Principal components analysis

Principal components 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).

PCA is mostly used as a tool in exploratory data analysis and for making predictive models. PCA involves the calculation of the eigenvalue decomposition or Singular value decomposition of a data set, usually after mean centering the data for each attribute. The results of a PCA are usually discussed in terms of component scores and loadings.

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 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 most optimum transform for a given data in least square terms.

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. But this is not necessarily the case, depending on the application.

For a data matrix, XT, 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:


 * $$\mathbf{Y^T}=\mathbf{X^T}\mathbf{W}$$
 * $$ = \mathbf{V}\mathbf{\Sigma} $$

where V Σ WT is the singular value decomposition (svd) of XT.

In this article we shall adopt the other convention, so that each column is made up of results for a different subject, and each row the results from a different probe. This will mean that the PCA for our data matrix X will be given by:


 * $$\mathbf{Y}=\mathbf{W}^T\mathbf{X}$$
 * $$ = \mathbf{\Sigma}\mathbf{V^T} $$

where W Σ VT is the svd of X.

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, PCA does not have a fixed set of basis vectors. Its basis vectors depend on the data set.

Discussion
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:


 * $$\mathbf{w}_1

= \arg\max_{\Vert \mathbf{w} \Vert = 1} \operatorname{var}\{ \mathbf{w}^T \mathbf{x} \} = \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$$ principal components from x:
 * $$\mathbf{\hat{x}}_{k - 1}

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


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

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, WL:


 * $$\mathbf{Y}=\mathbf{W_L}^T\mathbf{X} = \mathbf{\Sigma_L}\mathbf{V_L}^T$$

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


 * $$\mathbf{X}\mathbf{X}^T = \mathbf{W}\mathbf{\Sigma}\mathbf{\Sigma}^T\mathbf{W}^T$$

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

PCA is equivalent to empirical orthogonal functions (EOF).

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.

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.

Algorithm #1: 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:


 * $$ \mathbf{Y} = \mathbb{KLT} \{ \mathbf{X} \} $$

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


 * Write $$\mathbf{x}_1 \ldots \mathbf{x}_N$$ as column vectors, each of which has M rows.
 * Place the column vectors into a single matrix X of dimensions M &times; 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 &times; 1.


 * $$u[m] = {1 \over N} \sum_{n=1}^N X[m,n] $$

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 &times; N matrix B.


 * $$\mathbf{B} = \mathbf{X} - \mathbf{u} \cdot \mathbf{h} $$
 * where h is a 1 x N row vector of all 1's:


 * $$h[n] = 1 \, \qquad \qquad \mathrm{for \ } n = 1 \ldots N $$

Find the covariance matrix

 * Find the M &times; M empirical covariance matrix C from the outer product of matrix B with itself:
 * $$\mathbf{C} = \mathbb{ E } \left[ \mathbf{B} \otimes \mathbf{B} \right] = \mathbb{ E } \left[ \mathbf{B} \cdot \mathbf{B}^{*} \right] = { 1 \over N } \mathbf{B} \cdot \mathbf{B}^{*}$$
 * where
 * $$\mathbb{E} $$ is the expected value operator,
 * $$ \otimes $$ 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.

Find the eigenvectors and eigenvalues of the covariance matrix

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


 * $$\mathbf{V}^{-1} \mathbf{C} \mathbf{V} = \mathbf{D} $$


 * 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 &times; M diagonal matrix, where
 * $$D[p,q] = \lambda_m \qquad \mathrm{for} \qquad p = q = m$$


 * is the mth eigenvalue of the covariance matrix C, and


 * $$D[p,q] = 0 \qquad \mathrm{for} \qquad p \ne q.$$


 * Matrix V, also of dimension M &times; 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:


 * $$g[m] = \sum_{q=1}^m D[p,q] \qquad \mathrm{for} \qquad p = q \qquad \mathrm{and} \qquad m = 1...M $$

Select a subset of the eigenvectors as basis vectors

 * Save the first L columns of V as the M &times; L matrix W:


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


 * where


 * $$1 \leq L \leq M.$$


 * 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


 * $$ g[m=L] \ge 90% $$

Convert the source data to z-scores

 * Create an M &times; 1 empirical standard deviation vector s from the square root of each element along the main diagonal of the covariance matrix C:
 * $$ \mathbf{s} = \{ s[m] \} = \sqrt{C[p,q]} \qquad \mathrm{for \ } p = q = m = 1 \ldots M $$


 * Calculate the M &times; N z-score matrix:
 * $$ \mathbf{Z} = { \mathbf{B} \over \mathbf{s} \cdot \mathbf{h} } $$ (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!

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

 * The projected vectors are the columns of the matrix


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


 * 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

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

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 $$d \times d$$ orthonormal transformation matrix P such that


 * $$\mathbf{Y} = \mathbf{P}^\top \mathbf{X}$$

with the constraint that


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

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{P}^\top \mathbf{X}) (\mathbf{X}^\top \mathbf{P})] \\ \ &=& \mathbf{P}^\top \mathbb{E}[\mathbf{X} \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 $$d \times 1$$ column vectors, so


 * $$\mathbf{P} = [P_1, P_2, \ldots, P_d]$$

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



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

Substituting into equation above, we obtain:


 * $$[\lambda_1 P_1, \lambda_2 P_2, \ldots, \lambda_d P_d] =

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

Notice that in $$\lambda_i P_i = \operatorname{cov}(X)P_i$$, Pi is an eigenvector of X&prime;s covariance matrix. Therefore, by finding the eigenvectors of X&prime;s covariance matrix, we find a projection matrix P that satisfies the original constraints.

Relation to K-means clustering
It has been shown recently that the relaxed solution of K-means clustering, specified by the cluster indicators, are given by the PCA (principal component analysis) 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.

Correspondence analysis
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

 * Computer Vision Library
 * Multivariate Data Analysis Software
 * in Matlab, the function "princomp" gives the principal component
 * in the open source statistical package R, the function "princomp" can be used for principal component analysis.
 * In XLMiner, the Principles Component tab can be used for principal component analysis.