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Calibration (statistics) : Calibration – Check on the accuracy of measurement devices Calibrated probability assessment – Subjective probabilities assigned in a way that historically represents their uncertainty Conformal prediction == References ==
Soft independent modelling of class analogies : Soft independent modelling by class analogy (SIMCA) is a statistical method for supervised classification of data. The method requires a training data set consisting of samples (or objects) with a set of attributes and their class membership. The term soft refers to the fact the classifier can identify samples as belonging to multiple classes and not necessarily producing a classification of samples into non-overlapping classes.
Soft independent modelling of class analogies : In order to build the classification models, the samples belonging to each class need to be analysed using principal component analysis (PCA); only the significant components are retained. For a given class, the resulting model then describes either a line (for one Principal Component or PC), plane (for two PCs) or hyper-plane (for more than two PCs). For each modelled class, the mean orthogonal distance of training data samples from the line, plane, or hyper-plane (calculated as the residual standard deviation) is used to determine a critical distance for classification. This critical distance is based on the F-distribution and is usually calculated using 95% or 99% confidence intervals. New observations are projected into each PC model and the residual distances calculated. An observation is assigned to the model class when its residual distance from the model is below the statistical limit for the class. The observation may be found to belong to multiple classes and a measure of goodness of the model can be found from the number of cases where the observations are classified into multiple classes. The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant.
Soft independent modelling of class analogies : SIMCA as a method of classification has gained widespread use especially in applied statistical fields such as chemometrics and spectroscopic data analysis.
Soft independent modelling of class analogies : Wold, Svante, and Sjostrom, Michael, 1977, SIMCA: A method for analyzing chemical data in terms of similarity and analogy, in Kowalski, B.R., ed., Chemometrics Theory and Application, American Chemical Society Symposium Series 52, Wash., D.C., American Chemical Society, p. 243-282.
Statistical classification : When classification is performed by a computer, statistical methods are normally used to develop the algorithm. Often, the individual observations are analyzed into a set of quantifiable properties, known variously as explanatory variables or features. These properties may variously be categorical (e.g. "A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure). Other classifiers work by comparing observations to previous observations by means of a similarity or distance function. An algorithm that implements classification, especially in a concrete implementation, is known as a classifier. The term "classifier" sometimes also refers to the mathematical function, implemented by a classification algorithm, that maps input data to a category. Terminology across fields is quite varied. In statistics, where classification is often done with logistic regression or a similar procedure, the properties of observations are termed explanatory variables (or independent variables, regressors, etc.), and the categories to be predicted are known as outcomes, which are considered to be possible values of the dependent variable. In machine learning, the observations are often known as instances, the explanatory variables are termed features (grouped into a feature vector), and the possible categories to be predicted are classes. Other fields may use different terminology: e.g. in community ecology, the term "classification" normally refers to cluster analysis.
Statistical classification : Classification and clustering are examples of the more general problem of pattern recognition, which is the assignment of some sort of output value to a given input value. Other examples are regression, which assigns a real-valued output to each input; sequence labeling, which assigns a class to each member of a sequence of values (for example, part of speech tagging, which assigns a part of speech to each word in an input sentence); parsing, which assigns a parse tree to an input sentence, describing the syntactic structure of the sentence; etc. A common subclass of classification is probabilistic classification. Algorithms of this nature use statistical inference to find the best class for a given instance. Unlike other algorithms, which simply output a "best" class, probabilistic algorithms output a probability of the instance being a member of each of the possible classes. The best class is normally then selected as the one with the highest probability. However, such an algorithm has numerous advantages over non-probabilistic classifiers: It can output a confidence value associated with its choice (in general, a classifier that can do this is known as a confidence-weighted classifier). Correspondingly, it can abstain when its confidence of choosing any particular output is too low. Because of the probabilities which are generated, probabilistic classifiers can be more effectively incorporated into larger machine-learning tasks, in a way that partially or completely avoids the problem of error propagation.
Statistical classification : Early work on statistical classification was undertaken by Fisher, in the context of two-group problems, leading to Fisher's linear discriminant function as the rule for assigning a group to a new observation. This early work assumed that data-values within each of the two groups had a multivariate normal distribution. The extension of this same context to more than two groups has also been considered with a restriction imposed that the classification rule should be linear. Later work for the multivariate normal distribution allowed the classifier to be nonlinear: several classification rules can be derived based on different adjustments of the Mahalanobis distance, with a new observation being assigned to the group whose centre has the lowest adjusted distance from the observation.
Statistical classification : Unlike frequentist procedures, Bayesian classification procedures provide a natural way of taking into account any available information about the relative sizes of the different groups within the overall population. Bayesian procedures tend to be computationally expensive and, in the days before Markov chain Monte Carlo computations were developed, approximations for Bayesian clustering rules were devised. Some Bayesian procedures involve the calculation of group-membership probabilities: these provide a more informative outcome than a simple attribution of a single group-label to each new observation.
Statistical classification : Classification can be thought of as two separate problems – binary classification and multiclass classification. In binary classification, a better understood task, only two classes are involved, whereas multiclass classification involves assigning an object to one of several classes. Since many classification methods have been developed specifically for binary classification, multiclass classification often requires the combined use of multiple binary classifiers.
Statistical classification : Most algorithms describe an individual instance whose category is to be predicted using a feature vector of individual, measurable properties of the instance. Each property is termed a feature, also known in statistics as an explanatory variable (or independent variable, although features may or may not be statistically independent). Features may variously be binary (e.g. "on" or "off"); categorical (e.g. "A", "B", "AB" or "O", for blood type); ordinal (e.g. "large", "medium" or "small"); integer-valued (e.g. the number of occurrences of a particular word in an email); or real-valued (e.g. a measurement of blood pressure). If the instance is an image, the feature values might correspond to the pixels of an image; if the instance is a piece of text, the feature values might be occurrence frequencies of different words. Some algorithms work only in terms of discrete data and require that real-valued or integer-valued data be discretized into groups (e.g. less than 5, between 5 and 10, or greater than 10).
Statistical classification : A large number of algorithms for classification can be phrased in terms of a linear function that assigns a score to each possible category k by combining the feature vector of an instance with a vector of weights, using a dot product. The predicted category is the one with the highest score. This type of score function is known as a linear predictor function and has the following general form: score ⁡ ( X i , k ) = β k ⋅ X i , (\mathbf _,k)=_\cdot \mathbf _, where Xi is the feature vector for instance i, βk is the vector of weights corresponding to category k, and score(Xi, k) is the score associated with assigning instance i to category k. In discrete choice theory, where instances represent people and categories represent choices, the score is considered the utility associated with person i choosing category k. Algorithms with this basic setup are known as linear classifiers. What distinguishes them is the procedure for determining (training) the optimal weights/coefficients and the way that the score is interpreted. Examples of such algorithms include Logistic regression – Statistical model for a binary dependent variable Multinomial logistic regression – Regression for more than two discrete outcomes Probit regression – Statistical regression where the dependent variable can take only two valuesPages displaying short descriptions of redirect targets The perceptron algorithm Support vector machine – Set of methods for supervised statistical learning Linear discriminant analysis – Method used in statistics, pattern recognition, and other fields
Statistical classification : Since no single form of classification is appropriate for all data sets, a large toolkit of classification algorithms has been developed. The most commonly used include: Artificial neural networks – Computational model used in machine learning, based on connected, hierarchical functionsPages displaying short descriptions of redirect targets Boosting (machine learning) – Method in machine learning Random forest – Tree-based ensemble machine learning method Genetic programming – Evolving computer programs with techniques analogous to natural genetic processes Gene expression programming – Evolutionary algorithm Multi expression programming Linear genetic programming – type of genetic programming algorithmPages displaying wikidata descriptions as a fallback Kernel estimation – Window functionPages displaying short descriptions of redirect targets k-nearest neighbor – Non-parametric classification methodPages displaying short descriptions of redirect targets Learning vector quantization Linear classifier – Statistical classification in machine learning Fisher's linear discriminant – Method used in statistics, pattern recognition, and other fieldsPages displaying short descriptions of redirect targets Logistic regression – Statistical model for a binary dependent variable Naive Bayes classifier – Probabilistic classification algorithm Perceptron – Algorithm for supervised learning of binary classifiers Quadratic classifier – used in machine learning to separate measurements of two or more classes of objectsPages displaying wikidata descriptions as a fallback Support vector machine – Set of methods for supervised statistical learning Least squares support vector machine Choices between different possible algorithms are frequently made on the basis of quantitative evaluation of accuracy.
Statistical classification : Classification has many applications. In some of these, it is employed as a data mining procedure, while in others more detailed statistical modeling is undertaken. Biological classification – The science of identifying, describing, defining and naming groups of biological organisms Biometric – Metrics related to human characteristicsPages displaying short descriptions of redirect targets identification Computer vision – Computerized information extraction from images Medical image analysis and medical imaging – Technique and process of creating visual representations of the interior of a body Optical character recognition – Computer recognition of visual text Video tracking – Locating a moving object by analyzing frames of a video Credit scoring – Numerical expression representing a person's creditworthinessPages displaying short descriptions of redirect targets Document classification – Process of categorizing documents Drug discovery and development – Process of bringing a new pharmaceutical drug to the market Toxicogenomics – branch of toxicology and genomicsPages displaying wikidata descriptions as a fallback Quantitative structure-activity relationship – Predictive chemical modelPages displaying short descriptions of redirect targets Geostatistics – Branch of statistics focusing on spatial data sets Handwriting recognition – Ability of a computer to receive and interpret intelligible handwritten input Internet search engines Micro-array classification Pattern recognition – Automated recognition of patterns and regularities in data Recommender system – System to predict users' preferences Speech recognition – Automatic conversion of spoken language into text Statistical natural language processing – Field of linguistics and computer sciencePages displaying short descriptions of redirect targets
Variable kernel density estimation : In statistics, adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied depending upon either the location of the samples or the location of the test point. It is a particularly effective technique when the sample space is multi-dimensional.
Variable kernel density estimation : Given a set of samples, _\rbrace , we wish to estimate the density, P ( x → ) ) , at a test point, x → : P ( x → ) ≈ W n h D )\approx W = ∑ i = 1 n w i ^w_ w i = K ( x → − x → i h ) =K\left(-_\right) where n is the number of samples, K is the "kernel", h is its width and D is the number of dimensions in x → . The kernel can be thought of as a simple, linear filter. Using a fixed filter width may mean that in regions of low density, all samples will fall in the tails of the filter with very low weighting, while regions of high density will find an excessive number of samples in the central region with weighting close to unity. To fix this problem, we vary the width of the kernel in different regions of the sample space. There are two methods of doing this: balloon and pointwise estimation. In a balloon estimator, the kernel width is varied depending on the location of the test point. In a pointwise estimator, the kernel width is varied depending on the location of the sample. For multivariate estimators, the parameter, h, can be generalized to vary not just the size, but also the shape of the kernel. This more complicated approach will not be covered here.
Variable kernel density estimation : A common method of varying the kernel width is to make it inversely proportional to the density at the test point: h = k [ n P ( x → ) ] 1 / D )\right]^ where k is a constant. If we back-substitute the estimated PDF, and assuming a Gaussian kernel function, we can show that W is a constant: W = k D ( 2 π ) D / 2 (2\pi )^ A similar derivation holds for any kernel whose normalising function is of the order hD, although with a different constant factor in place of the (2 π)D/2 term. This produces a generalization of the k-nearest neighbour algorithm. That is, a uniform kernel function will return the KNN technique. There are two components to the error: a variance term and a bias term. The variance term is given as: e 1 = P ∫ K 2 n h D = . The bias term is found by evaluating the approximated function in the limit as the kernel width becomes much larger than the sample spacing. By using a Taylor expansion for the real function, the bias term drops out: e 2 = h 2 n ∇ 2 P =\nabla ^P An optimal kernel width that minimizes the error of each estimate can thus be derived.
Variable kernel density estimation : The method is particularly effective when applied to statistical classification. There are two ways we can proceed: the first is to compute the PDFs of each class separately, using different bandwidth parameters, and then compare them as in Taylor. Alternatively, we can divide up the sum based on the class of each sample: P ( j , x → ) ≈ 1 n ∑ i = 1 , c i = j n w i )\approx \sum _=j^w_ where ci is the class of the ith sample. The class of the test point may be estimated through maximum likelihood.
Variable kernel density estimation : akde1d.m - Matlab m-file for one-dimensional adaptive kernel density estimation. libAGF - A C++ library for multivariate adaptive kernel density estimation. akde.m - Matlab function for multivariate (high-dimensional) variable kernel density estimation. == References ==
(1+ε)-approximate nearest neighbor search : (1+ε)-approximate nearest neighbor search is a variant of the nearest neighbor search problem. A solution to the (1+ε)-approximate nearest neighbor search is a point or multiple points within distance (1+ε) R from a query point, where R is the distance between the query point and its true nearest neighbor. Reasons to approximate nearest neighbor search include the space and time costs of exact solutions in high-dimensional spaces (see curse of dimensionality) and that in some domains, finding an approximate nearest neighbor is an acceptable solution. Approaches for solving (1+ε)-approximate nearest neighbor search include kd-trees, Locality Sensitive Hashing and brute force search. == References ==
Alternating decision tree : An alternating decision tree (ADTree) is a machine learning method for classification. It generalizes decision trees and has connections to boosting. An ADTree consists of an alternation of decision nodes, which specify a predicate condition, and prediction nodes, which contain a single number. An instance is classified by an ADTree by following all paths for which all decision nodes are true, and summing any prediction nodes that are traversed.
Alternating decision tree : ADTrees were introduced by Yoav Freund and Llew Mason. However, the algorithm as presented had several typographical errors. Clarifications and optimizations were later presented by Bernhard Pfahringer, Geoffrey Holmes and Richard Kirkby. Implementations are available in Weka and JBoost.
Alternating decision tree : Original boosting algorithms typically used either decision stumps or decision trees as weak hypotheses. As an example, boosting decision stumps creates a set of T weighted decision stumps (where T is the number of boosting iterations), which then vote on the final classification according to their weights. Individual decision stumps are weighted according to their ability to classify the data. Boosting a simple learner results in an unstructured set of T hypotheses, making it difficult to infer correlations between attributes. Alternating decision trees introduce structure to the set of hypotheses by requiring that they build off a hypothesis that was produced in an earlier iteration. The resulting set of hypotheses can be visualized in a tree based on the relationship between a hypothesis and its "parent." Another important feature of boosted algorithms is that the data is given a different distribution at each iteration. Instances that are misclassified are given a larger weight while accurately classified instances are given reduced weight.
Alternating decision tree : An alternating decision tree consists of decision nodes and prediction nodes. Decision nodes specify a predicate condition. Prediction nodes contain a single number. ADTrees always have prediction nodes as both root and leaves. An instance is classified by an ADTree by following all paths for which all decision nodes are true and summing any prediction nodes that are traversed. This is different from binary classification trees such as CART (Classification and regression tree) or C4.5 in which an instance follows only one path through the tree.
Alternating decision tree : The inputs to the alternating decision tree algorithm are: A set of inputs ( x 1 , y 1 ) , … , ( x m , y m ) ,y_),\ldots ,(x_,y_) where x i is a vector of attributes and y i is either -1 or 1. Inputs are also called instances. A set of weights w i corresponding to each instance. The fundamental element of the ADTree algorithm is the rule. A single rule consists of a precondition, a condition, and two scores. A condition is a predicate of the form "attribute <comparison> value." A precondition is simply a logical conjunction of conditions. Evaluation of a rule involves a pair of nested if statements: 1 if (precondition) 2 if (condition) 3 return score_one 4 else 5 return score_two 6 end if 7 else 8 return 0 9 end if Several auxiliary functions are also required by the algorithm: W + ( c ) (c) returns the sum of the weights of all positively labeled examples that satisfy predicate c W − ( c ) (c) returns the sum of the weights of all negatively labeled examples that satisfy predicate c W ( c ) = W + ( c ) + W − ( c ) (c)+W_(c) returns the sum of the weights of all examples that satisfy predicate c The algorithm is as follows: 1 function ad_tree 2 input Set of m training instances 3 4 wi = 1/m for all i 5 a = 1 2 ln W + ( t r u e ) W − ( t r u e ) (true)(true) 6 R0 = a rule with scores a and 0, precondition "true" and condition "true." 7 P = =\ 8 C = = the set of all possible conditions 9 for j = 1 … T 10 p ∈ P , c ∈ C ,c\in get values that minimize z = 2 ( W + ( p ∧ c ) W − ( p ∧ c ) + W + ( p ∧ ¬ c ) W − ( p ∧ ¬ c ) ) + W ( ¬ p ) (p\wedge c)W_(p\wedge c)+(p\wedge \neg c)W_(p\wedge \neg c)\right)+W(\neg p) 11 P + = p ∧ c + p ∧ ¬ c +=p\wedge c+p\wedge \neg c 12 a 1 = 1 2 ln W + ( p ∧ c ) + 1 W − ( p ∧ c ) + 1 =(p\wedge c)+1(p\wedge c)+1 13 a 2 = 1 2 ln W + ( p ∧ ¬ c ) + 1 W − ( p ∧ ¬ c ) + 1 =(p\wedge \neg c)+1(p\wedge \neg c)+1 14 Rj = new rule with precondition p, condition c, and weights a1 and a2 15 w i = w i e − y i R j ( x i ) =w_e^R_(x_) 16 end for 17 return set of Rj The set P grows by two preconditions in each iteration, and it is possible to derive the tree structure of a set of rules by making note of the precondition that is used in each successive rule.
Alternating decision tree : Figure 6 in the original paper demonstrates that ADTrees are typically as robust as boosted decision trees and boosted decision stumps. Typically, equivalent accuracy can be achieved with a much simpler tree structure than recursive partitioning algorithms.
Alternating decision tree : An introduction to Boosting and ADTrees (Has many graphical examples of alternating decision trees in practice). JBoost software implementing ADTrees.
Analogical modeling : Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and other categorization tasks. Analogical modeling is related to connectionism and nearest neighbor approaches, in that it is data-based rather than abstraction-based; but it is distinguished by its ability to cope with imperfect datasets (such as caused by simulated short term memory limits) and to base predictions on all relevant segments of the dataset, whether near or far. In language modeling, AM has successfully predicted empirically valid forms for which no theoretical explanation was known (see the discussion of Finnish morphology in Skousen et al. 2002).
Analogical modeling : Analogy has been considered useful in describing language at least since the time of Saussure. Noam Chomsky and others have more recently criticized analogy as too vague to really be useful (Bańko 1991), an appeal to a deus ex machina. Skousen's proposal appears to address that criticism by proposing an explicit mechanism for analogy, which can be tested for psychological validity.
Analogical modeling : Analogical modeling has been employed in experiments ranging from phonology and morphology (linguistics) to orthography and syntax.
Analogical modeling : Though analogical modeling aims to create a model free from rules seen as contrived by linguists, in its current form it still requires researchers to select which variables to take into consideration. This is necessary because of the so-called "exponential explosion" of processing power requirements of the computer software used to implement analogical modeling. Recent research suggests that quantum computing could provide the solution to such performance bottlenecks (Skousen et al. 2002, see pp 45–47).
Analogical modeling : Computational Linguistics Connectionism Instance-based learning k-nearest neighbor algorithm
Analogical modeling : Royal Skousen (1989). Analogical Modeling of Language (hardcover). Dordrecht: Kluwer Academic Publishers. xii+212pp. ISBN 0-7923-0517-5. Miroslaw Bańko (June 1991). "Review: Analogical Modeling of Language" (PDF). Computational Linguistics. 17 (2): 246–248. Archived from the original (PDF) on 2003-08-02. Royal Skousen (1992). Analogy and Structure. Dordrect: Kluwer Academic Publishers. ISBN 0-7923-1935-4. Royal Skousen; Deryle Lonsdale; Dilworth B. Parkinson, eds. (2002). Analogical Modeling: An exemplar-based approach to language (Human Cognitive Processing vol. 10). Amsterdam/Philadelphia: John Benjamins Publishing Company. p. x+417pp. ISBN 1-58811-302-7. Skousen, Royal. (2003). Analogical Modeling: Exemplars, Rules, and Quantum Computing. Presented at the Berkeley Linguistics Society conference.
Analogical modeling : Analogical Modeling Research Group Homepage LINGUIST List Announcement of Analogical Modeling, Skousen et al. (2002)
Averaged one-dependence estimators : Averaged one-dependence estimators (AODE) is a probabilistic classification learning technique. It was developed to address the attribute-independence problem of the popular naive Bayes classifier. It frequently develops substantially more accurate classifiers than naive Bayes at the cost of a modest increase in the amount of computation.
Averaged one-dependence estimators : AODE seeks to estimate the probability of each class y given a specified set of features x1, ... xn, P(y \| x1, ... xn). To do so it uses the formula P ^ ( y ∣ x 1 , … x n ) = ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y , x i ) ∏ j = 1 n P ^ ( x j ∣ y , x i ) ∑ y ′ ∈ Y ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y ′ , x i ) ∏ j = 1 n P ^ ( x j ∣ y ′ , x i ) (y\mid x_,\ldots x_)=)\geq m(y,x_)\prod _^(x_\mid y,x_)\in Y\sum _)\geq m(y^,x_)\prod _^(x_\mid y^,x_) where P ^ ( ⋅ ) (\cdot ) denotes an estimate of P ( ⋅ ) , F ( ⋅ ) is the frequency with which the argument appears in the sample data and m is a user specified minimum frequency with which a term must appear in order to be used in the outer summation. In recent practice m is usually set at 1.
Averaged one-dependence estimators : We seek to estimate P(y \| x1, ... xn). By the definition of conditional probability P ( y ∣ x 1 , … x n ) = P ( y , x 1 , … x n ) P ( x 1 , … x n ) . ,\ldots x_)=,\ldots x_),\ldots x_). For any 1 ≤ i ≤ n , P ( y , x 1 , … x n ) = P ( y , x i ) P ( x 1 , … x n ∣ y , x i ) . ,\ldots x_)=P(y,x_)P(x_,\ldots x_\mid y,x_). Under an assumption that x1, ... xn are independent given y and xi, it follows that P ( y , x 1 , … x n ) = P ( y , x i ) ∏ j = 1 n P ( x j ∣ y , x i ) . ,\ldots x_)=P(y,x_)\prod _^P(x_\mid y,x_). This formula defines a special form of One Dependence Estimator (ODE), a variant of the naive Bayes classifier that makes the above independence assumption that is weaker (and hence potentially less harmful) than the naive Bayes' independence assumption. In consequence, each ODE should create a less biased estimator than naive Bayes. However, because the base probability estimates are each conditioned by two variables rather than one, they are formed from less data (the training examples that satisfy both variables) and hence are likely to have more variance. AODE reduces this variance by averaging the estimates of all such ODEs.
Averaged one-dependence estimators : Like naive Bayes, AODE does not perform model selection and does not use tuneable parameters. As a result, it has low variance. It supports incremental learning whereby the classifier can be updated efficiently with information from new examples as they become available. It predicts class probabilities rather than simply predicting a single class, allowing the user to determine the confidence with which each classification can be made. Its probabilistic model can directly handle situations where some data are missing. AODE has computational complexity O ( l n 2 ) ) at training time and O ( k n 2 ) ) at classification time, where n is the number of features, l is the number of training examples and k is the number of classes. This makes it infeasible for application to high-dimensional data. However, within that limitation, it is linear with respect to the number of training examples and hence can efficiently process large numbers of training examples.
Averaged one-dependence estimators : The free Weka machine learning suite includes an implementation of AODE.
Averaged one-dependence estimators : Cluster-weighted modeling == References ==
Automated Pain Recognition : Automated Pain Recognition (APR) is a method for objectively measuring pain and at the same time represents an interdisciplinary research area that comprises elements of medicine, psychology, psychobiology, and computer science. The focus is on computer-aided objective recognition of pain, implemented on the basis of machine learning. Automated pain recognition allows for the valid, reliable detection and monitoring of pain in people who are unable to communicate verbally. The underlying machine learning processes are trained and validated in advance by means of unimodal or multimodal body signals. Signals used to detect pain may include facial expressions or gestures and may also be of a (psycho-)physiological or paralinguistic nature. To date, the focus has been on identifying pain intensity, but visionary efforts are also being made to recognize the quality, site, and temporal course of pain. However, the clinical implementation of this approach is a controversial topic in the field of pain research. Critics of automated pain recognition argue that pain diagnosis can only be performed subjectively by humans.
Automated Pain Recognition : Pain diagnosis under conditions where verbal reporting is restricted - such as in verbally and/or cognitively impaired people or in patients who are sedated or mechanically ventilated - is based on behavioral observations by trained professionals. However, all known observation procedures (e.g., Zurich Observation Pain Assessment (ZOPA)); Pain Assessment in Advanced Dementia Scale (PAINAD) require a great deal of specialist expertise. These procedures can be made more difficult by perception- and interpretation-related misjudgments on the part of the observer. With regard to the differences in design, methodology, evaluation sample, and conceptualization of the phenomenon of pain, it is difficult to compare the quality criteria of the various tools. Even if trained personnel could theoretically record pain intensity several times a day using observation instruments, it would not be possible to measure it every minute or second. In this respect, the goal of automated pain recognition is to use valid, robust pain response patterns that can be recorded multimodally for a temporally dynamic, high-resolution, automated pain intensity recognition system.
Automated Pain Recognition : For automated pain recognition, pain-relevant parameters are usually recorded using non-invasive sensor technology, which captures data on the (physical) responses of the person in pain. This can be achieved with camera technology that captures facial expressions, gestures, or posture, while audio sensors record paralinguistic features. (Psycho-)physiological information such as muscle tone and heart rate can be collected via biopotential sensors (electrodes). Pain recognition requires the extraction of meaningful characteristics or patterns from the data collected. This is achieved using machine learning techniques that are able to provide an assessment of the pain after training (learning), e.g., "no pain," "mild pain," or "severe pain."
Automated Pain Recognition : Although the phenomenon of pain comprises different components (sensory discriminative, affective (emotional), cognitive, vegetative, and (psycho-)motor), automated pain recognition currently relies on the measurable parameters of pain responses. These can be divided roughly into the two main categories of "physiological responses" and "behavioral responses".
Automated Pain Recognition : After the recording, pre-processing (e.g., filtering), and extraction of relevant features, an optional information fusion can be performed. During this process, modalities from different signal sources are merged to generate new or more precise knowledge. The pain is classified using machine learning processes. The method chosen has a significant influence on the recognition rate and depends greatly on the quality and granularity of the underlying data. Similar to the field of affective computing, the following classifiers are currently being used: Support Vector Machine (SVM): The goal of an SVM is to find a clearly defined optimal hyperplane with the greatest minimal distance to two (or more) classes to be separated. The hyperplane acts as a decision function for classifying an unknown pattern. Random Forest (RF): RF is based on the composition of random, uncorrelated decision trees. An unknown pattern is judged individually by each tree and assigned to a class. The final classification of the patterns by the RF is then based on a majority decision. k-Nearest Neighbors (k-NN): The k-NN algorithm classifies an unknown object using the class label that most commonly classifies the k neighbors closest to it. Its neighbors are determined using a selected similarity measure (e.g., Euclidean distance, Jaccard coefficient, etc.). Artificial neural networks (ANNs): ANNs are inspired by biological neural networks and model their organizational principles and processes in a very simplified manner. Class patterns are learned by adjusting the weights of the individual neuronal connections.
Automated Pain Recognition : In order to classify pain in a valid manner, it is necessary to create representative, reliable, and valid pain databases that are available to the machine learner for training. An ideal database would be sufficiently large and would consist of natural (not experimental), high-quality pain responses. However, natural responses are difficult to record and can only be obtained to a limited extent; in most cases they are characterized by suboptimal quality. The databases currently available therefore contain experimental or quasi-experimental pain responses, and each database is based on a different pain model. The following list shows a selection of the most relevant pain databases (last updated: April 2020): UNBC-McMaster Shoulder Pain BioVid Heat Pain EmoPain SenseEmotion X-ITE Pain
Automated Pain Recognition : Automated Pain Research Group at the University of Ulm, Germany
C4.5 algorithm : C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date". It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008.
C4.5 algorithm : C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set S = s 1 , s 2 , . . . ,s_,... of already classified samples. Each sample s i consists of a p-dimensional vector ( x 1 , i , x 2 , i , . . . , x p , i ) ,x_,...,x_) , where the x j represent attribute values or features of the sample, as well as the class in which s i falls. At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists. This algorithm has a few base cases. All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class. None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class. Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value.
C4.5 algorithm : J48 is an open source Java implementation of the C4.5 algorithm in the Weka data mining tool.
C4.5 algorithm : C4.5 made a number of improvements to ID3. Some of these are: Handling both continuous and discrete attributes - In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it. Handling training data with missing attribute values - C4.5 allows attribute values to be marked as ? for missing. Missing attribute values are simply not used in gain and entropy calculations. Handling attributes with differing costs. Pruning trees after creation - C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes.
C4.5 algorithm : Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are: Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude) Memory usage - C5.0 is more memory efficient than C4.5 Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees. Support for boosting - Boosting improves the trees and gives them more accuracy. Weighting - C5.0 allows you to weight different cases and misclassification types. Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful. Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL).
C4.5 algorithm : ID3 algorithm Modifying C4.5 to generate temporal and causal rules
C4.5 algorithm : Original implementation on Ross Quinlan's homepage: http://www.rulequest.com/Personal/ See5 and C5.0
Chi-square automatic interaction detection : Chi-square automatic interaction detection (CHAID) is a decision tree technique based on adjusted significance testing (Bonferroni correction, Holm-Bonferroni testing).
Chi-square automatic interaction detection : CHAID is based on a formal extension of AID (Automatic Interaction Detection) and THAID (THeta Automatic Interaction Detection) procedures of the 1960s and 1970s, which in turn were extensions of earlier research, including that performed by Belson in the UK in the 1950s. In 1975, the CHAID technique itself was developed in South Africa. It was published in 1980 by Gordon V. Kass, who had completed a PhD thesis on the topic. A history of earlier supervised tree methods can be found in Ritschard, including a detailed description of the original CHAID algorithm and the exhaustive CHAID extension by Biggs, De Ville, and Suen.
Chi-square automatic interaction detection : CHAID can be used for prediction (in a similar fashion to regression analysis, this version of CHAID being originally known as XAID) as well as classification, and for detection of interaction between variables. In practice, CHAID is often used in the context of direct marketing to select groups of consumers to predict how their responses to some variables affect other variables, although other early applications were in the fields of medical and psychiatric research. Like other decision trees, CHAID's advantages are that its output is highly visual and easy to interpret. Because it uses multiway splits by default, it needs rather large sample sizes to work effectively, since with small sample sizes the respondent groups can quickly become too small for reliable analysis. One important advantage of CHAID over alternatives such as multiple regression is that it is non-parametric.
Chi-square automatic interaction detection : Bonferroni correction Chi-squared distribution Decision tree learning Latent class model Market segment Multiple comparisons Structural equation modeling
Chi-square automatic interaction detection : Press, Laurence I.; Rogers, Miles S.; & Shure, Gerald H.; An interactive technique for the analysis of multivariate data, Behavioral Science, Vol. 14 (1969), pp. 364–370 Hawkins, Douglas M.; and Kass, Gordon V.; Automatic Interaction Detection, in Hawkins, Douglas M. (ed), Topics in Applied Multivariate Analysis, Cambridge University Press, Cambridge, 1982, pp. 269–302 Hooton, Thomas M.; Haley, Robert W.; Culver, David H.; White, John W.; Morgan, W. Meade; & Carroll, Raymond J.; The Joint Associations of Multiple Risk Factors with the Occurrence of Nosocomial Infections, American Journal of Medicine, Vol. 70, (1981), pp. 960–970 Brink, Susanne; & Van Schalkwyk, Dirk J.; Serum ferritin and mean corpuscular volume as predictors of bone marrow iron stores, South African Medical Journal, Vol. 61, (1982), pp. 432–434 McKenzie, Dean P.; McGorry, Patrick D.; Wallace, Chris S.; Low, Lee H.; Copolov, David L.; & Singh, Bruce S.; Constructing a Minimal Diagnostic Decision Tree, Methods of Information in Medicine, Vol. 32 (1993), pp. 161–166 Magidson, Jay; The CHAID approach to segmentation modeling: chi-squared automatic interaction detection, in Bagozzi, Richard P. (ed); Advanced Methods of Marketing Research, Blackwell, Oxford, GB, 1994, pp. 118–159 Hawkins, Douglas M.; Young, S. S.; & Rosinko, A.; Analysis of a large structure-activity dataset using recursive partitioning, Quantitative Structure-Activity Relationships, Vol. 16, (1997), pp. 296–302
Chi-square automatic interaction detection : Luchman, J.N.; CHAID: Stata module to conduct chi-square automated interaction detection, Available for free download, or type within Stata: ssc install chaid. Luchman, J.N.; CHAIDFOREST: Stata module to conduct random forest ensemble classification based on chi-square automated interaction detection (CHAID) as base learner, Available for free download, or type within Stata: ssc install chaidforest. IBM SPSS Decision Trees grows exhaustive CHAID trees as well as a few other types of trees such as CART. An R package CHAID is available on R-Forge.
Classifier chains : Classifier chains is a machine learning method for problem transformation in multi-label classification. It combines the computational efficiency of the binary relevance method while still being able to take the label dependencies into account for classification.
Classifier chains : Several problem transformation methods exist. One of them is the Binary Relevance method (BR). Given a set of labels L \, and a data set with instances of the form ( x , Y ) \, where x \, is a feature vector and Y ⊆ L is a set of labels assigned to the instance. BR transforms the data set into \| L \| data sets and learns \| L \| binary classifiers H : X → for each label l ∈ L . During this process the information about dependencies between labels is not preserved. This can lead to a situation where a set of labels is assigned to an instance although these labels never co-occur together in the data set. Thus, information about label co-occurrence can help to assign correct label combinations. Loss of this information can in some cases lead to a decrease in classification performance. Another approach, which takes into account label correlations, is the Label Powerset method (LP). Each combination of labels in a data set is considered to be a single label. After transformation a single-label classifier H : X → P ( L ) (L) is trained where P ( L ) (L) is the power set of all labels in L . The main drawback of this approach is that the number of label combinations grows exponentially with the number of labels. For example, a multi-label data set with 10 labels can have up to 2 10 = 1024 =1024 label combinations. This increases the run-time of classification. The Classifier Chains method is based on the BR method and it is efficient even on a big number of labels. Furthermore, it considers dependencies between labels.
Classifier chains : For a given set of labels L \, the Classifier Chain model (CC) learns \| L \| classifiers as in the Binary Relevance method. All classifiers are linked in a chain through feature space. Given a data set where the i -th instance has the form ( x i , Y i ) ,Y_)\, where Y i \, is a subset of labels, x i \, is a set of features. The data set is transformed in \| L \| data sets where instances of the j -th data set has the form ( ( x i , l 1 , . . . , l j − 1 ) , l j ) , l j ∈ ,l_,...,l_),l_),l_\in \ . If the j -th label was assigned to the instance then l j \, is 1 , otherwise it is 0 . Thus, classifiers build a chain where each of them learns binary classification of a single label. The features given to each classifier are extended with binary values that indicate which of previous labels were assigned to the instance. By classifying new instances the labels are again predicted by building a chain of classifiers. The classification begins with the first classifier C 1 \, and proceeds to the last one C \| L \| \, by passing label information between classifiers through the feature space. Hence, the inter-label dependency is preserved. However, the result can vary for different order of chains. For example, if a label often co-occur with some other label, then only instances of the label which comes later in the chain will have information about the other one in its feature vector. In order to solve this problem and increase accuracy it is possible to use ensemble of classifiers. In Ensemble of Classifier Chains (ECC) several CC classifiers can be trained with random order of chains (i.e. random order of labels) on a random subset of data set. Labels of a new instance are predicted by each classifier separately. After that, the total number of predictions or "votes" is counted for each label. The label is accepted if it was predicted by a percentage of classifiers that is bigger than some threshold value.
Classifier chains : There is also regressor chains, which themselves can resemble vector autoregression models if the order of the chain makes sure temporal order is respected.
Classifier chains : Better Classifier Chains for Multi-label Classification Presentation on Classifier Chains by Jesse Read and Fernando Pérez Cruz
Co-training : Co-training is a machine learning algorithm used when there are only small amounts of labeled data and large amounts of unlabeled data. One of its uses is in text mining for search engines. It was introduced by Avrim Blum and Tom Mitchell in 1998.
Co-training : Co-training is a semi-supervised learning technique that requires two views of the data. It assumes that each example is described using two different sets of features that provide complementary information about the instance. Ideally, the two views are conditionally independent (i.e., the two feature sets of each instance are conditionally independent given the class) and each view is sufficient (i.e., the class of an instance can be accurately predicted from each view alone). Co-training first learns a separate classifier for each view using any labeled examples. The most confident predictions of each classifier on the unlabeled data are then used to iteratively construct additional labeled training data. The original co-training paper described experiments using co-training to classify web pages into "academic course home page" or not; the classifier correctly categorized 95% of 788 web pages with only 12 labeled web pages as examples. The paper has been cited over 1000 times, and received the 10 years Best Paper Award at the 25th International Conference on Machine Learning (ICML 2008), a renowned computer science conference. Krogel and Scheffer showed in 2004 that co-training is only beneficial if the data sets are independent; that is, if one of the classifiers correctly labels a data point that the other classifier previously misclassified. If the classifiers agree on all unlabeled data, i.e. they are dependent, labeling the data does not create new information. In an experiment where dependence of the classifiers was greater than 60%, results worsened.
Co-training : Co-training has been used to classify web pages using the text on the page as one view and the anchor text of hyperlinks on other pages that point to the page as the other view. Simply put, the text in a hyperlink on one page can give information about the page it links to. Co-training can work on "unlabeled" text that has not already been classified or tagged, which is typical for the text appearing on web pages and in emails. According to Tom Mitchell, "The features that describe a page are the words on the page and the links that point to that page. The co-training models utilize both classifiers to determine the likelihood that a page will contain data relevant to the search criteria." Text on websites can judge the relevance of link classifiers, hence the term "co-training". Mitchell claims that other search algorithms are 86% accurate, whereas co-training is 96% accurate. Co-training was used on FlipDog.com, a job search site, and by the U.S. Department of Labor, for a directory of continuing and distance education. It has been used in many other applications, including statistical parsing and visual detection.
Co-training : Notes Chakrabarti, Soumen (2002). Mining the Web: Discovering Knowledge from Hypertext Data. Morgan-Kaufmann Publishers. p. 352. ISBN 978-1-55860-754-5. Nigam, Kamal; Rayid Ghani (2000). "Analyzing the Effectiveness and Applicability of Co-training". Proceedings of the Ninth International Conference on Information and Knowledge Management. NY, USA: ACM: 86–93. CiteSeerX 10.1.1.37.4669. Abney, Steven (2007). Semisupervised Learning for Computational Linguistics. CRC Computer Science & Data Analysis. Chapman & Hall. p. 308. ISBN 978-1-58488-559-7. Wang, William Yang; Kapil Thadani; Kathleen McKeown (2011). Identifying Event Descriptions using Co-training with Online News Summaries (PDF). the 5th International Joint Conference on Natural Language Processing (IJCNLP 2011). AFNLP & ACL.
Co-training : Lecture by Tom Mitchell introducing co-training and other semi-supervised machine learning for use on unlabeled data Lecture by Avrim Blum on semi-supervised learning, including co-training Co-Training group at Pittsburgh Science of Learning Center
CoBoosting : CoBoost is a semi-supervised training algorithm proposed by Collins and Singer in 1999. The original application for the algorithm was the task of named-entity recognition using very weak learners, but it can be used for performing semi-supervised learning in cases where data features may be redundant. It may be seen as a combination of co-training and boosting. Each example is available in two views (subsections of the feature set), and boosting is applied iteratively in alternation with each view using predicted labels produced in the alternate view on the previous iteration. CoBoosting is not a valid boosting algorithm in the PAC learning sense.
CoBoosting : CoBoosting was an attempt by Collins and Singer to improve on previous attempts to leverage redundancy in features for training classifiers in a semi-supervised fashion. CoTraining, a seminal work by Blum and Mitchell, was shown to be a powerful framework for learning classifiers given a small number of seed examples by iteratively inducing rules in a decision list. The advantage of CoBoosting to CoTraining is that it generalizes the CoTraining pattern so that it could be used with any classifier. CoBoosting accomplishes this feat by borrowing concepts from AdaBoost. In both CoTrain and CoBoost the training and testing example sets must follow two properties. The first is that the feature space of the examples can separated into two feature spaces (or views) such that each view is sufficiently expressive for classification. Formally, there exist two functions f 1 ( x 1 ) (x_) and f 2 ( x 2 ) (x_) such that for all examples x = ( x 1 , x 2 ) ,x_) , f 1 ( x 1 ) = f 2 ( x 2 ) = f ( x ) (x_)=f_(x_)=f(x) . While ideal, this constraint is in fact too strong due to noise and other factors, and both algorithms instead seek to maximize the agreement between the two functions. The second property is that the two views must not be highly correlated.
CoBoosting : Input: i = 1 n ,x_)\_^ , i = 1 m \_^ Initialize: ∀ i , j : g j 0 ( x i ) = 0 ^()=0 . For t = 1 , . . . , T and for j = 1 , 2 : Set pseudo-labels: y i ^ = =\left\y_,1\leq i\leq m\\sign(g_^()),m<i\leq n\end\right. Set virtual distribution: D t j ( i ) = 1 Z t j e − y i ^ g j t − 1 ( x j , i ) ^(i)=^e^g_^() where Z t j = ∑ i = 1 n e − y i ^ g j t − 1 ( x j , i ) ^=\sum _^e^g_^() Find the weak hypothesis h t j ^ that minimizes expanded training error. Choose value for α t that minimizes expanded training error. Update the value for current strong non-thresholded classifier: ∀ i : g j t ( x j , i ) = g j t − 1 ( x j , i ) + α t h t j ( x j , i ) ^()=g_^()+\alpha _h_^() The final strong classifier output is f ( x ) = s i g n ( ∑ j = 1 2 g j T ( x j ) ) )=sign\left(\sum _^g_^()\right)
CoBoosting : CoBoosting builds on the AdaBoost algorithm, which gives CoBoosting its generalization ability since AdaBoost can be used in conjunction with many other learning algorithms. This build up assumes a two class classification task, although it can be adapted to multiple class classification. In the AdaBoost framework, weak classifiers are generated in series as well as a distribution over examples in the training set. Each weak classifier is given a weight and the final strong classifier is defined as the sign of the sum of the weak classifiers weighted by their assigned weight. (See AdaBoost Wikipedia page for notation). In the AdaBoost framework Schapire and Singer have shown that the training error is bounded by the following equation: 1 m ∑ i = 1 m e ( − y i ( ∑ t = 1 T α t h t ( x i ) ) ) = ∏ t Z t \sum _^e^\left(\sum _^\alpha _h_()\right)\right)=\prod _Z_ Where Z t is the normalizing factor for the distribution D t + 1 . Solving for Z t in the equation for D t ( i ) (i) we get: Z t = ∑ i : x t ∉ x i D t ( i ) + ∑ i : x t ∈ x i D t ( i ) e − y i α i h t ( x i ) =\sum _\notin x_D_(i)+\sum _\in x_D_(i)e^\alpha _h_() Where x t is the feature selected in the current weak hypothesis. Three equations are defined describing the sum of the distributions for in which the current hypothesis has selected either correct or incorrect label. Note that it is possible for the classifier to abstain from selecting a label for an example, in which the label provided is 0. The two labels are selected to be either -1 or 1. W 0 = ∑ i : h t ( x i ) = 0 D t ( i ) =\sum _(x_)=0D_(i) W + = ∑ i : h t ( x i ) = y i D t ( i ) =\sum _(x_)=y_D_(i) W − = ∑ i : h t ( x i ) = − y i D t ( i ) =\sum _(x_)=-y_D_(i) Schapire and Singer have shown that the value Z t can be minimized (and thus the training error) by selecting α t to be as follows: α t = 1 2 ln ⁡ ( W + W − ) =\ln \left(\right) Providing confidence values for the current hypothesized classifier based on the number of correctly classified vs. the number of incorrectly classified examples weighted by the distribution over examples. This equation can be smoothed to compensate for cases in which W − is too small. Deriving Z t from this equation we get: Z t = W 0 + 2 W + W − =W_+2W_ The training error thus is minimized by selecting the weak hypothesis at every iteration that minimizes the previous equation.
CoBoosting : CoBoosting extends this framework in the case where one has a labeled training set (examples from 1... m ) and an unlabeled training set (from m 1 . . . n ...n ), as well as satisfy the conditions of redundancy in features in the form of x i = ( x 1 , i , x 2 , i ) =(x_,x_) . The algorithm trains two classifiers in the same fashion as AdaBoost that agree on the labeled training sets correct labels and maximizes the agreement between the two classifiers on the unlabeled training set. The final classifier is the sign of the sum of the two strong classifiers. The bounded training error on CoBoost is extended as follows, where Z C O is the extension of Z t : Z C O = ∑ i = 1 m e − y i g 1 ( x 1 , i ) + ∑ i = 1 m e − y i g 2 ( x 2 , i ) + ∑ i = m + 1 n e − f 2 ( x 2 , i ) g 1 ( x 1 , i ) + ∑ i = m + 1 n e − f 1 ( x 1 , i ) g 2 ( x 2 , i ) =\sum _^e^g_()+\sum _^e^g_()+\sum _^e^()g_()+\sum _^e^()g_() Where g j is the summation of hypotheses weight by their confidence values for the j t h view (j = 1 or 2). f j is the sign of g j . At each iteration of CoBoost both classifiers are updated iteratively. If g j t − 1 ^ is the strong classifier output for the j t h view up to the t − 1 iteration we can set the pseudo-labels for the jth update to be: y i ^ = =\left\y_1\leq i\leq m\\sign(g_^())m<i\leq n\end\right. In which 3 − j selects the other view to the one currently being updated. Z C O is split into two such that Z C O = Z C O 1 + Z C O 2 =Z_^+Z_^ . Where Z C O j = ∑ i = 1 n e − y i ^ ( g j t − 1 ( x i ) + α t j g t j ( x j , i ) ) ^=\sum _^e^(g_^()+\alpha _^g_^()) The distribution over examples for each view j at iteration t is defined as follows: D t j ( i ) = 1 Z t j e − y i ^ g j t − 1 ( x j , i ) ^(i)=^e^g_^() At which point Z C O j ^ can be rewritten as Z C O j = ∑ i = 1 n D t j e − y i ^ α t j g t j ( x j , i ) ^=\sum _^D_^e^\alpha _^g_^() Which is identical to the equation in AdaBoost. Thus the same process can be used to update the values of α t j ^ as in AdaBoost using y i ^ and D t j ^ . By alternating this, the minimization of Z C O 1 ^ and Z C O 2 ^ in this fashion Z C O is minimized in a greedy fashion.
CoBoosting : === Footnotes ===
Decision boundary : In a statistical-classification problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each class. The classifier will classify all the points on one side of the decision boundary as belonging to one class and all those on the other side as belonging to the other class. A decision boundary is the region of a problem space in which the output label of a classifier is ambiguous. If the decision surface is a hyperplane, then the classification problem is linear, and the classes are linearly separable. Decision boundaries are not always clear cut. That is, the transition from one class in the feature space to another is not discontinuous, but gradual. This effect is common in fuzzy logic based classification algorithms, where membership in one class or another is ambiguous. Decision boundaries can be approximations of optimal stopping boundaries. The decision boundary is the set of points of that hyperplane that pass through zero. For example, the angle between a vector and points in a set must be zero for points that are on or close to the decision boundary. Decision boundary instability can be incorporated with generalization error as a standard for selecting the most accurate and stable classifier.
Decision boundary : In the case of backpropagation based artificial neural networks or perceptrons, the type of decision boundary that the network can learn is determined by the number of hidden layers the network has. If it has no hidden layers, then it can only learn linear problems. If it has one hidden layer, then it can learn any continuous function on compact subsets of Rn as shown by the universal approximation theorem, thus it can have an arbitrary decision boundary. In particular, support vector machines find a hyperplane that separates the feature space into two classes with the maximum margin. If the problem is not originally linearly separable, the kernel trick can be used to turn it into a linearly separable one, by increasing the number of dimensions. Thus a general hypersurface in a small dimension space is turned into a hyperplane in a space with much larger dimensions. Neural networks try to learn the decision boundary which minimizes the empirical error, while support vector machines try to learn the decision boundary which maximizes the empirical margin between the decision boundary and data points.
Decision boundary : Discriminant function Hyperplane separation theorem
Decision boundary : Duda, Richard O.; Hart, Peter E.; Stork, David G. (2001). Pattern Classification (2nd ed.). New York: Wiley. pp. 215–281. ISBN 0-471-05669-3.
Decision tree learning : Decision tree learning is a supervised learning approach used in statistics, data mining and machine learning. In this formalism, a classification or regression decision tree is used as a predictive model to draw conclusions about a set of observations. Tree models where the target variable can take a discrete set of values are called classification trees; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically real numbers) are called regression trees. More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences. Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity. In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data (but the resulting classification tree can be an input for decision making).
Decision tree learning : Decision tree learning is a method commonly used in data mining. The goal is to create a model that predicts the value of a target variable based on several input variables. A decision tree is a simple representation for classifying examples. For this section, assume that all of the input features have finite discrete domains, and there is a single target feature called the "classification". Each element of the domain of the classification is called a class. A decision tree or a classification tree is a tree in which each internal (non-leaf) node is labeled with an input feature. The arcs coming from a node labeled with an input feature are labeled with each of the possible values of the target feature or the arc leads to a subordinate decision node on a different input feature. Each leaf of the tree is labeled with a class or a probability distribution over the classes, signifying that the data set has been classified by the tree into either a specific class, or into a particular probability distribution (which, if the decision tree is well-constructed, is skewed towards certain subsets of classes). A tree is built by splitting the source set, constituting the root node of the tree, into subsets—which constitute the successor children. The splitting is based on a set of splitting rules based on classification features. This process is repeated on each derived subset in a recursive manner called recursive partitioning. The recursion is completed when the subset at a node has all the same values of the target variable, or when splitting no longer adds value to the predictions. This process of top-down induction of decision trees (TDIDT) is an example of a greedy algorithm, and it is by far the most common strategy for learning decision trees from data. In data mining, decision trees can be described also as the combination of mathematical and computational techniques to aid the description, categorization and generalization of a given set of data. Data comes in records of the form: ( x , Y ) = ( x 1 , x 2 , x 3 , . . . , x k , Y ) ,Y)=(x_,x_,x_,...,x_,Y) The dependent variable, Y , is the target variable that we are trying to understand, classify or generalize. The vector x is composed of the features, x 1 , x 2 , x 3 ,x_,x_ etc., that are used for that task.
Decision tree learning : Decision trees used in data mining are of two main types: Classification tree analysis is when the predicted outcome is the class (discrete) to which the data belongs. Regression tree analysis is when the predicted outcome can be considered a real number (e.g. the price of a house, or a patient's length of stay in a hospital). The term classification and regression tree (CART) analysis is an umbrella term used to refer to either of the above procedures, first introduced by Breiman et al. in 1984. Trees used for regression and trees used for classification have some similarities – but also some differences, such as the procedure used to determine where to split. Some techniques, often called ensemble methods, construct more than one decision tree: Boosted trees Incrementally building an ensemble by training each new instance to emphasize the training instances previously mis-modeled. A typical example is AdaBoost. These can be used for regression-type and classification-type problems. Committees of decision trees (also called k-DT), an early method that used randomized decision tree algorithms to generate multiple different trees from the training data, and then combine them using majority voting to generate output. Bootstrap aggregated (or bagged) decision trees, an early ensemble method, builds multiple decision trees by repeatedly resampling training data with replacement, and voting the trees for a consensus prediction. A random forest classifier is a specific type of bootstrap aggregating Rotation forest – in which every decision tree is trained by first applying principal component analysis (PCA) on a random subset of the input features. A special case of a decision tree is a decision list, which is a one-sided decision tree, so that every internal node has exactly 1 leaf node and exactly 1 internal node as a child (except for the bottommost node, whose only child is a single leaf node). While less expressive, decision lists are arguably easier to understand than general decision trees due to their added sparsity, permit non-greedy learning methods and monotonic constraints to be imposed. Notable decision tree algorithms include: ID3 (Iterative Dichotomiser 3) C4.5 (successor of ID3) CART (Classification And Regression Tree) OC1 (Oblique classifier 1). First method that created multivariate splits at each node. Chi-square automatic interaction detection (CHAID). Performs multi-level splits when computing classification trees. MARS: extends decision trees to handle numerical data better. Conditional Inference Trees. Statistics-based approach that uses non-parametric tests as splitting criteria, corrected for multiple testing to avoid overfitting. This approach results in unbiased predictor selection and does not require pruning. ID3 and CART were invented independently at around the same time (between 1970 and 1980), yet follow a similar approach for learning a decision tree from training tuples. It has also been proposed to leverage concepts of fuzzy set theory for the definition of a special version of decision tree, known as Fuzzy Decision Tree (FDT). In this type of fuzzy classification, generally, an input vector x is associated with multiple classes, each with a different confidence value. Boosted ensembles of FDTs have been recently investigated as well, and they have shown performances comparable to those of other very efficient fuzzy classifiers.
Decision tree learning : Algorithms for constructing decision trees usually work top-down, by choosing a variable at each step that best splits the set of items. Different algorithms use different metrics for measuring "best". These generally measure the homogeneity of the target variable within the subsets. Some examples are given below. These metrics are applied to each candidate subset, and the resulting values are combined (e.g., averaged) to provide a measure of the quality of the split. Depending on the underlying metric, the performance of various heuristic algorithms for decision tree learning may vary significantly.
Decision tree learning : James, Gareth; Witten, Daniela; Hastie, Trevor; Tibshirani, Robert (2017). "Tree-Based Methods" (PDF). An Introduction to Statistical Learning: with Applications in R. New York: Springer. pp. 303–336. ISBN 978-1-4614-7137-0.
Decision tree learning : Evolutionary Learning of Decision Trees in C++ A very detailed explanation of information gain as splitting criterion
Elastic matching : Elastic matching is one of the pattern recognition techniques in computer science. Elastic matching (EM) is also known as deformable template, flexible matching, or nonlinear template matching. Elastic matching can be defined as an optimization problem of two-dimensional warping specifying corresponding pixels between subjected images.
Elastic matching : Uchida, Seiichi (August 2005). "A Survey of Elastic Matching Techniques for Handwritten Character Recognition" (PDF). IEICE Transactions on Information and Systems. E88-D (8): 1781–1790. Bibcode:2005IEITI..88.1781U. doi:10.1093/ietisy/e88-d.8.1781.
Elastic matching : Dynamic time warping Graphical time warping
Generalization error : For supervised learning applications in machine learning and statistical learning theory, generalization error (also known as the out-of-sample error or the risk) is a measure of how accurately an algorithm is able to predict outcomes for previously unseen data. As learning algorithms are evaluated on finite samples, the evaluation of a learning algorithm may be sensitive to sampling error. As a result, measurements of prediction error on the current data may not provide much information about the algorithm's predictive ability on new, unseen data. The generalization error can be minimized by avoiding overfitting in the learning algorithm. The performance of machine learning algorithms is commonly visualized by learning curve plots that show estimates of the generalization error throughout the learning process.
Generalization error : In a learning problem, the goal is to develop a function f n ( x → ) () that predicts output values y for each input datum x → . The subscript n indicates that the function f n is developed based on a data set of n data points. The generalization error or expected loss or risk I [ f ] of a particular function f over all possible values of x → and y is the expected value of the loss function V ( f ) : I [ f ] = ∫ X × Y V ( f ( x → ) , y ) ρ ( x → , y ) d x → d y , V(f(),y)\rho (,y)ddy, where ρ ( x → , y ) ,y) is the unknown joint probability distribution for x → and y . Without knowing the joint probability distribution ρ , it is impossible to compute I [ f ] . Instead, we can compute the error on sample data, which is called empirical error (or empirical risk). Given n data points, the empirical error of a candidate function f is: I n [ f ] = 1 n ∑ i = 1 n V ( f ( x → i ) , y i ) [f]=\sum _^V(f(_),y_) An algorithm is said to generalize if: lim n → ∞ I [ f ] − I n [ f ] = 0 I[f]-I_[f]=0 Of particular importance is the generalization error I [ f n ] ] of the data-dependent function f n that is found by a learning algorithm based on the sample. Again, for an unknown probability distribution, I [ f n ] ] cannot be computed. Instead, the aim of many problems in statistical learning theory is to bound or characterize the difference of the generalization error and the empirical error in probability: P G = P ( I [ f n ] − I n [ f n ] ≤ ϵ ) ≥ 1 − δ n =P(I[f_]-I_[f_]\leq \epsilon )\geq 1-\delta _ That is, the goal is to characterize the probability 1 − δ n that the generalization error is less than the empirical error plus some error bound ϵ (generally dependent on δ and n ). For many types of algorithms, it has been shown that an algorithm has generalization bounds if it meets certain stability criteria. Specifically, if an algorithm is symmetric (the order of inputs does not affect the result), has bounded loss and meets two stability conditions, it will generalize. The first stability condition, leave-one-out cross-validation stability, says that to be stable, the prediction error for each data point when leave-one-out cross validation is used must converge to zero as n → ∞ . The second condition, expected-to-leave-one-out error stability (also known as hypothesis stability if operating in the L 1 norm) is met if the prediction on a left-out datapoint does not change when a single data point is removed from the training dataset. These conditions can be formalized as:
Generalization error : The concepts of generalization error and overfitting are closely related. Overfitting occurs when the learned function f S becomes sensitive to the noise in the sample. As a result, the function will perform well on the training set but not perform well on other data from the joint probability distribution of x and y . Thus, the more overfitting occurs, the larger the generalization error. The amount of overfitting can be tested using cross-validation methods, that split the sample into simulated training samples and testing samples. The model is then trained on a training sample and evaluated on the testing sample. The testing sample is previously unseen by the algorithm and so represents a random sample from the joint probability distribution of x and y . This test sample allows us to approximate the expected error and as a result approximate a particular form of the generalization error. Many algorithms exist to prevent overfitting. The minimization algorithm can penalize more complex functions (known as Tikhonov regularization), or the hypothesis space can be constrained, either explicitly in the form of the functions or by adding constraints to the minimization function (Ivanov regularization). The approach to finding a function that does not overfit is at odds with the goal of finding a function that is sufficiently complex to capture the particular characteristics of the data. This is known as the bias–variance tradeoff. Keeping a function simple to avoid overfitting may introduce a bias in the resulting predictions, while allowing it to be more complex leads to overfitting and a higher variance in the predictions. It is impossible to minimize both simultaneously.
Generalization error : Olivier, Bousquet; Luxburg, Ulrike; Rätsch, Gunnar, eds. (2004). Advanced Lectures on Machine Learning. Lecture Notes in Computer Science. Vol. 3176. pp. 169–207. doi:10.1007/b100712. ISBN 978-3-540-23122-6. S2CID 431437. Retrieved 10 December 2022. Bousquet, Olivier; Elisseeff, Andr´e (1 March 2002). "Stability and Generalization". The Journal of Machine Learning Research. 2: 499–526. doi:10.1162/153244302760200704. S2CID 1157797. Retrieved 10 December 2022. Mohri, M., Rostamizadeh A., Talwakar A., (2018) Foundations of Machine learning, 2nd ed., Boston: MIT Press. Moody, J.E. (1992), "The Effective Number of Parameters: An Analysis of Generalization and Regularization in Nonlinear Learning Systems Archived 2016-09-10 at the Wayback Machine", in Moody, J.E., Hanson, S.J., and Lippmann, R.P., Advances in Neural Information Processing Systems 4, 847–854. White, H. (1992b), Artificial Neural Networks: Approximation and Learning Theory, Blackwell.
Gesture Description Language : Gesture Description Language (GDL or GDL Technology) is a method of describing and automatic (computer) syntactic classification of gestures and movements created by doctor Tomasz Hachaj (PhD) and professor Marek R. Ogiela(PhD, DSc). GDL uses context-free formal grammar named GDLs (Gesture Description Language script). With GDLs it is possible to define rules that describe set of gestures. Those rules play similar role as rules in classic expert systems. With rules it is possible to define static body positions (so called key frames) and sequences of key frames that create together definitions of gestures or movements. The recognition is done by forward chaining inference engine. The latest GDL implementations utilize Microsoft Kinect controller and enable real time classification. The license for GDL-based software allows using those programs for educational and scientific purposes for free. == References ==
ID3 algorithm : In decision tree learning, ID3 (Iterative Dichotomiser 3) is an algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm, and is typically used in the machine learning and natural language processing domains.
ID3 algorithm : The ID3 algorithm begins with the original set S as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S and calculates the entropy H ( S ) or the information gain I G ( S ) of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch.
ID3 algorithm : Classification and regression tree (CART) C4.5 algorithm Decision tree learning Decision tree model
ID3 algorithm : Mitchell, Tom Michael (1997). Machine Learning. New York, NY: McGraw-Hill. pp. 55–58. ISBN 0070428077. OCLC 36417892. Grzymala-Busse, Jerzy W. (February 1993). "Selected Algorithms of Machine Learning from Examples" (PDF). Fundamenta Informaticae. 18 (2): 193–207 – via ResearchGate.
ID3 algorithm : Seminars – http://www2.cs.uregina.ca/ Description and examples – http://www.cise.ufl.edu/ Description and examples – http://www.cis.temple.edu/ Decision Trees and Political Party Classification
Information gain (decision tree) : In information theory and machine learning, information gain is a synonym for Kullback–Leibler divergence; the amount of information gained about a random variable or signal from observing another random variable. However, in the context of decision trees, the term is sometimes used synonymously with mutual information, which is the conditional expected value of the Kullback–Leibler divergence of the univariate probability distribution of one variable from the conditional distribution of this variable given the other one. The information gain of a random variable X obtained from an observation of a random variable A taking value A = a is defined as: I G X , A ( X , a ) = D KL ( P X ( x \| a ) ‖ P X ( x \| I ) ) =D_\\|P_\right) i.e. the Kullback–Leibler divergence of P X ( x \| I ) (the prior distribution for x ) from P X \| A ( x \| a ) (the posterior distribution for x given a ). The expected value of the information gain is the mutual information ⁠ I ( X ; A ) ⁠ of X and A – i.e. the reduction in the entropy of X achieved by learning the state of the random variable A . In machine learning, this concept can be used to define a preferred sequence of attributes to investigate to most rapidly narrow down the state of X. Such a sequence (which depends on the outcome of the investigation of previous attributes at each stage) is called a decision tree, and when applied in the area of machine learning is known as decision tree learning. Usually an attribute with high mutual information should be preferred to other attributes.

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