared deviation of the mean of the variance itself are derived in Sect. 6. The approach is used to discuss the modified Allan variance in Sect. 7, and the non-overlapping form of the Allan variance is treated in Sect. 8. Appendix 1 discusses evaluation of a contour integral for the probability, first introduced in Sect. 4, in the general case.\n\n\\section{Discrete Time Series}\n\n\tWe imagine the noise amplitudes at Fourier frequencies $f_m$ are generated by a set of $N$ normally distributed random complex numbers $w_n$ having mean zero and variance $\\sigma$, that would by themselves generate a simulated spectrum for white phase noise. These random numbers are divided by a function of the frequency, $\\vert f_m \\vert^{\\lambda}$, producing a spectral density that has the desired frequency characteristics. For ordinary power law noise, the exponent $\\lambda$ is a multiple of $1\/2$, but it could be anything. The frequency noise is then transformed to the time domain, producing a time series with the statistical properties of the selected power law noise. The Allan variance, Modified Allan variance, Hadamard Variance, variances with dead time, and other quantities of interest can be calculated using either the frequency noise or the time series. \n\n\tIn the present paper we discuss applications to calculation of various versions of the Allan variance. Of considerable interest are results for the probability of observing a value of the Allan variance for particular values of the sampling time $\\tau$ and time series length $N$. The derivations in this paper are theoretical predictions. A natural frequency cutoff occurs at $f_h=1\/(2 \\tau_0)$, where $\\tau_0$ is the time between successive time deviations. This number is not necessarily related in an obvious way to some hardware bandwidth. The measurements are assumed to be made at the times $k\\tau_0$, and the time errors or residuals relative to the reference clock are denoted by $x_k$. The averaging or sampling time is denoted by $\\tau=s\\tau_0$, where $s$ is an integer. The total length of time of the entire measurement series is $T=N\\tau_0$. The possible frequencies that occur in the Fourier transform of the time residuals are\n\\be\nf_m=\\frac{m}{N\\tau_0}\\,.\\quad\\quad -\\frac{N}{2}+1 \\le m \\le \\frac{N}{2}\\,.\n\\ee \n\n\t{\\it Noise Sequences}\\\/. In order that a set of noise amplitudes in the frequency domain represent a real series in the time domain, the amplitudes must satisfy the reality condition\n\\be\nw_{-m}=(w_m)^{\\ast}.\n\\ee\n$N$ random numbers are placed in $N\/2$ real and $N\/2$ imaginary parts of the positive and negative frequency spectrum. Thus if $w_m=u_m+iv_m$ where $u_m$ and $v_m$ are independent uncorrelated random numbers, then $(w_m)^{\\ast}=u_m-iv_m$. Since the frequencies $\\pm 1\/2\\tau_0$ represent essentially the same contribution, $v_{N\/2}$ will not appear. We shall assume the variance of the noise amplitudes is such that\n\\be\\label{wmaverage}\n\\big<(w_m)^*w_n\\big>=\\big\\delta_{mn}=2\\sigma^2 \\delta_{mn};\\quad m\\ne 0,N\/2.\n\\ee \nAlso, $==0 {\\rm\\ for\\ }m\\ne N\/2.$\nThe index $m$ runs from $-N\/2+1$ to $N\/2$. In order to avoid division by zero, we shall always assume that the Fourier amplitude corresponding to zero frequency vanishes. This only means that the average of the time residuals in the time series will be zero, and has no effect on any variance that involves time differences.\n\n\tWe perform a discrete Fourier transform of the frequency noise and obtain the amplitude of the $k^{th}$ member of the time series for white PM:\n\\be\nx_k=\\frac{\\tau_0^2}{\\sqrt{N}}\\sum_{m=-N\/2+1}^{N\/2} e^{-\\frac{2\\pi i m k}{N}}w_m.\n\\ee\nThe factor $\\tau_0^2$ is inserted so that the time series will have the physical dimensions of time if $w_m$ has the dimensions of frequency. We then multiply each frequency component by $\\vert f_0\/f_m \\vert^{\\lambda}$. This will generate the desired power-law form of the spectral density. The time series will be represented by\n\\be\nX_k=\\frac{\\tau_0^2}{\\sqrt{N}}\\sum_{m=-N\/2+1}^{N\/2}\\frac{\\vert f_0 \\vert^{\\lambda}}{\\vert f_m \\vert^{\\lambda}} e^{-\\frac{2\\pi i m k}{N}}w_m.\n\\ee\nThe constant factor $\\vert f_0 \\vert^{\\lambda}$ has been inserted to maintain the physical units of the time series. The noise level is determined by $f_0$. For this to correspond to commonly used expressions for the one-sided spectral density, Eq. (\\ref{onesided}), we shall assume that\n\\be\n\\frac{\\tau_0^2\\vert f_0 \\vert^{\\lambda}}{\\sqrt{N}}=\\sqrt{\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0)}}\\,.\n\\ee\nWe shall show that if $2 \\lambda = 2-\\alpha$ the correct average spectral density is obtained. The simulated time series is \n\\be\\label{timeseries}\nX_k=\\sqrt{\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0)}}\n\\sum_m \\frac{e^{-\\frac{2\\pi i m k}{N}}}{\\vert f_m \\vert^{\\lambda}}w_m\\,.\n\\ee\nThe average (two-sided) spectral density of the time residuals is obtained from a single term in Eq. (\\ref{timeseries}):\n\\be\ns_x(f_m)=\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0) f_m^{2\\lambda}}\\bigg<\\frac{w_mw_m^*}{\\Delta f}\\bigg>=\\frac{h_{\\alpha}}{8\\pi^2 f_m^{2\\lambda}}\n\\ee\nwhere $\\Delta f = 1\/(N\\tau_0)$ is the spacing between successive allowed frequencies. The average (two-sided) spectral density of fractional frequency fluctuations is given by the well-known relation\n\\be\ns_y(f)=(2 \\pi f)^2 s_x(f)\\,,\n\\ee\nand the one-sided spectral density is\n\n\\be\nS_y(f)=\\cases{0, & $f < 0$;\\cr\n\t\t2 s_y(f)= h_{\\alpha}f^{\\alpha},& $f>0$\\,,\\cr}\n\\ee\nwhere $2 \\lambda = 2 -\\alpha$.\n\\section{Overlapping Allan Variance}\n\tConsider the second-difference operator defined by\n\\be\n\\Delta_{j,s}^{(2)}=\\frac{1}{\\sqrt{2 \\tau^2}}(X_{j+2s}-2X_{j+s}+X_j).\n\\ee\nThe fully overlapping Allan variance is formed by averaging the square of this quantity over all possible values of $j$ from 1 to $N-2s$. Thus\n\\be\\label{overlapvar}\n\\sigma_y^2(\\tau)=\\bigg<\\frac{1}{N-2s}\\sum_{j=1}^{N-2s}(\\Delta_{j,s}^{(2)})^2 \\bigg>.\n\\ee\nIn terms of the time series, Eq. (\\ref{timeseries}), the second difference can be reduced using elementary trigonometric identities:\n\\eject\n\\ba\\label{del2ave}\n\\Delta_{j,s}^{(2)}=\\sqrt{\\frac{h_{\\alpha}}{32 \\pi^2 \\tau^2 \\sigma^2 (N\\tau_0)}}\\hbox to 2.3in{}\\nonumber\\\\\n\\times \\sum_m\\frac{w_m}{\\vert f_m \\vert^{\\lambda}}\\bigg(e^{-\\frac{2 \\pi i m(j+2s)}{N}}-2e^{-\\frac{2 \\pi i m(j+s)}{N}}+e^{-\\frac{2 \\pi i m(j)}{N}}\\bigg)\\nonumber\\\\\n=-\\sqrt{\\frac{h_{\\alpha}}{2 \\pi^2 \\tau^2 \\sigma^2 (N \\tau_0)}}\\sum_m\\frac{w_m}{\\vert f_m \\vert^{\\lambda}}e^{-\\frac{2 \\pi i m(j+s)}{N}}\\bigg(\\sin\\frac{\\pi m s}{N}\\bigg)^2\\,.\n\\ea\nWe form the averaged square of $\\Delta_{j,s}^{(2)}$ by multiplying the real quantity times its complex conjugate, then averaging over all possible values of $j$.\n\\ba\n\\sigma_y^2(\\tau)=\\frac{h_{\\alpha}}{2 \\pi^2 \\tau^2 (N\\tau_0)(N-2s)}\n\\sum_{m,n,j}\\bigg<\\frac{w_m w_n^*}{\\sigma^2}\\bigg>\n\\frac{\\bigg(\\sin\\big(\\frac{\\pi m s}{N}\\big)\\sin\\big(\\frac{\\pi n s}{N}\\big)\\bigg)^2}{\\vert f_m f_n \\vert^{\\lambda}}\\nonumber\\\\\n\\hbox to -.5in{}\\times e^{-\\frac{2 \\pi i (m-n)(j+s)}{N}}\\,.\n\\ea\nThe average of the product of random variables only contributes $2 \\sigma^2$ when $m=n$ (see Eq. (\\ref{wmaverage}), except when $m=n=N\/2$ where $\\big<\\vert w_{N\/2} \\vert^2\\big>=\\sigma^2$.\nThe Allan variance reduces to\n\\be\n\\sigma_y^2(\\tau)=\\frac{h_{\\alpha}}{\\pi^2 \\tau^2(N\\tau_0)}\\bigg(\\sum_m \\frac{\\bigg(\\sin\\big(\\frac{\\pi m s}{N}\\big)\\bigg)^4}{\\vert f_m \\vert^{2\\lambda}} +\\frac{1}{2}\\frac{\\bigg(\\sin\\big(\\frac{\\pi s}{2}\\big)\\bigg)^4}{\\big(f_{N\/2}\\big)^{2\\lambda}}\\bigg)\\,,\n\\ee\nsince every term in the sum over $j$ contributes the same amount. The zero frequency term is excluded from the sum. For convenience we introduce the abbreviation\n\\be\nK=\\frac{2 h_{\\alpha}}{\\pi^2 \\tau^2 (N \\tau_0)}\\,. \n\\ee\nIf we sum over positive frequencies only, a factor of 2 comes in except for the most positive frequency and so\n\\be\n\\sigma_y^2(\\tau)=K\\bigg(\\sum_{m>0}^{N\/2-1}\\frac{\\bigg(\\sin\\big(\n\\frac{\\pi m s}{N}\\big)\\bigg)^4}\n{f_m^{2\\lambda}}+\\frac{1}{4}\\frac{\\bigg(\\sin\\frac{\\pi s}{2}\\bigg)^4}{(f_{N\/2})^{2\\lambda}} \\bigg).\n\\ee\nIf the frequencies are spaced densely enough to pass from the sum to an integral, then $\\Delta f= (N \\tau_0)^{-1}$ and\n\\be\n\\frac{1}{N\\tau_0}\\sum_mF(\\vert f_m \\vert)\\rightarrow\\int F(\\vert f \\vert)\\,df\n\\ee\nand we obtain the well-known result\\cite{barnes71}\n\\be\n\\sigma_y^2(\\tau)=2\\int_0^{f_h}\\frac{S_y(f)\\,df}{\\pi^2\\tau^2f^2}\\bigg(\\sin(\\pi f \\tau)\\bigg)^4\\,.\n\\ee\nSimilar arguments lead to known expressions for the non-overlapping version of the Allan variance as well as for the modified Allan variance. These will be discussed in later sections.\n\\section{Confidence Estimates}\n\tIn the present section we shall develop expressions for the probability of observing a particular value $A_o$ for the overlapping Allan variance in a single measurement, or in a single simulation run. $A_o$ is a random variable representing a possible value of the overlapping variance. We use a subscript ``o\" to denote the completely overlapping case. To save writing, we introduce the following abbreviations:\n\\ba\\label{FGdefs}\nF_m^j=\\frac{\\bigg(\\sin\\bigg(\\frac{\\pi m s}{N}\\bigg)\\bigg)^2}{\\vert f_m \\vert^{\\lambda}}\\cos\\bigg(\\frac{2\\pi m(j+s)}N{}\\bigg)\\nonumber\\\\\nG_m^j=\\frac{\\bigg(\\sin\\bigg(\\frac{\\pi m s}{N}\\bigg)\\bigg)^2}{\\vert f_m \\vert^{\\lambda}}\\sin\\bigg(\\frac{2\\pi m(j+s)}N{}\\bigg)\n\\ea\nThe dependence on $s$ is suppressed, but is to be understood. We write the second difference in terms of a sum over positive frequencies only, keeping in mind that the most positive and the most negative frequencies only contribute a single term since $\\sin(\\pi(j+s))=0$. The imaginary contributions cancel, and we obtain\n\\be\\label{diffFG}\n\\del2{j}=\\sqrt{K}\\sum_{m>0}\\big(F_m^j \\frac{u_m}{\\sigma}+G_m^j\\frac{v_m}{\\sigma}\\big)\\,.\n\\ee\nThere is no term in $v_{N\/2}$. It is easy to see that the overlapping Allan variance is given by\n\\be\\label{avarFG}\n\\sigma_y^2(\\tau)=\\frac{K}{N-2s}\\sum_j\\sum_{m>0}\\bigg((F_m^j)^2+(G_m^j)^2 \\bigg)\\,.\n\\ee\nTo compute the probability that a particular value $A_o$ is observed for the Allan variance, given all the possible values that the random variables $u_1,v_1,...u_{N\/2}$ can have, we form the integral\n\\be\n\\label{probdef}\nP(A_o)=\\int\\delta\\bigg(A_o-\\frac{1}{N-2s}\\sum_j \\big(\\Delta_{j,s}^{(2)}\\big)^2\\bigg)\\prod_{m>0}\\bigg(e^{-\\frac{u_m^2+v_m^2}{2\\sigma^2}}\\frac{du_m dv_m}{2\\pi \\sigma^2}\\bigg)\\,.\n\\ee\nThe delta function constrains the averaged second difference to the specific value $A_o$ while the random variables $u_1,v_1,...u_m,v_m,...u_{N\/2}$ range over their (normally distributed) values. There is no integral for $v_{N\/2}$. Inspecting this probability and the Eq. (\\ref{diffFG})for the second difference indicates that we can dispense with the factors of $\\sigma^{-1}$ and work with normally distributed random variables having variance unity. Henceforth we set $\\sigma=1$.\n\nThe exponent involving the random variables is a quadratic form that can be written in matrix form by introducing the $N-1$ dimension column vector $U$ (the zero frequency component is excluded)\n\\be\nU^T=[u_1\\,v_1\\,...u_m\\,v_m,...v_{N\/2-1},u_{N\/2}]\\,.\n\\ee\nThen\n\\be\n\\halb \\sum_{m>0}(u_m^2+v_m^2)=\\halb U^TU=\\halb U^T {\\bf 1} U,\n\\ee\nwhere ${\\bf 1}$ represents the unit matrix. The delta-function in Eq. (\\ref{probdef}) can be written in exponential form by introducing one of its well-known representations, an integral over all angular frequencies $\\omega$:\\cite{lighthill}\n\\be\nP(A_o)=\\int_{-\\infty}^{\\infty} \\frac{d \\omega}{2 \\pi}e^{i\\omega\\big(A_o-\\frac{1}{N-2s}\\sum_j \\big(\\Delta_{j,s}^{(2)}\\big)^2 \\big)}\\prod_{m>0}\\bigg(e^{-\\frac{u_m^2+v_m^2}{2\\sigma^2}}\\frac{du_m dv_m}{2\\pi \\sigma^2}\\bigg)\\,.\n\\ee\nThe contour of integration goes along the real axis in the complex $\\omega$ plane. \n\n\tThe squared second difference is a complicated quadratic form in the random variables $u_1,v_1,...u_m,v_m,...u_{N\/2}$. If this quadratic form could be diagonalized without materially changing the other quadratic terms in the exponent, then the integrals could be performed in spite of the imaginary factor $i$ in the exponent. To accomplish this we introduce a column vector $C^j$ that depends on $j,m,s,N$ and whose transpose is \n\\ba\n(C^j)^T=[F_1^j,G_1^j,...F_m^j,G_m^j,...G_{N\/2-1}^j,F_{N\/2}^j]\\,.\n\\ea\nDependence on $s$ is not written explicitly but is understood. The column vector has $N-1$ real components. It contains all the dependence of the second difference on frequency and on the particular power law noise. We use indices $\\{m,n\\}$ as matrix (frequency) indices. The (scalar) second difference operator can be written very compactly as a matrix product\n\\be\n\\del2{j}=\\sqrt{K} (C^j)^TU=\\sqrt{K}U^TC^j.\n\\ee\nThen \n\\be\n\\frac{1}{N-2s}\\sum_j\\bigg(\\Delta_{j,s}^{(2)} \\bigg)^2=U^T\\bigg(\\frac{K}{N-2s}\\sum_j C^j(C^j)^T\\bigg)U.\n\\ee\nThe matrix \n\\be\\label{hoverlapping}\nH_o=\\frac{K}{N-2s}\\sum_jC^j(C^j)^T\n\\ee\nis real and symmetric. $H_o$ is also Hermitian and therefore has real eigenvalues. A real symmetric matrix can be diagonalized by an orthogonal transformation,\\cite{Strang,Stoll} which we denote by $O$. Although we shall not need to determine this orthogonal transformation explicitly, it could be found by first finding the eigenvalues $\\epsilon$ and eigenvectors $\\psi_{\\epsilon}$ of $H_o$, by solving the equation \n\\be\nH_o\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nThe transformation $O$ is a matrix of dimension $(N-1)\\times (N-1)$ consisting of the components of the normalized eigenvectors placed in columns. Then\n\\be\nH_oO=OE\\,,\n\\ee\nwhere $E$ is a diagonal matrix with entries equal to the eigenvalues of the matrix $H_o$. Then since the transpose of an orthogonal matrix is the inverse of the matrix,\n\\be\nO^TH_oO=E\\,.\n\\ee\nThe matrix $H_o$ is thus diagonalized, at the cost of introducing a linear transformation of the random variables:\n\\be\n\\frac{K}{N-2s}\\sum_j\\big(\\del2{j}\\big)^2=U^TH_oU=U^TOO^TH_oOO^TU=(U^TO)E(O^TU)\\,.\n\\ee\nWe introduce $N-1$ new random variables by means of the transformation:\n\\be\nV=O^TU\\,.\n\\ee\nThen the term in the exponent representing the Gaussian distributions is\n\\be\n-\\halb U^T {\\bf 1 }U=-\\halb U^TO1O^TU=-\\halb V^T {\\bf 1}V=-\\halb \\sum_{n=1}^{N-1}V_n^2\\,.\n\\ee\nThe Gaussian distributions remain basically unchanged. \n\nFurther, the determinant of an orthogonal matrix is $\\pm 1$, because the transpose of the matrix is also the inverse:\n\\be\n\\det(O^{-1}O)=1=\\det(O^TO)=\\big(\\det(O)\\big)^2.\n\\ee\nTherefore, changes in the volume element are simple since the volume element for the new variables is\n\\ba\ndV_1 dV_2 ...dV_{N-1}=\\bigg|\\det \\bigg( \\frac{\\partial V_m}{\\partial U_n}\\bigg)\\bigg|dU_1dU_2...dU_{N-1}\\nonumber\\\\\n=\\vert \\det(O)\\vert dU_1 dU_2 ...dU_{N-1}\\nonumber\\\\ \n=dU_1 dU_2...dU_{N-1}.\n\\ea\nAfter completing the diagonalization, \n\\be\n\\frac{1}{N-2s}\\sum_j\\big(\\Delta_{j,s}^{(2)}\\big)^2=\\sum_i \\epsilon_i V_i^2\\,.\n\\ee\nThe probability is therefore\n\\be\nP(A_o)=\\int\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A_o-\\sum_k\\epsilon_k V_k^2\\big)}\\prod_i\\bigg( e^{-\\frac{V_i^2}{2}}\\frac{dV_i}{\\sqrt{2\\pi}}\\bigg)\\,.\n\\ee\nAn eigenvalue of zero will not contribute in any way to this probability since the random variable corresponding to a zero eigenvalue just integrates out. \n\nLet the eigenvalue $\\epsilon_i$ have multiplicity $\\mu_i$, by which is meant that the eigenvalue $\\epsilon_i$ is repeated $\\mu_i$ times. Integration over the random variables then gives a useful form for the probability:\n\\be\\label{probint}\nP(A_o)=\\int_{-\\infty}^{+\\infty} \\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A_o}}{\\prod_i(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ee\n\nFinally the contour integral may be deformed and closed in the upper half complex plane where it encloses the singularities of the integrand. This is discussed in Appendix 1. Knowing the probability, one may integrate with respect to the variance to find the cumulative distribution, and then find the limits corresponding to a 50\\% probability of observing the variance. \n\n\t{\\it Properties of the eigenvalues\\\/}. First, it is easily checked that the probability is correctly normalized by integrating over all $A_o$ and using properties of the delta-function:\n\\ba\n\\int P(A_o)dA_o=\\int_{-\\infty}^{+\\infty}\n\\frac{d \\omega}{2\\pi} \n\\frac{\\int e^{i\\omega A_o}dA_o}\n\t{\\prod_i(1+2 i \\epsilon_i \\omega)^{\\mu_i\/2}}\n=\\int_{-\\infty}^{+\\infty}\n\\frac{\\delta(\\omega) d\\omega }{\\prod_i(1+2 i \\epsilon_i \\omega)^{\\mu_i\/2}}\\nonumber\\\\\n=\\int_{-\\infty}^{+\\infty}d \\omega \\delta(\\omega)=1\\,.\n\\ea\nSecond, the eigenvalues are all either positive or zero. The eigenvalue equation for the eigenvector labeled by $\\epsilon$ is:\n\\be\n\\frac{K}{N-2s}\\sum_j C^j(C^j)^T\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nMultiply on the left by $\\psi_{\\epsilon}^T$; assuming the vector has been normalized, we obtain\n\\be\n\\epsilon=\\frac{K}{N-2s}\\sum_j\\bigg((C^j)^T\\psi_{\\epsilon}\\bigg)^2\\ge 0\\,.\n\\ee\nThus every eigenvalue must be positive or zero.\n\nNext let us calculate the trace of $H_o$. Since the trace is not changed by an orthogonal transformation,\n\\ba\n{\\rm Trace}(O^TH_oO)={\\rm Trace}(H_oOO^T)={\\rm Trace}(H_oOO^{-1})\\nonumber\\\\\n={\\rm Trace}(H_o)=\\sum_i \\epsilon_i\\,.\n\\ea\nThe sum of the diagonal elements of $H_o$ equals the sum of the eigenvalues of $H_o$. If we then explicitly evaluate the sum of the diagonal elements of $H_o$ we find\n\\ba\n\\sum_i \\epsilon_i=\\frac{K}{N-2s}\\sum_j{\\rm Trace}\\big(C^j (C^j)^T)\\nonumber\\\\\n=\\frac{K}{N-2s}\\sum_j\\sum_{m>0}\\bigg((F_m^j\\big)^2+\\big(G_m^j)^2)\\bigg)\\nonumber\\\\\n=K\\sum_{m>0}\n\\frac{\\bigg(\\sin\\frac{\\pi m s}{N}\\bigg)^4}\n{\\vert f_m \\vert^{2-2\\alpha}}=\\sigma_y^2(\\tau)\\,.\n\\ea\nEvery term labeled by $j$ contributes the same amount. We obtain the useful result that {\\it the overlapping Allan variance is equal to the sum of the eigenvalues of the matrix $H_o$.} Similar results can be established for many of the other types of variances.\n\n {\\it Distribution of eigenvalues\\\/}. The eigenvalue equation $H_o\\psi_{\\mu}=\\epsilon \\psi_{\\mu}$ produces many zero eigenvalues, especially when $\\tau$ is large. The dimension of the matrix $H_o$ is therefore much larger than necessary. Numerical calculation indicates that for the completely overlapping Allan variance, the eigenvalue equation has a total of $N-1$ eigenvalues, but only $N-2s$ non-zero eigenvalues; the number of significant eigenvalues is in fact equal to the number of terms in the sum over $j$ in the equations:\n\\be\n\\frac{K}{N-2s}\\sum_n\\sum_j^{N-2s}(C_m)^j(C_n)^j\\psi_{n}=\\epsilon \\psi_m\\,.\n\\ee\nThe factorized form of $H_o$, that arises on squaring a difference operator, permits the reduction of the size of the matrix that is to be diagonalized. We introduce the quantities\n\\be\n\\phi_{\\mu}^j=\\sum_n(C_n)^j\\psi_{n\\mu}\\,.\n\\ee\nWe are using the Greek index $\\mu$ to label a non-zero eigenvalue and the index $\\nu$ to label a zero eigenvalue. The eigenvalue equation becomes\n\\be\n\\frac{K}{N-2s} \\sum_j(C_m)^j\\phi_{\\mu}^j=\\epsilon \\psi_{m\\mu}\\,.\n\\ee\nMultiply by $(C_m)^l$ and sum over the frequency index $m$. Then\n\\be\\label{eigenoverlapJ}\n\\frac{K}{N-2s}\\sum_{m,j}(C_m)^l(C_m)^j\\phi_{\\mu}^j=\\epsilon\\phi_{\\mu}^l\\,.\n\\ee\nThis is an eigenvalue equation with reduced dimension $N-2s$ rather than $N-1$, since the number of possible values of $j$ is $N-2s$. The eigenvalue equation can be written in terms of a reduced matrix $H_{red}$, given by\n\\be\n(H_{red})^{lj}=\\frac{K}{N-2s}\\sum_m(C_m)^l(C_m)^j\\,.\n\\ee\nThe indices $l,j$ run from 1 to $N-2s$. Eigenvalues generated by Eq. (\\ref{eigenoverlapJ}) are all non-zero. To prove this, multiply Eq. (\\ref{eigenoverlapJ}) by $\\phi_{\\mu}^l$ and sum over $l$. We obtain\n\\be\n\\frac{K}{N-2s}\\sum_m\\bigg(\\sum_l (C_m)^l\\phi_{\\mu}^l \\bigg)^2=\\epsilon\\sum_l\\big(\\phi_{\\mu}^l \\big)^2\\,.\n\\ee\nThe eigenvalue cannot be zero unless\n\\be\\label{zerocondition}\n\\sum_l (C_m)^l\\phi_{\\mu}^l =0\n\\ee\nfor every $m$. The number of such conditions however is larger than the number $N-2s$ of variables, so the only way this can be satisfied is if $\\phi_{\\mu}^l=0$, a trivial solution. Therefore to obtain normalizable eigenvectors from Eq. (\\ref{eigenoverlapJ}), the corresponding eigenvalues must all be positive.\n This is true even through some of these conditions may be trivially satisfied if the factor $\\sin(\\pi m s\/N)$ vanishes, which happens sometimes when\n\\be\\label{lossequation}\nms=M N\n\\ee\nwhere M is an integer. Every time a solution of Eq. (\\ref{lossequation}) occurs, two equations relating components of $\\phi_{\\mu}^l$ are lost. Suppose there were $n$ solutions to Eq. (\\ref{lossequation}); then the number of conditions lost would be $2n$. The number of variables is $N-2s$ and the number of conditions left in Eq. (\\ref{zerocondition}) would be $N-1-2n$. The excess of conditions over variables is thus\n\\be\nN-1-2n-(N-2s)=2(s-n)-1\\,.\n\\ee\nIn Appendix 2 we prove that under all circumstances $2(s-n)-1>0$.\n\n\tWe temporarily drop the subscript $o$ since the remainder of the results in this section are valid for any of the variances. If the eigenvalues are found and the appropriate matrix is diagonalized, we may compute the probability for observing a value of the overlapping variance, denoted by the random variable $A$, by\n\n\\ba\nP(A)=\\int_{-\\infty}^{\\infty}\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A-V^TEV\\big)}\\prod_i\\bigg(\\frac{e^{-V_i^2\/2}dV_i}{\\sqrt{2\\pi}}\\bigg)\\nonumber\\\\\n=\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A}}{\\prod(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ea\n{\\it Case of a single eigenvalue\\\/}. If a single eigenvalue occurs once only, the general probability expression, Eq. (\\ref{probint}), has a single factor in the denominator:\n\\be\nP(A)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\\frac{d \\omega e^{i\\omega A}}{\\sqrt{1+2i\\omega\\epsilon}}\\,.\n\\ee\nThe integral is performed by closing the contour in the upper half complex $\\omega$ plane. There is a branch point on the imaginary axis at $\\omega=i\/(2\\epsilon)$, and a branch line from that point to infinity. Evaluation of the integral gives:\n\\be\\label{probmod}\nP(A)=\\frac{1}{\\sqrt{2\\pi\\sigma_y^2(\\tau)}}\\frac{e^{-A\/(2 \\sigma_y^2(\\tau))}}{\\sqrt{A}}\n\\ee\nThis is a chi-squared distribution with exactly one degree of freedom. The computation of the confidence interval for a given $s$ is simple. The cumulative probability obtained from Eq. (\\ref{probmod}) is\n\\be\n\\Phi(A)=\\int_0^{A}\\frac{1}{\\sqrt{2\\pi\\sigma_y^2(\\tau)}}\\frac{e^{-x\/(2 \\sigma_y^2(\\tau))}}{\\sqrt{x}}dx={\\rm erf}\\big(\\frac{A}{\\sqrt{2 \\sigma_y^2(\\tau)}} \\big)\\,.\n\\ee\nThe $\\pm 25$\\% limits on the probability of observing a value $A$ are then found to be $1.323 \\sigma_y^2$ and $0.1015 \\sigma_y^2$, respectively. An example of this is plotted in Figure 2. ($A$ is a variance, not a deviation.)\n\n{\\it Case of two distinct non-zero eigenvalues\\\/}. For the overlapping variance, when $s$ has its maximum value $N\/2-1$ there are two unequal eigenvalues. The probability integral can be performed by closing the contour in the upper half plane and gives the expression\n\\be\\label{twoeigenvalues}\nP(A)=\\frac{1}{2\\sqrt{\\epsilon_1 \\epsilon_2}}e^{-\\frac{A}{4}\\big(\\frac{1}{\\epsilon_1}+\\frac{1}{\\epsilon_2}\\big)}I_0\\bigg(\\frac{A}{4}\\big(\\frac{1}{\\epsilon_2}-\\frac{1}{\\epsilon_1} \\big) \\bigg)\\,.\n\\ee\nwhere $I_0$ is the modified Bessel function of order zero. The probability is correctly normalized. It differs from a chi-squared distribution in that the density does not have a singularity at $A=0$. This is illustrated in Figure 1. \n\nEvaluation of the contour integral when there are more than two distinct eigenvalues is discussed in Appendix 1. If an eigenvalue occurs an even number $2n$ times, the corresponding singularity becomes a pole of order $n$ and a chi-squared probability distribution may result; this has only been observed to occur for white PM.\n\n\n\\section{Variance of the Variance}\n\tOne of the goals of this investigation is to estimate the uncertainty in the variance when some value of the variance is measured. This can be rigorously defined if the probability $P(A_o)$ is available, but it may be difficult to evaluate the integral in Eq. (\\ref{probint}). In this case, one might be interested in a measure such as the rms value of the variance, measured with respect to the mean variance $\\sigma_y^2(\\tau)$. One method for obtaining this measure is obtained from the probability without introducing a Fourier integral representation for the delta function:\n\\ba\n\\int A_o^2 P(A_o)dA_o=\\int A_o^2\\delta(A_o-\\sum_i \\epsilon_i V_i^2)\n\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)dA_o \\nonumber\\\\\n=\\int \\bigg(\\sum_i \\epsilon_i V_i^2 \\bigg)^2\\delta(A_o-\\sum_i \\epsilon_i V_i^2)\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)dA_o \\nonumber\\\\\n=\\int \\bigg(\\sum_i \\epsilon_i V_i^2 \\bigg)^2\n\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)\\,.\n\\ea\nExpanding the square in the integrand, there are $N$ fourth-order terms and $N(N-1)$ cross terms so we get\n\\ba\n\\int A_o^2 P(A_o)dA_o=\\sum_i \\epsilon_i^2 \\big< V^4 \\big> + 2 \\sum_{i \\ne j}\\epsilon_i\\epsilon_j \\big^2\\nonumber\\\\\n=3\\sum_i \\epsilon^2+2 \\sum_{i \\ne j} \\epsilon_i \\epsilon_j\\,.\n\\ea\nTo obtain the variance of the variance, we must subtract\n\\be\n\\big^2=\\sum_i \\epsilon_i^2+2 \\sum_{i \\ne j} \\epsilon_i \\epsilon_j\\,.\n\\ee\nThe result is\n\\be\\label{varofvar}\n\\big-\\big^2=2\\sum_i \\epsilon_i^2\\,.\n\\ee\nTherefore {\\it the rms deviation of the variance from the mean variance equals $\\sqrt{2}$ times the square root of the sum of the squared eigenvalues.} This result is not very useful if there are only a small number of eigenvalues, since the factor $\\sqrt{2}$ can make the rms deviation from the mean of the variance larger than the variance. This is because the probability distribution of the variance is not a normal distribution, and can lead to unreasonably large estimates of confidence intervals.\n\nIf the eigenvalues or the probabilities are not readily available, a similar confidence estimate can be obtained from an alternative form of Eq. (\\ref{varofvar}) by considering the trace of the square of $H_o$:\n\\be\\label{varofvar2}\n2\\sum_i\\epsilon_i^2=2\\sum_{i,j}\\big(\\sum_{m>0}(F_m^iF_m^j+G_m^iG_m^j) \\big)^2\\,.\n\\ee\nTo prove this, consider diagonalization of $H_o$ by the orthogonal transformation $O$, and compute\n\\ba\n2\\sum_i\\epsilon_i^2=2{\\rm Trace}(E^2)=2{\\rm Trace}(O^TH_oOO^TH_oO)=2{\\rm Trace}H_o^2\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{m,n}\\bigg(\\sum_i\\big((C_m)^i(C_m)^i\\big)\\sum_j\\big((C_n)^j(C_n)^j\\big) \\bigg)\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{i,j}\\bigg(\\sum_m\\big((C_m)^i(C_m)^j\\big) \\sum_n\\big((C_n)^j(C_n)^i \\big) \\bigg)\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{i,j}\\bigg(\\sum_m\\big(F_m^iF_m^j+G_m^iG_m^j \\big) \\bigg)^2\\,.\n\\ea\nIf there are not too many terms in the sums over $i$ and $j$ then the sum over the frequency index in Eq. (\\ref{varofvar2}) becomes\n\\be\n\\sum_m\\big(F_m^iF_m^j+G_m^iG_m^j \\big)=\\sum_{m>0}\\frac{\\big(\\sin\\frac{\\pi m s}{N}\\big)^4}{f_m^{2\\lambda}}\\cos\\frac{2\\pi m(i-j)}{N}\\,.\n\\ee\nThis leads to some useful approximation schemes, but these will not be discussed further in this paper.\n\nUsually the variance of the variance is larger than the range of possible values of the variance computed from the $\\pm 25$\\% limits obtained from calculated probabilities. An example is given in Figure 1, where the confidence intervals are plotted for the non-overlapping variance for $N=1024$. For $s$ ranging between 342 and 511 there is only one eigenvalue, so the rms deviation of the variance from the variance is $\\sqrt{2}$ times the variance. This gives an upper limit on the confidence interval that is 1.414 times the variance, larger than that obtained from the actual probability distribution. The medium heavy lines in Figure 1 show the true $\\pm 25$\\% probability limits obtained from calculations such as Eq. (\\ref{probfunction})\n\\begin{figure}\\label{varsim}\n\\centering\n\\includegraphics[width=4.5 truein]{figure1.eps}\n\\caption{Comparisons of average non-overlapping Allan variance (light line) with rms deviations from the mean of the variance (heavy lines) and true $\\pm 25$\\% limits (medium heavy lines) for $N=1024$ data items in the time series. Results for the Allan variance for two independent simulation runs are plotted for comparison.}\n\\end{figure}\n\\section{Modified Allan Variance}\n\tThe modified Allan variance is defined so that averages over time are performed before squaring and averaging. We use a subscript $m$ to distinguish this form from the overlapping form of the variance. The definition is \n\\be\n\\sigma_m^2(\\tau)=\\bigg<\\bigg(\\frac{1}{s}\\sum_{j=1}^s \\del2{j} \\bigg)^2 \\bigg>\\,.\n\\ee\nSumming the expression for the second-difference given in Eq. (\\ref{del2ave}),\n\\ba\n\\frac{1}{s}\\sum_j^s \\del2{j} =-\\sqrt{\\frac{h_{\\alpha}}{2 s^2 \\pi^2 \\tau^2 \\sigma^2(N\\tau_0)}}\\sum_m\\frac{w_m}{\\vert f_m\\vert^{\\lambda}}\\big(\\sin\\frac{\\pi m s}{N} \\big)^2\\nonumber\\\\\n\\times\\bigg(\\frac{\\sin\\frac{\\pi m s}{N}}{\\sin \\frac{\\pi m}{N}} \\bigg)e^{-\\frac{\\pi i m (3s+1)}{N}}\\,.\n\\ea\nSquaring this, we write the complex conjugate of one factor and obtain\nthe ensemble average\n\\be\\\n\\sigma_m^2(\\tau)=\\frac{h_{\\alpha}}{2 s^2 \\pi^2 \\tau^2 \\sigma^2(N\\tau_0)}\n\\sum_{m,n}\\frac{\\big}{\\vert f_m f_n\\vert^{\\lambda}}\\frac{\\bigg(\\sin \\frac{\\pi m s}{N}\\bigg)^6}{\\big(\\sin \\frac{\\pi m}{N}\\big)^2}\\,.\n\\ee\nUsing Eq. (\\ref{wmaverage})and writing the result in terms of a sum over positive frequencies only,\n\\be\\label{modsigma}\n\\sigma_m^2(\\tau)=2\\sum_{m>0}\\frac{S_y(f_m)}{\\pi^2 s^2 \\tau^2 f_m^2 (N \\tau_0)}\n\\frac{\\bigg(\\sin \\frac{\\pi m s}{N}\\bigg)^6}{\\bigg(\\sin\\frac{\\pi m}{N}\\bigg)^2}\\,.\n\\ee\nPassing to an integral when the frequency spacing is sufficiently small,\n\\be\n\\sigma_m^2(\\tau)=2\\int_0^{f_h}\\frac{df S_y(f)}{\\pi^2 s^2 \\tau^2 f^2} \n\\frac{\\big(\\sin( \\pi f \\tau)\\big)^6}{\\big(\\sin(\\pi f\\tau_0)\\big)^2}\\,.\n\\ee\nThis agrees with previously derived expressions\\cite{sullivanetal}. The appearance of the sine function in the denominator of this expression makes the explicit evaluation of the integral difficult. In Table 1 we give the leading contributions to the modified Allan variance for very large values of the averaging time $\\tau$. In general there are additional oscillatory contributions with small amplitudes that are customarily neglected. These results do not agree however with those published in\\cite{lesage84}; the leading terms do agree with those published in\\cite{howe99}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\vbox to .3in{}\nNoise Type&$S_y(f)$& Mod $\\sigma_y^2(\\tau)$&$\\lambda$&$\\alpha$\\\\ \n\\vbox to -.3in{}&&&& \\\\ \\hline\n\\vbox to .0in{}&&&&\\\\ \n White PM& $h_2 f^2$ &$\\mathstrut$ $ \\frac{3h_2}{\\pi^2 \\tau^3}\\big(1 +\\frac{5}{18 s^3}\\big)$&0&2\\\\ \n\\vbox to .03in{}&&&& \\\\ \\hline\n\\vbox to .03in{}&&&&\\\\\nFlicker PM& $h_1 f $& $\\frac{3h_1 \\ln(256\/27)}{8\\pi^2\\tau^2}$&$\\frac{1}{2}$&1 \\\\\n\\vbox to .03in{}&&&& \\\\ \\hline\n\\vbox to .03in{}&&&&\\\\\n White FM&$h_0$& $\\frac{h_0}{4\\tau}\\big(1+\\frac{1}{2 s^2}\\big)$&1& 0 \\\\ \n\\vbox to .02in{}&&&& \\\\ \\hline\n\\vbox to .02in{}&&&&\\\\\n Flicker FM&$ h_{-1}f^{-1}$&$ 2 h_{-1}\\ln\\big(\\frac{3^{27\/16}}{4}\\big)$&$\\frac{3}{2}$& -1 \\\\ \n\\vbox to .02in{}&&&& \\\\ \\hline\n\\vbox to .02in{}&&&&\\\\\nRandom Walk&$h_{-2}f^{-2}$&$h_{-2}\\pi^2 \\tau\\big(\\frac{11}{20}+\\frac{1}{12 s^2}+\\frac{1}{40s^4} \\big) $&2& -2 \\\\ \n\\vbox to .2in{}&&&& \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic expressions for the Modified Allan Variance, in the limit of large sampling times $\\tau = s\\tau_0$.}\n\\end{center}\n\\end{table}\n\n{\\it Eigenvalue structure for modified Allan variance\\\/}.\n\tFor the modified case, instead of Eq. (\\ref{hoverlapping}) there are two summations; we use a subscript $m$ on $H$ to denote the modified form:\n\\be\nH_m=\\frac{K}{s^2}\\sum_j C^j\\sum_k(C^k)^T\\,.\n\\ee\nWe then seek solutions of the eigenvalue equation\n\\be\nH_m\\psi_{\\epsilon}=\\frac{K}{s^2}\\sum_j C^j\\sum_k( C^k)^T\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nMultiply by $\\sum_l(C^l)^T$ and sum over the frequency index. \n\\be\n\\frac{K}{s^2}\\sum_{l,j,m}\\big((C_m)^l)^T(C_m)^j\\big(\\sum_k(C^k)^T\\psi_{\\epsilon}\\big)=\\epsilon\\big(\\sum_l (C^l)^T \\psi_{\\epsilon}\\big)\\,.\n\\ee \nThe matrix equation has been reduced to a scalar equation for the quantity\n\\be\n\\phi=\\sum_k (C^k)^T\\psi_{\\epsilon}\\,.\n\\ee \nHere the matrix product of $(C^k)^T$ with $\\psi_{\\epsilon}$ entails a sum over all frequencies. Since we have a scalar eigenvalue equation for $\\epsilon$, there can be one and only one eigenvalue, which is easily seen to be the same as given by Eq. (\\ref{modsigma}) for each sampling time $\\tau $:\n\\be\n\\epsilon=\\frac{K}{s^2}\\sum_{l,k,m}\\big((C_m)^l\\big)^T (C_m)^k=\\sigma_m^2(\\tau)\\,.\n\\ee\nThe probability distribution will be of the form of Eq. (\\ref{probmod}), a chi-squared distribution with one degree of freedom.\n\n\\section{Non-overlapping Allan variance}\n\n\tIn the non-overlapping form of the Allan variance, the only change with respect to Eq. (\\ref{overlapvar}) is that the sum over $j$ goes from 1 in steps of $s$ up to $j_{max} \\le N-2s$. We denote the number of values of $j$ by $n_{max}$. The average non-overlapping variance is the same as that for the overlapping case, but the probability distributions are different. The matrix $H_{no}$ takes the form\n\\be\n(H_{no})_{mn}=\\frac{K}{n_{max}}\\sum_{j=1,1+2s..}^{j_{max}}(C_m)^j(C_n)^j\\,.\n\\ee\nThe subscript ``no\" labels a non-overlapping variance, and the indices $m$ and $n$ label frequencies. Here there are only $n_{max}$ terms in the sum since the values of $j$ skip by $s$. $n_{max}$ is given by:\n\\be\nn_{max}=\\lfloor \\frac{N-1}{s}\\rfloor -1\\,,\n\\ee\nwhere $\\lfloor Q \\rfloor $ denotes the largest integer less than or equal to $Q$. In order to diagonalize $H_{no}$ and compute probabilities, we look for eigenvalues by seeking solutions of:\n\\be\\label{eigenno}\n\\sum_n(H_{no})_{mn}\\psi_n=\\epsilon \\psi_m.\n\\ee\nWe define a reduced eigenvector by\n\\be\n\\phi_j=\\sum_n(C_n)^j\\psi_n.\n\\ee\nThe eigenvalue equation reduces to\n\\be\n\\frac{K}{n_{max}}\\sum_j(C_m)^j\\phi_j=\\epsilon \\psi_m.\n\\ee\nMultiply this equation by $(C_m)^l$ and sum over the frequency labels $m$. Then\n\\be\\label{eigennonoverlap}\n\\frac{K}{n_{max}}\\sum_j\\sum_m(\\big(C_m)^l(C_m)^j\\big) \\phi_j=\\epsilon \\phi_l\\,.\n\\ee\nLet\n\\be\nJ_{lj}=\\frac{K}{n_{max}}\\sum_m(C_m)^l (C_m)^j=\\frac{K}{n_{max}}(C^l)^T C^j;\n\\ee\nthe sum over frequencies is accomplished by the matrix multiplication.\nJust as for the overlapping case, if we calculate the trace of the matrix $J_{lj}$,\nwe find that since each term in the sum over $j$ contributes the same amount,\n\\be\n{\\rm Trace}(J_{lj})= \\frac{K}{n_{max}}\\sum_j C_j^TC_j=\\sigma_{no}^2(\\tau)\\,.\n\\ee\nBecause the trace remains unchanged under an orthogonal transformation, the non-overlapping Allan variance will be equal to the sum of the eigenvalues of Eq. (\\ref{eigennonoverlap}). The probability functions for a given $\\tau$ can differ from those for the overlapping or modified cases because the number of eigenvalues and their multiplicities may be different. The eigenvalues are still all greater than or equal to zero.\n\n\tIf the eigenvalues are found and the matrix $J_{lj}$ is diagonalized, we may compute the probability for observing a value of the overlapping variance, denoted by the random variable $A_{no}$, by\n\\ba\nP(A_{no})=\\int_{-\\infty}^{\\infty}\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A_{no}-U^TJU\\big)}\\prod_i\\frac{e^{-U_i^2\/2}}{\\sqrt{2\\pi}}\\nonumber\\\\\n=\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A_{no}}}{\\prod(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ea\nFor example, when there are only two distinct non-zero eigenvalues, The matrix $J_{lk}$ will have elements\n\\ba\nJ_{1,1}=J_{1+s,1+s}=\\frac{1}{2} \\sigma_{no}^2;\\hbox to 1in{}\\nonumber\\\\\nJ_{1,1+s}=J_{1+s,1}=\\int_0^{f_h} \\frac{S_y(f) df }{(\\pi \\tau f)^2} \\big(\\sin(\\pi \\tau f)\\big)^4\\cos(2\\pi \\tau f)\\,.\n \\ea\nThe eigenvalues are then\n\\ba\n\\epsilon_1=\\frac{1}{2}\\sigma_{no}^2+ |J_{1,1+s}|\\nonumber\\\\\n\\epsilon_2=\\frac{1}{2}\\sigma_{no}^2- |J_{1,1+s}|\\,.\n\\ea\nThe probability is of the form of Eq. (\\ref{twoeigenvalues}).\n\\begin{figure}\n\\label{flickersim}\n\\centering\n\\includegraphics[width=5 truein]{figure2.eps}\n\\caption{Simulation of Allan variance with $N=64$, $s=19$. For this case there are two distinct eigenvalues in the matrix for the non-overlapping case. The variance for $s=19$ was extracted from each of 4000 independent runs and a histogram of the values obtained was constructed for comparison with the probability, Eq. (\\ref{twoeigenvalues}). Chi-squared distributions with 1, 2, and 3 degrees of freedom are plotted for comparison.} \n\\end{figure}\nFigure 1 shows an example of this for flicker FM noise, for $N=64$ items in the time series. The variance corresponding to $\\tau=19 \\tau_0$ was extracted from each of 4000 independent simulation runs and a histogram of the resulting values was plotted. Good agreement with the predicted probability distribution can be seen.\n\n\\section{Other variances}\n\n\tThe analysis methods developed in this paper can be extended to other variances. For example, the Theo variance\\cite{Howe03} is defined by an overlapping average,\n\\ba\\label{theo}\n\\sigma_{Th}^2(s,\\tau_0,N)=\\frac{1}{N-s}\\sum_{i=1}^{N-m}\\frac{4}{3\\tau^2}\\times \\hbox to 1.5in{}\\nonumber\\\\\n\\sum_{\\delta=0}^{s\/2-1}\\frac{1}{s\/2-\\delta}\\bigg((X_i-X_{i-\\delta+s\/2})- \n\t(X_{i+s}-X_{i+\\delta+s\/2})\\bigg)^2\\,.\n\\ea\nwhere $s$ is assumed to be even, and $\\tau=3s\\tau_0\/4$. For any of the power law noises, and after passing to an integral, this expression can be transformed with the aid of elementary trigonometric identities to\n\\be\n\\sigma_{Th}^2(\\tau,\\tau_0,N)=\\frac{8}{3}\\int_0^{f_h}\\frac{S_y(f)df}{(\\pi \\tau f)^2}\\sum_{\\kappa=1}^{2\\tau\/3\\tau_0}\\frac{1}{\\kappa}\\bigg(\\sin(\\pi f \\kappa \\tau_0)\\sin\\big(\\pi f (\\frac{4}{3}\\tau-\\kappa \\tau_0)\\big)\\bigg)^2\\,.\n\\ee\nThe integer $\\kappa$ in the denominator makes this variance more difficult to evaluate, but a factorized quadratic form can still be constructed and the eigenvalue structure and resulting probabilities can be analyzed.\n\nThe Hadamard variance is defined in terms of a third difference, and is widely used to characterize clock stability when clock drift is a significant issue\\cite{hutsell95}. A third difference operator may be defined as\n\\be\n\\Delta_{j,s}^{(3)}=\\frac{1}{\\sqrt{6\\tau^2}}\\big(X_{j+3s}-3X_{j+2s}+X_{j+s}-X_j \\big)\\,.\n\\ee\nThe completely overlapping Hadamard variance is the average of the square of this third difference:\n\\ba\n\\sigma_H^2(\\tau)=\\frac{1}{N-3s}\\sum_{j=1}^{N-3s}\\big(\\Delta_{j,s}^{(3)}\\big)^2\\nonumber\\\\\n\\rightarrow\\frac{8}{3}\\int_0^{f_h}\\frac{S_y(f)df}{(\\pi \\tau f)^2}\\big(\\sin(\\pi \\tau f)\\big)^6\\,.\n\\ea\n\n\tThese methods can be applied to cases in which there is dead time between measurements of average frequency during the sampling intervals. Suppose for example that the measurements consist of intervals of length $\\tau=s\\tau_0$ during which an average frequency is measured, separated by dead time intervals of length $D-\\tau$ during which no measurements are available. Let the index $j$ label the measurement intervals with $j=1,2,...N$. An appropriate variance can be defined in terms of the difference between the average frequency in the $j^{th}$ interval and that in the interval labeled by $j+r$: \n\\be\n\\Delta_{j,r,s}^{(2)}=\\frac{1}{\\sqrt{2}}\\big(\\overline y_{j+r,s}-\\overline y_{j,s}\\big)\\,,\n\\ee\nwhere $\\overline y_{j,s}$ is the average frequency in the interval $j$ of length $s \\tau_0$.\nThen an appropriate variance can be defined as\n\\be\n\\Psi(\\tau,D)=\\bigg<\\big(\\Delta_{j,r,s}^{(2)} \\big)^2 \\bigg>\\,.\n\\ee\nIf the measurements are sufficiently densely spaced that it is possible to pass to an integral, this can be shown to reduce to\n\\be\n\\Psi(\\tau,D)=\\frac{2}{D\\tau}\\int_0^{f_h}df \\frac{S_y(f)}{(\\pi f)^2}\\big(\\sin(\\pi f r D)\\big)^2 \\big( \\sin(\\pi f \\tau)\\big)^2\\,.\n\\ee\nWhen $D=\\tau$ and $r=1$ there is no real dead time and this variance reduces to the ordinary Allan variance. \n\n\\section{Summary and Conclusion}\n\n\tIn this paper a method of simulating time series for the common power-law noises has been developed and applied to several variances used to characterize clock stability. These include overlapping and non-overlapping forms of the Allan variance, and the Modified Allan variance. Diagonalization of quadratic forms for the average variances leads to expressions for the probabilities of observing particular values of the variance for a given sampling time $\\tau=s \\tau_0$. The probabilities are expressed in terms of integrals depending on the eigenvalues of matrices formed from squares of the second differences that are used to define the variance. Generally speaking, the number of eigenvalues is equal to the number of terms occurring in the sum used to define averages of the second-difference operator, and this number gets smaller as the sampling time $\\tau$ gets larger. The probability distribution $P(A)$ for some variance $A$ is useful in estimating the $\\pm25$\\% confidence interval about the average variance. The eigenvalues are usually distinct; only for white PM have eigenvalues been observed to occur with multiplicities other than unity. This must happen in order that chi-square probability distributions result. Methods for computing the probabilities have been presented in a few useful cases. Results presented in this paper are confined to various forms of the Allan variance.\n\n\tOther methods of simulating power law noise have been published; the present approach differs from that of\\cite{kasdinwalter} in that no causality condition is imposed. The application of the methods developed here will be applied to other variances in future papers. \n\n\\section{Appendix 1. Evaluation of contour integrals for probability\\\/}\n\n\tIn almost all cases except for white PM, the eigenvalues are all distinct. We show here how then the probability function, Eq. (\\ref{probint}), can be reduced to a sum of real integrals. For each of the eigenvalues, we introduce the quantity\n\\be\nr_k=1\/(2\\epsilon_k)\\,.\n\\ee \nThe integral becomes\n\\be\\label{contour2}\nP(A)=\\prod_k\\big(\\frac{\\sqrt{r_k}}{e^{\\pi i\/4}} \\big)\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A}}{\\prod_k(\\omega-ir_k)^{1\/2}}\\,.\n\\ee\t\nBy Jordan's lemma\\cite{smirnov}, the contour of integration can be deformed into the upper half plane. The addition of a circular arcs at radius $\\vert\\omega\\vert=\\infty$ contributes nothing. Each of the square root factors in Eq. (\\ref{contour2}) has a branch point at $\\omega=ir_k$ with a branch line extending to $+i\\infty$ along the imaginary axis. We define the complex argument of each such factor by\n\\be\n-\\frac{3\\pi}{2} < \\arg(\\omega-ir_k)<\\frac{\\pi}{2}\\,.\n\\ee\n\\begin{figure}\n\\label{contourplot}\n\\centering\n\\includegraphics[width=3. truein]{figure3.eps}\n\\caption{Contour deformed to run along branch line on the $y-$axis. A pair of segments is shown with an odd number of singularities below the segments. The branch line required by square root factors of the form $\\sqrt{\\omega-r_k}$ is defined by the angles--either $\\pi\/2$ or $-3\\pi\/2$--of the directed segments relative to the Re$(\\omega)$ axis. Contributions from pairs of segments that are above an even number of branch points cancel out.}\n\\end{figure}\n\n\nAll branch lines extend along the positive $y-$axis to infinity. The largest eigenvalues give singularities closest to the real axis. Fig. 3 illustrates the resulting contour. Around each branch point is a circular portion which contributes nothing because as the radius $\\delta$ of each circle approaches zero, the contribution to the integral approaches zero as $\\sqrt{\\delta}$. The integral then consists of straight segments where $\\omega = iy \\pm \\delta$, where $\\delta$ approaches zero. Two such straight segments are illustrated in Fig. 3, one on each side of the branch line along the $y-$axis. Suppose the interval of interest is placed so that out of a total of $M$ eigenvalues, $n$ of them are below the interval (three are shown in the figure). For the contribution to the integral on the left, $y$ is decreasing and we can account for this by reversing the limits and introducing a minus sign. The phase factor contributed by $n$ factors with branch points below the interval is\n\\be\n\\frac{1}{\\big(e^{-\\frac{3\\pi i}{4}}\\big)^n}=\\bigg(e^{\\frac{3\\pi i}{4}} \\bigg)^n\\,.\n\\ee\nThe contribution from branch points above the interval of interest is the same on both sides of the $y-$axis, and is\n\\be\n\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\,.\n\\ee\nThe factor in front of the integral sign in Eq. (\\ref{contour2}) is the same on both sides of the $y-$axis and is\n\\be\n\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\,.\n\\ee\nThe total phase factor of this contribution, including a factor $i$ that comes from setting $\\omega=iy$ is thus\n\\be\n-i\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\bigg(e^{\\frac{3\\pi i}{4}} \\bigg)^n=-i\\bigg(e^{\\frac{\\pi i}{2}} \\bigg)^n\\,.\n\\ee\nFor the integral along the segment on the positive side of the $y-$axis, the only difference is that the phase of each of the $n$ contributions from branch points below the interval changes from $3\\pi\/4$ to $-\\pi\/4$. The phase factor for this part of the contour is thus\n\\be\n+i\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\bigg(e^{\\frac{-\\pi i}{4}} \\bigg)^n=+i\\bigg(e^{\\frac{-\\pi i}{2}} \\bigg)^n\\,.\n\\ee\nIf $n$ is even, the two contributions cancel. If $n=2m+1$ is odd, then the contributions add up with a factor $2(-1)^m$. The probability is thus always real and consists of contributions with alternating signs, with every other interval left out.\n\nIn summary, the contour integral contributions from portions of the imaginary axis in the complex $\\omega$ plane that have an even number of branch points below the interval will not contribute to the integral. For example, if there are four distinct eigenvalues the probability will reduce to \n\\ba\\label{probfunction}\nP(A)=\\frac{\\sqrt{r_1 r_2 r_3 r_4}}{\\pi}\\int_{r_1}^{r_2}\\frac{e^{-yA}dy}{\\sqrt{(y-r_1)(r_2-y)(r_3-y)(r_4-y)}}\\nonumber\\\\\n-\\frac{\\sqrt{r_1 r_2 r_3 r_4}}{\\pi}\\int_{r_3}^{r_4}\\frac{e^{-yA}dy}{\\sqrt{(y-r_1)(y-r_2)(y-r_3)(r_4-y)}}\\,.\n\\ea\nSuch results have been used to evaluate the probabilities for certain sampling intervals for flicker fm noise in Sect. 4.\n\n\\section{Appendix 2. Proof that Eq. (\\ref{eigenoverlapJ}) generates positive eigenvalues}\n\nIn this Appendix we show that under all circumstances Eq. (\\ref{lossequation}) gives rise to positive eigenvalues. \n\nObviously if $N$ is a prime number Eq. (\\ref{lossequation}) can never be satisfied ($n=0$). Consider the case $s=N-1$. Then $m=MN$ and there are no solutions since $m \\le N\/2$. In general both $m$ and $s$ may contain factors that divide $M$ or $N$. Suppose $s=ab$ where at least one of the factors $a,b$ is greater than 1, where $a$ divides $M$ and $b$ divides $N$. Then let\n\\be\nM=am_a,\\quad N=bn_b\\,.\n\\ee\nThen $m=m_a n_b$. $m=m_a n_b>N\/2=bn_b\/2$ cannot happen because $m \\le N\/2$.\nIf $m_a n_b=N\/2=b n_b$, then the number of solutions is $n=1$ and\n$2(s-n)-1=2(ab-1)-1 \\ge 2(2-1)-1>0$. If $m=m_a n_b < N\/2=b n_b\/2$, then there may be solutions $m=m_a n_b, m=2 m_a n_b, m=3m_a n_b,...$up to $b n_b\/2$. The number of such solutions is\n\\be\nn=\\lfloor \\frac{bn_b}{m_a n_b}\\rfloor=\\lfloor \\frac{b}{m_a}\\rfloor\n\\ee\nwhere $\\lfloor x \\rfloor$ means the largest integer less than or equal to $x$. The excess of conditions is then\n\\be\n2(s-n)-1=2(a b -\\lfloor \\frac{b}{m_a} \\rfloor) > 1\\,,\n\\ee\nwhich proves the assertion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nIn traditional drug development, safety and efficacy of potential new drugs have been evaluated separately. A phase I trial first finds the maximum tolerated dose, the dose associated with some predetermined probability of observing a Dose-Limiting Toxicity (DLT). This dose is then carried forward to phase II, where efficacy is evaluated, with limited borrowing of information between the two evaluations of safety and efficacy. An alternative option is a seamless Phase I\/II trial where safety and efficacy are evaluated simultaneously, with the aim to find the Optimum Biological Dose (OBD). The main advantage of collecting information on both is the increase in chances of finding a dose that is both safe and efficacious, by allowing for more sharing of information. \\\\\n\nThere are multiple methods that have been proposed to design such a trial that range in complexity, for example the relatively simple model-assisted toxicity and efficacy interval design of Lin and Yin (STEIN, \\citep{Lin2017}), or more complex model-based designs that can jointly model the efficacy-dose and toxicity-dose relationship such as the utility contour of Thall and Cook \\citep{Thall2004}, or the approach based on toxicity and efficacy odds ratios by Lin et al. \\citep{Yin2006}, or the bivariate CRM for competing outcomes of Braun \\citep{Braun2002} amongst others. These methods differ in both their approaches to inference of the bivariate (or trinary) outcomes, and the decision criteria based on this inference. A standard trial setting assumes that as each cohort of patients enters the trial, responses from all previous cohorts are available to inform the decision of the next dose assignment. The next cohort is assigned the dose in order to collect more information at the current estimate of the OBD.\\\\\n\nHowever, it is not always the case that the complete outcomes of all previous cohorts are available before the next cohort is assigned. Particularly in Oncology, where there are multiple cycles of treatment, both efficacy and safety outcomes may have a delayed onset. There are a limited number of phase I designs that account for late onset toxicities, but an even more limited number of designs that account for late onset outcomes in both safety and efficacy. The main contributions to dose-finding trial designs incorporating late onset toxicities include the interval censored approach of Sinclair and Whitehead (ISCDP, \\citep{Sinclair2014}), the approach of including a cycle effect in a proportional odds mixed effects model by Dousseau et al. (POMM, \\citep{Doussau2013}) and the Time-to-Event version of the Continual Reassessment Method by Cheung and Chappell (TITE-CRM, \\citep{YingKuenCheung2000}). When including late-onset efficacy as well, there is a limited number of approaches, for example the method jointly modelling the time-to-event of efficacy and toxicity of Yuan and Yin ($A_T\/A_E$, \\citep{Yuan2009}), or the model assisted approach of Liu and Johnson \\citep{Liu2016}.\\\\\n\nThe setting of late onset toxicities and efficacies is challenging, hence the very few methods that exist for such a trial, yet extremely relevant when multiple cycles of treatment are given. Using one of the many available methods for binary outcomes will give a much longer trial duration, as all cycles of treatment must be completed before the next dose assignment.\\\\\n\nIn this paper we investigate a joint time to event CRM for the implementation of a phase I\/II trial design with delayed onset outcomes for both safety and efficacy, in comparison to the existing methods. In Section~\\ref{sec:methods} we introduce the methods and setting, presenting results of their application in Section \\ref{sec:results} before concluding with a discussion in Section~\\ref{sec:discussion}.\\\\\n\\section{Methods} \\label{sec:methods}\nWe consider the setting where there are $J$ dose levels with values $d_1, \\ldots , d_j, \\ldots , d_J$. Patients enter the trial in cohorts. Each cohort is assigned to a dose, and followed up for $\\tau$ cycles of treatment. A DLT may occur at any time during the follow up period, in which case the patient leaves the trial. The patient may also observe an efficacy response at any time during their follow up, which is censored if the patient observes a DLT.\\\\\n\nThe choice of dose to assign to the next cohort, and the final dose recommendation, are chosen based on the design. Here we outline the proposed Joint TITE-CRM design, the $AT\/AE$ design of Yuan \\& Yin \\citep{Yuan2009} and the model-assisted design of Liu \\& Johnson \\citep{Liu2016}. For comparability, in each case, the trial proceeds in the following way\\\\\n\\begin{enumerate}\n\\item The first cohort of patients is assigned to the lowest dose.\n\\item After one cycle of follow up, if no DLT is observed, escalate to next highest dose and continue after each cycle until a DLT is observed or the highest dose is reached.\n\\item The relevant models are fitted to the currently observed responses (which will be from one cycle of follow up for the last cohort, two cycles for the last but one cohort etc.) and posterior distributions are updated. \\label{item:pp}\n\\item A set of `admissible' doses are calculated based on the models fitted.\n\\item The best dose of the `admissible' set according to some criterion of the design is chosen to assign to the next cohort (subject to certain prespecified rules discussed in Section~\\ref{sec:rules}). \\label{item:crit}\n\\item After one cycle of follow up, return to step \\ref{item:pp}. The trial is stopped when one of the pre-specified stopping rules is triggered. \n\\end{enumerate}\n\nIt is worthwhile to note that the form of the prior and posterior in step \\ref{item:pp} and criterion in step \\ref{item:crit} are the only aspects unique to each design.\n\n\\subsection{Joint Model TITE-CRM} \\label{sec:j_titecrm}\nBased on the TITE-CRM \\citep{YingKuenCheung2000} and similar to the bCRM of Braun \\cite{Braun2002}, we modify the procedure to include both safety and efficacy, whilst keeping the structure of the TITE-CRM.\\\\\n\nWe use a two-parameter logistic model for both efficacy and toxicity outcomes:\n\\[\nF(d,\\mathbf{\\beta_E})= \\frac{exp(\\beta_{E,0} + \\beta_{E,1} d)}{1+exp(\\beta_{E,0} + \\beta_{E,1} d)},\n\\]\n\\[\nF(d,\\mathbf{\\beta_T})= \\frac{exp(\\beta_{T,0} + \\beta_{T,1} d)}{1+exp(\\beta_{T,0} + \\beta_{T,1} d)},\n\\]\n\nWe then must give weights to both efficacy and toxicity observations based on their follow ups, and a weighted dose response model is used for each:\n\\[ \n{G(d,w{(E)},\\beta_E) = w^{(E)} F(d,\\beta_E)},\n\\]\n\\[ \n{G(d,w{(T)},\\beta_T) = w^{(T)} F(d,\\beta_T)},\n\\]\nwhere {$0 \\leq w^{(E)}, w^{(T)} \\leq 1$} are functions of time-to-event of a patient response. We use $w^{(T)}_{i,n}=u_{i,n}\/\\tau$, where $u_{i,n}$ is the current number of cycles patient $i$ has been observed for, unless a DLT is observed in which case $w^{(T)}_{i,n}=1$. In a similar fashion, we use $w^{(E)}_{i,n}=u_{i,n}\/\\tau$, where $u_{i,n}$ is the current number of cycles patient $i$ has been observed for, unless a DLT is observed before an efficacy outcome can be observed in which case $w^{(E)}_{i,n}=(\\mbox{DLT time - entry time})\/\\tau$, and $w^{(E)}_{i,n}=1$ if an efficacy outcome is observed.\\\\\n\nWe consider all binary combinations of efficacy and toxicity outcomes (as opposed to trinary sometimes used in such applications), related by a Gumbel Model \\citep{Thall2004}\n\n\\begin{align*}\n\\pi_{a,b}= &(G(d,w{(E)},\\beta_E) )^a (1-G(d,w{(E)},\\beta_E) )^{1-a} (G(d,w{(T)},\\beta_T) )^b (1-G(d,w{(T)},\\beta_T) )^{1-b} + \\\\ &(-1)^{a+b} (G(d,w{(E)},\\beta_E) ) (1-G(d,w{(E)},\\beta_E) ) G(d,w{(T)},\\beta_T) (1-G(d,w{(T)},\\beta_T) ) \\left( \\frac{e^{\\psi}-1}{e^{\\psi}+1} \\right),\n\\end{align*}\n\n\nwhere $\\pi_{a,b}$ is the probability of observing the binary combination of efficacy outcome ($a=0$ for no efficacy observed, $a=1$ for efficacy observed) and toxicity outcome ($b=0$ for no toxicity observed, $b=1$ for toxicity observed). \\\\\n\nThe likelihood is then based on the categorical variable $\\pi_{a,b}$ which can be in one of four states,\n\n\n\\begin{equation} \\label{eq:LIK}\n\\mathcal{L}(\\beta) = \\prod_{i=1}^{n} \\prod_{a=0}^{1} \\prod_{b=0}^{1} \\{\\pi_{a,b}(d_{[i]},\\beta)\\}^{\\mathbb{I}(Y=(a,b))}.\n\\end{equation}\n\nPriors are elicited on $\\psi$, $\\beta_{E,0}$, $\\beta_{E,1}$, $\\beta_{T,0}$, $\\beta_{T,1}$, and the likelihood (\\ref{eq:LIK}) is used to update the joint posterior for all of the parameters using a Gibbs Sampler.The next dose is then chosen based on a utility function of $\\pi_E$ and $\\pi_T$. \\\\\n\nThe two main components of the design are the model (either dose-response, survival or assisting model) and the utility function. The Joint TITE-CRM and the Model-assisted method in our implementations use the same utility function, as a result of a sensitivity analysis conducted on this choice. \\\\\n\nWe vary the weights $w_1$ and $w_2$ in the linear utility function, and also consider the more complex utility contour method suggested by Thall and Cook \\citep{Thall2004}. \\\\\n\nIt was found that there was negligible difference in the outcomes for any of the alternative utilities. This led to use using the simple linear utility used by Liu \\& Johnson \\citep{Liu2016} in our implementation of the Joint TITE-CRM, with the same weights. \\\\\n\n\\begin{equation}\nU(\\pi_E, \\pi_T) = \\pi_E - \\omega_1 \\pi_T- \\omega_2 \\pi_T \\mathds{I}(\\pi_T > \\phi_T). \\label{eq:utility}\n\\end{equation}\n\nThis uses two weights, $\\omega_1$ and $\\omega_2$ and a toxicity threshold $\\phi_T$, above which we penalise doses that have a higher probability of DLT than this threshold. The next dose is chosen to maximise the utility out of a set of admissible doses.\\\\\n\nThis utility is simple to implement and to interpret. Using the same utility function for the two different designs also allows the results to be interpreted in terms of the inference used.\n\n\\subsection{Joint Model CRM} \\label{sec:j_crm}\nFor comparison, we also consider a non-time-to-event method. For this implementation, all $\\tau$ cycles must be observed from the previous cohort before the next cohort is assigned.\n\nThis is equivalent to setting $ w^{(E)}= w^{(T)}=1$ for each patient in the above Joint TITE-CRM, effectively removing the time-to-event element of the design.\n\n\n\\subsection{$A_T$\/$A_E$ Design} \\label{sec:ATAE}\nWe here give an overview of the $A_T$\/$A_E$ design, for further details we refer the reader to the original proposal by Yuan \\& Yin \\citep{Yuan2009}.\\\\\n\nThis design fits survival models to the time to event data for both efficacy and toxicity outcomes, assuming a Weibull distribution.\n\n\\[\nS_T(t|d) = \\exp\\{-\\lambda_T t^{\\alpha_T} \\exp (\\beta_T d) \\},\n\\]\n\n\\[\nS_E(t|d) = \\exp\\{-\\lambda_E t^{\\alpha_E} \\exp (\\beta_E d) \\},\n\\]\n\n\\[\nS^*_E(t|d) = 1-\\pi + \\pi S_E(t|d).\n\\]\n\nThe bivariate Time to Event data is then modelled as\n\n\\[\nS(t_T,t_E|d)= \\{S_T(t_T|d)^{-1\/\\phi} + S_E(t_E|d)^{-1\/\\phi} -1\\}^{-\\phi}.\n\\]\nIn order to compute the likelihood, define $y_T=\\min (t_T,c_T)$ and $\\Delta_T = \\mathbb{I}(t_T \\leq c_T)$ where $c_T$ is the censoring time, and define $y_E$ and $\\Delta_E$ similarly for efficacy. The likelihood is then\n\\[\nL(\\theta | data_i) = L_1^{\\Delta_T \\Delta_E} L_2^{\\Delta_T (1-\\Delta_E)} L_3^{(1-\\Delta_T) \\Delta_E} L_4^{(1-\\Delta_T) (1-\\Delta_E)},\n\\]\nwhere\n\\[\nL_1=\\pi \\frac{\\partial^2 S(y_T,y_E|d)}{\\partial y_T \\partial y_E},\n\\]\n\n\\[\nL_2=-(1-\\pi) \\frac{\\partial S_T(y_T|d)}{\\partial y_T} - \\pi \\frac{\\partial S(y_T,y_E|d)}{\\partial y_T},\n\\]\n\n\\[\nL_3= -\\pi \\frac{\\partial S(y_T,y_E|d)}{\\partial y_E},\n\\]\n\n\\[\nL_4=(1-\\pi) \\partial S_T(y_T|d) + \\pi S(y_T,y_E|d).\n\\]\n\n\n$\\lambda_E$, $\\alpha_E$, $\\beta_E$, $\\lambda_T$, $\\alpha_T$ and $\\beta_T$ are assigned independent gamma priors and $\\pi$ and $\\phi$ uniform priors. The joint posterior distribution is then updated using a Gibbs Sampler.\\\\\n\nThe criterion used for decision making is a ratio of the area under the curve (AUC) for the survival for toxicity and efficacy:\n\n\\[\n\\frac{A_T}{A_E}= \\frac{\\alpha_T^{-1}\\{\\lambda_T \\exp (\\beta_T d)\\}^{-1\/\\alpha_T} \\Gamma \\{ \\alpha_T^{-1},\\lambda_T \\exp (\\beta_T d) \\tau^{\\alpha_T} \\}}{(1-\\pi)\\tau + \\pi \\alpha_E^{-1}\\{\\lambda_E \\exp (\\beta_E d)\\}^{-1\/\\alpha_E} \\Gamma \\{ \\alpha_E^{-1},\\lambda_E \\exp (\\beta_E d) \\tau^{\\alpha_E} \\}},\n\\] \n\nwhere $\\Gamma (a,b)$ is the incomplete gamma function and $\\tau$ is the follow up time. The dose in the admissible dose set that maximises the $A_T$\/$A_E$ is chosen for the next cohort of patients.\n\\subsection{Model-Assisted}\nThe model-assisted method by Liu and Johnson \\citep{Liu2016} uses a Bayesian dynamic model as follows:\n\\begin{align*}\ny_{T,i}|d=d_j &\\sim \\mbox{Bern} (p_{T,j}) \\\\\np_{T,j} &= p_{T,j-1} + ( 1-p_{T,j-1} )\\beta_{T,j} , \\hspace{10pt} j=2,\\ldots J \\\\\np_{T,1} &= \\beta_{T,1} \\\\\n\\beta_{T,j} &\\sim \\mbox{Beta} (a_{T,j},b_{T,j}) \\hspace{10pt} j=1,\\ldots J.\n\\end{align*}\nThe subscript $T$ indicates toxicity, with the equivalent for efficacy labelled $E$. The specification of the $p_{T,j}$ ensures monotonicity, since there is no dose-response model.\\\\\n\nThe likelihood is given as \n\\[\nL(\\mathbf{y} | \\mathbf{\\beta_T},\\mathbf{\\beta_E}) = \\prod_{k=\\{E,T\\}} \\prod_{i=1}^{n} \\left\\{ 1- \\prod_{r=1}^{j[i]} (1-\\beta_{k,r}) \\right\\}^{y_{k,j}} \\prod_{r=1}^{j[i]} (1 - \\beta_{k,r})^{1-y_{k,j}},\n\\]\nwhich is used to update the posterior distribution. The same utility function as is used in the Joint TITE-CRM (Equation~\\ref{eq:utility}) is used to assign the next dose.\n\n\\subsection{Rules} \\label{sec:rules}\n\\subsubsection{Admissible Dose Set} \\label{sec:admis}\nThe admissible dose set is calculated as all doses satisfying the following criteria:\n\\[\n\\mathbb{P} (\\pi_T < \\pi_T^*) > q_T \n\\hspace{15pt} \\mbox{\\&} \\hspace{15pt} \n\\mathbb{P} (\\pi_E > \\pi_E^*) > q_E.\n\\]\nwhere $\\pi_T^*$ and $\\pi_E^*$ are the target probabilities for efficacy and toxicity in the entire follow-up $\\tau$.\\\\\n\nThese constraints ensure that for both designs, the escalation proceeds to a promising dose that is considered neither futile or unsafe.\n\\subsubsection{Stopping \\& Enforcement Rules} \\label{sec:stop_enf_rules}\nAs well as ensuring any assigned dose is admissible, we also apply a set of enforcement and stopping rules.\nFor any given dose $d_j$, $p_{1,d_j}$ is the $P(DLT)$ in the first cycle, $p_{\\tau,d_j}$ is the $P(DLT)$ in the full follow up of $\\tau$.\\\\\nEnforcement rules are to ensure the safety in the first cycle of treatment.\n\\textbf{\\textit{Enforcement Rules:}}\n\\begin{enumerate}\n\\item \\textbf{Hard Safety}: This rule ensures that if there is a very high probability that the toxicity of an experimented dose exceeds 0.3 in the first cycle, then that dose and all above are excluded from any further experimentation.(i.e. dose $d_j$ and all above are excluded when $P(p_{1,d_j}>0.3)>\\zeta$ for some threshold $\\zeta$). In this implementation we use a threshold for excessive toxicity of $\\zeta =0.95$, with a $Beta(1,1)$ prior for Binomial responses. This is equivalent to if there are at least 3 DLT responses out of 3 patients, at least 4 DLT responses out of 6 patients, or at least 5 DLT responses out of 9 patients, then all dose assignments must be strictly lower than that dose for the rest of the study. If the lowest dose is excluded then the trial stops with no dose recommendation made.\n\\item \\textbf{K-fold Skipping Doses}: The next dose assignment must be no more than a 2-fold-rise in the value of the highest experimented dose so far.\n\\end{enumerate}\n\nStopping rules are to define when we may stop the trial, either for futility\/safety issues or because we have a level of certainty about the estimated OBD.\n\n\\textbf{\\textit{Stopping Rules:}}\n\\begin{enumerate}\n\\item \\textbf{No Admissible Doses}: If no doses satisfy the two constraints in Section~\\ref{sec:admis}, then the trial is stopped, either for futility or safety.\n\\item \\textbf{Lowest Dose Deemed Unsafe}: If $P(p_{1,d_1}>30\\%)>0.80$ and at least one cohort of patients has been assigned to dose $d_1$, the trial is stopped. \\label{rule:low_unsafe}\n\\item \\textbf{Highest Dose Deemed Very Safe}: If $P(p_{1,d_J}\\leq 30\\%)>0.80$ and at least one cohort of patients has been assigned to dose $d_J$, the trial is stopped. \\label{rule:hi_safe}\n\\item \\textbf{Sufficient Information}: If a dose is recommended for the next cohort on which $C_{suff}$ cohorts have already been assigned in the escalation, excluding backfilled cohorts, the trial is stopped. \\label{rule:suff}\n\\item \\textbf{Precision}: If the safety and efficacy profile are both estimated precisely enough, the trial is stopped. This precision is defined as $CV(MTD)<30\\%$ and $CV(d_{[\\pi^*_E]})<30\\%$, with the coefficient of variation calculated as an adjusted median absolute deviation divided by the median and $d_{[\\pi^*_E]}$ is the dose associated with the target efficacy. Both of these must be satisfied before the stopping rule is enforced. This stopping rule is only used once at least $C_{suff}$ cohorts of patients have had at least one cycle of treatment in the escalation, on any dose. This rule is not applicable for the model-assisted method. \\label{rule:prec}\n\\item \\textbf{Hard Safety}: If the lowest dose is considered unsafe according to the hard safety enforcement rule, the trial is stopped.\n\\item \\textbf{Maximum Patients}: If the maximum number of patients ($n=n_{\\mbox{max}}$) have been recruited, the trial is stopped.\n\\end{enumerate}\n\nWe consider two settings; a realistic setting where all enforcement and stopping rules are applied, and a theoretical setting where stopping rules \\ref{rule:low_unsafe}, \\ref{rule:hi_safe}, \\ref{rule:suff}, \\ref{rule:prec} are not applied. This second setting is to investigate the behaviour of the designs without restrictions.\n\n\\section{Simulations} \\label{sec:sims}\nIn order to compare the joint TITE-CRM to both the $A_T\/A_E$ design and the model-assisted design, we conduct simulation studies in a range of scenarios.\n\n\\subsection{Setting} \\label{sec:setting}\nWe consider the setting of six doses to investigate: 1.5MBq , 2.5MBq , 3.5MBq , 4.5MBq , 6.0MBq , 7.0MBq, with a follow-up period of $\\tau=3$ cycles. Each cycle lasts 6 weeks, although for simplicity of time-to-event outcomes, we measure the time to event in units of cycles. The variables for the rules are $n_{max}=60$ and $C_{suff}=30$.\n\\\\\n\nWe use a combination of five safety scenarios and four efficacy scenarios, giving 20 scenarios in total. We then extend to consider different efficacy patterns in Section~\\ref{sec:eff_time_trends}. The five safety scenarios represent four scenarios where there is at least one safe dose and one where no doses are safe. We give the probability of DLT in cycle 1 and in the full follow up period of 3 cycles, with $\\pi_T^*=0.391$. It is assumed that the probability of DLT decreases by a factor of 1\/3 in subsequent cycles, conditional on survival through the previous cycles. In T.1, dose levels 4 and above are unsafe; in T.2, the highest dose is unsafe; in T.3, all doses are unsafe; in T.4, dose levels 2 and above are unsafe; and in T.5 all doses are well below target safety. The four efficacy scenarios are described by the probability of efficacy in the whole follow up period, with the lower bound on target efficacy as $\\pi_T^*=0.2$. In E.1, dose levels 2 and above are efficacious, with efficacy increasing with dose; in E.2, all dose levels are efficacious with a plateau reached at dose level 3; in E.3, no dose levels are efficacious; and in E.4, dose levels 4 and above are efficacious.\\\\\n\nIn the main implementations, it is assumed the probability of efficacy in the total follow up is three times the probability of efficacy in the first cycle. In Section~\\ref{sec:eff_time_trends}, we alter this assumption.\\\\\n\nScenarios are referred to as Ex.Ty where x is the efficacy scenario, y is the safety scenario. Table~\\ref{tab:scens} gives the individual safety and efficacy scenarios, while Table~\\ref{tab:scens_ut} gives the utility and $AT\/AE$ values for each of the 20 combination scenarios.\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{lllllll}\n \\hline\n\\hline\nSafety & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \n \\hline\nT1 (cycle 1)& 0.100 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 \\\\ \nT1 (full follow up)& 0.140 & 0.270 & 0.391 & 0.503 & 0.606 & 0.701 \\\\ \n T2 (cycle 1)& 0.100 & 0.130 & 0.160 & 0.200 & 0.250 & 0.400 \\\\ \n T2 (full follow up) & 0.140 & 0.180 & 0.219 & 0.270 & 0.332 & 0.503 \\\\ \n T3 (cycle 1)& 0.400 & 0.450 & 0.500 & 0.550 & 0.600 & 0.650 \\\\ \n T3 (full follow up) & 0.503 & 0.556 & 0.606 & 0.655 & 0.701 & 0.746 \\\\ \n T4 (cycle 1)& 0.300 & 0.400 & 0.450 & 0.500 & 0.550 & 0.600 \\\\ \n T4 (full follow up)& 0.391 & 0.503 & 0.556 & 0.606 & 0.655 & 0.701 \\\\ \n T5 (cycle 1)&0.100 &0.120 &0.140 &0.160 &0.180 &0.200 \\\\ \n T5 (full follow up)&0.140 &0.166 &0.193 &0.219 &0.245& 0.270 \\\\ \n \\hline\n\\hline\nEfficacy & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \n \\hline\nE1 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 & 0.700 \\\\ \n E2 & 0.300 & 0.400 & 0.500 & 0.500 & 0.500 & 0.500 \\\\ \n E3 & 0.100 & 0.120 & 0.140 & 0.160 & 0.180 & 0.200 \\\\ \n E4 & 0.100 & 0.150 & 0.200 & 0.300 & 0.500 & 0.700 \\\\ \n \\hline\n\\end{tabular}\n\\caption{Scenario definitions for efficacy and toxicity.}\n\\label{tab:scens}\n\\end{table}\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{rrrrrrr} \\hline\nScenario & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \\hline \\hline\n E1.T1 Utility & \\textit{0.15} & \\textit{0.21} & \\textbf{0.27 }& -0.22 & -0.26 & -0.30 \\\\ \n E1.T1 $AT\/AE$ & \\textit{0.99} & \\textit{0.93} & \\textbf{0.86} & 0.79 & 0.71 & 0.61 \\\\ \n \\hline\nE1.T2 Utility & \\textit{0.15} & \\textit{0.24} & \\textit{0.33} & \\textit{0.41} & \\textbf{0.49} & -0.02 \\\\ \n E1.T2 $AT\/AE$ & \\textit{0.99} & \\textit{1.02} & \\textit{1.05} & \\textit{1.06} & \\textbf{1.07} & 0.92 \\\\ \n \\hline\nE1.T3 Utility & -0.52 & -0.49 & -0.46 & -0.43 & -0.40 & -0.36 \\\\ \n E1.T3 $AT\/AE$ & 0.65 & 0.63 & 0.61 & 0.59 & 0.57 & 0.54 \\\\ \n \\hline\nE1.T4 Utility & \\textit{0.07} & -0.42 & -0.39 & -0.36 & -0.33 & -0.30 \\\\ \n E1.T4 $AT\/AE$ & \\textit{0.76} & 0.69 & 0.67 & 0.66 & 0.64 & 0.61 \\\\ \n \\hline\nE1.T5 Utility & \\textit{0.15} & \\textit{0.25} & \\textit{0.34} & \\textit{0.43} & \\textit{0.52} & \\textit{0.61} \\\\ \n E1.T5 $AT\/AE$ & \\textit{0.99} & \\textit{1.03} & \\textit{1.07} & \\textit{1.12} & \\textit{1.17} & \\textit{1.23} \\\\ \n \\hline\nE2.T1 Utility & \\textit{0.25} & \\textit{0.31} & \\textbf{0.37} & -0.22 & -0.36 & -0.50 \\\\ \n E2.T1 $AT\/AE$ & \\textit{1.05} & \\textit{0.99} & \\textbf{0.92} & 0.79 & 0.66 & 0.53 \\\\ \n \\hline\nE2.T2 Utility & \\textit{0.25} & \\textit{0.34} &\\textbf{ 0.43} & 0.41 & 0.39 & -0.22 \\\\ \n E2.T2 $AT\/AE$ & \\textit{1.05} & \\textit{1.08} & \\textbf{1.12} & 1.06 & 0.99 & 0.79 \\\\ \n \\hline\nE2.T3 Utility & -0.42 & -0.39 & -0.36 & -0.43 & -0.50 & -0.56 \\\\ \n E2.T3 $AT\/AE$ & 0.69 & 0.67 & 0.66 & 0.59 & 0.53 & 0.46 \\\\ \n \\hline\nE2.T4 Utility & \\textbf{0.17} & -0.32 & -0.29 & -0.36 & -0.43 & -0.50 \\\\ \n E2.T4 $AT\/AE$ & \\textbf{0.81} & 0.74 & 0.72 & 0.66 & 0.59 & 0.53 \\\\ \n \\hline\nE2.T5 Utility & \\textit{0.25} & \\textit{0.35} & \\textit{0.44 }& \\textit{0.43} & \\textit{0.42} & \\textit{0.41} \\\\ \n E2.T5 $AT\/AE$ & \\textit{1.05} & \\textit{1.10} & \\textit{1.14} & \\textit{1.12} & \\textit{1.09} & \\textit{1.06} \\\\ \n \\hline\nE3.T1 Utility & 0.05 & 0.03 & 0.01 & -0.56 & -0.68 & -0.80 \\\\ \n E3.T1 $AT\/AE$ & 0.94 & 0.84 & 0.74 & 0.64 & 0.54 & 0.44 \\\\ \n \\hline\nE3.T2 Utility & 0.05 & 0.06 & 0.07 & 0.07 & 0.07 & -0.52 \\\\ \n E3.T2 $AT\/AE$ & 0.94 & 0.92 & 0.89 & 0.86 & 0.81 & 0.65 \\\\ \n \\hline\nE3.T3 Utility & -0.62 & -0.67 & -0.72 & -0.77 & -0.82 & -0.86 \\\\ \n E3.T3 $AT\/AE$ & 0.62 & 0.57 & 0.53 & 0.48 & 0.43 & 0.38 \\\\ \n \\hline\nE3.T4 Utility & -0.03 & -0.60 & -0.65 & -0.70 & -0.75 & -0.80 \\\\ \n E3.T4 $AT\/AE$ & 0.72 & 0.63 & 0.58 & 0.53 & 0.48 & 0.44 \\\\ \n \\hline\nE3.T5 Utility & 0.05 & 0.07 & 0.08 & 0.09 & 0.10 & \\textit{0.11} \\\\ \n E3.T5 $AT\/AE$ & 0.94 & 0.93 & 0.92 & 0.90 & 0.89 & \\textit{0.88} \\\\ \n \\hline\nE4.T1 Utility & 0.05 & 0.06 & \\textit{0.07} & -0.42 & -0.36 & -0.30 \\\\ \n E4.T1 $AT\/AE$ & 0.94 & 0.85 & \\textit{0.76} & 0.69 & 0.66 & 0.61 \\\\ \n \\hline\nE4.T2 Utility & 0.05 & 0.09 & \\textit{0.13} & \\textit{0.21} & \\textbf{0.39} & -0.02 \\\\ \n E4.T2 $AT\/AE$ & 0.94 & 0.93 & \\textit{0.92} & \\textit{0.93} & \\textbf{0.99} & 0.92 \\\\ \n \\hline\nE4.T3 Utility & -0.62 & -0.64 & -0.66 & -0.63 & -0.50 & -0.36 \\\\ \n E4.T3 $AT\/AE$ & 0.62 & 0.58 & 0.54 & 0.52 & 0.53 & 0.54 \\\\ \n \\hline\nE4.T4 Utility & -0.03 & -0.57 & -0.59 & -0.56 & -0.43 & -0.30 \\\\ \n E4.T4 $AT\/AE$ & 0.72 & 0.64 & 0.60 & 0.58 & 0.59 & 0.61 \\\\ \n \\hline\nE4.T5 Utility & 0.05 & 0.10 & \\textit{0.14} & \\textit{0.23} &\\textit{ 0.42} & \\textit{0.61} \\\\ \n E4.T5 $AT\/AE$ & 0.94 & 0.94 & \\textit{0.95} & \\textit{0.98} & \\textit{1.09} & \\textit{1.23} \\\\ \n \\hline\n\\end{tabular}\n\\caption{Utility for the twenty scenarios, with OBD highlighted in \\textbf{bold} and acceptable doses highlighted in \\textit{italics}. Note that the $AT\/AE$ criteria has no penalty for exceeding a safety threshold and hence may give higher values for unsafe doses.}\n \\label{tab:scens_ut}\n\\end{table}\n\nIn Table~\\ref{tab:scens_ut}, it can be seen that the utility criterion and $AT\/AE$ criterion do not always agree on the OBD. For example in scenarios E1.T1 and E2.T1, the 3.5MBq dose is the OBD according to the utility criterion, and the 1.5MBq dose is the OBD according to the $AT\/AE$ criterion. We have chosen to designate the 3.5MBq dose as the true OBD as this is more realistic, with a P(DLT) equal to target and a higher P(efficacy) than the lower dose. This does however highlight an issue concerning the $AT\/AE$ criteria, in that in maximizing the ratio between the two areas under the curves, it does not target any specific safety level. Hence when ratios are similar across doses, even when values are not, there may be difficulties in selection of the optimal dose.\n\n\\subsection{Data Generation} \\label{sec:data_gen}\nTo generate the event times for toxicities and efficacies, we use a model that is not based on the assumptions of any of the methods. We use a bivariate log-normal distribution for survival times, with parameters matched to the first cycle and full follow up probabilities of efficacy and toxicity, with a correlation parameter of -1\/2. This indicates that there is a negative association between the event times. This is a simple yet effective mechanism to generate data that allows for easy specification of probabilities at dose levels that do not themselves follow any specific parametric dose-response model.\\\\\n\\[\n\\begin{pmatrix}\nt_T \\\\\nt_E\n\\end{pmatrix} \\sim Lognormal_2 \\left( \\begin{pmatrix}\n\\mu_T\\\\\n\\mu_E\n\\end{pmatrix} , \\begin{pmatrix}\n\\sigma^2_T & -\\frac{\\sigma_T \\sigma_E}{2}\\\\\n-\\frac{\\sigma_T \\sigma_E}{2} & \\sigma^2_E\n\\end{pmatrix} \\right).\n\\]\n\n\n\\subsection{Prior distributions} \\label{sec:prior}\nSince all methods are Bayesian, we must specify prior distributions for the parameters of the models.\\\\\n\\subsubsection{Joint Model TITE-CRM \\& Joint Model CRM}\nHere we use priors\n\\[\n\\begin{pmatrix}\n\\beta_0 \\\\\n\\log (\\beta_1)\n\\end{pmatrix} \\sim N_2 \\left( \\begin{pmatrix}\nc_1\\\\\nc_2\n\\end{pmatrix} , \\begin{pmatrix}\nv_1 & 0 \\\\\n0 & v_2 \n\\end{pmatrix} \\right).\n\\]\nfor both toxicity and efficacy. Values of hyper-parameters used are:\n$c_{1,T}=\\log (1\/16) $, $c_{2,T}=\\log (1\/4)$, $v_{1,T}=1$, $v_{2,T}=2$ and $c_{1,E}= -3 $, $c_{2,E}=-0.2$, $v_{1,E}=1$, $v_{2,E}=1$. The values for toxicity are calibrated over a range of scenarios. The values for efficacy are chosen so that all doses have prior mean $(\\pi_E)>$ 0.3 and this increases with dose, with average effective prior sample size of one patient per dose level. We used a vague prior for $\\phi \\sim N(0,100)$.\n\n\\subsubsection{$A_T$\/$A_E$ Design}\nHere, we follow the priors implemented by Yuan \\& Yin \\citep{Yuan2009}, with some minor modification to fit our setting. $\\lambda_E$, $\\alpha_E$, $\\beta_E$, $\\lambda_T$, $\\alpha_T$ and $\\beta_T$ are assigned independent Gamma priors with shape and rate parameters of 0.1. These are constrained to be less than 3, 4, 1, 2, 3 and 1 respectively. The modification is of the upper bound of the $\\beta$ to account for the differing values of doses in our implementation to the original implementation. The uniform priors for $\\pi$ and $\\phi$ are $\\pi\\sim U(0.6,1)$ and $\\phi\\sim U(0,5)$.\n\\subsubsection{Model-Assisted}\nHere we use the priors suggested by Liu \\& Johnson \\cite{Liu2016} of $\\mathbf{a_E}=(0.2,0.3,0.4,0.45,0.5,0.6)$, $\\mathbf{b_E}=(0.95,0.9,0.8,0.75,0.7,0.65)$, $\\mathbf{a_T}=(0.05,0.1,0.2,0.25,0.3,0.35)$ and $\\mathbf{b_T}=(0.95,0.9,0.8,0.75,0.7,0.65)$.\n\n\\subsection{Results} \\label{sec:results}\nWe conducted 1000 simulations in each of the 20 scenarios. Figure~\\ref{fig:correct} shows the percentage of simulations that select the true OBD if there is one, or the trial is stopped correctly when no true OBD exists in the experimental dose set. The most noticeable feature is that the model-assisted method gives a lower proportion of the correct recommendations than the two model-based methods. In scenarios where all doses are too safe, the model-assisted method performs poorly, a reflection on the cautious escalation so often apparent in model-assisted designs. The overall performance of the two model based methods is similar, although with some differences in individual scenarios. For example the performance in E2.T4, where the lowest dose is the OBD, the Joint TITE-CRM method vastly outperforms the $AT\/AE$ design, which too often stops early for no admissible doses. \\\\\n\nInterestingly, the percentage of acceptable selections does not follow such a similar pattern. Figure~\\ref{fig:acceptable} displays these results. Here we define an `acceptable' selection as one that is both safe and efficacious, regardless if it the best in terms of utility criterion. The Joint TITE-CRM is the best performing overall. The Model-assisted method performs poorly for example in E3.T5, where the acceptable selections are either the highest dose or stopping because the highest dose is too safe.\\\\\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Correct_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Proportion of correct selections across scenarios} \\label{fig:correct}\n \\end{figure}\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Acceptable_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Proportion of acceptable selections across scenarios} \\label{fig:acceptable}\n \\end{figure}\n\n\nIn terms of sample size, the model assisted method has a higher sample sample size on average across scenarios, as shown in Figure~\\ref{fig:ssize}. The two model-based methods have very similar sample sizes.\\\\\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{SSize_comparison_YYLJall_TTEv4noTITE.pdf}\n \\caption{Mean sample size across scenarios} \\label{fig:ssize}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Duration_comparison_YYLJall_TTEv4noTITE.pdf}\n \\caption{Mean trial duration across scenarios} \\label{fig:dur}\n \\end{figure}\nHowever, although the patterns are similar between the two model-based methods in terms of total sample sizes, correct and acceptable selections, there is a difference in the number of patients treated at unsafe doses, with the $AT\/AE$ method assigning more patients to unsafe doses than both other methods in almost all scenarios. The model assisted method has much fewer patients assigned to unsafe doses, again due to the cautious escalation behaviour of model-assisted methods. \\\\\n\nAlthough the performance of the Joint CRM is similar to that of the Joint TITE-CRM, with slight improvement, this is at the cost of a much greater average trial length. Figure~\\ref{fig:dur} indicated this relationship, with an average trial duration of more than twice that of the Joint TITE-CRM. This is a sure indication of the need for trial designs that can use partial information on cohorts effectively.\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Unsafe_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Number of patients assigned to unsafe doses across scenarios} \\label{fig:unsafe}\n \\end{figure}\n\n\nThe overall performance of the three methods in the setting of realistic stopping and enforcement rules can be summarised as follows. The cautious escalation of the model-assisted method gives rise to a poorer performance in terms of selections of correct and acceptable doses, with larger sample sizes and fewer patients exposed to unsafe doses. The two model based designs perform better, with similar percentages of correct and acceptable recommendations, but with the $AT\/AE$ design assigning more patients to unsafe doses, showing a more aggressive escalation that is not rewarded with better selection performance. Additionally, the $AT\/AE$ design is substantially more complex and computationally intensive than both other methods, a cost that does not increase the performance.\\\\\n\nWhen the stopping rules are relaxed to investigate the operating characteristics of the designs more closely, note that since stopping rule \\ref{rule:hi_safe} is no longer enforced, there is no longer a correct selection in safety scenario 5 and so all methods give 0\\% correct selection. We see that in most scenarios where there is a true OBD (e.g. E1.T2, E2.T4, E4.T2) the Joint TITE-CRM gives a better performance of correct selections. When there are no admissible doses (e.g. E3.T1, E3.T2) the Joint TITE-CRM performs worse than the other two time-to-event methods. \n\nIt is noticeable that when considering acceptable selections, the Joint TITE-CRM performs the best out of the selection in most scenarios. In the absence of the early stopping rules, this provides evidence that the Joint TITE-CRM is benefiting from the model-based inference that the model-assisted method lacks. However, the $AT\/AE$ method also performs worse than the Joint TITE-CRM. This may be due to the deviations from assumptions on the survival model used in the inference and data generation.\n\nThe sample size is of course considerably larger when the stopping rules are relaxed, with many scenarios reaching nearly the maximum sample size on average. The Joint TITE-CRM has a larger sample size on average, which tallies up to the observation that this design had fewer correct stoppings for no admissible doses. This suggests that in general, this method is more willing to label a dose as admissible than the other methods.\\\\\n\nThe mean trial duration is also drastically increased for all methods, with the Joint CRM having an average duration of up to seven years, compared to the other methods giving less than three years.\\\\\n\nWith the relaxed stopping rules, the $AT\/AE$ method once again has a large number of patients assigned to unsafe doses, indicating a more aggressive escalation. The model-assisted method has much fewer patients assigned to unsafe doses, due to the more cautious escalation.\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Correct_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Proportion of correct selections across scenarios, lesser stopping} \\label{fig:correctnoSTOP}\n \\end{figure}\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Acceptable_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Proportion of acceptable selections across scenarios lesser stopping} \\label{fig:acceptablenoSTOP}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{SSize_comparison_YYLJall_TTEv4noTITEnoSTOP.pdf}\n \\caption{Mean sample size across scenarios lesser stopping} \\label{fig:ssizenoSTOP}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Duration_comparison_YYLJall_TTEv4noTITEnoSTOP.pdf}\n \\caption{Mean trial duration across scenarios lesser stopping} \\label{fig:durnoSTOP}\n \\end{figure}\n \n \n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Unsafe_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Number of patients assigned to unsafe doses across scenarios lesser stopping} \\label{fig:unsafenoSTOP}\n \\end{figure}\n\n\n\\subsection{Efficacy Time Trend} \\label{sec:eff_time_trends}\nIn the main implementations, the data generation of the efficacy times assumed that the probability of efficacy in the first cycle is one third of the probability of efficacy in the entire follow up of three cycles. However, it may be the case that this is actually higher or lower than one third.\\\\\n\nTo investigate the effect of efficacy occurring at various points in the follow up, we consider three efficacy patterns. In efficacy pattern 1, the probability of efficacy in cycle 1 is 1\/3 of the total probability; in efficacy pattern 2, it is 1\/6 and in efficacy pattern 3 in is 1\/2. This is to investigate the effect of varying time trends in efficacy.\\\\\n\nIt is possible that this change in efficacy time trend will affect the performance of the three methods that us the time-to-event outcomes, and so we investigate the difference that this can make in a selection of scenarios.\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Eff_patternsALL.pdf}\n \\caption{Operating characteristics for the three efficacy time trends} \\label{fig:eff_trends}\n \\end{figure}\n\nFigure~\\ref{fig:eff_trends} shows the operating characteristics of the three time to event methods when the time trend is varied. There is no difference in the number of patients assigned to unsafe doses, and very little difference to the overall sample size. In terms of selections, the $AT\/AE$ methods of Yuan \\& Yin is most variable to the time trend, since this method is more heavily dependent on the survival curve. The Joint TITE-CRM and the model-assisted method of Liu \\& Johnson are more robust to efficacy time trends. \n\n\n\n\\section{Discussion} \\label{sec:discussion}\nIn this work, we have compared three designs for a dose-finding trial that uses both efficacy and toxicity outcomes in the form of time-to-event responses. The comparison has highlighted the impact of both the inference of dose-response relationship and the decision criteria. The challenge presented by a trial using both toxicity and efficacy outcomes is compounded by the late onset nature of the responses.\\\\\n\nThe decision criteria used in the $AT\/AE$ design, whilst intuitive for survival outcomes, does somewhat deviate from the objective of the dose-finding trial. Without penalty for unsafe doses, it gave an aggressive escalation path in many scenarios. This leads on from the point made regarding Table~\\ref{tab:scens_ut}, that without targeting any safety level, scenarios with similar true ratios across levels present very challenging for this design. Additionally, the increased computational intensity of this design is not rewarded by an increase in performance.\\\\\n\nThe model-assisted method and the Joint TITE-CRM implemented both used the same utility criterion, which allowed a comparison between the simple inference of Liu \\& Johnson \\cite{Liu2016} and the joint logistic model. Here the increase in complexity is rewarded with increased performance. The Joint TITE-CRM is recommended as an alternative that balances complexity and performance across a range of scenarios.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLight is a very peculiar form of energy that constantly travels from one point to another, which makes it difficult to store or `freeze' it in one place. This, however, can be effectively overcome by using optical resonators that utilize feedback mechanisms together with wave interference effects to recycle light along periodic trajectories. Depending on the resonator's geometry, these trajectories may intercept each other in opposite directions forming standing-wave patterns with a vanishing Poynting vector. Alternatively, they may form closed loops that support degenerate circulating traveling-wave modes (clockwise (CW) or counterclockwise (CCW)) with a non-vanishing Poynting vector along the loop direction. This ability to confine and trap light has enabled several scientific breakthroughs over the past few decades. Importantly, recent technological progress in micro- and nano-fabrication has enabled the realization of on-chip optical resonators with spatial dimensions comparable to the wavelength of the trapped light, or even smaller \\citep{Iwanaga-PR} with a wide range of applications including microlasers \\cite{Loncar2002APL,Wu2004APL,McCall1992APL,Sandoghdar1996PRA,Miao2016S,Zhang2020LSA,Peng2016PNAS} and sensing \\cite{Foreman2015AOP,Yalcin2006IEEE,Wiersig2016PRA,Chen2017N,Hodaei2017N,Zhong2019PRL}, just to mention few examples. Despite the large variety in their designs (microrings, microdisks, photonic crystals, Bragg structures, etc.), sizes, and material systems, optical resonators are typically classified into one of the aforementioned categories, i.e. standing- or traveling-wave devices \\cite{Vahala-OM,Vahala2003N,Saleh-FP}. This classification is generally accepted as complete. Thus, research in the field of optical resonators has focused on applications of these resonators and implementations of novel designs with unique features. In particular, standing-wave resonators can be engineered to support small mode volumes and high quality factors, which makes them perfect choice for engineering quantum light-matter interactions \\cite{John1990PRL,Chang2006PRL,Louyer2005PRA,Moreau2001APL,Vuckovic2003APL,Bennett2005APL,Strauf2007NPhot,Pelton2002PRL,Zhong2021PRR}. On the other hand, traveling-wave resonators are the preferred platform for (classical and quantum) nonlinear optics due to the ability to engineer the interaction between different wave components and the unidirectional propagation properties of these modes which facilitates the input and output coupling \\cite{Kues2019NPhot,Kippenberg2018S}. In addition, it was shown previously that the interaction between an atom and a standing-wave pattern of light depends on whether the latter is generated in a standing-wave resonator or as a result of interference between counter-propagating waves in a traveling-wave resonator \\cite{Shore1991JOSAB}.\n\nIn this work we show that this classification scheme for optical resonators (as standing- or traveling-wave resonators) is not complete. Instead, we reveal a new type of optical modes supported by certain resonator structures that represents a missing link between these two categories. Specifically, we propose a new resonator concept that supports an optical mode exhibiting hybrid standing- and traveling-wave patterns simultaneously.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=6.2in]{Fig-Schematic.png}\n\t\\caption{The concept of a hybrid-wave resonator. A resonator structure that supports two degenerate standing-wave modes such that (A) mode $M_1$ resides in domain $D_0 \\cup D_1$, and (B) mode $M_2$ resides in domain $D_0 \\cup D_2$. (C) It is possible that a proper linear superposition between $M_1$ and $M_2$ can result in new modes that preserve the standing-wave character in domains $D_{1,2}$ while at the same time form a traveling wave in domain $D_0$. In the figure, standing and traveling waves are schematically represented by interference fringes and uniform field distributions respectively.}\n\t\\label{Fig-Schematic}\n\\end{figure*}\n\nThis article is organized as follows. First, we introduce a general concept outlining the behavior of the proposed resonator without a reference to a particular structure. Afterwards, we discuss an implementation based on standard chip-scale photonics technology. To gain insight into the modal structure of the proposed resonator, we present a detailed analysis of its modal features using a scattering matrix approach. Next, we confirm our results by using full-wave finite element simulations. Finally, we investigate the effect of local perturbation due to a small scatterer on the eigenmodes of the proposed hybrid-wave resonator.\n\n\\section{General concept} \nWe start by considering a generic concept of optical resonators that supports modes overlapping only partially with the physical structure of the resonator. For simplicity, assume that such a resonator can be divided into three domains (extension to more domains is straightforward): $D_0$, $D_1$ and $D_2$ (the exact definition of the domain boundaries is not important). Furthermore, assume that it supports two degenerate standing-wave modes $M_{1,2}$ such that the field distribution of $M_1$ mainly resides in $D_0 \\cup D_1$, and similarly the field associated with mode $M_2$ resides in $D_0 \\cup D_2$ as shown schematically in Figure \\ref{Fig-Schematic}A and \\ref{Fig-Schematic}B respectively. The continuity conditions for the electromagnetic field across domain boundaries dictate that the field distributions associated with modes $M_{1,2}$ must be different across domain $D_0$. Let us now consider a general superposition of these two modes. It may be anticipated that, a new set of basis (owing to the degeneracy, there are infinitely many bases) can be constructed such that the standing-wave nature of modes $M_{1,2}$ in domains $D_{1,2}$ remain unaltered and yet the field distribution in $D_0$ forms a traveling-wave pattern due to a particular linear superposition of modes $M_{1,2}$ (Figure \\ref{Fig-Schematic}C), akin to the relation $\\cos(kz)+i \\sin(kz)=\\exp(ikz)$. Such a `mutant' resonator, if it exists, will support a `mutant' optical mode that, in some properly chosen basis, exhibits purely standing- and traveling-wave patterns at the same time---a feature that, to the best of our knowledge, has never been discussed before. We will refer to such a resonator as a hybrid-wave resonator. So far we have kept the discussion abstract. In what follows, we show that this abstract concept can be implemented in realistic optical resonator designs. In the main text, we focus on chip-scale devices but we note that it is straightforward to extend the discussion to implementations using free-space optics. \n\n\\section{Integrated photonics implementation} \n\nTo demonstrate that the concept discussed above can be realized by standard optical components, here we consider an implementation based on integrated photonics as shown in Figure \\ref{Fig-Resonator}A (possible realizations based on free-space optics are discussed in Appendix A). The structure consists of three open ring sections connected by two beam splitters (labeled as $\\text{BS}_{1,2}$). The outer rings here act as Sagnac loop reflectors \\cite{Li2019LPR}. We note that even though variants of this geometry have been considered before for building various optical devices for different applications such as sensing, lasing, and information processing \\cite{Simova2005JOSAB, Das2003OE, Parared2022APR,Shamy2022P,Moslehi1984IEEE}, the peculiar feature that we highlight in this work has escaped attention. \n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=5.4in]{Fig-Resonator.png}\n\t\\caption{An implementation of a hybrid-wave resonator in integrated photonic platforms. (A) A hybrid-wave resonator can be constructed by deforming a ring resonator to introduce two 50\/50 beam splitters, BS$_{1,2}$. The resonator can be divided into three domains: $D_0=D_0' \\cup D_0''$ is the union of the left and right side rings, and domains $D_1$ and $D_2$ represent the top and bottom middle sections, respectively. The field amplitudes at each location and their traveling directions are indicated on the figure and labeled as $a_i$ and $b_i$, $i=1,2,3,4$. (B) Resonant frequencies (horizontal axis) as a function of the beam splitter coupling coefficient $\\kappa$ (vertical axis). Note that each $\\kappa$ value represents an independent resonator structure. Horizontal dashed lines indicate values of $\\kappa$ for which the spectrum is doubly degenerate. The limit $\\kappa=0$ corresponds to conventional microring resonator with two degenerate traveling-wave modes in clockwise and counterclockwise direction while the limit $\\kappa=1$ corresponds to degenerate `knotted' modes. Hybrid-wave modes exist for $\\kappa=1\/\\sqrt{2}$ as discussed in detail in the main text.}\n\t\\label{Fig-Resonator}\n\\end{figure*}\n\nTo investigate the modal structure of this resonator, we will employ a scattering matrix analysis along the junctions indicated in Figure \\ref{Fig-Resonator}A. Away from the beam splitter junctions, the field amplitudes can be decomposed into two traveling waves in opposite directions as shown in Figure \\ref{Fig-Resonator}A. Within the context of scattering matrix formalism \\cite{Yariv2000EL,Van-OMR,Saleh-FP}, the relations between these amplitudes are given by:\n\\begin{equation}\\label{Eq-Matrix_equation}\n\t\\begin{aligned}\n\\relax [a_1, b_1]^T = S_c [a_4, b_2]^T, \\\\\n\t\t[a_4, b_2]^T = S_c [a_3, b_3]^T, \\\\\n\t\t[a_3, b_3]^T = S_c [a_2, b_4]^T, \\\\\n\t\t[a_2, b_4]^T = S_c [a_1, b_1]^T,\n\t\\end{aligned}\n\\end{equation} \nwhere $S_c=\\exp{(i \\phi\/4)} S_b$ and $S_b =\\begin{bmatrix}\n\t\\tau & i \\kappa\\\\\n\ti \\kappa & \\tau\n\\end{bmatrix}$ is the scattering matrix of each beam splitter. Here, $\\tau$ and $\\kappa$ are the field transmission and coupling coefficient of each beam splitter, and in the absence of any loss, they satisfy $\\tau^2+\\kappa^2=1$. For the special case of a 50\/50 beam splitter, which is relevant to our discussion later, $\\tau=\\kappa=1\/\\sqrt{2}$. The phase term $\\phi$ is defined as $\\phi=2\\pi n_\\text{eff}Lf\/c$, where $n_\\text{eff}$ is the effective guiding index, $L$ is total length across the perimeter of the resonator, $f$ is the frequency and $c$ is the speed of light in vacuum. The numerical factor $1\/4$ in the expression of $S_c$ arises because each wave component travels one quarter the length structure between any two consecutive junctions. In the absence of dispersion, $\\phi$ is a linear function of $f$ and thus can be used directly to determine the resonant frequencies (the effect of dispersion is considered later in the full-wave simulations). \n\n\n\\begin{table*}[!t]\n\t\\caption{Field components associated with the degenerate eigenmodes of the structure shown in Figure \\ref{Fig-Resonator} as expressed in the three different bases $B_{1,2,3}$. } \n\t\\centering\n\t\\begin{threeparttable}\n\t\\begin{tabular}{|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|}\n\t\t\\hline\n\t\\multicolumn{2}{|p{1.0in}<{\\centering}|}{Domains} & \\multicolumn{2}{c|}{$D_0'$} & \\multicolumn{2}{c|}{$D_1$} & \\multicolumn{2}{c|}{$D_0''$} & \\multicolumn{2}{c|}{$D_2$} \\\\ \\hline\n\t\t\n\t\\multicolumn{2}{|p{1.0in}<{\\centering}|}{Field components} & $a_1$ & $b_1$ & $a_2$ & $b_2$ & $a_3$ & $b_3$ & $a_4$ & $b_4$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_1$} &$M_1^{(1)}$ & $1$ & $1$ & $e^{i\\frac{\\phi_m+\\pi}{4}}$ & $ie^{i\\frac{3\\phi_m+\\pi}{4}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{3\\phi_m+\\pi}{4}}$ & $e^{i\\frac{\\phi_m+\\pi}{4}}$ \\\\ \\cline{2-10}\n\t\t\n\t&$M_2^{(1)}$ & $1$ & $-1 $& $e^{i\\frac{\\phi_m-\\pi}{4}}$ & $e^{i\\frac{3\\phi_m+\\pi}{4}}$ & $-ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $-e^{i\\frac{3\\phi_m+\\pi}{4}}$ & $-e^{i\\frac{\\phi_m-\\pi}{4}}$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_2$} &$M_1^{(2)}$ & $1$& $-i$ & $\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $i \\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $e^{i\\frac{\\phi_m}{2}}$ & $i e^{i\\frac{\\phi_m}{2}}$ &\\cellcolor{zcolor} $0$ & \\cellcolor{zcolor}$0$ \\\\ \\hhline{|~|---------|}\n\t\t\n\t&$M_2^{(2)}$ & $1$ & $i$ &\\cellcolor{zcolor}$0$& \\cellcolor{zcolor} $0$ & $-e^{i\\frac{\\phi_m}{2}}$ & $i e^{i\\frac{\\phi_m}{2}}$ & $-\\sqrt{2}e^{i\\frac{3\\phi_m}{4}}$ & $i \\sqrt{2}e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_3$} &$M_1^{(3)}$ &\\cellcolor{tcolor} $2$ & \\cellcolor{tcolor} $0$ &$\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $i\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & \\cellcolor{tcolor} $0$ & \\cellcolor{tcolor} $2ie^{i\\frac{\\phi_m}{2}}$ & $-\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $i\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hhline{|~|---------|}\n\t\t\n\t&$M_2^{(3)}$ & \\cellcolor{tcolor} $0$ & \\cellcolor{tcolor} $2$ & $i\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $-\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & \\cellcolor{tcolor} $2ie^{i\\frac{\\phi_m}{2}}$ &\\cellcolor{tcolor} $0$ & $i\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hline\n\t\\end{tabular}\n\t\\begin{tablenotes}\n \\item Note: In this table, pink cells indicate the regions with traveling waves while uncolored cells denote standing waves, and blue cells indicate the regions where the field vanishes. And $\\phi_m=(2m+1)\\pi$, where $m$ is an integer.\n \\end{tablenotes}\n\\end{threeparttable}\n\t\\label{Table1}\n\\end{table*}\n\nBy successive substitution of the right side of each line in Eq. (\\ref{Eq-Matrix_equation}) from the expression in the next line, we find:\n\\begin{equation} \\label{Eq-Consistency}\n\t[a_1,b_1]^T= S_c^4 [a_1,b_1]^T= e^{i \\phi} S_b^4 [a_1,b_1]^T.\n\\end{equation} \nThe resonant modes can then be obtained by imposing a consistency condition requiring the operator $ \\exp (i \\phi) S_b^4$ to have an eigenvalue equal to unity \\cite{Hecht-Optics}. In order to find the values of $\\phi$ that satisfy the consistency condition and hence obtain the eigenfrequencies, we first note that the eigenvalues of $S_b$ are given by $\\lambda_{1,2}=\\exp(\\pm i \\theta)$ and the corresponding eigenvectors are $\\vec{v}_{1,2}=[1,\\pm1]^T$. Here, $\\theta=\\arcsin(\\kappa) \\in [0,\\pi\/2]$ since $\\tau$ and $\\kappa$ take only positive values. The consistency condition then reduces to $\\exp(i\\phi) \\exp(\\pm i4\\theta)=1$, which has two solutions given by $\\phi_m^\\pm=\\pm 4\\theta+2m\\pi$, where $m$ is an integer. In order to better understand this result, we recall that in the absence of beam splitters ($\\kappa=0$) the eigenfrequencies are doubly degenerate (for each resonant frequency there are two modes, one propagating in the CW and the other in the CCW direction) and are given by $\\phi_m=2m\\pi$. Introducing the beam splitters results in coupling between CW and CCW modes and thus lifts the degeneracy. As a result, each degenerate pair described by $\\phi_m$ splits into two modes: blue-shifted $\\phi_m^+$ and red-shifted $\\phi_m^-$. Interestingly, for identical 50\/50 beam splitters, i.e. when $\\kappa=1\/\\sqrt{2}$ corresponding to $\\theta=\\pi\/4$, the eigenmodes associated with $S_b^4$ become degenerate (this is not the case for the eigenvectors of $S_b$) and hence the eigenmodes of the resonators form degenerate pairs satisfying the resonant conditions $\\phi_m^+=\\phi_{m+1}^-=(2m+1)\\pi \\equiv\\phi_m$, as shown in Figure \\ref{Fig-Resonator}B. Before we proceed, we emphasize that the above-predicted degeneracy is not a result of a particular geometric symmetry. For instance, the length of any of the curved sections in the four domains ( $D_0'$, $D_0''$, $D_1$ and $D_2$) can be increased by a multiple of the operation wavelength without affecting the degeneracy despite the fact that it will break part of the geometric symmetries of the structure. \n\nWe now investigate the eigenmode structure associated with these newly-formed degenerate modes. In principle, these eigenmodes can be expressed in any basis of the eigenvectors of $S_b^4$. Choosing a particular basis fixes the vector $[a_1,b_1]^T$ which can be then used to obtain all other field components through Eq. (\\ref{Eq-Matrix_equation}). Table \\ref{Table1} lists the field values for three different bases given by: (1) $B_1=\\{\\vec{v}_{1,2}\\}$, $\\vec{v}_{1,2}=[1,\\pm 1]^T$; (2) $B_2=\\{\\vec{v}_{3,4}\\}$, $\\vec{v}_{3,4}=[1,\\mp i]^T$; and (3) $B_3=\\{\\vec{v}_{5,6}\\}$, $\\vec{v}_{5}=[2,0]^T$, $\\vec{v}_{6}=[0,2]^T$. These bases are related via the linear transformations: $\\vec{v}_{3,4}=\\left[(1\\mp i)\\vec{v}_1+(1\\pm i)\\vec{v}_2 \\right]\/2$ and $\\vec{v}_{5,6}=\\vec{v}_1\\pm\\vec{v}_2$. Expressed differently, $\\vec{v}_{5,6}$ can be also written as $\\vec{v}_{5}=\\vec{v}_3 + \\vec{v}_4$ and $\\vec{v}_{6}=i(\\vec{v}_3 - \\vec{v}_4)$.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=6.5in]{Fig-Eigenmode.png}\n\t\\caption{Eigenmodes of the hybrid-wave resonator presented in Figure \\ref{Fig-Resonator}. (A) and (B) are plots of electric field component perpendicular to the resonator's plane ($|E_z|$) associated with the two degenerate standing-wave modes $M_{1,2}^{(2)}$ which resides in domain $D_0 \\cup D_1$ and $D_0 \\cup D_2$, respectively, where $D_0=D_0' \\cup D_0''$. These corresponds to basis $B_2$. On the other hand, (C) and (D) depict the field distribution corresponding to $M_{1}^{(3)}=M_1^{(2)} + M_2^{(2)}$ and $M_{2}^{(3)}=M_1^{(2)} - M_2^{(2)}$, which feature a hybrid standing- and traveling-wave character. The white arrows in (C) and (D) indicate the traveling direction of traveling wave. The Poynting vectors (black arrows in inset of (C)) circulate around the loop in domain $D_0$ and vanish in domain $D_{1,2}$. The details of the geometry and material parameters used in these simulations are presented in Appendix C.}\n\t\\label{Fig-Eigenmode}\n\\end{figure*}\n\nTable \\ref{Table1} lists the field components associated with the degenerate eigenmodes of the structure shown in Figure \\ref{Fig-Resonator} as expressed in the three different bases $B_{1,2,3}$. The modes are expressed by the vector $M_i^{(j)} \\equiv [a_1, b_1, a_2, b_2, a_3,$ $b_3, a_4, b_4]$ in each basis, where $i$ represents the mode number and $j$ denote the basis number. Note that there is a pure standing wave whenever the field components belonging to any domain have the same amplitude. On the other hand, if one of the field components vanishes, the wave is traveling. Evidently, in basis $B_3$ the eigenmodes exhibit a hybrid-wave character with both standing and traveling waves coexisting as part of the same mode. Obviously, modes $M_{1,2}^{(1)}$ represent a standing wave that extends all over the structure. On the other hand, modes $M_{1,2}^{(2)}$ represent a standing wave that covers only part of the structure. Thus modes $M_{1,2}^{(3)}$ represent potential candidates for satisfying the conditions necessary for generating hybrid-wave modes. Indeed, this is confirmed by the field distribution of modes $M_{1,2}^{(3)}$ which exhibits the dual character of traveling and standing waves covering different domains of the resonators at the same time.\n\nAn important observation here is that for the perfectly closed resonator with no loss, the hybrid-wave modes occur only when the beam splitter is $50\/50$. A slight deviation from this condition removes the degeneracy and destroys the hybrid nature of the modes. This may seem to pose a challenge for experimentally observing these modes. However, realistic resonators are not perfectly closed but rather have losses due to optical absorption, radiation to free space, or due to the coupling to input and output channels. In turn, this will introduce an upper limit on the resonator's quality factor and result in a finite bandwidth of operation, which relaxes the above constraint as discussed in detail in Appendix B and C.\n\nIn order to verify the above predictions, we perform a finite element method (FEM) full-wave simulation (using COMSOL software package) of a realistic implementation for the structure shown in Figure \\ref{Fig-Eigenmode}. The details of the geometry and material parameters used in our simulations are presented in Appendix C. The eigenmodes generated by COMSOL package are in the basis $B_1$ and are shown in Appendix C. Figure \\ref{Fig-Eigenmode}A and \\ref{Fig-Eigenmode}B show the electric field distributions associated with the modes as represented in basis $B_2$ which are generated via linear superposition of the degenerate modes $M_1^{(1)}$ and $M_2^{(1)}$. On the other hand, the field distributions in basis $B_3$ are depicted in Figure \\ref{Fig-Eigenmode}C and \\ref{Fig-Eigenmode}D. The nature of the waves can be deduced from the field distribution. Standing waves are visible through their interference pattern while traveling waves are characterized by uniform fields without interference. These plots are in agreement with the field distributions expected from Table \\ref{Table1} and indeed confirm the results obtained using the scattering matrix analysis above. The field distribution of modes $M_{1,2}^{(3)}$ deserves more attention. At the center of the middle sections (domains $D_{1,2}$), the field features a standing wave while at the center of domains $D_0'$ and $D_0''$, they feature traveling waves. At the beam splitter junction, however, the field is neither a pure standing nor traveling wave. These regions represent transition domains where the wave gradually changes its character. From the Poynting vector point of view, this remarkable mode structure is enabled by the beam splitter junctions acting as interferometric mirrors for domains $D_{1,2}$ while at the same time recirculating the power incident on them from domains $D_0$ in a closed loop. Importantly, the standing-traveling wave nature observed in Figure \\ref{Fig-Eigenmode}C and \\ref{Fig-Eigenmode}D are characteristic of the eigenmodes of a single resonator structure and not associated with a particular steady state solution under certain engineered excitation\\cite{Schmid2011PRA}. Equally important is the fact that the traveling waves in the presented structure are part of the quasi-bound state within the resonator's boundaries and not part of the leaked radiation waves outside the resonator as in the case of a finite Fabry--Perot or photonic crystal geometry for example. From a practical point of view, mapping these field distributions experimentally can be done only by using near-field probes which is possible but not an easy task. In Appendix E, we discuss a more practical scheme for accessing these modes by using input\/output waveguides ports evanescently coupled to the various sections of the resonator.\t\n\n\\section{Local perturbations and sensing applications} \nIn this section, we investigate the effect of local perturbation due to a small scatterer on the eigenmodes of the proposed hybrid-wave resonator---a problem relevant to sensing applications \\cite{Foreman2015AOP}. Given that the modes of the resonator shown in Figure \\ref{Fig-Resonator}A can be written in various bases, only one of which demonstrates the hybrid-wave character, one may wonder if this feature will have any consequences under more general conditions where the particular mode is not selectively excited. This section demonstrates that this is indeed the case. To illustrate this, we consider the situation where a scatterer (nanoparticle or a fiber tip for instance) is located within the evanescent field of the hybrid-wave resonator. In particular, we investigate the two scenarios where the scatterer is located either in the traveling- or standing-wave domains, as shown in Figure \\ref{Fig-Particle-frequency}A.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=5.5in]{Fig-Particle-frequency.png}\n\t\\caption{Effect of local perturbation. (A) A schematic of the resonator structure with nanoparticles scatterers added in the traveling wave (case 1) or standing waves (case 2) regions. In case 1, the presence of the scatterer generates two new optical modes that exhibit either a node (B) or an antinode (C) at the location of the particle with corresponding frequency shifts as shown in (D). In this case, the frequency shifts are independent of the particle location as long as it resides in the traveling wave domain. In case 2, the scatter leaves one of the modes intact with zero frequency shift (E) while at the same time introduces a perturbation to the second mode (F) with a frequency shift that varies with the location of the particle as expected (G). In both cases, $\\delta=2\\pi$ corresponds to a distance of 0.5 $\\mu$m along the perimeter. A rigorous derivation of these results as well as their intuitive explanations are discussed in the main text. In the figure, $\\Delta f_j=f_{j,m}-f_m$ with $j=1,2$. The scatterers in panels (B), (C), (E) and (F) are indicated by small white circles. }\n\t\\label{Fig-Particle-frequency}\n\\end{figure*}\n\nBefore we proceed, it is useful to review the situation for purely traveling-wave resonators (such as microring and microdisk arrangements) and purely standing-wave resonators (such as Bragg and photonic crystal arrangements). In the former, the scatterer breaks the rotational symmetry of the geometry and introduces coupling between the clockwise and counterclockwise modes, leading to a splitting of the eigenfrequency \\cite{Zhu2017PR,Zhu2010NP,Ozdemir2014PNAS}. Importantly, this behavior is independent of the location of the scatterer. In the latter case, however, the situation is quite different. An optical mode that has an electric field node at the scatterer location along the resonator direction will not be affected by its presence. On the other hand, a mode that exhibits an antinode at the location of the scatterer will experience a shift in its eigenfrequency \\cite{Lalouat2007PRB,He2013NJP}. \n\nTo this end, we consider a small perturbation caused by a scatterer having a scattering matrix (Appendix F): \n\\begin{equation} \\label{Eq-Sp}\n\tS_p=e^{i\\phi_p}\\begin{bmatrix}\n\t\tt & i r\\\\\n\t\ti r & t\n\t\\end{bmatrix},\n\\end{equation}\nwhere $r$ and $t$ are reflection and transmission coefficients which are taken to be real numbers satisfying $r^2+t^2=1$ (i.e. no loss); and $\\phi_p=\\arcsin(r)$ is an overall additional phase.\n\n\\textbf{Scatterer located along the traveling-wave domain:}\nHere, we assume a scatterer located along the traveling-wave domain, say $D''_0$, at a fixed distance from the resonator waveguide edge, as shown case 1 in Figure \\ref{Fig-Particle-frequency}A. The angular position of the scatterer is defined by distance $l$ (Figure \\ref{Fig-Particle-frequency}A) and the corresponding phase shift $\\delta\\equiv 2\\pi n_\\text{eff} l f\/c$. By following the same approach used in deriving Eq. (\\ref{Eq-Matrix_equation}), we find that the resonant frequencies (see Appendix G):\n\\begin{equation}\\label{Eq-Particle-traveling}\n\t\\begin{cases}\n\t\t\\phi_{1,m}=\\phi_m, \\\\\n\t\t\\phi_{2,m}=\\phi_m-2\\phi_p.\\\\\n\t\\end{cases}\n\\end{equation}\nwhere $\\phi_{j,m}$ with $j=1,2$ indicates the new eigenfrequencies (due to the perturbation introduced by the scatterer) branched from the unperturbed eigenfrequency $\\phi_m$. In other words, the perturbation shifts the frequency of only one mode while leaving that of the other unchanged. This can be explained by the fact that the new modes, arising because of the perturbation, both exhibit a standing-wave pattern, with the node of one mode and the antinode of the other located at the position of the scatterer. Moreover, Eq. (\\ref{Eq-Particle-traveling}) does not depend on the location of the scatterer along the perimeter of the resonator, as long as it lies in the traveling-wave domain. This behavior is exactly identical to the case of a scatterer introduced in the vicinity of traveling-wave resonator such as a microring or microdisk geometry \\cite{He2013NJP,Zhu2010OE}. \n\nThese predictions are confirmed by performing full-wave simulations, where the scattering was introduced via a nanoparticle. Figure \\ref{Fig-Particle-frequency}B and \\ref{Fig-Particle-frequency}C depict the field distribution of the perturbed modes around the particle. Note that, as expected, the particle modifies the field distribution and creates a standing-wave pattern. Moreover, the node of the first mode and the antinode of the second mode coincide with the particle location along the perimeter of the resonator, which is consistent with our theoretical predictions. As a result, the eigenfrequency of the first mode remains unchanged while that of the second mode experiences a constant shift that does not depend on the particle location, as shown in Figure \\ref{Fig-Particle-frequency}D. As a side note, we remark that the blue dots representing the simulation data in Figure \\ref{Fig-Particle-frequency}D do not exactly coincide with the zero axis as predicted by our scattering matrix analysis but rather exhibit a small shift. This can be explained by recalling that, in our analysis, we treat the particle as a Rayleigh scatterer, whereas in reality higher-order multipole terms must be considered in order to obtain more accurate results. To confirm this, we have performed additional numerical simulations for different particle sizes and indeed observed that this frequency shift decreases as the particle size is reduced (in fact we could not resolve the frequency shift for particles with radii less than 30 nm).\n\n\\textbf{Scatterer located along the standing-wave domain:} Next, we consider the case when the scatterer is located in the standing-wave region, for instance in domain $D_2$, as shown by case 2 in Figure \\ref{Fig-Particle-frequency}A. The resonant frequencies are given by (see Appendix G):\n\\begin{equation}\\label{Eq-Particle-standing}\n\t\\begin{cases}\n\t\t\\phi_{1,m}=\\phi_m, \\\\\n\t\t\\phi_{2,m}=\\phi_m-2\\phi_p [1+(-1)^{m+1}\\cos2\\delta].\\\\\n\t\\end{cases}\n\\end{equation}\nThe mode corresponding to the eigenfrequency $\\phi_{1,m}$ (Figure \\ref{Fig-Particle-frequency}E) is associated with $M_1^{(2)}$ in Figure \\ref{Fig-Eigenmode}A, in which the electric field is zero at domain $D_2$, leading to an unperturbed resonant frequency after adding the particle at domain $D_2$. On the other hand, the eigenfrequency $\\phi_{2,m}$ corresponds to a perturbation of mode $M_2^{(2)}$ in Figure \\ref{Fig-Eigenmode}A. Since mode $M_2^{(2)}$ is a pure standing wave at domain $D_2$, the eigenfrequency $\\phi_{2,m}$ varies with the angular position of the scatterer (Figure \\ref{Fig-Particle-frequency}F and \\ref{Fig-Particle-frequency}G). In Eq. (\\ref{Eq-Particle-standing}), when $m$ is an odd number, it is an antinode at the middle of $D_2$ domain ($\\delta=0$), $\\phi_{2,m}$ experiences the maximum frequency shift $-4\\phi_p$ from $\\phi_m$; when $m$ is an even number, it is a node at the middle of $D_2$ domain, the scatterer will not alter the field much and the resonant frequency will stay the same, i.e., $\\phi_{2,m}=\\phi_m$ at $\\delta=0$. In both scenarios, $\\phi_{2,m}$ will oscillate between $\\phi_m$ an $\\phi_m-4\\phi_p$ as a function of $\\delta$. In our simulation, the fact that $m$ is an odd number can be determined from Figure \\ref{Fig-Eigenmode}, and it is also verified by the electric field around the particle are shown in Figure \\ref{Fig-Particle-frequency}F. The two eigenfrequencies varying with $\\delta$ are shown in Figure \\ref{Fig-Particle-frequency}G, consistent with Eq. (\\ref{Eq-Particle-standing}). \n\nFrom the above analysis, it is clear that a resonator exhibiting hybrid-wave modes will respond very differently to perturbations affecting the standing- or traveling-wave zones. In terms of applications, this can be useful in a number of ways. For instance, the larger splitting in the location of the field maxima in the standing-wave zone can be utilized for selective sensing by functionalizing \\cite{Vollmer2021NatRev} this exact location with receptors that can bind only to a particular molecule while at the same time use the traveling-wave zone for excitation and collection. On the other hand, one can instead use both zones for detecting the presence of more than one molecule (but only one at a time). This can be achieved by attaching different receptors to each zone and inferring the presence of a particular substance by measuring the degree of splitting. We plan to investigate these possibility in future works.\n\n\\section{Conclusion}\nIn conclusion, we have proposed a new concept for optical resonators that exhibit simultaneously co-existing standing and traveling waves as part of the field distribution of the same optical mode but occupying different locations along the resonator geometry. In addition, we have presented a specific example of a structure that implements this concept and verified its standing and traveling wave nature by using scattering matrix analysis and FEM full-wave simulations. We have investigated the robustness of the hybrid-wave feature and shown that the openness of the system allows for a larger bandwidth of operation and thus facilitates experimental observation. In addition, we have described a practical experiential scheme for probing the hybrid-wave nature by using several waveguide channels attached to various sections of the resonator geometry. Furthermore, we have discussed the implication of the hybrid-wave nature for sensing applications by investigating how the eigenmodes of such hybrid-wave resonator interact with a small scatterer located at different sections of the structure, demonstrating that the system's response can be very different depending on the location of the scatterer along the standing- or traveling-wave sections. Another arena where hybrid-wave modes may prove useful is optical manipulation and trapping of particles. For instance, it is expected that a nanoparticle located in the traveling-wave zone will experience radiation pressure and lateral force acting in the direction towards the resonator, while a similar particle located in the standing-wave zone will in addition be subject to a trapping force along the perimeter of the resonator. Furthermore, actively tuning the beam-splitting values may allow for controlling the behavior of the resonator in real time and thus controlling its interaction with nanoparticles.\t\n\nIn addition, the existence of the hybrid-wave modes identified above, which to the best of our knowledge has not been known before, raises several fundamental questions in photonics, nonlinear and quantum optics applications. For instance, it is not a priori clear how such a resonator will behave under nonlinear conditions. Does it exhibit different nonlinear bistability responses than those observed in conventional ring resonators \\cite{Braginsky1989PLA}? Does it provide any new features in terms of frequency comb generation? Can soliton crystals \\cite{Cole2017NPhot} form in the presence of hybrid-wave modes? Along similar lines, it is not clear to what extent the presence of the hybrid-wave modes will impact the dynamics and instability features of laser devices made of such resonators. In the quantum domain, it would be interesting to explore how quantum emitters located inside or in the vicinity of such resonators will behave. How would spontaneous emission and superradiance scale in different sections of the resonators? It has been shown previously that the interaction between atoms and electromagnetic waves featuring a standing field pattern depends on the type of resonator \\cite{Shore1991JOSAB} (standing or traveling wave resonator). What makes these exploratory questions particularly interesting is that the proposed resonator exhibit transition regions (the beam splitter regions in Figure \\ref{Fig-Resonator}A that interpolates between the traveling- and standing-wave domains). Light-matter interaction in this region is expected to differ from its typical behavior in standard traveling- and standing-wave resonators, which may lead to interesting new effects. At the engineering level, our work also raises interesting questions. For instance, is there a fundamental size limit on building hybrid-wave resonators? Can one implement a small volume hybrid-wave mode? What would be the modes of these structures when implemented in material platforms that support plasmonic resonances? We plan to investigate these open questions as well as implementations in other platforms such as acoustics \\cite{Ingard1953JASA} and microwave \\cite{Pozar1990Microwave} in future works.\n\n\\begin{acknowledgements}\nThis project is supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative\t(MURI) Award on Programmable systems with non-Hermitian quantum dynamics\t(Award No. FA9550-21-1-0202). R.E. also acknowledges support from Army Research Office (ARO) (Grant No. W911NF-17-1-0481), National Science Foundation (NSF) (Grant No. ECCS 1807552), Henes Center for Quantum Phenomena, and the Alexander-von-Humboldt Foundation. S.K.O. also acknowledges support from ARO (Grant No. W911NF-16-1-0013, W911NF-17-1-0481), NSF (Grant No. ECCS 1454531, DMR-1420620, ECCS 1807552, ECCS 1757025, ECCS 1807485).\n\\end{acknowledgements}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}