Principal component analysis
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Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on various numerical values) into a set of values of linearly uncorrelated variables called principal components. If there are observations with variables, then the number of distinct principal components is . This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors (each being a linear combination of the variables and containing n observations) are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.
PCA was invented in 1901 by Karl Pearson,^{[1]} as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s.^{[2]} 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 (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of X^{T}X 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^{[3]}), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.
PCA is mostly used as a tool in exploratory data analysis and for making predictive models. It is often used to visualize genetic distance and relatedness between populations. PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition of a data matrix, usually after a normalization step of the initial data. The normalization of each attribute consists of mean centering – subtracting each data value from its variable’s measured mean so that its empirical mean (average) is zero – and, possibly, normalizing each variable’s variance to make it equal to 1; see Zscores.^{[4]} The results of a PCA are usually discussed in terms of component scores, sometimes called factor scores (the transformed variable values corresponding to a particular data point), and loadings (the weight by which each standardized original variable should be multiplied to get the component score).^{[5]} If component scores are standardized to unit variance loadings must contain the data variance in them (and that is the magnitude of eigenvalues). If component scores are not standardized (therefore they contain the data variance) then loadings must be unitscaled, ("normalized") and these weights are called eigenvectors; they are the cosines of orthogonal rotation of variables into principal components or back.
PCA is the simplest of the true eigenvectorbased multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way that best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a highdimensional data space (1 axis per variable), PCA can supply the user with a lowerdimensional picture, a projection of this object when viewed from its most informative viewpoint. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.
PCA 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 crosscovariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.^{[6]}^{[7]}
Intuition
PCA can be thought of as fitting an ndimensional 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, and by omitting that axis and its corresponding principal component from our representation of the dataset, we lose only a commensurately small amount of information.
To find the axes of the ellipsoid, we must first subtract the mean of each variable from the dataset to center the data around the origin. 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 become 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 our covariance matrix into a diagonalised form with the diagonal elements representing 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.
This procedure is sensitive to the scaling of the data, and there is no consensus as to how to best scale the data to obtain optimal results.
Details
PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some 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.^{[3]}
Consider a data matrix, X, with columnwise 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 pdimensional vectors of weights or coefficients that map each row vector of X to a new vector of principal component scores , given by
in such a way that the individual variables 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.
First component
In order to maximize variance, the first weight vector w_{(1)} thus has to satisfy
Equivalently, writing this in matrix form gives
Since w_{(1)} has been defined to be a unit vector, it equivalently also satisfies
The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as X^{T}X is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector.
With w_{(1)} found, the first principal component of a data vector x_{(i)} can then be given as a score t_{1(i)} = x_{(i)} ⋅ w_{(1)} in the transformed coordinates, or as the corresponding vector in the original variables, {x_{(i)} ⋅ w_{(1)}} w_{(1)}.
Further components
The kth component can be found by subtracting the first k − 1 principal components from X:
and then finding the weight vector which extracts the maximum variance from this new data matrix
It turns out that this gives the remaining eigenvectors of X^{T}X, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of X^{T}X.
The kth principal component of a data vector x_{(i)} can therefore be given as a score t_{k(i)} = x_{(i)} ⋅ w_{(k)} in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x_{(i)} ⋅ w_{(k)}} w_{(k)}, where w_{(k)} is the kth eigenvector of X^{T}X.
The full principal components decomposition of X can therefore be given as
where W is a pbyp matrix of weights whose columns are the eigenvectors of X^{T}X. The transpose of W is sometimes called the whitening or sphering transformation. Columns of W multiplied by the square root of corresponding eigenvalues, i.e. eigenvectors scaled up by the variances, are called loadings in PCA or in Factor analysis.
Covariances
X^{T}X itself can be recognised as proportional to the empirical sample covariance matrix of the dataset X.
The sample covariance Q between two of the different principal components over the dataset is given by:
where the eigenvalue property of w_{(k)} has been used to move from line 2 to line 3. However eigenvectors w_{(j)} and w_{(k)} corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.
Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.
In matrix form, the empirical covariance matrix for the original variables can be written
The empirical covariance matrix between the principal components becomes
where Λ is the diagonal matrix of eigenvalues λ_{(k)} of X^{T}X
(λ_{(k)} being equal to the sum of the squares over the dataset associated with each component k: λ_{(k)} = Σ_{i} t_{k}^{2}_{(i)} = Σ_{i} (x_{(i)} ⋅ w_{(k)})^{2})
Dimensionality reduction
The transformation T = X W maps a data vector x_{(i)} from an original space of p variables to a new space of p variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first L principal components, produced by using only the first L eigenvectors, gives the truncated transformation
where the matrix T_{L} now has n rows but only L columns. In other words, PCA learns a linear transformation where the columns of p × L matrix W form an orthogonal basis for the L features (the components of representation t) that are decorrelated.^{[8]} By construction, of all the transformed data matrices with only L columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error or .
Such dimensionality reduction can be a very useful step for visualising and processing highdimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting L = 2 and keeping only the first two principal components finds the twodimensional plane through the highdimensional dataset in which the data is most spread out, so if the data contains clusters these too may be most spread out, and therefore most visible to be plotted out in a twodimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more likely to substantially overlay each other, making them indistinguishable.
Similarly, in regression analysis, the larger the number of explanatory variables allowed, the greater is the chance of overfitting the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called principal component regression.
Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix W, which can be thought of as a highdimensional rotation of the coordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signaltonoise ratio. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss.
Singular value decomposition
The principal components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X,
Here Σ is an nbyp rectangular diagonal matrix of positive numbers σ_{(k)}, called the singular values of X; U is an nbyn matrix, the columns of which are orthogonal unit vectors of length n called the left singular vectors of X; and W is a pbyp whose columns are orthogonal unit vectors of length p and called the right singular vectors of X.
In terms of this factorization, the matrix X^{T}X can be written
where is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies . Comparison with the eigenvector factorization of X^{T}X establishes that the right singular vectors W of X are equivalent to the eigenvectors of X^{T}X, while the singular values σ_{(k)} of are equal to the squareroot of the eigenvalues λ_{(k)} of X^{T}X.
Using the singular value decomposition the score matrix T can be written
so each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T.
Efficient algorithms exist to calculate the SVD of X without having to form the matrix X^{T}X, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix, unless only a handful of components are required.
As with the eigendecomposition, a truncated n × L score matrix T_{L} can be obtained by considering only the first L largest singular values and their singular vectors:
The truncation of a matrix M or T using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of rank L to the original matrix, in the sense of the difference between the two having the smallest possible Frobenius norm, a result known as the Eckart–Young theorem [1936].
Further considerations
Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line. The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points. The singular values (in Σ) are the square roots of the eigenvalues of the matrix X^{T}X. 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 DCTII 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 positively 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.) Note that 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.^{[9]}
Meancentering 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 crossproduct of two standard scores (Zscores) or statistical moments (hence the name: Pearson ProductMoment Correlation). Also see the article by Kromrey & FosterJohnson (1998) on "Meancentering in Moderated Regression: Much Ado About Nothing".
An autoencoder neural network with a linear hidden layer is similar 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, yet the principal components can easily be recovered from them using singular value decomposition.^{[10]}
PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability.^{[11]} 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.^{[12]} The linear discriminant analysis is an alternative which is optimized for class separability.
Table of symbols and abbreviations
Symbol  Meaning  Dimensions  Indices 

data matrix, consisting of the set of all data vectors, one vector per row 
 
the number of row vectors in the data set  scalar  
the number of elements in each row vector (dimension)  scalar  
the number of dimensions in the dimensionally reduced subspace,  scalar  
vector of empirical means, one mean for each column j of the data matrix  
vector of empirical standard deviations, one standard deviation for each column j of the data matrix  
vector of all 1's  
deviations from the mean of each column j of the data matrix 
 
zscores, 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 subset of those in V 
 
matrix consisting of n row 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. 

Properties and limitations of PCA
Properties
Some properties of PCA include:^{[13]}
 Property 1: For any integer q, 1 ≤ q ≤ p, consider the orthogonal linear transformation
 where is a qelement vector and is a (q × p) matrix, and let be the variancecovariance matrix for . Then the trace of , denoted , is maximized by taking , where consists of the first q columns of is the transposition of .
 Property 2: Consider again the orthonormal transformation
 with and defined as before. Then is minimized by taking where consists of the last q columns of .
The statistical implication of this property is that the last few PCs are not simply unstructured leftovers after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected nearconstant linear relationships between the elements of x, and they may also be useful in regression, in selecting a subset of variables from x, and in outlier detection.
 Property 3: (Spectral Decomposition of Σ)
Before we look at its usage, we first look at diagonal elements,
Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of x into decreasing contributions due to each PC, but we can also decompose the whole covariance matrix into contributions from each PC. Although not strictly decreasing, the elements of will tend to become smaller as increases, as is nonincreasing for increasing , whereas the elements of tend to stay about the same size because of the normalization constraints: .
Limitations
As noted above, the results of PCA depend on the scaling of the variables. A scaleinvariant form of PCA has been developed.^{[14]}
The applicability of PCA is limited by certain assumptions^{[15]} made in its derivation.
The other limitation is the meanremoval process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are nonnegative, and the meanremoval process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes,^{[16]} and forward modeling has to be performed to recover the true magnitude of the signals.^{[17]} As an alternative method, nonnegative matrix factorization focusing only on the nonnegative elements in the matrices, which is wellsuited for astrophysical observations.^{[18]}^{[19]}^{[20]} See more at Relation between PCA and Nonnegative Matrix Factorization.
PCA and information theory
Dimensionality reduction loses information, in general. PCAbased dimensionality reduction tends to minimize that information loss, under certain signal and noise models.
Under the assumption that
i.e., that the data vector is the sum of the desired informationbearing signal and a noise signal one can show that PCA can be optimal for dimensionality reduction, from an informationtheoretic pointofview.
In particular, Linsker showed that if is Gaussian and is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information between the desired information and the dimensionalityreduced output .^{[21]}
If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (i.e., the components of the vector are iid), but the informationbearing signal is nonGaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as^{[22]}^{[23]}
The optimality of PCA is also preserved if the noise is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the informationbearing signal .^{[24]} In general, even if the above signal model holds, PCA loses its informationtheoretic optimality as soon as the noise becomes dependent.
Computing PCA using the covariance method
The following is a detailed description of PCA using the covariance method (see also here) as opposed to the correlation method.^{[25]}
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:
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 with each representing a single grouped observation of the p variables.
 Write as row vectors, each of which has p columns.
 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.
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.^{[26]} Hence we proceed by centering the data as follows:
 Subtract the empirical mean vector from each row of the data matrix X.
 Store meansubtracted data in the n × p matrix B.
 where h is an n × 1 column vector of all 1s:
Find the covariance matrix
 Find the p × p empirical covariance matrix C from the outer product of matrix B with itself:
 where 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.
 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:
 where D is the diagonal matrix of eigenvalues of C. This step will typically involve the use of a computerbased algorithm for computing eigenvectors and eigenvalues. These algorithms are readily available as subcomponents of most matrix algebra systems, such as SAS,^{[27]} R, MATLAB,^{[28]}^{[29]} Mathematica,^{[30]} 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
 is the jth eigenvalue of the covariance matrix C, and
 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:
Select a subset of the eigenvectors as basis vectors
 Save the first L columns of V as the p × L matrix W:
 where
 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
Project the zscores of the data onto the new basis
 The projected vectors are the columns of the matrix
 The rows of matrix T represent the KosambiKarhunen–Loève transforms (KLT) of the data vectors in the rows of matrix X.
Derivation of PCA using the covariance method
Let X be a ddimensional 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 (i.e. PX is a random vector with all its distinct components pairwise uncorrelated).
A quick computation assuming were unitary yields:
Hence holds if and only if were diagonalisable by .
This is very constructive, as cov(X) is guaranteed to be a nonnegative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.
Iterative computation
In practical implementations especially with high dimensional data (large p), the covariance method is rarely used because it is not efficient. One way to compute the first principal component efficiently^{[31]} is shown in the following pseudocode, for a data matrix X with zero mean, without ever computing its covariance matrix.
r = a random vector of length p do c times: s = 0 (a vector of length p) for each row exit if return
This power iteration algorithm simply calculates the vector X^{T}(X r), normalizes, and places the result back in r. The eigenvalue is approximated by r^{T} (X^{T}X) r, which is the Rayleigh quotient on the unit vector r for the covariance matrix X^{T}X . The algorithm avoids the np^{2} operations of explicitly calculating and storing the covariance matrix X^{T}X , i.e. is one of matrixfree methods based on the function evaluating the product X^{T}(X r) at the cost of 2np operations. If the largest singular value is well separated from the next largest one, the vector r gets close to the first principal component of X within the number of iterations c, which is small relative to p, at the total cost 2cnp. The power iteration convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced matrixfree methods, such as the Lanczos algorithm or the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.
Subsequent principal components can be computed onebyone via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation. The latter approach in the block power method replaces singlevectors r and s with blockvectors, matrices R and S. Every column of R approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product X^{T}(X R). Implemented, e.g., in LOBPCG, efficient blocking eliminates the accumulation of the errors, allows using highlevel BLAS matrixmatrix product functions, and typically leads to faster convergence, compared to the singlevector onebyone technique.
The NIPALS method
Nonlinear iterative partial least squares (NIPALS) is an algorithm for computing the first few components in a principal component or partial least squares analysis. For veryhighdimensional datasets, such as those generated in the *omics sciences (e.g., genomics, metabolomics) it is usually only necessary to compute the first few PCs. The nonlinear iterative partial least squares (NIPALS) algorithm calculates t_{1} and w_{1}^{T} from X. The outer product, t_{1}w_{1}^{T} can then be subtracted from X leaving the residual matrix E_{1}. This can be then used to calculate subsequent PCs.^{[32]} This results in a dramatic reduction in computational time since calculation of the covariance matrix is avoided.
However, for large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality due to machine precision limitations accumulated in each iteration step.^{[33]} A Gram–Schmidt (GS) reorthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.^{[34]}
Online/sequential estimation
In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.^{[35]}
PCA and qualitative variables
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 e.g. 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, pvalue 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.
Applications
Quantitative finance
In quantitative finance, principal component analysis can be directly applied to the risk management of interest rate derivatives portfolios.^{[36]} Trading multiple swap instruments which are usually a function of 30500 other market quotable swap instruments is sought to be reduced to usually 3 or 4 principal components, representing the path of interest rates on a macro basis. Converting risks to be represented as those to factor loadings (or multipliers) provides assessments and understanding beyond that available to simply collectively viewing risks to individual 30500 buckets.
PCA has also been applied to share portfolios in a similar fashion.^{[37]} One application is to reduce portfolio risk, where allocation strategies are applied to the "principal portfolios" instead of the underlying stocks.^{[38]} A second is to enhance portfolio return, using the principal components to select stocks with upside potential.^{[39]}
Neuroscience
A variant of principal components analysis is used in neuroscience to identify the specific properties of a stimulus that increase a neuron's probability of generating an action potential.^{[40]} This technique is known as spiketriggered covariance analysis. In a typical application an experimenter presents a white noise process as a stimulus (usually either as a sensory input to a test subject, or as a current injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the covariance matrix of the spiketriggered ensemble, the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The eigenvectors of the difference between the spiketriggered covariance matrix and the covariance matrix of the prior stimulus ensemble (the set of all stimuli, defined over the same length time window) then indicate the directions in the space of stimuli along which the variance of the spiketriggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spiketriggered ensemble showed the largest positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.
In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs clustering analysis to associate specific action potentials with individual neurons.
PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, i.e. order parameters, during phase transitions in the brain.^{[41]}
Relation with other methods
Correspondence analysis
Correspondence analysis (CA) was developed by JeanPaul Benzécri^{[42]} and is conceptually similar to PCA, but scales the data (which should be nonnegative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. CA decomposes the chisquared statistic associated to this table into orthogonal factors.^{[43]} Because CA is a descriptive technique, it can be applied to tables for which the chisquared statistic is appropriate or not. Several variants of CA are available including detrended correspondence analysis and canonical correspondence analysis. One special extension is multiple correspondence analysis, which may be seen as the counterpart of principal component analysis for categorical data.^{[44]}
Factor analysis
Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a preprocessing step before clustering. PCA is a variancefocused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (i.e., translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.
Factor analysis is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlationfocused approach seeking to reproduce the intercorrelations among variables, in which the factors "represent the common variance of variables, excluding unique variance".^{[45]} In terms of the correlation matrix, this corresponds with focusing on explaining the offdiagonal terms (i.e. shared covariance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the ondiagonal terms, PCA also tends to fit relatively well the offdiagonal correlations.^{[46]} Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (i.e., latent constructs or factors) or causal modeling.
Kmeans clustering
It was asserted in ^{[47]}^{[48]} that the relaxed solution of kmeans clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of kmeans clustering was not a new result (see, for example,^{[49]}), and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.^{[50]}
Nonnegative matrix factorization
Nonnegative matrix factorization (NMF) is a dimension reduction method where only nonnegative elements in the matrices are used, which is therefore a promising method in astronomy,^{[18]}^{[19]}^{[20]} in the sense that astrophysical signals are nonnegative. The PCA components are orthogonal to each other, while the NMF components are all nonnegative and therefore constructs a nonorthogonal basis.
In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data.^{[16]} For NMF, its components are ranked based only on the empirical FRV curves.^{[20]} The residual fractional eigenvalue plots, i.e., as a function of component number given a total of components, for PCA has a flat plateau, where no data is captured to remove the quasistatic noise, then the curves dropped quickly as an indication of overfitting and captures random noise.^{[16]} The FRV curves for NMF is decreasing continuously ^{[20]} when the NMF components are constructed sequentially,^{[19]} indicating the continuous capturing of quasistatic noise; then converge to higher levels than PCA,^{[20]} indicating the less overfitting property of NMF.
Generalizations
Nonlinear generalizations
Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or Kmeans. Pearson's original 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^{[54]} 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, as it is illustrated by Fig. See also the elastic map algorithm and principal geodesic analysis. Another popular generalization is kernel PCA, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.
Multilinear generalizations
In multilinear subspace learning,^{[55]} PCA is generalized to multilinear PCA (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, nonnegative MPCA and robust MPCA.
Higher order
Nway principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, coinertia analysis, STATIS, and DISTATIS.
Robustness – weighted PCA
While PCA finds the mathematically optimal method (as in minimizing the squared error), it is sensitive to outliers in the data that produce large errors PCA tries to avoid. It therefore is common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in data mining algorithms like correlation clustering, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA^{[56]} based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.
Robust PCA via decomposition in lowrank and sparse matrices
Robust principal component analysis (RPCA) via decomposition in lowrank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations.^{[57]}^{[58]}^{[59]}^{[60]}
Robustness – L1 weighting
Outlierresistant versions of PCA have also been proposed on L1norm formulations (L1PCA).^{[61]}
Sparse PCA
A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables.
Similar techniques
Independent component analysis
Independent component analysis (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.
Network component analysis
Given a matrix , it tries to decompose it into two matrices such that . A key difference from techniques such as PCA and ICA is that some of the entries of are constrained to be 0. Here is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :
 has full column rank
 Each column of must have at least zeroes where is the number of columns of (or alternatively the number of rows of ). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.
 must have full row rank.
then the decomposition is unique up to multiplication by a scalar.^{[62]}
Software/source code
 ALGLIB  a C++ and C# library that implements PCA and truncated PCA
 Analytica – The builtin EigenDecomp function computes principal components.
 ELKI – includes PCA for projection, including robust variants of PCA, as well as PCAbased clustering algorithms.
 Julia – Supports PCA with the
pca
function in the MultivariateStats package  KNIME – A java based nodal arrenging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily.
 Mathematica – Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods.
 MATLAB Statistics Toolbox – The functions
princomp
andpca
(R2012b) give the principal components, while the functionpcares
gives the residuals and reconstructed matrix for a lowrank PCA approximation.  Matplotlib – Python library have a PCA package in the .mlab module.
 MLPACK – Provides an implementation of principal component analysis in C++.
 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 valueSVDS_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
andprcomp
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
, andFactoMineR
.  SAS  Proprietary software; for example, see ^{[63]}
 Scikitlearn – Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module.
 Weka – Java library for machine learning which contains modules for computing principal components.
See also
 Correspondence analysis (for contingency tables)
 Multiple correspondence analysis (for qualitative variables)
 Factor analysis of mixed data (for quantitative and qualitative variables)
 Canonical correlation
 CUR matrix approximation (can replace of lowrank SVD approximation)
 Detrended correspondence analysis
 Dynamic mode decomposition
 Eigenface
 Exploratory factor analysis (Wikiversity)
 Factorial code
 Functional principal component analysis
 Geometric data analysis
 Independent component analysis
 Kernel PCA
 L1norm principal component analysis
 Lowrank approximation
 Matrix decomposition
 Nonnegative matrix factorization
 Nonlinear dimensionality reduction
 Oja's rule
 Point distribution model (PCA applied to morphometry and computer vision)
 Principal component analysis (Wikibooks)
 Principal component regression
 Singular spectrum analysis
 Singular value decomposition
 Sparse PCA
 Transform coding
 Weighted least squares
References
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 ↑ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon (Dec 2001). "Spectral Relaxation for Kmeans Clustering" (PDF). Neural Information Processing Systems vol.14 (NIPS 2001). Vancouver, Canada: 1057–1064.
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 ↑ Wang, Y.; Klijn, J. G.; Zhang, Y.; Sieuwerts, A. M.; Look, M. P.; Yang, F.; Talantov, D.; Timmermans, M.; Meijervan Gelder, M. E.; Yu, J.; et al. (2005). "Gene expression profiles to predict distant metastasis of lymphnodenegative primary breast cancer". The Lancet. 365 (9460): 671–679. doi:10.1016/S01406736(05)179471. Data online
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 ↑ A.N. Gorban, B. Kegl, D.C. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007. ISBN 9783540737490
 ↑ Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. doi:10.1016/j.patcog.2011.01.004.
 ↑ Kriegel, H. P.; Kröger, P.; Schubert, E.; Zimek, A. (2008). "A General Framework for Increasing the Robustness of PCABased Correlation Clustering Algorithms". Scientific and Statistical Database Management. Lecture Notes in Computer Science. 5069: 418–435. doi:10.1007/9783540694977_27. ISBN 9783540694762.
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 ↑ T. Bouwmans; E. Zahzah (2014). "Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance". Special Issue on Background Models Challenge, Computer Vision and Image Understanding.
 ↑ T. Bouwmans; A. Sobral; S. Javed; S. Jung; E. Zahzah (2015). "Decomposition into Lowrank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a LargeScale Dataset". Computer Science Review. 23: 1. arXiv:1511.01245. doi:10.1016/j.cosrev.2016.11.001.
 ↑ N. Vaswani; T. Bouwmans, S. Javed, P. Narayanamurthy (2017). "Robust PCA and Robust Subspace Tracking". Preprint. arXiv:1711.09492. Bibcode:2018ISPM...35...32V. doi:10.1109/MSP.2018.2826566.
 ↑ Markopoulos, Panos P.; Karystinos, George N.; Pados, Dimitris A. (October 2014). "Optimal Algorithms for L1subspace Signal Processing". IEEE Transactions on Signal Processing. 62 (19): 5046–5058. arXiv:1405.6785. Bibcode:2014ITSP...62.5046M. doi:10.1109/TSP.2014.2338077.
 ↑ "Network component analysis: Reconstruction of regulatory signals in biological systems" (PDF). Retrieved February 10, 2015.
 ↑ "Principal Components Analysis". Institute for Digital Research and Education. UCLA. Retrieved 29 May 2018.
Further reading
 Jackson, J.E. (1991). A User's Guide to Principal Components (Wiley).
 Jolliffe, I. T. (1986). Principal Component Analysis. SpringerVerlag. p. 487. doi:10.1007/b98835. ISBN 9780387954424.
 Jolliffe, I.T. (2002). Principal Component Analysis, second edition (Springer).
 Husson François, Lê Sébastien & Pagès Jérôme (2009). Exploratory Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. 224p. ISBN 9782753509382
 Pagès Jérôme (2014). Multiple Factor Analysis by Example Using R. Chapman & Hall/CRC The R Series London 272 p
External links
Wikimedia Commons has media related to Principal component analysis. 
 University of Copenhagen video by Rasmus Bro on YouTube
 Stanford University video by Andrew Ng on YouTube
 A Tutorial on Principal Component Analysis
 A layman's introduction to principal component analysis on YouTube (a video of less than 100 seconds.)
 StatQuest: Principal Component Analysis (PCA) clearly explained on YouTube
 See also the list of Software implementations