n displaystyle n observations with p displaystyle p variables, then the number of distinct principal components is min ( n − 1 , p ) displaystyle min(n-1,p) . 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 is orthogonal to the preceding
components. The resulting vectors are an uncorrelated orthogonal basis
set. PCA is sensitive to the relative scaling of the original
variables.
PCA was invented in 1901 by Karl Pearson,[1] as an analogue of the
principal axis theorem in mechanics; it was later independently
developed and named by
Contents 1 Intuition 2 Details 2.1 First component 2.2 Further components 2.3 Covariances 2.4 Dimensionality reduction 2.5 Singular value decomposition 3 Further considerations 4 Table of symbols and abbreviations 5 Properties and limitations of PCA 5.1 Properties 5.2 Limitations 5.3 PCA and information theory 6 Computing PCA using the covariance method 6.1 Organize the data set 6.2 Calculate the empirical mean 6.3 Calculate the deviations from the mean 6.4 Find the covariance matrix 6.5 Find the eigenvectors and eigenvalues of the covariance matrix 6.6 Rearrange the eigenvectors and eigenvalues 6.7 Compute the cumulative energy content for each eigenvector 6.8 Select a subset of the eigenvectors as basis vectors 6.9 Project the z-scores of the data onto the new basis 7 Derivation of PCA using the covariance method 7.1 Iterative computation 7.2 The NIPALS method 7.3 Online/sequential estimation 8 PCA and qualitative variables 9 Applications 9.1 Quantitative finance 9.2 Neuroscience 10 Relation with other methods 10.1 Correspondence analysis 10.2 Factor analysis 10.3 K-means clustering 10.4 Non-negative matrix factorization 11 Generalizations 11.1 Nonlinear generalizations 11.2 Multilinear generalizations 11.3 Higher order 11.4 Robustness – weighted PCA 11.5 Robust PCA via decomposition in low-rank and sparse matrices 11.6 Sparse PCA 12 Similar techniques 12.1 Independent component analysis 12.2 Network component analysis 13 Software/source code 14 See also 15 References 16 Further reading 17 External links Intuition[edit] PCA can be thought of as fitting an n-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small, and by omitting that axis and its corresponding principal component from our representation of the dataset, we lose only a commensurately small amount of information. To find the axes of the ellipsoid, we must first subtract the mean of each variable from the dataset to center the data around the origin. Then, we compute the covariance matrix of the data, and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to become unit vectors. Once this is done, each of the mutually orthogonal, unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform our covariance matrix into a diagonalised form with the diagonal elements representing the variance of each axis . The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. This procedure is sensitive to the scaling of the data, and there is no consensus as to how to best scale the data to obtain optimal results. Details[edit] PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some 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.[3] Consider a data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of p-dimensional vectors of weights or loadings w ( k ) = ( w 1 , … , w p ) ( k ) displaystyle mathbf w _ (k) =(w_ 1 ,dots ,w_ p )_ (k) that map each row vector x ( i ) displaystyle mathbf x _ (i) of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t m ) ( i ) displaystyle mathbf t _ (i) =(t_ 1 ,dots ,t_ m )_ (i) , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , m displaystyle t_ k _ (i) =mathbf x _ (i) cdot mathbf w _ (k) qquad mathrm for qquad i=1,dots ,nqquad k=1,dots ,m in such a way that the individual variables t 1 , … , t m displaystyle t_ 1 ,dots ,t_ m of t considered over the data set successively inherit the maximum possible variance from x, with each loading vector w constrained to be a unit vector. First component[edit] In order to maximize variance, the first loading vector w(1) thus has to satisfy w ( 1 ) = arg m a x ‖ w ‖ = 1 ∑ i ( t 1 ) ( i ) 2 = arg m a x ‖ w ‖ = 1 ∑ i ( x ( i ) ⋅ w ) 2 displaystyle mathbf w _ (1) = underset Vert mathbf w Vert =1 operatorname arg ,max ,left sum _ i left(t_ 1 right)_ (i) ^ 2 right = underset Vert mathbf w Vert =1 operatorname arg ,max ,left sum _ i left(mathbf x _ (i) cdot mathbf w right)^ 2 right Equivalently, writing this in matrix form gives w ( 1 ) = arg m a x ‖ w ‖ = 1 ‖ X w ‖ 2 = arg m a x ‖ w ‖ = 1 w T X T X w displaystyle mathbf w _ (1) = underset Vert mathbf w Vert =1 operatorname arg ,max , Vert mathbf Xw Vert ^ 2 = underset Vert mathbf w Vert =1 operatorname arg ,max ,left mathbf w ^ T mathbf X ^ T mathbf Xw right Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg m a x w T X T X w w T w displaystyle mathbf w _ (1) = operatorname arg ,max ,left frac mathbf w ^ T mathbf X ^ T mathbf Xw mathbf w ^ T mathbf w right The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector x(i) can then be given as a score t1(i) = x(i) ⋅ w(1) in the transformed co-ordinates, or as the corresponding vector in the original variables, x(i) ⋅ w(1) w(1). Further components[edit] The kth component can be found by subtracting the first k − 1 principal components from X: X ^ k = X − ∑ s = 1 k − 1 X w ( s ) w ( s ) T displaystyle mathbf hat X _ k =mathbf X -sum _ s=1 ^ k-1 mathbf X mathbf w _ (s) mathbf w _ (s) ^ rm T and then finding the loading vector which extracts the maximum variance from this new data matrix w ( k ) = a r g m a x ‖ w ‖ = 1 ‖ X ^ k w ‖ 2 = arg m a x w T X ^ k T X ^ k w w T w displaystyle mathbf w _ (k) = underset Vert mathbf w Vert =1 operatorname arg,max left Vert mathbf hat X _ k mathbf w Vert ^ 2 right = operatorname arg ,max ,left tfrac mathbf w ^ T mathbf hat X _ k ^ T mathbf hat X _ k mathbf w mathbf w ^ T mathbf w right It turns out that this gives the remaining eigenvectors of XTX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the loading vectors are eigenvectors of XTX. The kth principal component of a data vector x(i) can therefore be given as a score tk(i) = x(i) ⋅ w(k) in the transformed co-ordinates, or as the corresponding vector in the space of the original variables, x(i) ⋅ w(k) w(k), where w(k) is the kth eigenvector of XTX. The full principal components decomposition of X can therefore be given as T = X W displaystyle mathbf T =mathbf X mathbf W where W is a p-by-p matrix whose columns are the eigenvectors of XTX. The transpose of W is sometimes called the whitening or sphering transformation. Covariances[edit] XTX itself can be recognised as proportional to the empirical sample covariance matrix of the dataset X. The sample covariance Q between two of the different principal components over the dataset is given by: Q ( P C ( j ) , P C ( k ) ) ∝ ( X w ( j ) ) T ( X w ( k ) ) = w ( j ) T X T X w ( k ) = w ( j ) T λ ( k ) w ( k ) = λ ( k ) w ( j ) T w ( k ) displaystyle begin aligned Q(mathrm PC _ (j) ,mathrm PC _ (k) )&propto (mathbf X mathbf w _ (j) )^ T (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) end aligned where the eigenvalue property of w(k) has been used to move from line 2 to line 3. However eigenvectors w(j) and w(k) corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset. Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix. In matrix form, the empirical covariance matrix for the original variables can be written Q ∝ X T X = W Λ W T displaystyle mathbf Q propto mathbf X ^ T mathbf X =mathbf W mathbf Lambda mathbf W ^ T The empirical covariance matrix between the principal components becomes W T Q W ∝ W T W Λ W T W = Λ displaystyle mathbf W ^ T mathbf Q mathbf W propto mathbf W ^ T mathbf W ,mathbf Lambda ,mathbf W ^ T mathbf W =mathbf Lambda where Λ is the diagonal matrix of eigenvalues λ(k) of XTX (λ(k) being equal to the sum of the squares over the dataset associated with each component k: λ(k) = Σi tk2(i) = Σi (x(i) ⋅ w(k))2) Dimensionality reduction[edit] The transformation T = X W maps a data vector x(i) from an original space of p variables to a new space of p variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first L principal components, produced by using only the first L loading vectors, gives the truncated transformation T L = X W L displaystyle mathbf T _ L =mathbf X mathbf W _ L where the matrix TL now has n rows but only L columns. In other words, PCA learns a linear transformation t = W T x , x ∈ R p , t ∈ R L , displaystyle t=W^ T x,xin R^ p ,tin R^ L , where the columns of p × L matrix W form an orthogonal basis for the L features (the components of representation t) that are decorrelated.[8] 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 W T − T L W L T ‖ 2 2 displaystyle mathbf T mathbf W ^ T -mathbf T _ L mathbf W _ L ^ T _ 2 ^ 2 or ‖ X − X L ‖ 2 2 displaystyle mathbf X -mathbf X _ L _ 2 ^ 2 . A principal components analysis scatterplot of
Such dimensionality reduction can be a very useful step for
visualising and processing high-dimensional datasets, while still
retaining as much of the variance in the dataset as possible. For
example, selecting L = 2 and keeping only the first two
principal components finds the two-dimensional plane through the
high-dimensional dataset in which the data is most spread out, so if
the data contains clusters these too may be most spread out, and
therefore most visible to be plotted out in a two-dimensional diagram;
whereas if two directions through the data (or two of the original
variables) are chosen at random, the clusters may be much less spread
apart from each other, and may in fact be much more likely to
substantially overlay each other, making them indistinguishable.
Similarly, in regression analysis, the larger the number of
explanatory variables allowed, the greater is the chance of
overfitting the model, producing conclusions that fail to generalise
to other datasets. One approach, especially when there are strong
correlations between different possible explanatory variables, is to
reduce them to a few principal components and then run the regression
against them, a method called principal component regression.
X = U Σ W T displaystyle mathbf X =mathbf U mathbf Sigma mathbf W ^ T Here Σ is an n-by-p rectangular diagonal matrix of positive numbers σ(k), called the singular values of X; U is an n-by-n 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 factorization, the matrix XTX can be written X T X = W Σ T U T U Σ W T = W Σ T Σ W T = W Σ ^ 2 W T displaystyle begin aligned mathbf X ^ T mathbf X &=mathbf W mathbf Sigma ^ T mathbf U ^ T mathbf U mathbf Sigma mathbf W ^ T \&=mathbf W mathbf Sigma ^ T mathbf Sigma mathbf W ^ T \&=mathbf W mathbf hat Sigma ^ 2 mathbf W ^ T end aligned where Σ ^ displaystyle mathbf hat Sigma is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies Σ ^ 2 = Σ T Σ displaystyle mathbf hat Sigma ^ 2 =mathbf Sigma ^ T mathbf Sigma . Comparison with the eigenvector factorization of XTX establishes that the right singular vectors W of X are equivalent to the eigenvectors of XTX, while the singular values σ(k) of Σ displaystyle mathbf Sigma are equal to the squareroot of the eigenvalues λ(k) of XTX. Using the singular value decomposition the score matrix T can be written T = X W = U Σ W T W = U Σ displaystyle begin aligned mathbf T &=mathbf X mathbf W \&=mathbf U mathbf Sigma mathbf W ^ T mathbf W \&=mathbf U mathbf Sigma end aligned so each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T. 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[citation needed], unless only a handful of components are required. As with the eigen-decomposition, a truncated n × L score matrix TL can be obtained by considering only the first L largest singular values and their singular vectors: T L = U L Σ L = X W L displaystyle mathbf T _ L =mathbf U _ L mathbf Sigma _ L =mathbf X mathbf W _ L The truncation of a matrix M or T using a truncated singular value
decomposition in this way produces a truncated matrix that is the
nearest possible matrix of rank L to the original matrix, in the sense
of the difference between the two having the smallest possible
Frobenius norm, a result known as the
Symbol Meaning Dimensions Indices X = X i j displaystyle mathbf X = X_ ij data matrix, consisting of the set of all data vectors, one vector per row n × p displaystyle ntimes p i = 1 … n displaystyle i=1ldots n j = 1 … p displaystyle j=1ldots p n displaystyle n, the number of row vectors in the data set 1 × 1 displaystyle 1times 1 scalar p displaystyle p, the number of elements in each row vector (dimension) 1 × 1 displaystyle 1times 1 scalar L displaystyle L, the number of dimensions in the dimensionally reduced subspace, 1 ≤ L ≤ p displaystyle 1leq Lleq p 1 × 1 displaystyle 1times 1 scalar u = u j displaystyle mathbf u = u_ j vector of empirical means, one mean for each column j of the data matrix p × 1 displaystyle ptimes 1 j = 1 … p displaystyle j=1ldots p s = s j displaystyle mathbf s = s_ j vector of empirical standard deviations, one standard deviation for each column j of the data matrix p × 1 displaystyle ptimes 1 j = 1 … p displaystyle j=1ldots p h = h i displaystyle mathbf h = h_ i vector of all 1's 1 × n displaystyle 1times n i = 1 … n displaystyle i=1ldots n B = B i j displaystyle mathbf B = B_ ij deviations from the mean of each column j of the data matrix n × p displaystyle ntimes p i = 1 … n displaystyle i=1ldots n j = 1 … p displaystyle j=1ldots p Z = Z i j displaystyle mathbf Z = Z_ ij z-scores, computed using the mean and standard deviation for each row m of the data matrix n × p displaystyle ntimes p i = 1 … n displaystyle i=1ldots n j = 1 … p displaystyle j=1ldots p C = C j j ′ displaystyle mathbf C = C_ jj' covariance matrix p × p displaystyle ptimes p j = 1 … p displaystyle j=1ldots p j ′ = 1 … p displaystyle j'=1ldots p R = R j j ′ displaystyle mathbf R = R_ jj' correlation matrix p × p displaystyle ptimes p j = 1 … p displaystyle j=1ldots p j ′ = 1 … p displaystyle j'=1ldots p V = V j j ′ displaystyle mathbf V = V_ jj' matrix consisting of the set of all eigenvectors of C, one eigenvector per column p × p displaystyle ptimes p j = 1 … p displaystyle j=1ldots p j ′ = 1 … p displaystyle j'=1ldots p D = D j j ′ displaystyle mathbf D = D_ jj' diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements p × p displaystyle ptimes p j = 1 … p displaystyle j=1ldots p j ′ = 1 … p displaystyle j'=1ldots p W = W j l displaystyle mathbf W = W_ jl 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 × L displaystyle ptimes L j = 1 … p displaystyle j=1ldots p l = 1 … L displaystyle l=1ldots L T = T i l displaystyle mathbf T = T_ il 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 × L displaystyle ntimes L i = 1 … n displaystyle i=1ldots n l = 1 … L displaystyle l=1ldots L Properties and limitations of PCA[edit] Properties[edit] Some properties of PCA include:[12] Property 1: For any integer q, 1 ≤ q ≤ p, consider the orthogonal linear transformation y = B ′ x displaystyle y=mathbf B' x where y displaystyle y is a q-element vector and B ′ displaystyle mathbf B' is a (q × p) matrix, and let Σ y = B ′ Σ B displaystyle mathbf Sigma _ y =mathbf B' mathbf Sigma mathbf B be the variance-covariance matrix for y displaystyle y . Then the trace of Σ y displaystyle mathbf Sigma _ y , denoted tr ( Σ y ) displaystyle text tr (mathbf Sigma _ y ) , is maximized by taking B = A q displaystyle mathbf B =mathbf A _ q , where A q displaystyle mathbf A _ q consists of the first q columns of A displaystyle mathbf A ( B ′ displaystyle (mathbf B' is the transposition of B ) displaystyle mathbf B ) . Property 2: Consider again the orthonormal transformation y = B ′ x displaystyle y=mathbf B' x with x , B , A displaystyle x,mathbf B ,mathbf A and Σ y displaystyle mathbf Sigma _ y defined as before. Then tr ( Σ y ) displaystyle text tr (mathbf Sigma _ y ) is minimized by taking B = A q ∗ , displaystyle mathbf B =mathbf A _ q ^ * , where A q ∗ displaystyle mathbf A _ q ^ * consists of the last q columns of A displaystyle mathbf A . The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of x, and they may also be useful in regression, in selecting a subset of variables from x, and in outlier detection. Property 3: (Spectral Decomposition of Σ) Σ = λ 1 α 1 α 1 ′ + ⋯ + λ p α p α p ′ displaystyle mathbf Sigma =lambda _ 1 alpha _ 1 alpha _ 1 '+cdots +lambda _ p alpha _ p alpha _ p ' Before we look at its usage, we first look at diagonal elements, Var ( x j ) = ∑ k = 1 P λ k α k j 2 displaystyle text Var (x_ j )=sum _ k=1 ^ P lambda _ k alpha _ kj ^ 2 Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of x into decreasing contributions due to each PC, but we can also decompose the whole covariance matrix into contributions λ k α k α k ′ displaystyle lambda _ k alpha _ k alpha _ k ' from each PC. Although not strictly decreasing, the elements of λ k α k α k ′ displaystyle lambda _ k alpha _ k alpha _ k ' will tend to become smaller as k displaystyle k increases, as λ k α k α k ′ displaystyle lambda _ k alpha _ k alpha _ k ' is nonincreasing for increasing k displaystyle k , whereas the elements of α k displaystyle alpha _ k tend to stay 'about the same size'because of the normalization constraints: α k ′ α k = 1 , k = 1 , ⋯ , p displaystyle alpha _ k 'alpha _ k =1,k=1,cdots ,p Limitations[edit]
As noted above, the results of PCA depend on the scaling of the
variables. A scale-invariant form of PCA has been developed.[13]
The applicability of PCA is limited by certain assumptions[14] made in
its derivation.
The other limitation is the mean-removal process before constructing
the covariance matrix for PCA. In fields such as astronomy, all the
signals are non-negative, and the mean-removal process will force the
mean of some astrophysical exposures to be zero, which consequently
creates unphysical negative fluxes,[15] and forward modeling has to be
performed to recover the true magnitude of the signals.[16] As an
alternative method, non-negative matrix factorization focusing only on
the non-negative elements in the matrices, which is well-suited for
astrophysical observations.[17][18][19] See more at Relation between
PCA and Non-negative Matrix Factorization.
PCA and information theory[edit]
x = s + n displaystyle mathbf x =mathbf s +mathbf n i.e., that the data vector x displaystyle mathbf x is the sum of the desired information-bearing signal s displaystyle mathbf s and a noise signal n displaystyle mathbf n one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view. In particular, Linsker showed that if s displaystyle mathbf s is Gaussian and n displaystyle mathbf n is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information I ( y ; s ) displaystyle I(mathbf y ;mathbf s ) between the desired information s displaystyle mathbf s and the dimensionality-reduced output y = W L T x displaystyle mathbf y =mathbf W _ L ^ T mathbf x .[20] If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (i.e., the components of the vector n displaystyle mathbf n are iid), but the information-bearing signal s displaystyle 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[21][22] I ( x ; s ) − I ( y ; s ) . displaystyle I(mathbf x ;mathbf s )-I(mathbf y ;mathbf s ). The optimality of PCA is also preserved if the noise n displaystyle mathbf n is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal s displaystyle mathbf s .[23] In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise n displaystyle 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) as opposed to the correlation method.[24] 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: Y = K L T X displaystyle 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 x 1 … x n displaystyle mathbf x _ 1 ldots mathbf x _ n with each x i displaystyle mathbf x _ i representing a single grouped observation of the p variables. Write x 1 … x n displaystyle mathbf x _ 1 ldots mathbf x _ n as row vectors, each of which has p columns. Place the row vectors into a single matrix X of dimensions n × p. Calculate the empirical mean[edit] Find the empirical mean along each column j = 1, ..., p. Place the calculated mean values into an empirical mean vector u of dimensions p × 1. u j = 1 n ∑ i = 1 n X i j displaystyle u_ j = 1 over n sum _ i=1 ^ n X_ ij Calculate the deviations from the mean[edit]
Subtract the empirical mean vector u from each row of the data matrix X. Store mean-subtracted data in the n × p matrix B. B = X − h u T displaystyle mathbf B =mathbf X -mathbf h mathbf u ^ T where h is an n × 1 column vector of all 1s: h i = 1 for i = 1 , … , n displaystyle h_ i =1,qquad qquad text for i=1,ldots ,n Find the covariance matrix[edit] Find the p × p empirical covariance matrix C from the outer product of matrix B with itself: C = 1 n − 1 B ∗ ⊗ B displaystyle mathbf C = 1 over n-1 mathbf B ^ * otimes mathbf B where ∗ displaystyle * 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. 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] Compute the matrix V of eigenvectors which diagonalizes the covariance matrix C: V − 1 C V = D displaystyle 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 SAS,[26] R,
MATLAB,[27][28] Mathematica,[29] SciPy, IDL (Interactive Data
Language), or
Matrix D will take the form of an p × p diagonal matrix, where D k l = λ k for k = l displaystyle D_ kl =lambda _ k qquad text for k=l is the jth eigenvalue of the covariance matrix C, and D k l = 0 for k ≠ l . displaystyle D_ kl =0qquad text for kneq l. Matrix V, also of dimension p × p, contains p column vectors, each of length p, which represent the p eigenvectors of the covariance matrix C. The eigenvalues and eigenvectors are ordered and paired. The jth eigenvalue corresponds to the jth eigenvector. Matrix V denotes the matrix of right eigenvectors (as opposed to left eigenvectors). In general, the matrix of right eigenvectors need not be the (conjugate) transpose of the matrix of left eigenvectors. Rearrange the eigenvectors and eigenvalues[edit] 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[edit] The eigenvalues represent the distribution of the source data's energy[clarification needed] 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 j = ∑ k = 1 j D k k f o r j = 1 , … , p displaystyle g_ j =sum _ k=1 ^ j D_ kk qquad mathrm for qquad j=1,dots ,p [citation needed] Select a subset of the eigenvectors as basis vectors[edit] Save the first L columns of V as the p × L matrix W: W k l = V k l f o r k = 1 , … , p l = 1 , … , L displaystyle W_ kl =V_ kl qquad mathrm for qquad k=1,dots ,pqquad l=1,dots ,L where 1 ≤ L ≤ p . displaystyle 1leq Lleq p. Use the vector g as a guide in choosing an appropriate value for L. The goal is to choose a value of L as small 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 L g p ≥ 0.9 displaystyle frac g_ L g_ p geq 0.9, Project the z-scores of the data onto the new basis[edit] The projected vectors are the columns of the matrix T = Z ⋅ W = K L T X . displaystyle mathbf T =mathbf Z cdot mathbf W =mathbb KLT mathbf X . The rows of matrix T represent the Kosambi-Karhunen–Loève transforms (KLT) of the data vectors in the rows of matrix X. Derivation of PCA using the covariance method[edit] 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 ( ∗ ) displaystyle (ast ), a d × d orthonormal transformation matrix P so that PX has a diagonal covariance matrix (i.e. PX is a random vector with all its distinct components pairwise uncorrelated). A quick computation assuming P displaystyle P were unitary yields: cov ( P X ) = E [ P X ( P X ) ∗ ] = E [ P X X ∗ P ∗ ] = P E [ X X ∗ ] P ∗ = P cov ( X ) P − 1 displaystyle begin aligned operatorname cov (PX)&=mathbb E [PX~(PX)^ * ]\&=mathbb E [PX~X^ * P^ * ]\&=P~mathbb E [XX^ * ]P^ * \&=P~operatorname cov (X)P^ -1 \end aligned Hence ( ∗ ) displaystyle (ast ), holds if and only if cov ( X ) displaystyle operatorname cov (X) were diagonalisable by P displaystyle 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. Iterative computation[edit] 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[30] is shown in the following pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix. r = a random vector of length p do c times: s = 0 (a vector of length p) for each row x ∈ X displaystyle mathbf x in mathbf X s = s + ( x ⋅ r ) x displaystyle mathbf s =mathbf s +(mathbf x cdot mathbf r )mathbf x r = s
s
displaystyle mathbf r = frac mathbf s mathbf s return r This algorithm is simply an efficient way of calculating XTX r,
normalizing, and placing the result back in r (power iteration). It
avoids the np2 operations of calculating the covariance matrix. r will
typically get close to the first principal component of X within a
small number of iterations, c. (The magnitude of s 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 r from X (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 mean-squared-distance
between the rows and the centroid, 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[edit]
Identification, on the factorial planes, of the different species e.g. using different colors. Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species. For each center of gravity and each axis, p-value to judge the significance of the difference between the center of gravity and origin. These results are what is called introducing a qualitative variable as
supplementary element. This procedure is detailed in and Husson, Lê
& Pagès 2009 and Pagès 2013. Few software offer this option in
an "automatic" way. This is the case of SPAD that historically,
following the work of Ludovic Lebart, was the first to propose this
option, and the R package FactoMineR.
Applications[edit]
Quantitative finance[edit]
See also: Portfolio optimization
In quantitative finance, principal component analysis can be directly
applied to the risk management of interest rate derivatives
portfolios.[35] Trading multiple swap instruments which are usually a
function of 30-500 other market quotable swap instruments is sought to
be reduced to usually 3 or 4 principal components, representing the
path of interest rates on a macro basis. Converting risks to be
represented as those to factor loadings (or multipliers) provides
assessments and understanding beyond that available to simply
collectively viewing risks to individual 30-500 buckets.
PCA has also been applied to share portfolios in a similar
fashion.[36] One application is to reduce portfolio risk, where
allocation strategies are applied to the "principal portfolios"
instead of the underlying stocks.[37] A second is to enhance portfolio
return, using the principal components to select stocks with upside
potential.[38]
Neuroscience[edit]
A variant of principal components analysis is used in neuroscience to
identify the specific properties of a stimulus that increase a
neuron's probability of generating an action potential.[39] This
technique is known as spike-triggered covariance analysis. In a
typical application an experimenter presents a white noise process as
a stimulus (usually either as a sensory input to a test subject, or as
a current injected directly into the neuron) and records a train of
action potentials, or spikes, produced by the neuron as a result.
Presumably, certain features of the stimulus make the neuron more
likely to spike. In order to extract these features, the experimenter
calculates the covariance matrix of the spike-triggered ensemble, the
set of all stimuli (defined and discretized over a finite time window,
typically on the order of 100 ms) that immediately preceded a spike.
The eigenvectors of the difference between the spike-triggered
covariance matrix and the covariance matrix of the prior stimulus
ensemble (the set of all stimuli, defined over the same length time
window) then indicate the directions in the space of stimuli along
which the variance of the spike-triggered ensemble differed the most
from that of the prior stimulus ensemble. Specifically, the
eigenvectors with the largest positive eigenvalues correspond to the
directions along which the variance of the spike-triggered ensemble
showed the largest positive change compared to the variance of the
prior. Since these were the directions in which varying the stimulus
led to a spike, they are often good approximations of the sought after
relevant stimulus features.
In neuroscience, PCA is also used to discern the identity of a neuron
from the shape of its action potential.
Fractional residual variance (FRV) plots for PCA and NMF;[19] for PCA, the theoretical values are the contribution from the residual eigenvalues. In comparison, the FRV curves for PCA reaches a flat plateau where no signal are captured effectively; while the NMF FRV curves are declining continuously, indicating a better ability to capture signal. The FRV curves for NMF also converges to higher levels than PCA, indicating the less-overfitting property of NMF.
1 − ∑ i = 1 k λ i / ∑ k = 1 n λ k displaystyle 1-sum _ i=1 ^ k lambda _ i /sum _ k=1 ^ n lambda _ k as a function of component number k displaystyle k given a total of n displaystyle n components, for PCA has a flat plateau, where no data is captured to remove the quasi-static noise, then the curves dropped quickly as an indication of over-fitting and captures random noise.[15] The FRV curves for NMF is decreasing continuously [19] when the NMF components are constructed sequentially,[18] indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA,[19] indicating the less over-fitting property of NMF. Generalizations[edit] Nonlinear generalizations[edit] Linear PCA versus nonlinear Principal Manifolds[50] for visualization of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The "basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition.[51] Software is available for free non-commercial use.[52] 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[53] 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. Another popular generalization is kernel
PCA, which corresponds to PCA performed in a reproducing kernel
Hilbert space associated with a positive definite kernel.
Multilinear generalizations[edit]
In multilinear subspace learning,[54] 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[edit]
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[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[55] based on a weighted PCA increases
robustness by assigning different weights to data objects based on
their estimated relevancy.
Robust PCA via decomposition in low-rank and sparse matrices[edit]
E displaystyle E , it tries to decompose it into two matrices such that E = A P displaystyle E=AP . A key difference from techniques such as PCA and ICA is that some of the entries of A displaystyle A are constrained to be 0. Here P displaystyle P is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :- A displaystyle A has full column rank Each column of A displaystyle A must have at least L − 1 displaystyle L-1 zeroes where L displaystyle L is the number of columns of A displaystyle A (or alternatively the number of rows of P displaystyle P ). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank. P displaystyle P must have full row rank. then the decomposition is unique up to multiplication by a scalar.[60] Software/source code[edit]
See also[edit]
References[edit] ^ Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems
of Points in Space" (PDF). Philosophical Magazine. 2 (11): 559–572.
doi:10.1080/14786440109462720.
^ 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. 28: 321–377. doi:10.2307/2333955.
^ a b 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
^ Abdi. H., & Williams, L.J. (2010). "Principal component
analysis" (PDF). Wiley Interdisciplinary Reviews: Computational
Statistics. 2 (4): 433–459. doi:10.1002/wics.101.
^ Shaw P.J.A. (2003)
Further reading[edit] Jackson, J.E. (1991). A User's Guide to Principal Components (Wiley). Jolliffe, I. T. (1986). Principal Component Analysis. Springer-Verlag. p. 487. doi:10.1007/b98835. ISBN 978-0-387-95442-4. Jolliffe, I.T. (2002). Principal Component Analysis, second edition (Springer). Husson François, Lê Sébastien & Pagès Jérôme (2009). Exploratory Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. 224p. ISBN 978-2-7535-0938-2 Pagès Jérôme (2014). Multiple Factor Analysis by Example Using R. Chapman & Hall/CRC The R Series London 272 p External links[edit] Wikimedia Commons has media related to Principal component analysis. University of Copenhagen video by Rasmus Bro on YouTube
Stanford University video by Andrew Ng on YouTube
A Tutorial on Principal Component Analysis
A layman's introduction to principal component analysis on
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