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Principal component analysis (PCA) is a technique used to reduce multidimensional data sets to lower dimensions for analysis. Depending on the field of application, it is also named the discrete Karhunen-Loève transform, the Hotelling transform or proper orthogonal decomposition (POD). A data set (or dataset) is a collection of data, usually presented in tabular form. ... In the theory of stochastic processes, the Karhunen-Loève theorem (named after Kari Karhunen and Michel Loève) states that a centered stochastic process {Xt}t (where centered means that the expectations E(Xt) are defined and equal to 0 for all t) satisfying a technical continuity condition, admits... Harold Hotelling (Fulda, Minnesota, September 29, 1895 - December 26, 1973) was a mathematical statistician. ...

PCA was invented in 1901 by Karl Pearson^[1]. Now it is mostly used as a tool in exploratory data analysis and for making predictive models. PCA involves the calculation of the eigenvalue decomposition of a data covariance matrix or singular value decomposition of a data matrix, usually after mean centering the data for each attribute. The results of a PCA are usually discussed in terms of component scores and loadings^{[citation needed]}. Year 1901 (MCMI) was a common year starting on Tuesday (link will display calendar) of the Gregorian calendar (or a common year starting on Monday [1] of the 13-day-slower Julian calendar). ... Karl Pearson FRS (March 27, 1857 – April 27, 1936) established the discipline of mathematical statistics. ... Exploratory data analysis (EDA) is that part of statistical practice concerned with reviewing, communicating and using data where there is a low level of knowledge about its cause system. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

Details

PCA is mathematically defined^[2] as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any 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. PCA is theoretically the optimum transform for a given data in least square terms. In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

PCA can be used for dimensionality reduction in a data set by retaining those characteristics of the data set that contribute most to its variance, by keeping lower-order principal components and ignoring higher-order ones. Such low-order components often contain the "most important" aspects of the data. However, depending on the application this may not always be the case. In statistics, dimensionality reduction is mapping a multidimensional space into a space of fewer dimensions. ... This article is about mathematics. ...

For a data matrix, X^T, with zero empirical mean (the empirical mean of the distribution has been subtracted from the data set), where each row represents a different repetition of the experiment, and each column gives the results from a particular probe, the PCA transformation is given by: In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...

where V Σ W^T is the singular value decomposition (svd) of X^T. In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

PCA has the distinction of being the optimal linear transformation for keeping the subspace that has largest variance. This advantage, however, comes at the price of greater computational requirement if compared, for example, to the discrete cosine transform. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ...

Discussion

Though most derivations and implementations fail to identify the importance of mean subtraction, data centering is carried out because it is part of the solution towards finding a basis that minimizes the mean square error of approximating the data^[3]. Assuming zero empirical mean (the empirical mean of the distribution has been subtracted from the data set), the principal component w₁ of a data set x can be defined as: In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...

(See arg max for the notation.) With the first k − 1 components, the k-th component can be found by subtracting the first k − 1 principal components from x: In mathematics, arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument for which the value of the given expression attains its maximum value: This is well-defined only if the maximum is reached at a single value. ...

and by substituting this as the new data set to find a principal component in

The Karhunen-Loève transform is therefore equivalent to finding the singular value decomposition of the data matrix X, In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

and then obtaining the reduced-space data matrix Y by projecting X down into the reduced space defined by only the first L singular vectors, W_L:

The matrix W of singular vectors of X is equivalently the matrix W of eigenvectors of the matrix of observed covariances C = X X^T,

The eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the data set (see Rayleigh quotient). In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... Several sets of (x, y) points, with the correlation coefficient of x and y for each set. ... In mathematics, for a given real symmetric matrix A and real nonzero vector x, the Rayleigh quotient R(A,x) is defined as: Note that R(A,c·x) = R(A,x) for any real scalar c. ...

PCA is equivalent to empirical orthogonal functions (EOF). In statistics and signal processing, the method of empirical orthogonal functions is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. ...

An autoencoder neural network with a linear hidden layer is also equivalent to PCA. Upon convergence, the weight vectors of the K neurons in the hidden layer will form a basis for the space spanned by the first K principal components. Unlike PCA, this technique will not necessarily produce orthogonal vectors. An auto-encoder is an artificial neural network used for learning efficient codings. ... An artificial neural network (ANN), often just called a neural network (NN), is a mathematical model or computational model based on biological neural networks. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

PCA is a popular technique in pattern recognition. But it is not optimized for class separability^[4]. An alternative is the linear discriminant analysis, which does take this into account. PCA optimally minimizes reconstruction error under the L2 norm. Pattern recognition is a field within the area of machine learning. ... Linear discriminant analysis (LDA) and the related Fishers linear discriminant are methods used in statistics and machine learning to find the linear combination of features which best separate two or more classes of object or event. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

Table of symbols and abbreviations

Symbol	Meaning	Dimensions	Indices
	data matrix, consisting of the set of all data vectors, one vector per column
	the number of column vectors in the data set		scalar
	the number of elements in each column vector (dimension)		scalar
	the number of dimensions in the dimensionally reduced subspace,		scalar
	vector of empirical means, one mean for each row m of the data matrix
	vector of empirical standard deviations, one standard deviation for each row m of the data matrix
	vector of all 1's
	deviations from the mean of each row m of the data matrix
	z-scores, computed using the mean and standard deviation for each row m of the data matrix
	covariance matrix
	correlation matrix
	matrix consisting of the set of all eigenvectors of C, one eigenvector per column
	diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements
	matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of C, and where the vectors in W are a sub-set of those in V
	matrix consisting of N column vectors, where each vector is the projection of the corresponding data vector from matrix X onto the basis vectors contained in the columns of matrix W.

This article is about mathematical mean. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In radio, frequency deviation is the result of frequency modulation. ... In statistics, a standard score (z) is a dimensionless quantity derived by subtracting the sample mean from an individual (raw) score and then dividing the difference by the sample standard deviation: The quantity z represents the number of standard deviations between the raw score and the mean; it is negative... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In probability theory and statistics, correlation, also called correlation coefficient, indicates the strength and direction of a linear relationship between two random variables. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Properties and Limitations of PCA

PCA is theoretically the optimal linear scheme, in terms of least mean square error, for compressing a set of high dimensional vectors into a set of lower dimensional vectors and then reconstructing the original set. It is a non-parametric analysis and the answer is unique and independent of any hypothesis about data probability distribution. However, the latter two properties are regarded as weakness as well as strength, in that being non-parametric, no prior knowledge can be incorporated and that PCA compressions often incur loss of information. Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean squared error possible. ... A probability distribution describes the values and probabilities that a random event can take place. ...

The applicability of PCA is limited by the assumptions^[5] made in its derivation. These assumptions are:

• Assumption on Linearity

We assumed the observed data set to be linear combinations of certain basis. Non-linear methods such as kernel PCA have been developed without assuming linearity.

• Assumption on the statistical importance of mean and covariance

PCA uses the eigenvectors of the covariance matrix and it only finds the independent axes of the data under the Gaussian assumption. For non-Gaussian or multi-modal Gaussian data, PCA simply de-correlates the axes. When PCA is used for clustering, its main limitation is that it does not account for class separability since it makes no use of the class label of the feature vector. There is no guarantee that the directions of maximum variance will contain good features for discrimination. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... For the physics topics, see covariant transformation; about the mathematics example for groupoids, see covariance in special relativity; for the computer science topic see covariance and contravariance (computer science), in mathematics and theoretical physics see covariance and contravariance of vectors, in category theory see covariance and contravariance of functors. ...

• Assumption that large variances have important dynamics

PCA simply performs a coordinate rotation that aligns the transformed axes with the directions of maximum variance. It is only when we believe that the observed data has a high signal-to-noise ratio that the principal components with larger variance correspond to interesting dynamics and lower ones correspond to noise.

Essentially, PCA involves only rotation and scaling. The above assumptions are made in order to simplify the algebraic computation on the data set. Some other methods have been developed without one or more of these assumptions; these are briefly described below.

Computing PCA using the Covariance Method

Following is a detailed description of PCA using the covariance method. The goal is to transform a given data set X of dimension M to an alternative data set Y of smaller dimension L. Equivalently, we are seeking to find the matrix Y, where Y is the Karhunen-Loeve transform (KLT) of matrix X:

Organize the data set

Suppose you have data comprising a set of observations of M variables, and you want to reduce the data so that each observation can be described with only L variables, L < M. Suppose further, that the data are arranged as a set of N data vectors with each representing a single grouped observation of the M variables.

• Write as column vectors, each of which has M rows.
• Place the column vectors into a single matrix X of dimensions M × N.

Calculate the empirical mean

• Find the empirical mean along each dimension m = 1...M.
• Place the calculated mean values into an empirical mean vector u of dimensions M × 1.

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^[6]. Hence we proceed by centering the data as follows:

• Subtract the empirical mean vector u from each column of the data matrix X.
• Store mean-subtracted data in the M × N matrix B.

where h is a 1 x N row vector of all 1's:

Find the covariance matrix

• Find the M × M empirical covariance matrix C from the outer product of matrix B with itself:

where

is the expected value operator,

is the outer product operator, and

is the conjugate transpose operator. Note that if B consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular transpose.

• Please note that the information in this section is indeed a bit fuzzy. See the covariance matrix sections on the discussion page for more information.

Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...

Find the eigenvectors and eigenvalues of the covariance matrix

• Compute the matrix V of eigenvectors which diagonalizes the covariance matrix C:

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 MATLAB, Mathematica, SciPy, or IDL(Interactive Data Language). See, for example, the eig function.

• Matrix D will take the form of an M × M diagonal matrix, where

is the mth eigenvalue of the covariance matrix C, and

• Matrix V, also of dimension M × M, contains M column vectors, each of length M, which represent the M eigenvectors of the covariance matrix C.
• The eigenvalues and eigenvectors are ordered and paired. The mth eigenvalue corresponds to the mth eigenvector.

In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... For other uses, see Mathematica (disambiguation). ... SciPy is an open source library of high-quality scientific tools for the Python programming language. ... Wikipedia does not yet have an article with this exact name. ... IDL, short for interactive data language, is a programming language which has been a popular choice for data analysis among scientists in the past 15 years. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

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 mth eigenvector is the sum of the energy content across all of the eigenvectors from 1 through m:

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

Select a subset of the eigenvectors as basis vectors

• Save the first L columns of V as the M × L matrix W:

where

• Use the vector g as a guide in choosing an appropriate value for L. The goal is to choose as small a value of L 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

Convert the source data to z-scores

• Create an M × 1 empirical standard deviation vector s from the square root of each element along the main diagonal of the covariance matrix C:

• Calculate the M × N z-score matrix:

(divide element-by-element)

• Note: While this step is useful for various applications as it normalizes the data set with respect to its variance, it is not integral part of PCA/KLT!

Compares the various grading methods in a normal distribution. ...

Project the z-scores of the data onto the new basis

• The projected vectors are the columns of the matrix

• The columns of matrix Y represent the Karhunen-Loeve transforms (KLT) of the data vectors in the columns of matrix X.

Derivation of PCA using the covariance method

Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero empirical mean. We want to find a orthonormal transformation matrix P such that In mathematics, an orthonormal basis of an inner product space V(i. ...

with the constraint that

is a diagonal matrix and

By substitution, and matrix algebra, we obtain: In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

We now have:

Rewrite P as d column vectors, so

and as:

Substituting into equation above, we obtain:

Notice that in , P_i is an eigenvector of X′s covariance matrix. Therefore, by finding the eigenvectors of X′s covariance matrix, we find a projection matrix P that satisfies the original constraints. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Relation to K-means clustering

It has been shown recently ^{[7] [8]} that the relaxed solution of K-means clustering, specified by the cluster indicators, are given by the PCA principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace specified by the between-class scatter matrix. The K-means algorithm is an algorithm to cluster objects based on attributes into k partitions. ... It has been suggested that this article or section be merged into sample mean and sample covariance. ...

Correspondence analysis

Correspondence analysis was developed by J.-P. Benzecri^[9] is conceptually similar to PCA, but scales the data (which must be positive) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. Correspondence analysis decomposes the Chi-square statistic associated to this table into orthogonal factors. Because correspondence analysis is a descriptive technique, it can be applied to tables for which the Chi-square statistic is appropriate or not.

Spectral Map Analysis

Spectral Map Analysis,or spectral mapping was devised by Lewi ^{[10] [11]} and is an alternative approach for the graphical analysis of multivariate data. Basically, the method consists of logarithmic transformation and double centering of the data followed by factorization by means of a (weighted) singular value decomposition and a biplot of the resulting factor scores and loadings. The advantage over correspondence analysis is that its scope is not limited to contingency tables and cross-tabulations. The method has more recently also been referred to as Log-Ratio Analysis. In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

Nonlinear generalizations

Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or K-means. The original Pearson's idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. Principal curves and manifolds give the natural geometric framework for PCA generalization and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold for data approximation, and by encoding using standard geometric projection onto the manifold^[12]. High dimensional data can be difficult to interpret. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... It has been suggested that this article or section be merged with estimation. ... In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ...

Software/source code

• Computer Vision Library
• Multivariate Data Analysis Software
• in Matlab, the function "princomp" gives the principal component
• in the open source statistical package R, the functions "princomp" and "prcomp" can be used for principal component analysis; prcomp uses singular value decomposition which generally gives better numerical accuracy.
• "spm" is a generic package developed in R for multivariate projection methods that allows principal components analysis, correspondence analysis, and spectral map analysis
• In XLMiner, the Principles Component tab can be used for principal component analysis.
• SciLab

Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... For the database, see IBM System R The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and graphics. ... In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ... For the database, see IBM System R The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and graphics. ...

References

1. ^ Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space". Philosophical Magazine 2 (6): 559–572. 
2. ^ Jolliffe I.T. Principal Component Analysis, Series: Springer Series in Statistics, 2nd ed., Springer, NY, 2002, XXIX, 487 p. 28 illus. ISBN 978-0-387-95442-4
3. ^ A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
4. ^ Fukunaga, Keinosuke (1990). Introduction to Statistical Pattern Recognition. Elsevier. 
5. ^ Jon Shlens, A Tutorial on Principal Component Analysis.
6. ^ A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
7. ^ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon. "Spectral Relaxation for K-means Clustering", Neural Information Processing Systems vol.14 (NIPS 2001). pp. 1057-1064, Vancouver, Canada. Dec. 2001.
8. ^ Chris Ding and Xiaofeng He. "K-means Clustering via Principal Component Analysis". Proc. of Int'l Conf. Machine Learning (ICML 2004), pp 225-232. July 2004.
9. ^ Benzécri, J.-P. (1973). L'Analyse des Données. Volume II. L'Analyse des Correspondences. Paris, France: Dunod. 
10. ^ Lewi, P.J. (1976). "Spectral Mapping, a technique for classifying biological activity profiles of chemical compounds". Arzneimittel Forschung (Drug Research) 26: 1295-1300. 
11. ^ Lewi, P.J. (2005). "Spectral Mapping, a personal and historical account of an adventure in multivariate data analysis". Chemometrics and Intelligent Laboratory Systems 77: 215-223. 
12. ^ A. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer, Berlin - Heidelberg - New York, 2007. ISBN 978-3-540-73749-0

See also

Some eigenfaces from AT&T Laboratories Cambridge. ... Factor analysis is a statistical method used to explain variability among observed random variables in terms of fewer unobserved random variables called factors. ... Most real world data sets consist of data vectors whose individual components are not statistically independent, that is, they are redundant in the statistical sense. ... Independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents supposing the mutual statistical independence of the non-Gaussian source signals. ... Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. ... In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ... High dimensional data can be difficult to interpret. ... The Point Distribution Model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. ... In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ... Transform coding is a type of data compression for natural data like audio signals or photographic images. ...

External links

• Spectroscopy and PCA
• An introductory explanation of PCA from StatSoft
• A Tutorial on Principal Component Analysis (PDF)
• A tutorial on PCA by Lindsay I. Smith (PDF)
• A layman's explanation from Umetrics
• Principal Component Analysis using Hebbian learning tutorial
• Presentation of Principal Component Analysis used in Biomedical Engineering
• Application to microarray and other biomedical data
• PCA in functional neuroimaging, free software
• Uncertainty estimation for PCA
• FactoMineR, an R package dedicated to exploratory multivariate analysis
• A web-site with presentations and open source software on exploratory multivariate data analysis
• EasyPCA, a very simple and small PCA program under the GPL license