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Independent component analysis : What is independent component analysis? by Aapo Hyvärinen Independent Component Analysis: A Tutorial by Aapo Hyvärinen A Tutorial on Independent Component Analysis FastICA as a package for Matlab, in R language, C++ ICALAB Toolboxes for Matlab, developed at RIKEN High Performance Signal Analysis Toolkit provides C++ implementations of FastICA and Infomax ICA toolbox Matlab tools for ICA with Bell-Sejnowski, Molgedey-Schuster and mean field ICA. Developed at DTU. Demonstration of the cocktail party problem Archived 2010-03-13 at the Wayback Machine EEGLAB Toolbox ICA of EEG for Matlab, developed at UCSD. FMRLAB Toolbox ICA of fMRI for Matlab, developed at UCSD MELODIC, part of the FMRIB Software Library. Discussion of ICA used in a biomedical shape-representation context FastICA, CuBICA, JADE and TDSEP algorithm for Python and more... Group ICA Toolbox and Fusion ICA Toolbox Tutorial: Using ICA for cleaning EEG signals
Locality-sensitive hashing : In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. (The number of buckets is much smaller than the universe of possible input items.) Since similar items end up in the same buckets, this technique can be used for data clustering and nearest neighbor search. It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items. Hashing-based approximate nearest-neighbor search algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as locality-preserving hashing (LPH). Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion.
Locality-sensitive hashing : A finite family F of functions h : M → S is defined to be an LSH family for a metric space M = ( M , d ) =(M,d) , a threshold r > 0 , an approximation factor c > 1 , and probabilities p 1 > p 2 >p_ if it satisfies the following condition. For any two points a , b ∈ M and a hash function h chosen uniformly at random from F : If d ( a , b ) ≤ r , then h ( a ) = h ( b ) (i.e., a and b collide) with probability at least p 1 , If d ( a , b ) ≥ c r , then h ( a ) = h ( b ) with probability at most p 2 . Such a family F is called ( r , c r , p 1 , p 2 ) ,p_) -sensitive.
Locality-sensitive hashing : LSH has been applied to several problem domains, including: Near-duplicate detection Hierarchical clustering Genome-wide association study Image similarity identification VisualRank Gene expression similarity identification Audio similarity identification Nearest neighbor search Audio fingerprint Digital video fingerprinting Shared memory organization in parallel computing Physical data organization in database management systems Training fully connected neural networks Computer security Machine Learning
Locality-sensitive hashing : One of the main applications of LSH is to provide a method for efficient approximate nearest neighbor search algorithms. Consider an LSH family F . The algorithm has two main parameters: the width parameter k and the number of hash tables L. In the first step, we define a new family G of hash functions g, where each function g is obtained by concatenating k functions h 1 , … , h k ,\ldots ,h_ from F , i.e., g ( p ) = [ h 1 ( p ) , … , h k ( p ) ] (p),\ldots ,h_(p)] . In other words, a random hash function g is obtained by concatenating k randomly chosen hash functions from F . The algorithm then constructs L hash tables, each corresponding to a different randomly chosen hash function g. In the preprocessing step we hash all n d-dimensional points from the data set S into each of the L hash tables. Given that the resulting hash tables have only n non-zero entries, one can reduce the amount of memory used per each hash table to O ( n ) using standard hash functions. Given a query point q, the algorithm iterates over the L hash functions g. For each g considered, it retrieves the data points that are hashed into the same bucket as q. The process is stopped as soon as a point within distance cR from q is found. Given the parameters k and L, the algorithm has the following performance guarantees: preprocessing time: O ( n L k t ) , where t is the time to evaluate a function h ∈ F on an input point p; space: O ( n L ) , plus the space for storing data points; query time: O ( L ( k t + d n P 2 k ) ) ^)) ; the algorithm succeeds in finding a point within distance cR from q (if there exists a point within distance R) with probability at least 1 − ( 1 − P 1 k ) L ^)^ ; For a fixed approximation ratio c = 1 + ϵ and probabilities P 1 and P 2 , one can set k = ⌈ log ⁡ n log ⁡ 1 / P 2 ⌉ \right\rceil and L = ⌈ P 1 − k ⌉ = O ( n ρ P 1 − 1 ) ^\rceil =O(n^P_^) , where ρ = log ⁡ P 1 log ⁡ P 2 . Then one obtains the following performance guarantees: preprocessing time: O ( n 1 + ρ P 1 − 1 k t ) P_^kt) ; space: O ( n 1 + ρ P 1 − 1 ) P_^) , plus the space for storing data points; query time: O ( n ρ P 1 − 1 ( k t + d ) ) P_^(kt+d)) ;
Locality-sensitive hashing : Bloom filter – Data structure for approximate set membership Curse of dimensionality – Difficulties arising when analyzing data with many aspects ("dimensions") Feature hashing – Vectorizing features using a hash function Fourier-related transforms Geohash – Public domain geocoding invented in 2008 Multilinear subspace learning – Approach to dimensionality reduction Principal component analysis – Method of data analysis Random indexing Rolling hash – Type of hash function Singular value decomposition – Matrix decomposition Sparse distributed memory – Mathematical model of memory Wavelet compression – Mathematical technique used in data compression and analysisPages displaying short descriptions of redirect targets
Locality-sensitive hashing : Samet, H. (2006) Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann. ISBN 0-12-369446-9 Indyk, Piotr; Motwani, Rajeev; Raghavan, Prabhakar; Vempala, Santosh (1997). "Locality-preserving hashing in multidimensional spaces". Proceedings of the twenty-ninth annual ACM symposium on Theory of computing. STOC '97. pp. 618–625. CiteSeerX 10.1.1.50.4927. doi:10.1145/258533.258656. ISBN 978-0-89791-888-6. S2CID 15693787. Chin, Andrew (1994). "Locality-preserving hash functions for general purpose parallel computation" (PDF). Algorithmica. 12 (2–3): 170–181. doi:10.1007/BF01185209. S2CID 18108051.
Locality-sensitive hashing : Alex Andoni's LSH homepage LSHKIT: A C++ Locality Sensitive Hashing Library A Python Locality Sensitive Hashing library that optionally supports persistence via redis Caltech Large Scale Image Search Toolbox: a Matlab toolbox implementing several LSH hash functions, in addition to Kd-Trees, Hierarchical K-Means, and Inverted File search algorithms. Slash: A C++ LSH library, implementing Spherical LSH by Terasawa, K., Tanaka, Y LSHBOX: An Open Source C++ Toolbox of Locality-Sensitive Hashing for Large Scale Image Retrieval, Also Support Python and MATLAB. SRS: A C++ Implementation of An In-memory, Space-efficient Approximate Nearest Neighbor Query Processing Algorithm based on p-stable Random Projection TLSH open source on Github JavaScript port of TLSH (Trend Micro Locality Sensitive Hashing) bundled as node.js module Java port of TLSH (Trend Micro Locality Sensitive Hashing) bundled as maven package
Low-rank approximation : In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent semantic analysis, orthogonal regression, and dynamic mode decomposition.
Low-rank approximation : Given structure specification S : R n p → R m × n :\mathbb ^\to \mathbb ^ , vector of structure parameters p ∈ R n p ^ , norm ‖ ⋅ ‖ , and desired rank r , minimize over p ^ ‖ p − p ^ ‖ subject to rank ⁡ ( S ( p ^ ) ) ≤ r . \quad \quad \\|p-\\|\quad \quad \operatorname ()\leq r.
Low-rank approximation : Linear system identification, in which case the approximating matrix is Hankel structured. Machine learning, in which case the approximating matrix is nonlinearly structured. Recommender systems, in which cases the data matrix has missing values and the approximation is categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation is nonnegative. Computer algebra, in which case the approximation is Sylvester structured.
Low-rank approximation : The unstructured problem with fit measured by the Frobenius norm, i.e., minimize over D ^ ‖ D − D ^ ‖ F subject to rank ⁡ ( D ^ ) ≤ r \quad \quad \\|D-\\|_\quad \quad \operatorname \leq r has an analytic solution in terms of the singular value decomposition of the data matrix. The result is referred to as the matrix approximation lemma or Eckart–Young–Mirsky theorem. This problem was originally solved by Erhard Schmidt in the infinite dimensional context of integral operators (although his methods easily generalize to arbitrary compact operators on Hilbert spaces) and later rediscovered by C. Eckart and G. Young. L. Mirsky generalized the result to arbitrary unitarily invariant norms. Let D = U Σ V ⊤ ∈ R m × n , m ≥ n \in \mathbb ^,\quad m\geq n be the singular value decomposition of D , where Σ =: diag ⁡ ( σ 1 , … , σ m ) (\sigma _,\ldots ,\sigma _) is the m × n rectangular diagonal matrix with the singular values σ 1 ≥ … ≥ σ m \geq \ldots \geq \sigma _ . For a given r ∈ , partition U , Σ , and V as follows: U =: [ U 1 U 2 ] , Σ =: [ Σ 1 0 0 Σ 2 ] , and V =: [ V 1 V 2 ] , U_&U_\end,\quad \Sigma =:\Sigma _&0\\0&\Sigma _\end,\quad \quad V=:V_&V_\end, where U 1 is m × r , Σ 1 is r × r , and V 1 is n × r . Then the rank- r matrix, obtained from the truncated singular value decomposition D ^ ∗ = U 1 Σ 1 V 1 ⊤ , ^=U_\Sigma _V_^, is such that ‖ D − D ^ ∗ ‖ F = min rank ⁡ ( D ^ ) ≤ r ‖ D − D ^ ‖ F = σ r + 1 2 + ⋯ + σ m 2 . ^\\|_=\min _ ()\leq r\\|D-\\|_=^+\cdots +\sigma _^. The minimizer D ^ ∗ ^ is unique if and only if σ r + 1 ≠ σ r \neq \sigma _ .
Low-rank approximation : Let A ∈ R m × n ^ be a real (possibly rectangular) matrix with m ≤ n . Suppose that A = U Σ V ⊤ is the singular value decomposition of A . Recall that U and V are orthogonal matrices, and Σ is an m × n diagonal matrix with entries ( σ 1 , σ 2 , ⋯ , σ m ) ,\sigma _,\cdots ,\sigma _) such that σ 1 ≥ σ 2 ≥ ⋯ ≥ σ m ≥ 0 \geq \sigma _\geq \cdots \geq \sigma _\geq 0 . We claim that the best rank- k approximation to A in the spectral norm, denoted by ‖ ⋅ ‖ 2 , is given by A k := ∑ i = 1 k σ i u i v i ⊤ :=\sum _^\sigma _u_v_^ where u i and v i denote the i th column of U and V , respectively. First, note that we have ‖ A − A k ‖ 2 = ‖ ∑ i = 1 n σ i u i v i ⊤ − ∑ i = 1 k σ i u i v i ⊤ ‖ 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ 2 = σ k + 1 \\|_=\left\\|\sum _^\sigma _u_v_^-\sum _^\sigma _u_v_^\right\\|_=\left\\|\sum _^\sigma _u_v_^\right\\|_=\sigma _ Therefore, we need to show that if B k = X Y ⊤ =XY^ where X and Y have k columns then ‖ A − A k ‖ 2 = σ k + 1 ≤ ‖ A − B k ‖ 2 \\|_=\sigma _\leq \\|A-B_\\|_ . Since Y has k columns, then there must be a nontrivial linear combination of the first k + 1 columns of V , i.e., w = γ 1 v 1 + ⋯ + γ k + 1 v k + 1 , v_+\cdots +\gamma _v_, such that Y ⊤ w = 0 w=0 . Without loss of generality, we can scale w so that ‖ w ‖ 2 = 1 =1 or (equivalently) γ 1 2 + ⋯ + γ k + 1 2 = 1 ^+\cdots +\gamma _^=1 . Therefore, ‖ A − B k ‖ 2 2 ≥ ‖ ( A − B k ) w ‖ 2 2 = ‖ A w ‖ 2 2 = γ 1 2 σ 1 2 + ⋯ + γ k + 1 2 σ k + 1 2 ≥ σ k + 1 2 . \\|_^\geq \\|(A-B_)w\\|_^=\\|Aw\\|_^=\gamma _^\sigma _^+\cdots +\gamma _^\sigma _^\geq \sigma _^. The result follows by taking the square root of both sides of the above inequality.
Low-rank approximation : Let A ∈ R m × n ^ be a real (possibly rectangular) matrix with m ≤ n . Suppose that A = U Σ V ⊤ is the singular value decomposition of A . We claim that the best rank k approximation to A in the Frobenius norm, denoted by ‖ ⋅ ‖ F , is given by A k = ∑ i = 1 k σ i u i v i ⊤ =\sum _^\sigma _u_v_^ where u i and v i denote the i th column of U and V , respectively. First, note that we have ‖ A − A k ‖ F 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ F 2 = ∑ i = k + 1 n σ i 2 \\|_^=\left\\|\sum _^\sigma _u_v_^\right\\|_^=\sum _^\sigma _^ Therefore, we need to show that if B k = X Y ⊤ =XY^ where X and Y have k columns then ‖ A − A k ‖ F 2 = ∑ i = k + 1 n σ i 2 ≤ ‖ A − B k ‖ F 2 . \\|_^=\sum _^\sigma _^\leq \\|A-B_\\|_^. By the triangle inequality with the spectral norm, if A = A ′ + A ″ then σ 1 ( A ) ≤ σ 1 ( A ′ ) + σ 1 ( A ″ ) (A)\leq \sigma _(A')+\sigma _(A'') . Suppose A k ′ and A k ″ respectively denote the rank k approximation to A ′ and A ″ by SVD method described above. Then, for any i , j ≥ 1 σ i ( A ′ ) + σ j ( A ″ ) = σ 1 ( A ′ − A i − 1 ′ ) + σ 1 ( A ″ − A j − 1 ″ ) ≥ σ 1 ( A − A i − 1 ′ − A j − 1 ″ ) ≥ σ 1 ( A − A i + j − 2 ) ( since r a n k ( A i − 1 ′ + A j − 1 ″ ) ≤ i + j − 2 ) ) = σ i + j − 1 ( A ) . \sigma _(A')+\sigma _(A'')&=\sigma _(A'-A'_)+\sigma _(A''-A''_)\\&\geq \sigma _(A-A'_-A''_)\\&\geq \sigma _(A-A_)\qquad ((A'_+A''_)\leq i+j-2))\\&=\sigma _(A).\end Since σ k + 1 ( B k ) = 0 (B_)=0 , when A ′ = A − B k and A ″ = B k we conclude that for i ≥ 1 , j = k + 1 σ i ( A − B k ) ≥ σ k + i ( A ) . (A-B_)\geq \sigma _(A). Therefore, ‖ A − B k ‖ F 2 = ∑ i = 1 n σ i ( A − B k ) 2 ≥ ∑ i = k + 1 n σ i ( A ) 2 = ‖ A − A k ‖ F 2 , \\|_^=\sum _^\sigma _(A-B_)^\geq \sum _^\sigma _(A)^=\\|A-A_\\|_^, as required.
Low-rank approximation : The Frobenius norm weights uniformly all elements of the approximation error D − D ^ . Prior knowledge about distribution of the errors can be taken into account by considering the weighted low-rank approximation problem minimize over D ^ vec ⁡ ( D − D ^ ) ⊤ W vec ⁡ ( D − D ^ ) subject to rank ⁡ ( D ^ ) ≤ r , \quad \quad \operatorname (D-)^W\operatorname (D-)\quad \quad \operatorname ()\leq r, where vec ( A ) (A) vectorizes the matrix A column wise and W is a given positive (semi)definite weight matrix. The general weighted low-rank approximation problem does not admit an analytic solution in terms of the singular value decomposition and is solved by local optimization methods, which provide no guarantee that a globally optimal solution is found. In case of uncorrelated weights, weighted low-rank approximation problem also can be formulated in this way: for a non-negative matrix W and a matrix A we want to minimize ∑ i , j ( W i , j ( A i , j − B i , j ) ) 2 (W_(A_-B_))^ over matrices, B , of rank at most r .
Low-rank approximation : Let ‖ A ‖ p = ( ∑ i , j \| A i , j p \| ) 1 / p =\left(\sum _\|A_^\|\right)^ . For p = 2 , the fastest algorithm runs in n n z ( A ) + n ⋅ p o l y ( k / ϵ ) time. One of the important ideas been used is called Oblivious Subspace Embedding (OSE), it is first proposed by Sarlos. For p = 1 , it is known that this entry-wise L1 norm is more robust than the Frobenius norm in the presence of outliers and is indicated in models where Gaussian assumptions on the noise may not apply. It is natural to seek to minimize ‖ B − A ‖ 1 . For p = 0 and p ≥ 1 , there are some algorithms with provable guarantees.
Low-rank approximation : Let P = ,\ldots ,p_\ and Q = ,\ldots ,q_\ be two point sets in an arbitrary metric space. Let A represent the m × n matrix where A i , j = d i s t ( p i , q i ) =dist(p_,q_) . Such distances matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding. In an attempt to reduce their description size, one can study low rank approximation of such matrices.
Low-rank approximation : The low-rank approximation problems in the distributed and streaming setting has been considered in.
Low-rank approximation : Using the equivalences rank ⁡ ( D ^ ) ≤ r ⟺ there are P ∈ R m × r and L ∈ R r × n such that D ^ = P L ()\leq r\quad \iff \quad P\in \mathbb ^L\in \mathbb ^=PL and rank ⁡ ( D ^ ) ≤ r ⟺ there is full row rank R ∈ R m − r × m such that R D ^ = 0 ()\leq r\quad \iff \quad R\in \mathbb ^R=0 the weighted low-rank approximation problem becomes equivalent to the parameter optimization problems minimize over D ^ , P and L vec ⊤ ⁡ ( D − D ^ ) W vec ⁡ ( D − D ^ ) subject to D ^ = P L \quad ,PL\quad \operatorname ^(D-)W\operatorname (D-)\quad \quad =PL and minimize over D ^ and R vec ⊤ ⁡ ( D − D ^ ) W vec ⁡ ( D − D ^ ) subject to R D ^ = 0 and R R ⊤ = I r , \quad R\quad \operatorname ^(D-)W\operatorname (D-)\quad \quad R=0\quad \quad RR^=I_, where I r is the identity matrix of size r .
Low-rank approximation : The image representation of the rank constraint suggests a parameter optimization method in which the cost function is minimized alternatively over one of the variables ( P or L ) with the other one fixed. Although simultaneous minimization over both P and L is a difficult biconvex optimization problem, minimization over one of the variables alone is a linear least squares problem and can be solved globally and efficiently. The resulting optimization algorithm (called alternating projections) is globally convergent with a linear convergence rate to a locally optimal solution of the weighted low-rank approximation problem. Starting value for the P (or L ) parameter should be given. The iteration is stopped when a user defined convergence condition is satisfied. Matlab implementation of the alternating projections algorithm for weighted low-rank approximation:
Low-rank approximation : The alternating projections algorithm exploits the fact that the low rank approximation problem, parameterized in the image form, is bilinear in the variables P or L . The bilinear nature of the problem is effectively used in an alternative approach, called variable projections. Consider again the weighted low rank approximation problem, parameterized in the image form. Minimization with respect to the L variable (a linear least squares problem) leads to the closed form expression of the approximation error as a function of P f ( P ) = vec ⊤ ⁡ ( D ) ( W − W ( I n ⊗ P ) ( ( I n ⊗ P ) ⊤ W ( I n ⊗ P ) ) − 1 ( I n ⊗ P ) ⊤ W ) vec ⁡ ( D ) . ^(D)W-W(I_\otimes P)(I_\otimes P)^W(I_\otimes P)^(I_\otimes P)^W\operatorname (D). The original problem is therefore equivalent to the nonlinear least squares problem of minimizing f ( P ) with respect to P . For this purpose standard optimization methods, e.g. the Levenberg-Marquardt algorithm can be used. Matlab implementation of the variable projections algorithm for weighted low-rank approximation: The variable projections approach can be applied also to low rank approximation problems parameterized in the kernel form. The method is effective when the number of eliminated variables is much larger than the number of optimization variables left at the stage of the nonlinear least squares minimization. Such problems occur in system identification, parameterized in the kernel form, where the eliminated variables are the approximating trajectory and the remaining variables are the model parameters. In the context of linear time-invariant systems, the elimination step is equivalent to Kalman smoothing.
Low-rank approximation : Usually, we want our new solution not only to be of low rank, but also satisfy other convex constraints due to application requirements. Our interested problem would be as follows, minimize over p ^ ‖ p − p ^ ‖ subject to rank ⁡ ( S ( p ^ ) ) ≤ r and g ( p ^ ) ≤ 0 \quad \quad \\|p-\\|\quad \quad \operatorname ()\leq rg()\leq 0 This problem has many real world applications, including to recover a good solution from an inexact (semidefinite programming) relaxation. If additional constraint g ( p ^ ) ≤ 0 )\leq 0 is linear, like we require all elements to be nonnegative, the problem is called structured low rank approximation. The more general form is named convex-restricted low rank approximation. This problem is helpful in solving many problems. However, it is challenging due to the combination of the convex and nonconvex (low-rank) constraints. Different techniques were developed based on different realizations of g ( p ^ ) ≤ 0 )\leq 0 . However, the Alternating Direction Method of Multipliers (ADMM) can be applied to solve the nonconvex problem with convex objective function, rank constraints and other convex constraints, and is thus suitable to solve our above problem. Moreover, unlike the general nonconvex problems, ADMM will guarantee to converge a feasible solution as long as its dual variable converges in the iterations.
Low-rank approximation : CUR matrix approximation is made from the rows and columns of the original matrix
Low-rank approximation : M. T. Chu, R. E. Funderlic, R. J. Plemmons, Structured low-rank approximation, Linear Algebra and its Applications, Volume 366, 1 June 2003, Pages 157–172 doi:10.1016/S0024-3795(02)00505-0
Low-rank approximation : C++ package for structured-low rank approximation
Modes of variation : In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA).
Modes of variation : Modes of variation are a natural extension of PCA and FPCA.
Modes of variation : The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance.
Modes of variation : Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , and ± 3 , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality data, modes of variation of male mortality data shows higher mortality after around age 20, possibly related to the fact that life expectancy for women is higher than that for men. == References ==
Multidimensional scaling : Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n objects in a set into a configuration of n points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis.
Multidimensional scaling : MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:
Multidimensional scaling : The data to be analyzed is a collection of M objects (colors, faces, stocks, . . .) on which a distance function is defined, d i , j := := distance between i -th and j -th objects. These distances are the entries of the dissimilarity matrix D := ( d 1 , 1 d 1 , 2 ⋯ d 1 , M d 2 , 1 d 2 , 2 ⋯ d 2 , M ⋮ ⋮ ⋮ d M , 1 d M , 2 ⋯ d M , M ) . d_&d_&\cdots &d_\\d_&d_&\cdots &d_\\\vdots &\vdots &&\vdots \\d_&d_&\cdots &d_\end. The goal of MDS is, given D , to find M vectors x 1 , … , x M ∈ R N ,\ldots ,x_\in \mathbb ^ such that ‖ x i − x j ‖ ≈ d i , j -x_\\|\approx d_ for all i , j ∈ 1 , … , M , where ‖ ⋅ ‖ is a vector norm. In classical MDS, this norm is the Euclidean distance, but, in a broader sense, it may be a metric or arbitrary distance function. For example, when dealing with mixed-type data that contain numerical as well as categorical descriptors, Gower's distance is a common alternative. In other words, MDS attempts to find a mapping from the M objects into R N ^ such that distances are preserved. If the dimension N is chosen to be 2 or 3, we may plot the vectors x i to obtain a visualization of the similarities between the M objects. Note that the vectors x i are not unique: With the Euclidean distance, they may be arbitrarily translated, rotated, and reflected, since these transformations do not change the pairwise distances ‖ x i − x j ‖ -x_\\| . (Note: The symbol R indicates the set of real numbers, and the notation R N ^ refers to the Cartesian product of N copies of R , which is an N -dimensional vector space over the field of the real numbers.) There are various approaches to determining the vectors x i . Usually, MDS is formulated as an optimization problem, where ( x 1 , … , x M ) ,\ldots ,x_) is found as a minimizer of some cost function, for example, a r g m i n x 1 , … , x M ∑ i < j ( ‖ x i − x j ‖ − d i , j ) 2 . ,\ldots ,x_ \sum _(\\|x_-x_\\|-d_)^.\, A solution may then be found by numerical optimization techniques. For some particularly chosen cost functions, minimizers can be stated analytically in terms of matrix eigendecompositions.
Multidimensional scaling : There are several steps in conducting MDS research: Formulating the problem – What variables do you want to compare? How many variables do you want to compare? What purpose is the study to be used for? Obtaining input data – For example, :- Respondents are asked a series of questions. For each product pair, they are asked to rate similarity (usually on a 7-point Likert scale from very similar to very dissimilar). The first question could be for Coke/Pepsi for example, the next for Coke/Hires rootbeer, the next for Pepsi/Dr Pepper, the next for Dr Pepper/Hires rootbeer, etc. The number of questions is a function of the number of brands and can be calculated as Q = N ( N − 1 ) / 2 where Q is the number of questions and N is the number of brands. This approach is referred to as the “Perception data : direct approach”. There are two other approaches. There is the “Perception data : derived approach” in which products are decomposed into attributes that are rated on a semantic differential scale. The other is the “Preference data approach” in which respondents are asked their preference rather than similarity. Running the MDS statistical program – Software for running the procedure is available in many statistical software packages. Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS (which deals with ordinal data). Decide number of dimensions – The researcher must decide on the number of dimensions they want the computer to create. Interpretability of the MDS solution is often important, and lower dimensional solutions will typically be easier to interpret and visualize. However, dimension selection is also an issue of balancing underfitting and overfitting. Lower dimensional solutions may underfit by leaving out important dimensions of the dissimilarity data. Higher dimensional solutions may overfit to noise in the dissimilarity measurements. Model selection tools like AIC, BIC, Bayes factors, or cross-validation can thus be useful to select the dimensionality that balances underfitting and overfitting. Mapping the results and defining the dimensions – The statistical program (or a related module) will map the results. The map will plot each product (usually in two-dimensional space). The proximity of products to each other indicate either how similar they are or how preferred they are, depending on which approach was used. How the dimensions of the embedding actually correspond to dimensions of system behavior, however, are not necessarily obvious. Here, a subjective judgment about the correspondence can be made (see perceptual mapping). Test the results for reliability and validity – Compute R-squared to determine what proportion of variance of the scaled data can be accounted for by the MDS procedure. An R-square of 0.6 is considered the minimum acceptable level. An R-square of 0.8 is considered good for metric scaling and .9 is considered good for non-metric scaling. Other possible tests are Kruskal’s Stress, split data tests, data stability tests (i.e., eliminating one brand), and test-retest reliability. Report the results comprehensively – Along with the mapping, at least distance measure (e.g., Sorenson index, Jaccard index) and reliability (e.g., stress value) should be given. It is also very advisable to give the algorithm (e.g., Kruskal, Mather), which is often defined by the program used (sometimes replacing the algorithm report), if you have given a start configuration or had a random choice, the number of runs, the assessment of dimensionality, the Monte Carlo method results, the number of iterations, the assessment of stability, and the proportional variance of each axis (r-square).
Multidimensional scaling : ELKI includes two MDS implementations. MATLAB includes two MDS implementations (for classical (cmdscale) and non-classical (mdscale) MDS respectively). The R programming language offers several MDS implementations, e.g. base cmdscale function, packages smacof (mMDS and nMDS), and vegan (weighted MDS). scikit-learn contains function sklearn.manifold.MDS.
Multidimensional scaling : Data clustering t-distributed stochastic neighbor embedding Factor analysis Discriminant analysis Dimensionality reduction Distance geometry Cayley–Menger determinant Sammon mapping Iconography of correlations
Multifactor dimensionality reduction : Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data.
Multifactor dimensionality reduction : Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between the attributes (X1 and X2) and the class variable (Y) is defined by the XOR function such that Y = X1 XOR X2. Table 1 A machine learning algorithm would need to discover or approximate the XOR function in order to accurately predict Y using information about X1 and X2. An alternative strategy would be to first change the representation of the data using constructive induction to facilitate predictive modeling. The MDR algorithm would change the representation of the data (X1 and X2) in the following manner. MDR starts by selecting two attributes. In this simple example, X1 and X2 are selected. Each combination of values for X1 and X2 are examined and the number of times Y=1 and/or Y=0 is counted. In this simple example, Y=1 occurs zero times and Y=0 occurs once for the combination of X1=0 and X2=0. With MDR, the ratio of these counts is computed and compared to a fixed threshold. Here, the ratio of counts is 0/1 which is less than our fixed threshold of 1. Since 0/1 < 1 we encode a new attribute (Z) as a 0. When the ratio is greater than one we encode Z as a 1. This process is repeated for all unique combinations of values for X1 and X2. Table 2 illustrates our new transformation of the data. Table 2 The machine learning algorithm now has much less work to do to find a good predictive function. In fact, in this very simple example, the function Y = Z has a classification accuracy of 1. A nice feature of constructive induction methods such as MDR is the ability to use any data mining or machine learning method to analyze the new representation of the data. Decision trees, neural networks, or a naive Bayes classifier could be used in combination with measures of model quality such as balanced accuracy and mutual information.
Multifactor dimensionality reduction : As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine learning algorithms are good at finding patterns in completely random data. It is often difficult to determine whether a reported pattern is an important signal or just chance. One approach is to estimate the generalizability of a model to independent datasets using methods such as cross-validation. Models that describe random data typically don't generalize. Another approach is to generate many random permutations of the data to see what the data mining algorithm finds when given the chance to overfit. Permutation testing makes it possible to generate an empirical p-value for the result. Replication in independent data may also provide evidence for an MDR model but can be sensitive to difference in the data sets. These approaches have all been shown to be useful for choosing and evaluating MDR models. An important step in a machine learning exercise is interpretation. Several approaches have been used with MDR including entropy analysis and pathway analysis. Tips and approaches for using MDR to model gene-gene interactions have been reviewed.
Multifactor dimensionality reduction : Numerous extensions to MDR have been introduced. These include family-based methods, fuzzy methods, covariate adjustment, odds ratios, risk scores, survival methods, robust methods, methods for quantitative traits, and many others.
Multifactor dimensionality reduction : MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation, autism, bladder cancer, breast cancer, cardiovascular disease, hypertension, obesity, pancreatic cancer, prostate cancer and tuberculosis. It has also been applied to other biomedical problems such as the genetic analysis of pharmacology outcomes. A central challenge is the scaling of MDR to big data such as that from genome-wide association studies (GWAS). Several approaches have been used. One approach is to filter the features prior to MDR analysis. This can be done using biological knowledge through tools such as BioFilter. It can also be done using computational tools such as ReliefF. Another approach is to use stochastic search algorithms such as genetic programming to explore the search space of feature combinations. Yet another approach is a brute-force search using high-performance computing.
Multifactor dimensionality reduction : www.epistasis.org provides an open-source and freely-available MDR software package. An R package for MDR. An sklearn-compatible Python implementation. An R package for Model-Based MDR. MDR in Weka. Generalized MDR.
Multifactor dimensionality reduction : Data mining Dimensionality reduction Epistasis Feature Engineering Machine learning Multilinear subspace learning
Multifactor dimensionality reduction : Michalski, R. S., "Pattern Recognition as Knowledge-Guided Computer Induction," Department of Computer Science Reports, No. 927, University of Illinois, Urbana, June 1978.
Multilinear principal component analysis : Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear independent component analysis (MICA). The origin of MPCA can be traced back to the tensor rank decomposition introduced by Frank Lauren Hitchcock in 1927; to the Tucker decomposition; and to Peter Kroonenberg's "3-mode PCA" work. In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "Multilinear Singular Value Decomposition", (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R1, R2, ..., RN ) Approximation of Higher-order Tensors". Circa 2001, Vasilescu and Terzopoulos reframed the data analysis, recognition and synthesis problems as multilinear tensor problems. Tensor factor analysis is the compositional consequence of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in terms of their causal factors of data formation in the following works: Human Motion Signatures (CVPR 2001, ICPR 2002), face recognition – TensorFaces, (ECCV 2002, CVPR 2003, etc.) and computer graphics – TensorTextures (Siggraph 2004). Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics associated with each tensor mode/axis. Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as a collection of column/row observations, "data matrix" and concatenated into a data tensor. The main disadvantage of this approach is that, rather than computing all possible combinations, MPCA computes a set of orthonormal matrices associated with each mode of the data tensor which are analogous to the orthonormal row and column space of a matrix computed by the matrix SVD. This transformation aims to capture as high a variance as possible, accounting for as much of the variability in the data associated with each data tensor mode(axis).
Multilinear principal component analysis : The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent.
Multilinear principal component analysis : MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks.
Multilinear principal component analysis : Various extension of MPCA: Robust MPCA (RMPCA) Multi-Tensor Factorization, that also finds the number of components automatically (MTF)
Multilinear principal component analysis : Matlab code: MPCA. Matlab code: UMPCA (including data). R code: MTF
Multilinear subspace learning : Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space. Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA).
Multilinear subspace learning : Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. . Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction. These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor.
Multilinear subspace learning : MATLAB Tensor Toolbox by Sandia National Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab (data included).
Multilinear subspace learning : 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M);
Multilinear subspace learning : CP decomposition Dimension reduction Multilinear algebra Multilinear Principal Component Analysis Tensor Tensor decomposition Tensor software Tucker decomposition == References ==
Multiple correspondence analysis : In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.
Multiple correspondence analysis : MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display. In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues.
Multiple correspondence analysis : Since MCA is adapted to draw statistical conclusions from categorical variables (such as multiple choice questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles). When the dataset is completely represented as categorical variables, one is able to build the corresponding so-called complete disjunctive table. We denote this table X . If I persons answered a survey with J multiple choices questions with 4 answers each, X will have I rows and 4 J columns. More theoretically, assume X is the completely disjunctive table of I observations of K categorical variables. Assume also that the k -th variable have J k different levels (categories) and set J = ∑ k = 1 K J k ^J_ . The table X is then a I × J matrix with all coefficient being 0 or 1 . Set the sum of all entries of X to be N and introduce Z = X / N . In an MCA, there are also two special vectors: first r , that contains the sums along the rows of Z , and c , that contains the sums along the columns of Z . Note D r = diag ( r ) =(r) and D c = diag ( c ) =(c) , the diagonal matrices containing r and c respectively as diagonal. With these notations, computing an MCA consists essentially in the singular value decomposition of the matrix: M = D r − 1 / 2 ( Z − r c T ) D c − 1 / 2 ^(Z-rc^)D_^ The decomposition of M gives you P , Δ and Q such that M = P Δ Q T with P, Q two unitary matrices and Δ is the generalized diagonal matrix of the singular values (with the same shape as Z ). The positive coefficients of Δ 2 are the eigenvalues of Z . The interest of MCA comes from the way observations (rows) and variables (columns) in Z can be decomposed. This decomposition is called a factor decomposition. The coordinates of the observations in the factor space are given by F = D r − 1 / 2 P Δ ^P\Delta The i -th rows of F represent the i -th observation in the factor space. And similarly, the coordinates of the variables (in the same factor space as observations!) are given by G = D c − 1 / 2 Q Δ ^Q\Delta
Multiple correspondence analysis : In recent years, several students of Jean-Paul Benzécri have refined MCA and incorporated it into a more general framework of data analysis known as geometric data analysis. This involves the development of direct connections between simple correspondence analysis, principal component analysis and MCA with a form of cluster analysis known as Euclidean classification. Two extensions have great practical use. It is possible to include, as active elements in the MCA, several quantitative variables. This extension is called factor analysis of mixed data (see below). Very often, in questionnaires, the questions are structured in several issues. In the statistical analysis it is necessary to take into account this structure. This is the aim of multiple factor analysis which balances the different issues (i.e. the different groups of variables) within a global analysis and provides, beyond the classical results of factorial analysis (mainly graphics of individuals and of categories), several results (indicators and graphics) specific of the group structure.
Multiple correspondence analysis : In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu, notably in his books La Distinction, Homo Academicus and The State Nobility. Bourdieu argued that there was an internal link between his vision of the social as spatial and relational --– captured by the notion of field, and the geometric properties of MCA. Sociologists following Bourdieu's work most often opt for the analysis of the indicator matrix, rather than the Burt table, largely because of the central importance accorded to the analysis of the 'cloud of individuals'.
Multiple correspondence analysis : MCA can also be viewed as a PCA applied to the complete disjunctive table. To do this, the CDT must be transformed as follows. Let y i k denote the general term of the CDT. y i k is equal to 1 if individual i possesses the category k and 0 if not. Let denote p k , the proportion of individuals possessing the category k . The transformed CDT (TCDT) has as general term: x i k = y i k / p k − 1 =y_/p_-1 The unstandardized PCA applied to TCDT, the column k having the weight p k , leads to the results of MCA. This equivalence is fully explained in a book by Jérôme Pagès. It plays an important theoretical role because it opens the way to the simultaneous treatment of quantitative and qualitative variables. Two methods simultaneously analyze these two types of variables: factor analysis of mixed data and, when the active variables are partitioned in several groups: multiple factor analysis. This equivalence does not mean that MCA is a particular case of PCA as it is not a particular case of CA. It only means that these methods are closely linked to one another, as they belong to the same family: the factorial methods.
Multiple correspondence analysis : There are numerous software of data analysis that include MCA, such as STATA and SPSS. The R package FactoMineR also features MCA. This software is related to a book describing the basic methods for performing MCA . There is also a Python package for [1] which works with numpy array matrices; the package has not been implemented yet for Spark dataframes.
Multiple correspondence analysis : Le Roux, B. and H. Rouanet (2004), Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis at Google Books: [2] FactoMineR A R software devoted to exploratory data analysis.
NOMINATE (scaling method) : NOMINATE (an acronym for Nominal Three-Step Estimation) is a multidimensional scaling application developed by US political scientists Keith T. Poole and Howard Rosenthal in the early 1980s to analyze preferential and choice data, such as legislative roll-call voting behavior. In its most well-known application, members of the US Congress are placed on a two-dimensional map, with politicians who are ideologically similar (i.e. who often vote the same) being close together. One of these two dimensions corresponds to the familiar left–right political spectrum (liberal–conservative in the United States). As computing capabilities grew, Poole and Rosenthal developed multiple iterations of their NOMINATE procedure: the original D-NOMINATE method, W-NOMINATE, and most recently DW-NOMINATE (for dynamic, weighted NOMINATE). In 2009, Poole and Rosenthal were the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE. In 2016, the society awarded Poole its Career Achievement Award, stating that "the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores."
NOMINATE (scaling method) : The main procedure is an application of multidimensional scaling techniques to political choice data. Though there are important technical differences between these types of NOMINATE scaling procedures; all operate under the same fundamental assumptions. First, that alternative choices can be projected on a basic, low-dimensional (often two-dimensional) Euclidean space. Second, within that space, individuals have utility functions which are bell-shaped (normally distributed), and maximized at their ideal point. Because individuals also have symmetric, single-peaked utility functions which center on their ideal point, ideal points represent individuals' most preferred outcomes. That is, individuals most desire outcomes closest their ideal point, and will choose/vote probabilistically for the closest outcome. Ideal points can be recovered from observing choices, with individuals exhibiting similar preferences placed more closely than those behaving dissimilarly. It is helpful to compare this procedure to producing maps based on driving distances between cities. For example, Los Angeles is about 1,800 miles from St. Louis; St. Louis is about 1,200 miles from Miami; and Miami is about 2,700 miles from Los Angeles. From this (dis)similarities data, any map of these three cities should place Miami far from Los Angeles, with St. Louis somewhere in between (though a bit closer to Miami than Los Angeles). Just as cities like Los Angeles and San Francisco would be clustered on a map, NOMINATE places ideologically similar legislators (e.g., liberal Senators Barbara Boxer (D-Calif.) and Al Franken (D-Minn.)) closer to each other, and farther from dissimilar legislators (e.g., conservative Senator Tom Coburn (R-Okla.)) based on the degree of agreement between their roll call voting records. At the heart of the NOMINATE procedures (and other multidimensional scaling methods, such as Poole's Optimal Classification method) are algorithms they utilize to arrange individuals and choices in low dimensional (usually two-dimensional) space. Thus, NOMINATE scores provide "maps" of legislatures. Using NOMINATE procedures to study congressional roll call voting behavior from the First Congress to the present-day, Poole and Rosenthal published Congress: A Political-Economic History of Roll Call Voting in 1997 and the revised edition Ideology and Congress in 2007. In 2009, Poole and Rosenthal were named the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE, a recognition conferred to "individual(s) for developing statistical software that makes a significant research contribution". In 2016, Keith T. Poole was awarded the Society for Political Methodology's Career Achievement Award. The citation for this award reads, in part, "One can say perfectly correctly, and without any hyperbole: the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores. NOMINATE has produced data that entire bodies of our discipline—and many in the press—have relied on to understand the U.S. Congress."
NOMINATE (scaling method) : Poole and Rosenthal demonstrate that—despite the many complexities of congressional representation and politics—roll call voting in both the House and the Senate can be organized and explained by no more than two dimensions throughout the sweep of American history. The first dimension (horizontal or x-axis) is the familiar left-right (or liberal-conservative) spectrum on economic matters. The second dimension (vertical or y-axis) picks up attitudes on cross-cutting, salient issues of the day (which include or have included slavery, bimetallism, civil rights, regional, and social/lifestyle issues). Rosenthal and Poole have initially argued that the first dimension refers to socio-economic matters and the second dimension to race-relations. However, the often confusing and residual nature of the second dimension has led to the second dimension being largely ignored by other researchers. For the most part, congressional voting is uni-dimensional, with most of the variation in voting patterns explained by placement along the liberal-conservative first dimension. While the first dimension of the DW-NOMINATE score is able to predict results at 83% accuracy, the addition of the second dimension only increases accuracy to 85%. Furthermore, the second dimension only provided a significant increase in accuracy for Congresses 1-99. As late as the 1990s, the second dimension was able to measure partisan splits in abortion and gun rights issues. However, a 2017 analysis found that since 1987, the votes of the US Congress had best fit a one-dimensional model, suggesting increasing party polarization after 1987.
NOMINATE (scaling method) : For illustrative purposes, consider the following plots which use W-NOMINATE scores to scale members of Congress and uses the probabilistic voting model (in which legislators farther from the "cutting line" between "yea" and "nay" outcomes become more likely to vote in the predicted manner) to illustrate some major Congressional votes in the 1990s. Some of these votes, like the House's vote on President Clinton's welfare reform package (the Personal Responsibility and Work Opportunity Act of 1996) are best modeled through the use of the first (economic liberal-conservative) dimension. On the welfare reform vote, nearly all Republicans joined the moderate-conservative bloc of House Democrats in voting for the bill, while opposition was virtually confined to the most liberal Democrats in the House. The errors (those representatives on the "wrong" side of the cutting line which separates predicted "yeas" and predicted "nays") are generally close to the cutting line, which is what we would expect. A legislator directly on the cutting line is indifferent between voting "yea" and "nay" on the measure. All members are shown on the left panel of the plot, while only errors are shown on the right panel: Economic ideology also dominates the Senate vote on the Balanced Budget Amendment of 1995: On other votes, however, a second dimension (which has recently come to represent attitudes on cultural and lifestyle issues) is important. For example, roll call votes on gun control routinely split party coalitions, with socially conservative "blue dog" Democrats joining most Republicans in opposing additional regulation and socially liberal Republicans joining most Democrats in supporting gun control. The addition of the second dimension accounts for these inter-party differences, and the cutting line is more horizontal than vertical (meaning the cleavage is found on the second dimension rather than the first dimension on these votes) This pattern was evident in the 1991 House vote to require waiting periods on handguns:
NOMINATE (scaling method) : DW-NOMINATE scores have been used widely to describe the political ideology of political actors, political parties and political institutions. For instance, a score in the first dimension that is close to either pole means that such score is located at one of the extremes in the liberal-conservative scale. So, a score closer to 1 is described as conservative whereas a score closer to −1 can be described as liberal. Finally, a score at zero or close to zero is described as moderate.
NOMINATE (scaling method) : Poole and Rosenthal (beginning with their 1984 article "The Polarization of American Politics") have also used NOMINATE data to show that, since the 1970s, party delegations in Congress have become ideologically homogeneous and distant from one another (a phenomenon known as "polarization"). Using DW-NOMINATE scores (which permit direct comparisons between members of different Congresses across time), political scientists have demonstrated the expansion of ideological divides in Congress, which has spurred intense partisanship between Republicans and Democrats in recent decades. Contemporary political polarization has had important political consequences on American public policy, as Poole and Rosenthal (with fellow political scientist Nolan McCarty) show in their 2006 book Polarized America: The Dance of Ideology and Unequal Riches.
NOMINATE (scaling method) : NOMINATE has been used to test, refine, and/or develop wide-ranging theories and models of the United States Congress. In Ideology and Congress (pp. 270–271), Poole and Rosenthal agree that their findings are consistent with the "party cartel" model that Cox and McCubbins present in their 1993 book Legislative Leviathan. Keith Krehbiel utilizes NOMINATE scores to determine the ideological rank order of both chambers of Congress in developing his "pivotal politics" theory, as do Gary Cox and Matthew McCubbins in their tests of whether parties in Congress meet the conditions of responsible party government (RPG). NOMINATE scores have been used by popular media outlets like The New York Times and The Washington Post as a measure of the political ideology of political institutions and elected officials or candidates. Political blogger Nate Silver and his team at FiveThirtyEight have repeatedly used DW-NOMINATE scores to gauge the ideological location of major political figures and institutions. NOMINATE procedures and related roll call scaling techniques have also been applied to a number of other legislative bodies besides the United States Congress. These include the United Nations General Assembly, the European Parliament National Assemblies in Latin America, and the French Fourth Republic. Poole and Rosenthal note in Chapter 11 of Ideology and Congress (2007) that most of these analyses find that roll call voting is organized by only few dimensions (usually two): "These findings suggest that the need to form parliamentary majorities limits dimensionality."
NOMINATE (scaling method) : Multidimensional scaling (MDS) Scale analysis (statistics)
NOMINATE (scaling method) : Special issue of Studies in American Political Development on NOMINATE, 2016.
NOMINATE (scaling method) : Archive of K7MOA.com ("Permanent Archive of the Research of Poole and Rosenthal and Colleagues"). Includes "NOMINATE and American Political History: A Primer." A more extensive introduction to NOMINATE "NOMINATE: A Short Intellectual History" (by Keith T. Poole) W-NOMINATE in R: Software and Examples Jordan Ellenberg,"Growing Apart: The Mathematical Evidence for Congress' Growing Polarization," Slate Magazine, 26 December 2001
Nonlinear dimensionality reduction : Nonlinear dimensionality reduction, also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis.
Nonlinear dimensionality reduction : High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keep its essential features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns. The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32×32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information (the letter 'A') and recover only the varying information (rotation and scale). The image to the right shows sample images from this dataset (to save space, not all input images are shown), and a plot of the two-dimensional points that results from using a NLDR algorithm (in this case, Manifold Sculpting was used) to reduce the data into just two dimensions. By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner. It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold will represent the robot's position and orientation. Invariant manifolds are of general interest for model order reduction in dynamical systems. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of spectral submanifolds (SSM) gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques. Some of the more prominent nonlinear dimensionality reduction techniques are listed below.
Nonlinear dimensionality reduction : A method based on proximity matrices is one where the data is presented to the algorithm in the form of a similarity matrix or a distance matrix. These methods all fall under the broader class of metric multidimensional scaling. The variations tend to be differences in how the proximity data is computed; for example, isomap, locally linear embeddings, maximum variance unfolding, and Sammon mapping (which is not in fact a mapping) are examples of metric multidimensional scaling methods.
Nonlinear dimensionality reduction : Waffles is an open source C++ library containing implementations of LLE, Manifold Sculpting, and some other manifold learning algorithms. UMAP.jl implements the method for the programming language Julia. The method has also been implemented in Python (code available on GitHub)
Nonlinear dimensionality reduction : Manifold hypothesis Spectral submanifold Taken's theorem Whitney embedding theorem Discriminant analysis Elastic map Feature learning Growing self-organizing map (GSOM) Self-organizing map (SOM)
Nonlinear dimensionality reduction : Murphy, Kevin P. (2022). "Manifold Learning". Probabilistic Machine Learning. MIT Press. pp. 682–699. ISBN 978-0-262-04682-4.
Nonlinear dimensionality reduction : Isomap Generative Topographic Mapping Mike Tipping's Thesis Gaussian Process Latent Variable Model Locally Linear Embedding Relational Perspective Map DD-HDS homepage RankVisu homepage Short review of Diffusion Maps Nonlinear PCA by autoencoder neural networks
Ordination (statistics) : Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901.
Ordination (statistics) : Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods.
Ordination (statistics) : Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.
Ordination (statistics) : Multivariate statistics Principal components analysis Correspondence analysis Multiple correspondence analysis Detrended correspondence analysis Intrinsic dimension Latent space Latent variable model
Ordination (statistics) : Birks, H.J.B., 1998. An Annotated Bibliography Of Canonical Correspondence Analysis And Related Constrained Ordination Methods 1986–1996. Botanical Institute, University of Bergen. World Wide Web: http://www.bio.umontreal.ca/Casgrain/cca_bib/index.html Braak, C.J.F. ter & I.C. Prentice 1988 A theory of gradient analysis. Adv. Ecol. Res. 18:271-313. Gauch, H.G., Jr. 1982. Multivariate Analysis in Community Ecology. Cambridge University Press, Cambridge. Jongman et al., 1995. Data Analysis in Community and Landscape Ecology. Cambridge University Press, Cambridge. Pagani et al., 2015. Methodi Ordinatio: a proposed methodology to select and rank relevant scientific papers encompassing the impact factor, number of citation, and year of publication. Scientometrics, December 2015, Volume 105, Issue 3, pp 2109–2135.
Ordination (statistics) : General http://ordination.okstate.edu/ The Ordination Web Page - Ordination Methods for Ecologists https://www.davidzeleny.net/anadat-r/doku.php/en:start https://link.springer.com/article/10.1007/s11192-015-1744-x Specific Techniques http://www.statsoft.com/textbook/stcoran.html http://www.statsoft.com/textbook/stmulsca.html http://www.statsoft.com/textbook/glosfra.html https://link.springer.com/article/10.1007/s11192-015-1744-x Ordination method for articles, using year of publication, impact factor and number of citations. Software http://home.centurytel.net/~mjm/pcordwin.htm http://www.microcomputerpower.com/catalog/canoco.html http://www.brodgar.com http://www.VisuMap.com https://cran.r-project.org/web/packages/vegan/vegan.pdf R package for classical ordination methods https://cran.r-project.org/package=seriation R package for ordering objects https://cran.r-project.org/web/packages/gllvm/index.html R package for model-based ordination https://cran.r-project.org/web/packages/VGAM/index.html R package for model-based ordination https://cran.r-project.org/web/packages/boral/index.html R package for model-based ordination
Preference regression : Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps.
Preference regression : Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum. Like all regression methods, the computer fits weights to best predict data. The resultant regression line is referred to as an ideal vector because the slope of the vector is the ratio of the preferences for the two dimensions. If all the data is used in the regression, the program will derive a single equation and hence a single ideal vector. This tends to be a blunt instrument so researchers refine the process with cluster analysis. This creates clusters that reflect market segments. Separate preference regressions are then done on the data within each segment. This provides an ideal vector for each segment.
Preference regression : Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used.
Preference regression : Marketing Product management Positioning (marketing) Marketing research Perceptual mapping Multidimensional scaling Factor analysis Linear discriminant analysis#Marketing Preference-rank translation
Preference regression : Park, S. T.; Chu, W. (2009). "Pairwise preference regression for cold-start recommendation". Proceedings of the third ACM conference on Recommender systems - RecSys '09. p. 21. doi:10.1145/1639714.1639720. ISBN 9781605584355. Jarboe, G.R.; McDaniel, C.D.; Gates, R.H. (1992). "Preference regression modeling of multiple option healthcare delivery systems". Journal of Ambulatory Care Marketing, 5(1), p.71-82.
Principal component analysis : Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p unit vectors, where the i -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science.
Principal component analysis : When performing PCA, the first principal component of a set of p variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i -th principal component can be taken as a direction orthogonal to the first i − 1 principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed.
Principal component analysis : PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.
Principal component analysis : PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA.
Principal component analysis : PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l of p-dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) _=(w_,\dots ,w_)_ that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) _=(x_,\dots ,x_)_ of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) _=(t_,\dots ,t_)_ , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l _=\mathbf _\cdot \mathbf _\qquad \mathrm \qquad i=1,\dots ,n\qquad k=1,\dots ,l in such a way that the individual variables t 1 , … , t l ,\dots ,t_ of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector (where l is usually selected to be strictly less than p to reduce dimensionality). The above may equivalently be written in matrix form as T = X W =\mathbf \mathbf where T i k = t k ( i ) _=_ , X i j = x j ( i ) _=_ , and W j k = w j ( k ) _=_ .
Principal component analysis : The singular values (in Σ) are the square roots of the eigenvalues of the matrix XTX. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see below). PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the discrete cosine transform, and in particular to the DCT-II which is simply known as the "DCT". Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA. PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same sample variance and are completely correlated, then the PCA will entail a rotation by 45° and the "weights" (they are the cosines of rotation) for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance. Mean subtraction (a.k.a. "mean centering") is necessary for performing classical PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data. Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation). Also see the article by Kromrey & Foster-Johnson (1998) on "Mean-centering in Moderated Regression: Much Ado About Nothing". Since covariances are correlations of normalized variables (Z- or standard-scores) a PCA based on the correlation matrix of X is equal to a PCA based on the covariance matrix of Z, the standardized version of X. PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability. However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes. The linear discriminant analysis is an alternative which is optimized for class separability.
Principal component analysis : The following is a detailed description of PCA using the covariance method as opposed to the correlation method. The goal is to transform a given data set X of dimension p to an alternative data set Y of smaller dimension L. Equivalently, we are seeking to find the matrix Y, where Y is the Karhunen–Loève transform (KLT) of matrix X: Y = K L T =\mathbb \ \ Organize the data set Suppose you have data comprising a set of observations of p variables, and you want to reduce the data so that each observation can be described with only L variables, L < p. Suppose further, that the data are arranged as a set of n data vectors x 1 … x n _\ldots \mathbf _ with each x i _ representing a single grouped observation of the p variables. Write x 1 … x n _\ldots \mathbf _ as row vectors, each with p elements. Place the row vectors into a single matrix X of dimensions n × p. Calculate the empirical mean Find the empirical mean along each column j = 1, ..., p. Place the calculated mean values into an empirical mean vector u of dimensions p × 1. u j = 1 n ∑ i = 1 n X i j =\sum _^X_ Calculate the deviations from the mean Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data. Hence we proceed by centering the data as follows: Subtract the empirical mean vector u T ^ from each row of the data matrix X. Store mean-subtracted data in the n × p matrix B. B = X − h u T =\mathbf -\mathbf \mathbf ^ where h is an n × 1 column vector of all 1s: h i = 1 for i = 1 , … , n =1\,\qquad \qquad i=1,\ldots ,n In some applications, each variable (column of B) may also be scaled to have a variance equal to 1 (see Z-score). This step affects the calculated principal components, but makes them independent of the units used to measure the different variables. Find the covariance matrix Find the p × p empirical covariance matrix C from matrix B: C = 1 n − 1 B ∗ B =\mathbf ^\mathbf where ∗ is the conjugate transpose operator. If B consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular transpose. The reasoning behind using n − 1 instead of n to calculate the covariance is Bessel's correction. Find the eigenvectors and eigenvalues of the covariance matrix Compute the matrix V of eigenvectors which diagonalizes the covariance matrix C: V − 1 C V = D ^\mathbf \mathbf =\mathbf where D is the diagonal matrix of eigenvalues of C. This step will typically involve the use of a computer-based algorithm for computing eigenvectors and eigenvalues. These algorithms are readily available as sub-components of most matrix algebra systems, such as SAS, R, MATLAB, Mathematica, SciPy, IDL (Interactive Data Language), or GNU Octave as well as OpenCV. Matrix D will take the form of an p × p diagonal matrix, where D k ℓ = λ k for k = ℓ =\lambda _\qquad k=\ell is the jth eigenvalue of the covariance matrix C, and D k ℓ = 0 for k ≠ ℓ . =0\qquad k\neq \ell . Matrix V, also of dimension p × p, contains p column vectors, each of length p, which represent the p eigenvectors of the covariance matrix C. The eigenvalues and eigenvectors are ordered and paired. The jth eigenvalue corresponds to the jth eigenvector. Matrix V denotes the matrix of right eigenvectors (as opposed to left eigenvectors). In general, the matrix of right eigenvectors need not be the (conjugate) transpose of the matrix of left eigenvectors. Rearrange the eigenvectors and eigenvalues Sort the columns of the eigenvector matrix V and eigenvalue matrix D in order of decreasing eigenvalue. Make sure to maintain the correct pairings between the columns in each matrix. Compute the cumulative energy content for each eigenvector The eigenvalues represent the distribution of the source data's energy among each of the eigenvectors, where the eigenvectors form a basis for the data. The cumulative energy content g for the jth eigenvector is the sum of the energy content across all of the eigenvalues from 1 through j: g j = ∑ k = 1 j D k k for j = 1 , … , p =\sum _^D_\qquad j=1,\dots ,p Select a subset of the eigenvectors as basis vectors Save the first L columns of V as the p × L matrix W: W k l = V k ℓ for k = 1 , … , p ℓ = 1 , … , L =V_\qquad k=1,\dots ,p\qquad \ell =1,\dots ,L where 1 ≤ L ≤ p . Use the vector g as a guide in choosing an appropriate value for L. The goal is to choose a value of L as small as possible while achieving a reasonably high value of g on a percentage basis. For example, you may want to choose L so that the cumulative energy g is above a certain threshold, like 90 percent. In this case, choose the smallest value of L such that g L g p ≥ 0.9 \geq 0.9 Project the data onto the new basis The projected data points are the rows of the matrix T = B ⋅ W =\mathbf \cdot \mathbf That is, the first column of T is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc.
Principal component analysis : Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean. We want to find ( ∗ ) a d × d orthonormal transformation matrix P so that PX has a diagonal covariance matrix (that is, PX is a random vector with all its distinct components pairwise uncorrelated). A quick computation assuming P were unitary yields: cov ⁡ ( P X ) = E ⁡ [ P X ( P X ) ∗ ] = E ⁡ [ P X X ∗ P ∗ ] = P E ⁡ [ X X ∗ ] P ∗ = P cov ⁡ ( X ) P − 1 \operatorname (PX)&=\operatorname [PX~(PX)^]\\&=\operatorname [PX~X^P^]\\&=P\operatorname [XX^]P^\\&=P\operatorname (X)P^\\\end Hence ( ∗ ) holds if and only if cov ⁡ ( X ) (X) were diagonalisable by P . This is very constructive, as cov(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.
Principal component analysis : In practical implementations, especially with high dimensional data (large p), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the np2 operations of explicitly calculating and storing the covariance matrix XTX, instead utilizing one of matrix-free methods, for example, based on the function evaluating the product XT(X r) at the cost of 2np operations.
Principal component analysis : In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable species. For this, the following results are produced. Identification, on the factorial planes, of the different species, for example, using different colors. Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species. For each center of gravity and each axis, p-value to judge the significance of the difference between the center of gravity and origin. These results are what is called introducing a qualitative variable as supplementary element. This procedure is detailed in and Husson, Lê, & Pagès (2009) and Pagès (2013). Few software offer this option in an "automatic" way. This is the case of SPAD that historically, following the work of Ludovic Lebart, was the first to propose this option, and the R package FactoMineR.
Principal component analysis : ALGLIB – a C++ and C# library that implements PCA and truncated PCA Analytica – The built-in EigenDecomp function computes principal components. ELKI – includes PCA for projection, including robust variants of PCA, as well as PCA-based clustering algorithms. Gretl – principal component analysis can be performed either via the pca command or via the princomp() function. Julia – Supports PCA with the pca function in the MultivariateStats package KNIME – A java based nodal arranging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily. Maple (software) – The PCA command is used to perform a principal component analysis on a set of data. Mathematica – Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods. MathPHP – PHP mathematics library with support for PCA. MATLAB – The SVD function is part of the basic system. In the Statistics Toolbox, the functions princomp and pca (R2012b) give the principal components, while the function pcares gives the residuals and reconstructed matrix for a low-rank PCA approximation. Matplotlib – Python library have a PCA package in the .mlab module. mlpack – Provides an implementation of principal component analysis in C++. mrmath – A high performance math library for Delphi and FreePascal can perform PCA; including robust variants. NAG Library – Principal components analysis is implemented via the g03aa routine (available in both the Fortran versions of the Library). NMath – Proprietary numerical library containing PCA for the .NET Framework. GNU Octave – Free software computational environment mostly compatible with MATLAB, the function princomp gives the principal component. OpenCV Oracle Database 12c – Implemented via DBMS_DATA_MINING.SVDS_SCORING_MODE by specifying setting value SVDS_SCORING_PCA Orange (software) – Integrates PCA in its visual programming environment. PCA displays a scree plot (degree of explained variance) where user can interactively select the number of principal components. Origin – Contains PCA in its Pro version. Qlucore – Commercial software for analyzing multivariate data with instant response using PCA. R – Free statistical package, the functions princomp and prcomp can be used for principal component analysis; prcomp uses singular value decomposition which generally gives better numerical accuracy. Some packages that implement PCA in R, include, but are not limited to: ade4, vegan, ExPosition, dimRed, and FactoMineR. SAS – Proprietary software; for example, see scikit-learn – Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module. Scilab – Free and open-source, cross-platform numerical computational package, the function princomp computes principal component analysis, the function pca computes principal component analysis with standardized variables. SPSS – Proprietary software most commonly used by social scientists for PCA, factor analysis and associated cluster analysis. Weka – Java library for machine learning which contains modules for computing principal components.

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