PCA Tutorial

 Original: http://en.wikipedia.org/wiki/Principal_component_analysis

Example: http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf;

                  http://download.csdn.net/user/xlliu0226/

Principal component 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 of a data covariance matrix or singular value decomposition of a data matrix, 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 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. However, depending on the application this may not always be the case.

For a data matrix, X^T, 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:

where V Σ W^T is the singular value decomposition (svd) of X^T.

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:

(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:

and by substituting this as the new data set to find a principal component in

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

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:

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,

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^[1]. 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
	data matrix, consisting of the set of all data vectors, one vector per column
	the number of column vectors in the data set		scalar
	the number of elements in each column vector (dimension)		scalar
	the number of dimensions in the dimensionally reduced subspace,		scalar
	vector of empirical means, one mean for each row m of the data matrix
	vector of empirical standard deviations, one standard deviation for each row m of the data matrix
	vector of all 1's
	deviations from the mean of each row m of the data matrix
	z-scores, 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 sub-set of those in V
	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.

Properties and Limitations of PCA

PCA is theoretically the optimal linear scheme, in terms of least mean square error, for compressing a set of high dimensional vectors into a set of lower dimensional vectors and then reconstructing the original set. It is a non-parametric analysis and the answer is unique and independent of any hypothesis about data probability distribution. However, the latter two properties are regarded as weakness as well as strength, in that being non-parametric, no prior knowledge can be incorporated and that PCA compressions often incur loss of information.

The applicability of PCA is limited by the assumptions^[2] made in its derivation. These assumptions are:

• Assumption on Linearity

We assumed the observed data set to be linear combinations of certain basis. Non-linear methods such as kernel PCA have been developed without assuming linearity.

• Assumption that principal components are orthogonal

We assumed that principal components are orthogonal to each other. Methods such as Independent component analysis ( ICA ) have been developed to address this limitation.

• Assumption on the statistical importance of mean and covariance

PCA uses the eigenvectors of the covariance matrix and it only finds the independent axes of the data under the Gaussian assumption. For non-Gaussian or multi-modal Gaussian data, PCA simply de-correlates the axes. When PCA is used for clustering, its main limitation is that it does not account for class separability since it makes no use of the class label of the feature vector. There is no guarantee that the directions of maximum variance will contain good features for discrimination.

• Assumption that large variances have important dynamics

PCA simply performs a coordinate rotation that aligns the transformed axes with the directions of maximum variance. It is only when we believe that the observed data has a high signal-to-noise ratio that the principal components with larger variance correspond to interesting dynamics and lower ones correspond to noise.

Essentially, PCA involves only rotation and scaling. The above assumptions are made in order to simplify the algebraic computation on the data set. Some other methods have been developed without one or more of these assumptions; these are briefly described below.

Computing PCA using 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:

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 with each representing a single grouped observation of the M variables.

• Write 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 u of dimensions M × 1.

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.

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

Find the covariance matrix

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

where

is the expected value operator,

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:

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 × M diagonal matrix, where

is the mth eigenvalue of the covariance matrix C, and

• 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:

Select a subset of the eigenvectors as basis vectors

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

where

• 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

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:

• Calculate the M × N z-score matrix:

(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

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