Principal component analysis

Principal component analysis (PCA) is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has as high a variance as possible (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (uncorrelated with) the preceding components. Principal components are guaranteed to be independent only if the data set is jointly normally distributed. PCA is sensitive to the relative scaling of the original variables. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT), the Hotelling transform or proper orthogonal decomposition (POD).

PCA was invented in 1901 by Karl Pearson.[1] Now it is mostly used as a tool in exploratory data analysis and for making predictive models. PCA can be done by 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 (the transformed variable values corresponding to a particular data point) and loadings (the weight by which each standardized original variable should be multiplied to get the component score).[2]

PCA is the simplest of the true eigenvector-based multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way which best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture, a "shadow" of this object when viewed from its (in some sense) most informative viewpoint. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.

PCA is closely related to factor analysis; indeed, some statistical packages deliberately conflate the two techniques. True factor analysis makes different assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.

Contents

Details

PCA is mathematically defined[3] 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.

Define a data matrix, XT, with zero empirical mean (the empirical (sample) mean of the distribution has been subtracted from the data set), where each of the n rows represents a different repetition of the experiment, and each of the m columns gives a particular kind of datum (say, the results from a particular probe). (Note that XT is defined here and not X itself, and what we are calling XT is often alternatively denoted as X itself in the literature.) The singular value decomposition of X is X = WΣVT, where the m × m matrix W is the matrix of eigenvectors of XXT, the matrix Σ is an m × n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and the n × n matrix V is the matrix of eigenvectors of XTX. The PCA transformation that preserves dimensionality (that is, gives the same number of principal components as original variables) is then given by:

\begin{align} \mathbf{Y}^\top & = \mathbf{X}^\top\mathbf{W} \\ & = \mathbf{V}\mathbf{\Sigma}^\top\mathbf{W}^\top\mathbf{W} \\ & = \mathbf{V}\mathbf{\Sigma}^\top \end{align}

V is not uniquely defined in the usual case when m < n − 1, but Y will usually still be uniquely defined. Since W (by definition of the SVD of a real matrix) is an orthogonal matrix, each row of YT is simply a rotation of the corresponding row of XT. The first column of YT is made up of the "scores" of the cases with respect to the "principal" component, the next column has the scores with respect to the "second principal" component, and so on.

If we want a reduced-dimensionality representation, we can project X down into the reduced space defined by only the first L singular vectors, WL:

\mathbf{Y}=\mathbf{W_L}^\top\mathbf{X} = \mathbf{\Sigma_L}\mathbf{V}^\top where \mathbf{\Sigma_L}=\mathbf{I}_{L\times m}\mathbf{\Sigma} with \mathbf{I}_{L\times m} the L\times m rectangular identity matrix.

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}^\top = \mathbf{W}\mathbf{\Sigma}\mathbf{\Sigma}^\top\mathbf{W}^\top

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line. The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points. The singular values (in Σ) are the square roots of the eigenvalues of the matrix XXT. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is correlated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information. PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example and when applicable, to the discrete cosine transform. Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA.

PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same sample variance and are positively correlated, then the PCA will entail a rotation by 45° and the "loadings" for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Note that Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance.

Discussion

Mean subtraction (a.k.a. "mean centering") is necessary for performing PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data.[4]

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 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\,\operatorname{Var}\{ \mathbf{w}^\top \mathbf{X} \} = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\,E\left\{ \left( \mathbf{w}^\top \mathbf{X}\right)^2 \right\}

(See arg max for the notation.) With the first k − 1 components, the kth 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^\top \mathbf{X}

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

\mathbf{w}_k = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{arg\,max}}\,E\left\{ \left( \mathbf{w}^\top \mathbf{\hat{X}}_{k - 1} \right)^2 \right\}.

PCA is equivalent to empirical orthogonal functions (EOF), a name which is used in meteorology.

An autoencoder neural network with a linear hidden layer is similar 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 primary technique in pattern recognition. It is not, however, optimized for class separability.[5] An alternative is the linear discriminant analysis, which does take this into account.

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 (dimension) 1 \times 1 scalar
L \, the number of dimensions in the dimensionally reduced subspace, 1 \le L \le 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

Properties and limitations of PCA

As noted above, the results of PCA depend on the scaling of the variables.

The applicability of PCA is limited by certain assumptions[6] made in its derivation.

Computing PCA using the covariance method

The following is a detailed description of PCA using the covariance method (see also here). But note that it is better to use the singular value decomposition (using standard software).

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–Loève 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.

Calculate the empirical mean

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

Calculate the deviations from the mean

Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.[7] Hence we proceed by centering the data as follows:

\mathbf{B} = \mathbf{X} - \mathbf{u}\mathbf{h}
where h is a 1 × N row vector of all 1s:
h[n] = 1 \, \qquad \qquad \text{for } n = 1, \ldots, N

Find the covariance matrix

\mathbf{C} = \mathbb{ E } \left[ \mathbf{B} \otimes \mathbf{B} \right] = \mathbb{ E } \left[ \mathbf{B} \cdot \mathbf{B}^{*} \right] = { 1 \over N } \sum_{} \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.

Find the eigenvectors and eigenvalues of the covariance matrix

\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 R (programming language), MATLAB,[8][9] Mathematica,[10] SciPy, IDL(Interactive Data Language), or GNU Octave as well as OpenCV.
D[p,q] = \lambda_m \qquad \text{for } p = q = m
is the mth eigenvalue of the covariance matrix C, and
D[p,q] = 0 \qquad \text{for } p \ne q.

Rearrange the eigenvectors and eigenvalues

Compute the cumulative energy content for each eigenvector

g[m] = \sum_{q=1}^m D[q,q] \qquad \mathrm{for} \qquad m = 1,\dots,M

Select a subset of the eigenvectors as basis vectors

W[p,q] = V[p,q] \qquad \mathrm{for} \qquad p = 1,\dots,M \qquad q = 1,\dots,L
where
1 \leq L \leq M.
\frac{g[m=L]}{\sum_{q=1}^M D[q,q]} \ge 90%\,

Convert the source data to z-scores

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

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

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 mean.

We want to find (\ast)\, a d \times d orthonormal transformation matrix P so that PX has a diagonal covariant matrix (i.e. PX is a random vector with all its distinct components pairwise uncorrelated).

A quick computation assuming P were unitary yields:

\begin{array}[t]{rcl} \operatorname{cov}(PX) &= &\mathbb{E}[PX~(PX)^{\dagger}]\\ &= &\mathbb{E}[PX~X^{\dagger}P^{\dagger}]\\ &= &P~\mathbb{E}[XX^{\dagger}]P^{\dagger}\\ &= &P~\operatorname{cov}(X)P^{-1}\\ \end{array}

Hence (\ast)\, holds if and only if \operatorname{cov}(X) were diagonalisable by P.

This is very constructive, as cov(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.

Computing principal components iteratively

In practical implementations especially with high dimensional data (large m), the covariance method is rarely used because it is not efficient. One way to compute the first principal component efficiently[11] is shown in the following pseudo-code, for a data matrix XT with zero mean, without ever computing its covariance matrix. Note that here a zero mean data matrix means that the columns of XT should each have zero mean.

\mathbf{p} = a random vector
do c times:
      \mathbf{t} = 0 (a vector of length m)
      for each row \mathbf{x} \in \mathbf{X^T}
            \mathbf{t} = \mathbf{t} %2B (\mathbf{x} \cdot \mathbf{p})\mathbf{x}
      \mathbf{p} = \frac{\mathbf{t}}{|\mathbf{t}|}
return \mathbf{p}

This algorithm is simply an efficient way of calculating XXTp, normalizing, and placing the result back in p (Power iteration). It avoids the nm2 operations of calculating the covariance matrix. p will typically get close to the first principal component of XT within a small number of iterations, c. (The magnitude of t will be larger after each iteration. Convergence can be detected when it increases by an amount too small for the precision of the machine.)

Subsequent principal components can be computed by subtracting component p from XT (see Gram–Schmidt) and then repeating this algorithm to find the next principal component. However this simple approach is not numerically stable if more than a small number of principal components are required, because imprecisions in the calculations will additively affect the estimates of subsequent principal components. More advanced methods build on this basic idea, as with the closely related Lanczos algorithm.

One way to compute the eigenvalue that corresponds with each principal component is to measure the difference in sum-squared-distance between the rows and the mean, before and after subtracting out the principal component. The eigenvalue that corresponds with the component that was removed is equal to this difference.

The NIPALS method

Main article: Non-linear iterative partial least squares

For very high-dimensional datasets, such as those generated in the *omics sciences (e.g., genomics, metabolomics) it is usually only necessary to compute the first few PCs. The non-linear iterative partial least squares (NIPALS) algorithm calculates t1 and p1' from X. The outer product, t1p1' can then be subtracted from X leaving the residual matrix E1. This can be then used to calculate subsequent PCs.[12] This results in a dramatic reduction in computational time since calculation of the covariance matrix is avoided.

Relation between PCA and K-means clustering

It has been shown recently (2007)[13][14] that the relaxed solution of K-means clustering, specified by the cluster indicators, is given by the PCA 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. Thus PCA automatically projects to the subspace where the global solution of K-means clustering lies, and thus facilitates K-means clustering to find near-optimal solutions.

Correspondence analysis

Correspondence analysis (CA) was developed by Jean-Paul Benzécri[15] and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. CA decomposes the chi-squared statistic associated to this table into orthogonal factors.[16] Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not. Several variants of CA are available including detrended correspondence analysis and canonical correspondence analysis. One special extension is multiple correspondence analysis, which may be seen as the counterpart of principal component analysis for categorical data.[17]

Generalizations

Nonlinear generalizations

Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. Principal curves and manifolds[20] give the natural geometric framework for PCA generalization and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold for data approximation, and by encoding using standard geometric projection onto the manifold, as it is illustrated by Fig. See also the elastic map algorithm and principal geodesic analysis.

Multilinear generalizations

In multilinear subspace learning, PCA is generalized to multilinear PCA (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.

Higher order

N-way principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.

Robustness - Weighted PCA

While PCA finds the mathematically optimal method (as in minimizing the squared error), it is sensitive to outliers in the data that produce large errors PCA tries to avoid. It therefore is common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example in data mining algorithms like correlation clustering, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA [21] based on a Weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.

Software/source code

See also

Notes

  1. ^ Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space" (PDF). Philosophical Magazine 2 (6): 559–572. http://stat.smmu.edu.cn/history/pearson1901.pdf. 
  2. ^ Shaw PJA (2003) Multivariate statistics for the Environmental Sciences, Hodder-Arnold. ISBN 0-3408-0763-6.
  3. ^ Jolliffe I.T. Principal Component Analysis, Series: Springer Series in Statistics, 2nd ed., Springer, NY, 2002, XXIX, 487 p. 28 illus. ISBN 978-0-387-95442-4
  4. ^ A. A. Miranda, Y. A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
  5. ^ Fukunaga, Keinosuke (1990). Introduction to Statistical Pattern Recognition. Elsevier. ISBN 0122698517. http://books.google.com/books?visbn=0122698517. 
  6. ^ Jonathon Shlens, A Tutorial on Principal Component Analysis.
  7. ^ A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
  8. ^ eig function Matlab documentation
  9. ^ MATLAB PCA-based Face recognition software
  10. ^ Eigenvalues function Mathematica documentation
  11. ^ Roweis, Sam. "EM Algorithms for PCA and SPCA." Advances in Neural Information Processing Systems. Ed. Michael I. Jordan, Michael J. Kearns, and Sara A. Solla The MIT Press, 1998.
  12. ^ Geladi, Paul; Kowalski, Bruce (1986). "Partial Least Squares Regression:A Tutorial". Analytica Chimica Acta 185: 1–17. doi:10.1016/0003-2670(86)80028-9. 
  13. ^ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon. "Spectral Relaxation for K-means Clustering", http://ranger.uta.edu/~chqding/papers/Zha-Kmeans.pdf, Neural Information Processing Systems vol.14 (NIPS 2001). pp. 1057–1064, Vancouver, Canada. Dec. 2001.
  14. ^ C. Ding and X. He. "K-means Clustering via Principal Component Analysis". Proc. of Int'l Conf. Machine Learning (ICML 2004), pp 225–232. July 2004. http://ranger.uta.edu/~chqding/papers/KmeansPCA1.pdf
  15. ^ Benzécri, J.-P. (1973). L'Analyse des Données. Volume II. L'Analyse des Correspondances. Paris, France: Dunod. 
  16. ^ Greenacre, Michael (1983). Theory and Applications of Correspondence Analysis. London: Academic Press. ISBN 0-12-299050-1. 
  17. ^ Le Roux, Brigitte and Henry Rouanet (2004). Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis. Dordrecht: Kluwer. 
  18. ^ A. N. Gorban, A. Y. Zinovyev, Principal Graphs and Manifolds, In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28-59.
  19. ^ Wang, Y., Klijn, J.G., Zhang, Y., Sieuwerts, A.M., Look, M.P., Yang, F., Talantov, D., Timmermans, M., Meijer-van Gelder, M.E., Yu, J. et al.: Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet 365, 671-679 (2005); Data online
  20. ^ A. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007. ISBN 978-3-540-73749-0
  21. ^ Kriegel, H. P.; Kröger, P.; Schubert, E.; Zimek, A. (2008). "A General Framework for Increasing the Robustness of PCA-Based Correlation Clustering Algorithms". Scientific and Statistical Database Management. Lecture Notes in Computer Science 5069: 418. doi:10.1007/978-3-540-69497-7_27. ISBN 978-3-540-69476-2.  edit

References