Principal component analysis

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:


 * $$\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$$ principal components from x:
 * $$\mathbf{\hat{x}}_{k - 1}

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


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

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.

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 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:
 * $$\mathbf{C} = \mathbb{ E } \left[ \mathbf{B} \otimes \mathbf{B} \right] = \mathbb{ E } \left[ \mathbf{B} \cdot \mathbf{B}^{*} \right] = { 1 \over N-1 } \mathbf{B} \cdot \mathbf{B}^{*}$$
 * where
 * $$\mathbb{E} $$ is the expected value operator,
 * $$ \otimes $$ is the outer product operator, and
 * $$ * \ $$ is the conjugate transpose operator.

Find the eigenvectors and eigenvalues of the covariance matrix

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


 * $$\mathbf{C} \cdot \mathbf{V} = \mathbf{V} \cdot \mathbf{D} $$


 * 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. 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)

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 projection 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.

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