In statistics, principal components analysis (PCA) is a technique for simplifying a dataset. It is a 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 can be used for dimensionality reduction in a dataset while retaining those characteristics of the dataset 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. But this is not necessarily the case, depending on the application.

PCA is also called the (discrete) Karhunen-Loève transform (or KLT, named after Kari Karhunen and Michel Loève) or the Hotelling transform (in honor of Harold Hotelling). 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. Unlike other linear transforms, the PCA does not have a fixed set of basis vectors. Its basis vectors depend on the data set.

Assuming zero empirical mean (the empirical mean of the distribution has been subtracted from the data set), the principal component w₁ of a dataset x can be defined as:

$\backslash mathbf\{w\}\_1$

= \arg\max_{\Vert \mathbf{w} \Vert = 1} E\left\{ \left( \mathbf{w}^T \mathbf{x}\right)^2 \right\} (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:

$\backslash mathbf\{\backslash hat\{x\}\}\_\{k\; -\; 1\}$

= \mathbf{x} - \sum_{i = 1}^{k - 1} \mathbf{w}_i \mathbf{w}_i^T \mathbf{x} and by substituting this as the new dataset to find a principal component in

$\backslash mathbf\{w\}\_k$

= \arg\max_{\Vert \mathbf{w} \Vert = 1} E\left\{ \left( \mathbf{w}^T \mathbf{\hat{x}}_{k - 1} \right)^2 \right\}.

The Karhunen-Loève transform is therefore equivalent to finding the singular value decomposition of the data matrix X,

$\backslash mathbf\{X\}=\backslash mathbf\{W\}\backslash mathbf\{\backslash Sigma\}\backslash mathbf\{V\}^T,$

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:

$\backslash mathbf\{Y\}=\backslash mathbf\{W\_L\}^T\backslash mathbf\{X\}\; =\; \backslash mathbf\{\backslash Sigma\_L\}\backslash mathbf\{V\_L\}^T$

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,

$\backslash mathbf\{X\}\backslash mathbf\{X\}^T\; =\; \backslash mathbf\{W\}\backslash mathbf\{\backslash Sigma\}^2\backslash mathbf\{W\}^T$

The eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset (see Rayleigh quotient).

PCA is equivalent to empirical orthogonal functions (EOF).

PCA is a popular technique in pattern recognition. But it is not optimized for class separability. An alternative is the linear discriminant analysis, which does take this into account. PCA optimally minimizes reconstruction error under the L2 norm.

Table of symbols and abbreviations

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Symbol	Meaning	Dimensions	Indices
$\backslash mathbf\{X\}\; =\; \backslash \{\; X*\backslash \}$	data matrix, consisting of the set of all data vectors, one vector per column	$M\; \backslash times\; N$	$m\; =\; 1\; \backslash ldots\; M$ $n\; =\; 1\; \backslash ldots\; N$
$N\; \backslash ,$	the number of column vectors in the data set	$1\; \backslash times\; 1$	scalar
$M\; \backslash ,$	the number of elements in each column vector	$1\; \backslash times\; 1$	scalar
$L\; \backslash ,$	the number of dimensions in the dimensionally reduced subspace, $1\; \backslash le\; L\; \backslash le\; M$	$1\; \backslash times\; 1$	scalar
$\backslash mathbf\{u\}\; =\; \backslash \{\; u*\backslash \}$	vector of empirical means, one mean for each row m of the data matrix	$M\; \backslash times\; 1$	$m\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{s\}\; =\; \backslash \{\; s*\backslash \}$	vector of empirical standard deviations, one standard deviation for each row m of the data matrix	$M\; \backslash times\; 1$	$m\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{h\}\; =\; \backslash \{\; h*\backslash \}$	vector of all 1's	$1\; \backslash times\; N$	$n\; =\; 1\; \backslash ldots\; N$
$\backslash mathbf\{B\}\; =\; \backslash \{\; B*\backslash \}$	deviations from the mean of each row m of the data matrix	$M\; \backslash times\; N$	$m\; =\; 1\; \backslash ldots\; M$ $n\; =\; 1\; \backslash ldots\; N$
$\backslash mathbf\{Z\}\; =\; \backslash \{\; Z*\backslash \}$	z-scores, computed using the mean and standard deviation for each row m of the data matrix	$M\; \backslash times\; N$	$m\; =\; 1\; \backslash ldots\; M$ $n\; =\; 1\; \backslash ldots\; N$
$\backslash mathbf\{C\}\; =\; \backslash \{\; C*\backslash \}$	covariance matrix	$M\; \backslash times\; M$	$p\; =\; 1\; \backslash ldots\; M$ $q\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{R\}\; =\; \backslash \{\; R*\backslash \}$	correlation matrix	$M\; \backslash times\; M$	$p\; =\; 1\; \backslash ldots\; M$ $q\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{V\}\; =\; \backslash \{\; V*\backslash \}$	matrix consisting of the set of all eigenvectors of C, one eigenvector per column	$M\; \backslash times\; M$	$p\; =\; 1\; \backslash ldots\; M$ $q\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{D\}\; =\; \backslash \{\; D*\backslash \}$	diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements	$M\; \backslash times\; M$	$p\; =\; 1\; \backslash ldots\; M$ $q\; =\; 1\; \backslash ldots\; M$
$\backslash mathbf\{W\}\; =\; \backslash \{\; W*\backslash \}$	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	$M\; \backslash times\; L$	$p\; =\; 1\; \backslash ldots\; M$ $q\; =\; 1\; \backslash ldots\; L$
$\backslash mathbf\{Y\}\; =\; \backslash \{\; Y*\backslash \}$	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.	$L\; \backslash times\; N$	$m\; =\; 1\; \backslash ldots\; L$ $n\; =\; 1\; \backslash ldots\; N$

Algorithm #1: the covariance method

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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:

$\backslash mathbf\{Y\}\; =\; \backslash mathbb\{KLT\}\; \backslash \{\; \backslash mathbf\{X\}\; \backslash \}$

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 $\backslash mathbf\{x\}\_1\; \backslash ldots\; \backslash mathbf\{x\}\_N$ with each $\backslash mathbf\{x\}\_n$ representing a single grouped observation of the M variables.

• Write $\backslash mathbf\{x\}\_1\; \backslash ldots\; \backslash mathbf\{x\}\_N$ 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 vector u of dimensions M × 1.

$u=\; \{1\; \backslash over\; N\}\; \backslash sum\_\{n=1\}^N\; X[m,n$

Calculate the deviations from the mean

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

$\backslash mathbf\{B\}\; =\; \backslash mathbf\{X\}\; -\; \backslash mathbf\{u\}\; \backslash cdot\; \backslash mathbf\{h\}$

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

$h*=\; 1\; \backslash ,\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{for\; \backslash \; \}\; n\; =\; 1\; \backslash ldots\; N$

Find the covariance matrix

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

$\backslash mathbf\{C\}\; =\; \backslash mathbb\{\; E\; \}\; \backslash left\backslash mathbf\{B\}\; \backslash otimes\; \backslash mathbf\{B\}\; \backslash right=\; \backslash mathbb\{\; E\; \}\; \backslash left\backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{B\}^\{*\}\; \backslash right=\; \{\; 1\; \backslash over\; N-1\; \}\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{B\}^\{*\}$

where

$\backslash mathbb\{E\}$ is the expected value operator,

$\backslash otimes$ is the outer product operator, and

$*\; \backslash$ is the conjugate transpose operator.

Find the eigenvectors and eigenvalues of the covariance matrix

• Compute the eigenvalue matrix D and the eigenvector matrix V of the covariance matrix C:
  

$\backslash mathbf\{C\}\; \backslash cdot\; \backslash mathbf\{V\}\; =\; \backslash mathbf\{V\}\; \backslash cdot\; \backslash mathbf\{D\}$

This step will typically require 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 or Mathematica. See, for example, the eig function.

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

$D*=\; \backslash lambda\_m\; \backslash qquad\; \backslash mathrm\{for\}\; \backslash qquad\; p\; =\; q\; =\; m$

is the mth eigenvalue of the covariance matrix C, and

$D*=\; 0\; \backslash qquad\; \backslash mathrm\{for\}\; \backslash qquad\; p\; \backslash ne\; q.$

• 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.

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:

$g=\; \backslash sum\_\{q=1\}^m\; D[p,q\backslash qquad\; \backslash mathrm\{for\}\; \backslash qquad\; p\; =\; q\; \backslash qquad\; \backslash mathrm\{and\}\; \backslash qquad\; m\; =\; 1...M$

Select a subset of the eigenvectors as basis vectors

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

$W=\; V[p,q\backslash qquad\; \backslash mathrm\{for\}\; \backslash qquad\; p\; =\; 1...M\; \backslash qquad\; q\; =\; 1...L$

where

$1\; \backslash leq\; L\; \backslash leq\; M.$

• 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

$g*\backslash ge\; 90\%$

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:

$\backslash mathbf\{s\}\; =\; \backslash \{\; s\backslash \}\; =\; \backslash sqrt\{C[p,q\}\; \backslash qquad\; \backslash mathrm\{for\; \backslash \; \}\; p\; =\; q\; =\; m\; =\; 1\; \backslash ldots\; M$

• Calculate the M × N z-score matrix:

$\backslash mathbf\{Z\}\; =\; \{\; \backslash mathbf\{B\}\; \backslash over\; \backslash mathbf\{s\}\; \backslash cdot\; \backslash mathbf\{h\}\; \}$ (divide element-by-element)

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

• The projected vectors are the columns of the matrix

$\backslash mathbf\{Y\}\; =\; \backslash mathbf\{W\}^*\; \backslash cdot\; \backslash mathbf\{Z\}\; =\; \backslash mathbb\{KLT\}\; \backslash \{\; \backslash mathbf\{X\}\; \backslash \}.$

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

Algorithm #2: the correlation method

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Editor's note: This section is currently undergoing a major revision. See page history for previous revisions.

Derivation of PCA using the covariance method

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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 $d\; \backslash times\; d$ orthonormal projection matrix P such that

$\backslash mathbf\{Y\}\; =\; \backslash mathbf\{P\}^\backslash top\; \backslash mathbf\{X\}$

with the constraint that

$\backslash operatorname\{cov\}(\backslash mathbf\{Y\})$ is a diagonal matrix and $\backslash mathbf\{P\}^\{-1\}\; =\; \backslash mathbf\{P\}^\backslash top.$

By substitution, and matrix algebra, we obtain:

$$

\begin{matrix} \operatorname{cov}(\mathbf{Y}) &=& \mathbb{E}\mathbf{Y} \mathbf{Y}^\top\\ \ &=& \mathbb{E}\mathbf{P}^\top \mathbf{X} ) ( \mathbf{P}^\top \mathbf{X} )^\top\\ \ &=& \mathbb{E}\mathbf{X}) (\mathbf{X}^\top \mathbf{P}) \\ \ &=& \mathbf{P}^\top \mathbb{E}\mathbf{X}^\top \mathbf{P} \\ \ &=& \mathbf{P}^\top \operatorname{cov}(\mathbf{X}) \mathbf{P} \end{matrix}

We now have:

$$

\begin{matrix} \mathbf{P}\operatorname{cov}(\mathbf{Y}) &=& \mathbf{P} \mathbf{P}^\top \operatorname{cov}(\mathbf{X}) \mathbf{P}\\ \ &=& \operatorname{cov}(\mathbf{X}) \mathbf{P}\\ \end{matrix}

Rewrite P as d $d\; \backslash times\; 1$ column vectors, so

$\backslash mathbf\{P\}\; =P\_2,\; \backslash ldots,\; P\_d$

and $\backslash operatorname\{cov\}(\backslash mathbf\{Y\})$ as:

$$

\begin{bmatrix} \lambda_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_d \end{bmatrix}.

Substituting into equation above, we obtain:

$P\_1,\; \backslash lambda\_2\; P\_2,\; \backslash ldots,\; \backslash lambda\_d\; P\_d=$

[\operatorname{cov}(X)P_1, \operatorname{cov}(X)P_2, \ldots, \operatorname{cov}(X)P_d].

Notice that in $\backslash lambda\_i\; P\_i\; =\; \backslash operatorname\{cov\}(X)P\_i$, 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.

Correspondence analysis

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Correspondence analysis 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 where Pearson's chi-square test has shown a relationship between rows and columns.

Software/Source Code

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• Computer Vision Library
• Multivariate Data Analysis Software

References

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See also

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• eigenface
• transform coding
• independent components analysis
• singular value decomposition
• factor analysis
• kernel PCA
• PCA applied to yield curves

External links

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• A introductory explanation of PCA from StatSoft
• A tutorial on PCA by Lindsay I. Smith (PDF)
• A layman's explanation from Umetrics
• Principal Component Analysis using Hebbian learning tutorial

Statistics | Components

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