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Principal component analysis

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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 linearly 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 the largest possible variance (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 (i.e., uncorrelated with) the preceding components. Principal components are guaranteed to be independent 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) in signal processing, factor analysis, Eckart–Young theorem (Harman, 1960), or Schmidt–Mirsky theorem in psychometrics, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration.

PCA was invented in 1901 by Karl Pearson,[1] as an analogue of the principal axes theorem in mechanics; it was later independently developed (and named) by Harold Hotelling in the 1930s.[2] The method 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 (or correlation) matrix or singular value decomposition of a data matrix, usually after mean centering (and normalizing or using Z-scores) the data matrix for each attribute.[3] The results of a PCA are usually discussed in terms of component scores, sometimes called factor 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).[4]




Details Edit


Consider a data matrix, X, 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 p columns gives a particular kind of datum (say, the results from a particular probe).

Mathematically, the transformation is defined by a set of p-dimensional vectors of weights or loadings \mathbf{w}_{(k)} = (w_1, \dots, w_p)_{(k)} that map each row vector \mathbf{x}_{(i)} of X to new vector of principal component scores \mathbf{t}_{(i)} = (t_1, \dots, t_p)_{(i)}, given by

{t_{k}}_{(i)} = \mathbf{x}_{(i)} \cdot \mathbf{w}_{(k)}


First component Edit

The first loading vector w(1) thus has to satisfy

 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\,\{ \sum_i \left(t_{1}\right)^2_{(i)} \}
 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\, \sum_i \left(\mathbf{x}_{(i)} \cdot \mathbf{w} \right)^2

Equivalently, writing this in matrix form gives

 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\, \{ \Vert \mathbf{Xw} \Vert^2 \}
 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\, \{ \mathbf{w}^T \mathbf{X}^T \mathbf{X w} \}

Since w(1) has been defined to be a unit vector, it equally must also satisfy

  = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\, \left\{ \frac{\mathbf{w}^T\mathbf{X}^T \mathbf{X w}}{\mathbf{w}^T \mathbf{w}} \right\}



Further components Edit


\mathbf{\hat{X}}_{k - 1}
 = \mathbf{X} -
 \sum_{s = 1}^{k - 1}
 \mathbf{X} \mathbf{w}_{(s)} \mathbf{w}_{(s)}^{\rm T}


 = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{arg\,max}} \left\{
\Vert \mathbf{\hat{X}}_{k - 1} \mathbf{w} \Vert^2 \right\}
  = \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{\arg\,max}}\, \left\{ \tfrac{\mathbf{w}^T\mathbf{\hat{X}}_{k - 1}^T \mathbf{\hat{X}}_{k - 1} \mathbf{w}}{\mathbf{w}^T \mathbf{w}} \right\}



The full principal components decomposition of X can therefore be given as

\mathbf{T} = \mathbf{X} \mathbf{W}


Covariances Edit



Q(\mathrm{PC}_{(j)}, \mathrm{PC}_{(k)}) & \propto (\mathbf{X}\mathbf{w}_{(j)}) \cdot (\mathbf{X}\mathbf{w}_{(k)}) \\
& = \mathbf{w}_{(j)}^T \mathbf{X}^T \mathbf{X} \mathbf{w}_{(k)} \\
& = \mathbf{w}_{(j)}^T \lambda_{(k)} \mathbf{w}_{(k)} \\
& = \lambda_{(k)} \mathbf{w}_{(j)}^T \mathbf{w}_{(k)}




\mathbf{Q} \propto \mathbf{X}^T \mathbf{X} = \mathbf{W} \mathbf{\Lambda} \mathbf{W}^T

The empirical covariance matrix between the principal components becomes

\mathbf{W}^T \mathbf{Q} \mathbf{W} \propto \mathbf{W}^T \mathbf{W} \, \mathbf{\Lambda} \, \mathbf{W}^T \mathbf{W}
=  \mathbf{\Lambda}



Dimensionality reduction Edit


\mathbf{T}_L = \mathbf{X} \mathbf{W}_L

where the matrix TL now has n rows but only L columns. By construction, of all the transformed data matrices with only L columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error ||T − TL||2.

File:PCA of Haplogroup J using 37 STRs.png




Singular value decomposition Edit

The principal components transformation can also be associated with another matrix factorisation, the singular value decomposition (SVD) of X,

\mathbf{X} = \mathbf{U}\mathbf{\Sigma}\mathbf{W}^T

Here Σ is a p-by-p diagonal matrix of positive numbers σ(k), called the singular values of X; U is an n-by-p matrix, the columns of which are orthogonal unit vectors of length n called the left singular vectors of X; and W is a p-by-p whose columns are orthogonal unit vectors of length p and called the right singular vectors of X.

In terms of this factorisation, the matrix XTX can be written

\mathbf{X}^T\mathbf{X} & = \mathbf{W}\mathbf{\Sigma}\mathbf{U}^T \mathbf{U}\mathbf{\Sigma}\mathbf{W}^T \\
                       & = \mathbf{W}\mathbf{\Sigma}^2\mathbf{W}^T


Using the singular value decomposition the score matrix T can be written

\mathbf{T} & = \mathbf{X} \mathbf{W} \\
           & = \mathbf{U}\mathbf{\Sigma}\mathbf{W}^T \mathbf{W} \\


Efficient algorithms exist to calculate the SVD of X without having to form the matrix XTX, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix, unless only a handful of components are required.

As with the eigendecomposition, a truncated n-by-L score matrix TL can be obtained by considering only the first L largest singular values and their singular vectors:

\mathbf{T}_L = \mathbf{U}_L\mathbf{\Sigma}_L = \mathbf{X} \mathbf{W}_L


Further considerations Edit




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


PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability.[9] An alternative is the linear discriminant analysis, which does take this into account.

Table of symbols and abbreviations Edit

Symbol Meaning Dimensions Indices
\mathbf{X} = \{ X[i,j] \} data matrix, consisting of the set of all data vectors, one vector per row  n \times p  i = 1 \ldots n
 j = 1 \ldots p
n \, the number of row vectors in the data set 1 \times 1 scalar
p \, the number of elements in each row vector (dimension) 1 \times 1 scalar
L \, the number of dimensions in the dimensionally reduced subspace,  1 \le L \le p 1 \times 1 scalar
\mathbf{u} = \{ u[j] \} vector of empirical means, one mean for each column j of the data matrix  p \times 1  j = 1 \ldots p
\mathbf{s} = \{ s[j] \} vector of empirical standard deviations, one standard deviation for each column j of the data matrix  p \times 1  j = 1 \ldots p
\mathbf{h} = \{ h[i] \} vector of all 1's  1 \times n  i = 1 \ldots n
\mathbf{B} = \{ B[i,j] \} deviations from the mean of each column j of the data matrix  n \times p  i = 1 \ldots n
 j = 1 \ldots p
\mathbf{Z} = \{ Z[m,n] \} z-scores, computed using the mean and standard deviation for each row m of the data matrix  n \times p  i = 1 \ldots n
 j = 1 \ldots p
\mathbf{C} = \{ C[k, l] \} covariance matrix  p \times p  k = 1 \ldots p
 l = 1 \ldots p
\mathbf{R} = \{ R[k, l] \} correlation matrix  p \times p  k = 1 \ldots p
 l = 1 \ldots p
 \mathbf{V} = \{ V[j, k] \} matrix consisting of the set of all eigenvectors of C, one eigenvector per column  p \times p  j = 1 \ldots p
 k = 1 \ldots p
\mathbf{D} = \{ D[k, l] \} diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements  p \times p  k = 1 \ldots p
 l = 1 \ldots p
\mathbf{W} = \{ W[j, k] \} 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  p \times L  j = 1 \ldots p
 k = 1 \ldots L
\mathbf{T} = \{ T[i, k] \} matrix consisting of n row 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.  n \times L  i = 1 \ldots n
 k = 1 \ldots L

Properties and limitations of PCA Edit


The applicability of PCA is limited by certain assumptions[10] made in its derivation.

PCA and information theory Edit

The claim that the PCA used for dimensionality reduction preserves most of the information of the data is misleading. Indeed, without any assumption on the signal model, PCA cannot help to reduce the amount of information lost during dimensionality reduction, where information was measured using Shannon entropy.[11]

Under the assumption that


i.e., that the data vector \mathbf{x} is the sum of the desired information-bearing signal \mathbf{s} and a noise signal \mathbf{n} one can show that PCA can be optimal for dimensionality reduction also from an information-theoretic point-of-view.

In particular, Linsker showed that if \mathbf{s} is Gaussian and \mathbf{n} is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information I(\mathbf{y};\mathbf{s}) between the desired information \mathbf{s} and the dimensionality-reduced output \mathbf{y}=\mathbf{W}_L^T\mathbf{x}.[12]

If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (i.e., the components of the vector \mathbf{n} are iid), but the information-bearing signal \mathbf{s} is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as[13][14]


The optimality of PCA is also preserved if the noise \mathbf{n} is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal \mathbf{s}.[15] In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise \mathbf{n} becomes dependent.

Computing PCA using the covariance method Edit

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)Template:Cite quote.

The goal is to transform a given data set X of dimension p 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 Edit

Suppose you have data comprising a set of observations of p variables, and you want to reduce the data so that each observation can be described with only L variables, L < p. 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}_i representing a single grouped observation of the p variables.

  • Write \mathbf{x}_1 \ldots \mathbf{x}_n as row vectors, each of which has p columns.

Calculate the empirical mean Edit

  • Find the empirical mean along each dimension j = 1, ..., p.
u[j] = {1 \over N} \sum_{i=1}^n X[i,j]

Calculate the deviations from the mean Edit

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.[16] Hence we proceed by centering the data as follows:

\mathbf{B} = \mathbf{X} - \mathbf{h}\mathbf{u}^{T}
h[n] = 1 \, \qquad \qquad \text{for } i = 1, \ldots, n

Find the covariance matrix Edit

\mathbf{C} = { 1 \over {n-1} } \mathbf{B}^{*} \cdot \mathbf{B}
  • The reasoning behind using N-1 instead of N to calculate the covariance is Bessel's correction

Find the eigenvectors and eigenvalues of the covariance matrix Edit

\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, MATLAB,[17][18] Mathematica,[19] SciPy, IDL (Interactive Data Language), or GNU Octave as well as OpenCV.
  • Matrix D will take the form of an M × M diagonal matrix, where
D[k,l] = \lambda_k \qquad \text{for } k = l = j
is the jth eigenvalue of the covariance matrix C, and
D[k,l] = 0 \qquad \text{for } k \ne l.

Rearrange the eigenvectors and eigenvalues Edit


Compute the cumulative energy content for each eigenvector Edit

  • 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 jth eigenvector is the sum of the energy content across all of the eigenvalues from 1 through j:
g[m] = \sum_{k=1}^j D[k,k] \qquad \mathrm{for} \qquad j = 1,\dots,p [citation needed]

Select a subset of the eigenvectors as basis vectors Edit

 W[k,l] = V[k,l] \qquad \mathrm{for} \qquad k = 1,\dots,p \qquad l = 1,\dots,L
1 \leq L \leq p.
 \frac{g[L]}{g[p]} \ge 0.9\,

Convert the source data to z-scores (optional) Edit

 \mathbf{s} = \{ s[j] \} = \{ \sqrt{C[j,j]} \} \qquad \text{for } j = 1, \ldots, p
  • Calculate the n × p z-score matrix:
 \mathbf{Z} = { \mathbf{B} \over \mathbf{h} \cdot \mathbf{s}^{T} } (divide element-by-element)

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

  • The projected vectors are the columns of the matrix
 \mathbf{T} =  \mathbf{Z} \cdot \mathbf{W} = \mathbb{KLT} \{ \mathbf{X} \}.

Derivation of PCA using the covariance method Edit


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:

	&= &\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}\\

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


Iterative computationEdit

In practical implementations especially with high dimensional data (large p), the covariance method is rarely used because it is not efficient. One way to compute the first principal component efficiently[20] is shown in the following pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix

\mathbf{r} = a random vector of length p
do c times:
      \mathbf{s} = 0 (a vector of length p)
      for each row \mathbf{x} \in \mathbf{X}
            \mathbf{s} = \mathbf{s} + (\mathbf{x} \cdot \mathbf{r})\mathbf{x}
      \mathbf{r} = \frac{\mathbf{s}}{|\mathbf{s}|}
return \mathbf{r}




The NIPALS method Edit

Main article: Non-linear iterative partial least squares


However, for large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality due to machine precision limitations accumulated in each iteration step.[22] A Gram–Schmidt (GS) re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.[23]

Online/sequential estimation Edit


Relation between PCA and K-means clustering Edit

It has been shown recently (2001,2004)[25][26] 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 Edit

Correspondence analysis (CA) was developed by Jean-Paul Benzécri[27] 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.[28] 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.[29]

Generalizations Edit

Nonlinear generalizations Edit

File:Elmap breastcancer wiki.png


Multilinear generalizations Edit


Higher order Edit


Robustness – weighted PCA Edit

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[35] based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.

Software/source code Edit

  • Mathematica implements principal component analysis with the PrincipalComponents command[36] using both covariance and correlation methods.
  • In the NAG Library, principal components analysis is implemented via the g03aa routine (available in both the Fortran[37] and the C[38] versions of the Library).
  • In the MATLAB Statistics Toolbox, the functions princomp and pca (R2012b) give the principal components, while the function pcares gives the residuals and reconstructed matrix for a low-rank PCA approximation. An example MATLAB implementation of PCA is available.[39]
  • in GNU Octave, a free software computational environment mostly compatible with MATLAB, the function princomp[40] gives the principal component.
  • in the free statistical package R, the functions princomp[41] and prcomp[42] can be used for principal component analysis; prcomp uses singular value decomposition which generally gives better numerical accuracy. Recently there has been an explosion in implementations of principal component analysis in various R packages. Some packages that implement PCA in R, include, but are not limited to: ade4, vegan, ExPosition, and FactoMineR[43]
  • in SAS, PROC FACTOR offers principal components analysis.
  • In XLMiner, the Principal Components tab can be used for principal component analysis.[citation needed]
  • In Stata, the pca command provides principal components analysis.
  • Cornell Spectrum Imager – An open-source toolset built on ImageJ. Enables quick easy PCA analysis for 3D datacubes.[44]
  • imDEV – Free Excel addin to calculate principal components using R package[45][46]
  • "ViSta: The Visual Statistics System" – a free software that provides principal components analysis, simple and multiple correspondence analysis.[47]
  • "Spectramap" – software to create a biplot using principal components analysis, correspondence analysis or spectral map analysis.[48]
  • FinMath – a .NET numerical library containing an implementation of PCA.[49]
  • The Unscrambler is a multivariate analysis software enabling Principal Component Analysis (PCA) with PCA Projection.[citation needed]
  • OpenCV[50]
  • NMath, a proprietary numerical library containing PCA for the .NET Framework.[citation needed]
  • In IDL, the principal components can be calculated using the function pcomp.[51]
  • Weka computes principal components.[52]
  • Software for analyzing multivariate data with instant response using PCA[53]
  • Orange (software) supports PCA through its Linear Projection widget.[citation needed]
  • A version of PCA adapted for population genetics analysis can be found in the suite EIGENSOFT.[54]
  • PCA can also be performed by the statistical software Partek Genomics Suite.[55]
  • The libpca C++ library offers PCA and corresponding transformations

See also Edit

Notes Edit

  1. Pearson, K. (1901). On Lines and Planes of Closest Fit to Systems of Points in Space. Philosophical Magazine 2 (11): 559–572.
  2. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417-441, and 498-520.
    Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 27, 321-77
  3. Abdi. H., & Williams, L.J. (2010). Principal component analysis.. Wiley Interdisciplinary Reviews: Computational Statistics, 2: 433–459.
  4. Shaw P.J.A. (2003) Multivariate statistics for the Environmental Sciences, Hodder-Arnold. ISBN 0-340-80763-6. Template:Page needed
  5. Barnett, T. P., and R. Preisendorfer. (1987). Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis.. Monthly Weather Review 115.
  6. Hsu, Daniel, Sham M. Kakade, and Tong Zhang (2008). A spectral algorithm for learning hidden markov models.. arXiv preprint arXiv:0811.4413.
  9. Fukunaga, Keinosuke (1990). Introduction to Statistical Pattern Recognition, Elsevier.
  10. Jonathon Shlens, A Tutorial on Principal Component Analysis.
  11. Geiger, Bernhard, Kubin, Gernot (Sep 2012). Relative Information Loss in the PCA. Proc. IEEE Information Theory Workshop: 562–566.
  12. Linsker, Ralph (March 1988). Self-organization in a perceptual network. IEEE Computer 21 (3): 105–117.
  13. Deco & Obradovic (1996). An Information-Theoretic Approach to Neural Computing, New York, NY: Springer.
  14. Plumbley, Mark (1991). Information theory and unsupervised neural networks.Tech Note
  15. Geiger, Bernhard, Kubin, Gernot (January 2013). Signal Enhancement as Minimization of Relevant Information Loss. Proc. ITG Conf. on Systems, Communication and Coding.
  17. eig function Matlab documentation
  18. MATLAB PCA-based Face recognition software
  19. Eigenvalues function Mathematica documentation
  21. Geladi, Paul (1986). Partial Least Squares Regression:A Tutorial. Analytica Chimica Acta 185: 1–17.
  22. Kramer,R.,(1998) Chemometric Techniques for Quantitative Analysis (CRC Press, New York).
  23. M. Andrecut. Parallel GPU Implementation of Iterative PCA Algorithms. Journal of Computational Biology, 16(11), Nov. 2009.
  24. Warmuth, M. K. (2008). Randomized online PCA algorithms with regret bounds that are logarithmic in the dimension. Journal of Machine Learning Research 9: 2287–2320.
  25. H. Zha, C. Ding, M. Gu, X. He and H.D. Simon. "Spectral Relaxation for K-means Clustering",, Neural Information Processing Systems vol.14 (NIPS 2001). pp. 1057–1064, Vancouver, Canada. Dec. 2001.
  26. 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.
  27. Benzécri, J.-P. (1973). L'Analyse des Données. Volume II. L'Analyse des Correspondances, Paris, France: Dunod.
  28. Greenacre, Michael (1983). Theory and Applications of Correspondence Analysis, London: Academic Press.
  29. Le Roux, Brigitte and Henry Rouanet (2004). Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis, Dordrecht: Kluwer.
  31. 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
  32. A. Zinovyev, ViDaExpert – Multidimensional Data Visualization Tool (free for non-commercial use). Institut Curie, Paris.
  34. Lu, Haiping (2011). A Survey of Multilinear Subspace Learning for Tensor Data. Pattern Recognition 44: 1540–1551.
  35. DOI:10.1007/978-3-540-69497-7_27 10.1007/978-3-540-69497-7_27
  36. PrincipalComponents Mathematica Documentation
  37. The Numerical Algorithms Group NAG Library Routine Document: nagf_mv_prin_comp (g03aaf). NAG Library Manual, Mark 23. URL accessed on 2012-02-16.
  38. The Numerical Algorithms Group NAG Library Routine Document: nag_mv_prin_comp (g03aac). NAG Library Manual, Mark 9. URL accessed on 2012-02-16.
  39. PcaPress
  40. princomp
  41. princomp
  42. prcomp
  43. Multivariate
  44. Cornell Spectrum Imager
  45. imDEV
  46. pcaMethods
  47. "ViSta: The Visual Statistics System"
  48. "Spectramap"
  49. FinMath
  50. Computer Vision Library
  51. PCOMP (IDL Reference) | Exelis VIS Docs Center IDL online documentation
  52. javadoc
  53. Software for analyzing multivariate data with instant response using PCA
  55. Partek Genomics Suite

References Edit

  • Jolliffe, I. T. (1986). Principal Component Analysis, 487, Springer-Verlag.

External links Edit

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