Given a collection of points in two, three, or higher dimensional space, a "best fitting" line can be defined as one that minimizes the average squared distance from a point to the line. For a collection of points in and , a direction for the best-fitting line can be chosen from directions perpendicular to the first best-fitting lines. These directions comprise an orthonormal basis in which different individual dimensions of the data are uncorrelated. These basis vectors are called principal components and several related procedures principal component analysis (PCA). Usually, PCA refers to the process of computing the principal components and using them to perform a change of basis on the data, sometimes only using the first few principal components and ignoring the rest.
PCA is mostly used as a tool in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data's variation as possible. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The principal component can be taken as a direction orthogonal to the first principal components that maximizes the variance of the projected data.
From either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, principal components are often computed using eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.^{[1]}^{[2]}^{[3]}^{[4]}Robust and L1-norm-based variants of standard PCA have also been proposed.^{[5]}^{[6]}^{[4]}
PCA was invented in 1901 by Karl Pearson,^{[7]} as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s.^{[8]} Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of X^{T}X in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis),^{[9]} Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.
PCA can be thought of as fitting a p-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.
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 turn them into 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.
PCA is defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some scalar 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.^{[9]}
Consider an 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 size of p-dimensional vectors of weights or coefficients that map each row vector of X to a new vector of principal component scores , given by
in such a way that the individual variables of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector (where is usually selected to be less than to reduce dimensionality).
In order to maximize variance, the first weight vector w_{(1)} thus has to satisfy
Equivalently, writing this in matrix form gives
Since w_{(1)} has been defined to be a unit vector, it equivalently also satisfies
The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as X^{T}X 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 t_{1(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)}.
The kth component can be found by subtracting the first k − 1 principal components from X:
and then finding the weight vector which extracts the maximum variance from this new data matrix
It turns out that this gives the remaining eigenvectors of X^{T}X, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of X^{T}X.
The kth principal component of a data vector x_{(i)} can therefore be given as a score t_{k(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 X^{T}X.
The full principal components decomposition of X can therefore be given as
where W is a p-by-p matrix of weights whose columns are the eigenvectors of X^{T}X. The transpose of W is sometimes called the whitening or sphering transformation. Columns of W multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called loadings in PCA or in Factor analysis.
X^{T}X itself can be recognised as proportional to the empirical sample covariance matrix of the dataset X^{T}.
The sample covariance Q between two of the different principal components over the dataset is given by:
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
The empirical covariance matrix between the principal components becomes
where Λ is the diagonal matrix of eigenvalues λ_{(k)} of X^{T}X. λ_{(k)} is equal to the sum of the squares over the dataset associated with each component k, that is, λ_{(k)} = Σ_{i} t_{k}^{2}_{(i)} = Σ_{i} (x_{(i)} ⋅ w_{(k)})^{2}.
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 eigenvectors, gives the truncated transformation
where the matrix T_{L} now has n rows but only L columns. In other words, PCA learns a linear transformation where the columns of p × L matrix W form an orthogonal basis for the L features (the components of representation t) that are decorrelated.^{[10]} 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 or .
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.
Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix W, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signal-to-noise ratio. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss. If the dataset is not too large, the significance of the principal components can be tested using parametric bootstrap, as an aid in determining how many principal components to retain ^{[11]}.
The principal components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X,
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 X^{T}X can be written
where is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies . Comparison with the eigenvector factorization of X^{T}X establishes that the right singular vectors W of X are equivalent to the eigenvectors of X^{T}X, while the singular values σ_{(k)} of are equal to the square-root of the eigenvalues λ_{(k)} of X^{T}X.
Using the singular value decomposition the score matrix T can be written
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 X^{T}X, 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 T_{L} can be obtained by considering only the first L largest singular values and their singular vectors:
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 Eckart–Young theorem [1936].
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 X^{T}X. 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 associated 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 (see below). 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, and in particular to the DCT-II which is simply known as the "DCT". 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 "weights" (they are the cosines of rotation) 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 first 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.) 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, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.
Mean subtraction (a.k.a. "mean centering") is necessary for performing classical 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.^{[12]}
Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation). Also see the article by Kromrey & Foster-Johnson (1998) on "Mean-centering in Moderated Regression: Much Ado About Nothing".
PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability.^{[13]} However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes.^{[14]} The linear discriminant analysis is an alternative which is optimized for class separability.
Symbol | Meaning | Dimensions | Indices |
---|---|---|---|
data matrix, consisting of the set of all data vectors, one vector per row | | ||
the number of row vectors in the data set | scalar | ||
the number of elements in each row vector (dimension) | scalar | ||
the number of dimensions in the dimensionally reduced subspace, | scalar | ||
vector of empirical means, one mean for each column j of the data matrix | |||
vector of empirical standard deviations, one standard deviation for each column j of the data matrix | |||
vector of all 1's | |||
deviations from the mean of each column j of the data matrix | | ||
z-scores, computed using the mean and standard deviation for each row m of the data matrix | | ||
covariance matrix | | ||
correlation matrix | | ||
matrix consisting of the set of all eigenvectors of C, one eigenvector per column | | ||
diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements | | ||
matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of C, and where the vectors in W are a sub-set of those in V | | ||
matrix consisting of n 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. | |
Some properties of PCA include:^{[15]}
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.
Before we look at its usage, we first look at diagonal elements,
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 from each PC. Although not strictly decreasing, the elements of will tend to become smaller as increases, as is nonincreasing for increasing , whereas the elements of tend to stay about the same size because of the normalization constraints: .
As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.^{[16]}
The applicability of PCA as described above is limited by certain (tacit) assumptions^{[17]} made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see kernel PCA).
Another 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,^{[18]} and forward modeling has to be performed to recover the true magnitude of the signals.^{[19]} 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.^{[20]}^{[21]}^{[22]} See more at Relation between PCA and Non-negative Matrix Factorization.
Dimensionality reduction loses information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.
Under the assumption that
that is, that the data vector is the sum of the desired information-bearing signal and a noise signal one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.
In particular, Linsker showed that if is Gaussian and is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information between the desired information and the dimensionality-reduced output .^{[23]}
If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector are iid), but the information-bearing signal is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as^{[24]}^{[25]}
The optimality of PCA is also preserved if the noise is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal .^{[26]} In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise becomes dependent.
The following is a detailed description of PCA using the covariance method (see also here) as opposed to the correlation method.^{[27]}
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:
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 with each representing a single grouped observation of the p variables.
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.^{[28]} Hence we proceed by centering the data as follows:
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 a d × d orthonormal transformation matrix P so that PX has a diagonal covariance matrix (that is, PX is a random vector with all its distinct components pairwise uncorrelated).
A quick computation assuming were unitary yields:
Hence holds if and only if were diagonalisable by .
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.
In practical implementations, especially with high dimensional data (large p), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the np^{2} operations of explicitly calculating and storing the covariance matrix X^{T}X, instead utilizing one of matrix-free methods, for example, based on the function evaluating the product X^{T}(X r) at the cost of 2np operations.
One way to compute the first principal component efficiently^{[33]} 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 exit if return
This power iteration algorithm simply calculates the vector X^{T}(X r), normalizes, and places the result back in r. The eigenvalue is approximated by r^{T} (X^{T}X) r, which is the Rayleigh quotient on the unit vector r for the covariance matrix X^{T}X . If the largest singular value is well separated from the next largest one, the vector r gets close to the first principal component of X within the number of iterations c, which is small relative to p, at the total cost 2cnp. The power iteration convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced matrix-free methods, such as the Lanczos algorithm or the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.
Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation. The latter approach in the block power method replaces single-vectors r and s with block-vectors, matrices R and S. Every column of R approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product X^{T}(X R). Implemented, for example, in LOBPCG, efficient blocking eliminates the accumulation of the errors, allows using high-level BLAS matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.
Non-linear iterative partial least squares (NIPALS) is a variant the classical power iteration with matrix deflation by subtraction implemented for computing the first few components in a principal component or partial least squares analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (for example, genomics, metabolomics) it is usually only necessary to compute the first few PCs. The non-linear iterative partial least squares (NIPALS) algorithm updates iterative approximations to the leading scores and loadings t_{1} and r_{1}^{T} by the power iteration multiplying on every iteration by X on the left and on the right, that is, calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to X^{T}X, based on the function evaluating the product X^{T}(X r) = ((X r)^{T}X)^{T}.
The matrix deflation by subtraction is performed by subtracting the outer product, t_{1}r_{1}^{T} from X leaving the deflated residual matrix used to calculate the subsequent leading PCs.^{[34]} For large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality of PCs due to machine precision round-off errors accumulated in each iteration and matrix deflation by subtraction.^{[35]} A Gram–Schmidt re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.^{[36]} NIPALS reliance on single-vector multiplications cannot take advantage of high-level BLAS and results in slow convergence for clustered leading singular values—both these deficiencies are resolved in more sophisticated matrix-free block solvers, such as the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.
In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.^{[37]}
In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable species. For this, the following results are produced.
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.
See also: Portfolio optimization |
In quantitative finance, principal component analysis can be directly applied to the risk management of interest rate derivative portfolios.^{[38]} 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 equity portfolios in a similar fashion,^{[39]} both to portfolio risk and to risk return. One application is to reduce portfolio risk, where allocation strategies are applied to the "principal portfolios" instead of the underlying stocks.^{[40]} A second is to enhance portfolio return, using the principal components to select stocks with upside potential.^{[citation needed]}
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.^{[41]} 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. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs clustering analysis to associate specific action potentials with individual neurons.
PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, that is, order parameters, during phase transitions in the brain.^{[42]}
Correspondence analysis (CA) was developed by Jean-Paul Benzécri^{[43]} 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.^{[44]} 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.^{[45]}
Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a pre-processing step before clustering. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (that is, translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.
Factor analysis is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance".^{[46]} In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (that is, shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.^{[47]} Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or causal modeling. If the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.^{[48]}
It has been asserted that the relaxed solution of k-means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace.^{[49]}^{[50]} However, that PCA is a useful relaxation of k-means clustering was not a new result,^{[51]} and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.^{[52]}
Non-negative matrix factorization (NMF) is a dimension reduction method where only non-negative elements in the matrices are used, which is therefore a promising method in astronomy,^{[20]}^{[21]}^{[22]} in the sense that astrophysical signals are non-negative. The PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis.
In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data.^{[18]} For NMF, its components are ranked based only on the empirical FRV curves.^{[22]} The residual fractional eigenvalue plots, that is, as a function of component number given a total of 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.^{[18]} The FRV curves for NMF is decreasing continuously ^{[22]} when the NMF components are constructed sequentially,^{[21]} indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA,^{[22]} indicating the less over-fitting property of NMF.
Main article: Sparse PCA |
A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables. Several approaches have been proposed, including
The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies are recently reviewed in a survey paper.^{[59]}
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^{[63]} 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.
In multilinear subspace learning,^{[64]} 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.
N-way principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.
While PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to outliers in the data that produce large errors, something that the method tries to avoid in the first place. It is therefore 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^{[65]} based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.
Outlier-resistant variants of PCA have also been proposed, based on L1-norm formulations (L1-PCA).^{[5]}^{[3]}
Robust principal component analysis (RPCA) via decomposition in low-rank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations.^{[66]}^{[67]}^{[68]}
Independent component analysis (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.
Given a matrix , it tries to decompose it into two matrices such that . A key difference from techniques such as PCA and ICA is that some of the entries of are constrained to be 0. Here is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :
then the decomposition is unique up to multiplication by a scalar.^{[69]}
pca
command or via the princomp()
function.pca
function in the MultivariateStats packageprincomp
and pca
(R2012b) give the principal components, while the function pcares
gives the residuals and reconstructed matrix for a low-rank PCA approximation.g03aa
routine (available in both the Fortran versions of the Library).princomp
gives the principal component.DBMS_DATA_MINING.SVDS_SCORING_MODE
by specifying setting value SVDS_SCORING_PCA
princomp
and prcomp
can be used for principal component analysis; prcomp
uses singular value decomposition which generally gives better numerical accuracy. Some packages that implement PCA in R, include, but are not limited to: ade4
, vegan
, ExPosition
, dimRed
, and FactoMineR
.