Principal component analysis

In statistics, principal components analysis (PCA) is a technique that can be used to simplify a dataset. It is a linear transformation that chooses a new coordinate system for the data set such that the greatest variance by any projection of the data set comes to lie on the first axis (called the first principal component), the second greatest variance on the second axis, and so on. PCA can be used for reducing dimensionality 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. The idea is that such low-order components often contain the "most important" aspects of the data. However, 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 away 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, W_L:

$\mathbf {Y} =\mathbf {W_{L}} ^{T}\mathbf {X} =\mathbf {\Sigma _{L}} \mathbf {V_{L}} ^{T}$

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

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

By finding the eigenvalues and eigenvectors of the covariance matrix, the eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset.

PCA is equivalent to empirical orthogonal functions (EOF).

PCA is a popular technique in pattern recognition. However, PCA 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

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

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 N data vectors $\mathbf {x} _{1}\ldots \mathbf {x} _{N}$ each length M, written as $\mathbf {x} _{n}=(\mathbf {x} _{n}^{1}\ldots \mathbf {x} _{n}^{M})$, and you want to project your data into a L dimensional subspace.

• Organize the data into N column vectors, each of dimensions M × 1.
• 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.

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

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 × 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 × 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 × 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

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

• 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 eigenvalules 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 × 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 threshhold, like 90 percent. In this case, choose the smallest value ofL such that

$g[m=L]\geq 90\%$

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:

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

• Calculate the M × 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 $\lambda _{i}P_{i}=\operatorname {cov} (X)P_{i}$, 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

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 Libray
• Multivariate Data Analysis Software

References

• Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space" (PDF). Philosophical Magazine. 2 (6): 559–572.

See also

• eigenface
• transform coding
• independent components analysis
• singular value decomposition
• factor analysis

External links

• A introductory explanation of PCA from StatSoft
• A tutorial on PCA by Lindsay I. Smith
• A layman's explanation from Umetrics