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Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and po ...
,
microbiome A microbiome () is the community of microorganisms that can usually be found living together in any given habitat. It was defined more precisely in 1988 by Whipps ''et al.'' as "a characteristic microbial community occupying a reasonably wel ...
studies, and atmospheric science. The principal components of a collection of points in a
real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
are a sequence of p
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s, where the i-th vector is the direction of a line that best fits the data while being
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to the first i-1 vectors. Here, a best-fitting line is defined as one that minimizes the average squared
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
distance from the points to the line. These directions constitute an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. In data analysis, the first principal component of a set of p variables, presumed to be jointly normally distributed, is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. PCA is used 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 i-th principal component can be taken as a direction orthogonal to the first i-1 principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
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. Robust and L1-norm-based variants of standard PCA have also been proposed.


History

PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering,
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
(SVD) of X (invented in the last quarter of the 20th century), eigenvalue decomposition (EVD) of XTX 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''), 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.


Intuition

PCA can be thought of as fitting a ''p''-dimensional
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
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 center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
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 the covariance matrix into a diagonalized form, in which the diagonal elements represent 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. Biplots and scree plots (degree of explained variance) are used to explain findings of the PCA.


Details

PCA is defined as an
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
linear transformation that transforms the data to a new
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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. Consider an n \times p data
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, X, with column-wise zero
empirical mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
(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 l of ''p''-dimensional vectors of weights or coefficients \mathbf_ = (w_1, \dots, w_p)_ that map each row vector \mathbf_ of X to a new vector of principal component ''scores'' \mathbf_ = (t_1, \dots, t_l)_, given by :_ = \mathbf_ \cdot \mathbf_ \qquad \mathrm \qquad i = 1,\dots,n \qquad k = 1,\dots,l in such a way that the individual variables t_1, \dots, t_l 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 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
(where l is usually selected to be strictly less than p to reduce dimensionality).


First component

In order to maximize variance, the first weight vector w(1) thus has to satisfy :\mathbf_ = \arg\max_ \,\left\ = \arg\max_ \,\left\ Equivalently, writing this in matrix form gives :\mathbf_ = \arg\max_ \left\ = \arg\max_ \left\ Since w(1) has been defined to be a unit vector, it equivalently also satisfies :\mathbf_ = \arg\max \left\ The quantity to be maximised can be recognised as a
Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the ...
. A standard result for a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, ...
such as XTX is that the quotient's maximum possible value is the largest
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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, w(1).


Further components

The ''k''-th component can be found by subtracting the first ''k'' − 1 principal components from X: :\mathbf_k = \mathbf - \sum_^ \mathbf \mathbf_ \mathbf_^ and then finding the weight vector which extracts the maximum variance from this new data matrix :\mathbf_ = \mathop_ \left\ = \arg\max \left\ 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 weight vectors are eigenvectors of XTX. The ''k''-th 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 coordinates, or as the corresponding vector in the space of the original variables, w(''k''), where w(''k'') is the ''k''th eigenvector of XTX. The full principal components decomposition of X can therefore be given as :\mathbf = \mathbf \mathbf where W is a ''p''-by-''p'' matrix of weights whose columns are the eigenvectors of XTX. 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.


Covariances

XTX itself can be recognized as proportional to the empirical sample
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
of the dataset XT. The sample covariance ''Q'' between two of the different principal components over the dataset is given by: :\begin Q(\mathrm_, \mathrm_) & \propto (\mathbf\mathbf_)^\mathsf (\mathbf\mathbf_) \\ & = \mathbf_^\mathsf \mathbf^\mathsf \mathbf \mathbf_ \\ & = \mathbf_^\mathsf \lambda_ \mathbf_ \\ & = \lambda_ \mathbf_^\mathsf \mathbf_ \end 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 :\mathbf \propto \mathbf^\mathsf \mathbf = \mathbf \mathbf \mathbf^\mathsf The empirical covariance matrix between the principal components becomes :\mathbf^\mathsf \mathbf \mathbf \propto \mathbf^\mathsf \mathbf \, \mathbf \, \mathbf^\mathsf \mathbf = \mathbf where Λ is the diagonal matrix of eigenvalues ''λ''(''k'') of XTX. ''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.


Dimensionality reduction

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 :\mathbf_L = \mathbf \mathbf_L where the matrix TL now has ''n'' rows but only ''L'' columns. In other words, PCA learns a linear transformation t = W_L^\mathsf x, x \in \mathbb^p, t \in \mathbb^L, where the columns of matrix W_L form an orthogonal basis for the ''L'' features (the components of representation ''t'') that are decorrelated. 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 \, \mathbf\mathbf^T - \mathbf_L\mathbf^T_L\, _2^2 or \, \mathbf - \mathbf_L\, _2^2. 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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
, the larger the number of
explanatory variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
s allowed, the greater is the chance of
overfitting mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
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.


Singular value decomposition

The principal components transformation can also be associated with another matrix factorization, the
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
(SVD) of X, :\mathbf = \mathbf\mathbf\mathbf^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 :\begin \mathbf^T\mathbf & = \mathbf\mathbf^\mathsf \mathbf^\mathsf \mathbf\mathbf\mathbf^\mathsf \\ & = \mathbf\mathbf^\mathsf \mathbf \mathbf^\mathsf \\ & = \mathbf\mathbf^2 \mathbf^\mathsf \end where \mathbf is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies \mathbf=\mathbf^\mathsf \mathbf . 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 \mathbf are equal to the square-root of the eigenvalues ''λ''(''k'') of XTX. Using the singular value decomposition the score matrix T can be written :\begin \mathbf & = \mathbf \mathbf \\ & = \mathbf\mathbf\mathbf^\mathsf \mathbf \\ & = \mathbf\mathbf \end 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, unless only a handful of components are required. As with the eigen-decomposition, a truncated score matrix TL can be obtained by considering only the first L largest singular values and their singular vectors: :\mathbf_L = \mathbf_L\mathbf_L = \mathbf \mathbf_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 Eckart–Young theorem 936


Further considerations

The singular values (in Σ) are the square roots of the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the matrix XTX. 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 Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred 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 Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...
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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and are completely 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. 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"''. Since covariances are correlations of normalized variables ( Z- or standard-scores) a PCA based on the correlation matrix of X is equal to a PCA based on the covariance matrix of Z, the standardized version of X. PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability. 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. The
linear discriminant analysis Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features ...
is an alternative which is optimized for class separability.


Table of symbols and abbreviations


Properties and limitations of PCA


Properties

Some properties of PCA include: :''Property 1'': For any integer ''q'', 1 ≤ ''q'' ≤ ''p'', consider the orthogonal linear transformation ::y =\mathbfx :where y is a ''q-element'' vector and \mathbf is a (''q'' × ''p'') matrix, and let \mathbf_y = \mathbf\mathbf\mathbf be the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
-
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
matrix for y. Then the trace of \mathbf_y, denoted \operatorname (\mathbf_y), is maximized by taking \mathbf = \mathbf_q, where \mathbf_q consists of the first ''q'' columns of \mathbf (\mathbf is the transpose of \mathbf). :''Property 2'': Consider again the
orthonormal transformation In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
::y = \mathbfx :with x, \mathbf, \mathbf and \mathbf_y defined as before. Then \operatorname(\mathbf_y) is minimized by taking \mathbf = \mathbf_q^*, where \mathbf_q^* consists of the last ''q'' columns of \mathbf. 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 , and they may also be useful in regression, in selecting a subset of variables from , and in outlier detection. :''Property 3'': (Spectral decomposition of ) ::\mathbf = \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, :\operatorname(x_j) = \sum_^P \lambda_k\alpha_^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 into decreasing contributions due to each PC, but we can also decompose the whole
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
into contributions \lambda_k\alpha_k\alpha_k' from each PC. Although not strictly decreasing, the elements of \lambda_k\alpha_k\alpha_k' will tend to become smaller as k increases, as \lambda_k\alpha_k\alpha_k' is nonincreasing for increasing k, whereas the elements of \alpha_k tend to stay about the same size because of the normalization constraints: \alpha_'\alpha_=1, k=1, \dots, p.


Limitations

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.Leznik, M; Tofallis, C. 200
Estimating Invariant Principal Components Using Diagonal Regression.
/ref> The applicability of PCA as described above is limited by certain (tacit) assumptions 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, and forward modeling has to be performed to recover the true magnitude of the signals. 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. See more at Relation between PCA and Non-negative Matrix Factorization. PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were. They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss. PCA relies on a linear model. If a dataset has a pattern hidden inside it that is nonlinear, then PCA can actually steer the analysis in the complete opposite direction of progress. Researchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results. "If the number of subjects or blocks is smaller than 30, and/or the researcher is interested in PC's beyond the first, it may be better to first correct for the serial correlation, before PCA is conducted". The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".


PCA and information theory

Dimensionality reduction results in a loss of information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models. Under the assumption that :\mathbf=\mathbf+\mathbf, that is, that the data vector \mathbf is the sum of the desired information-bearing signal \mathbf and a noise signal \mathbf one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view. In particular, Linsker showed that if \mathbf is Gaussian and \mathbf is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
I(\mathbf;\mathbf) between the desired information \mathbf and the dimensionality-reduced output \mathbf=\mathbf_L^T\mathbf. If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector \mathbf are iid), but the information-bearing signal \mathbf is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the ''information loss'', which is defined as :I(\mathbf;\mathbf) - I(\mathbf;\mathbf). The optimality of PCA is also preserved if the noise \mathbf is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal \mathbf. In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise \mathbf becomes dependent.


Computing PCA using the covariance method

The following is a detailed description of PCA using the covariance method (see als
here
as opposed to the correlation method. 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 = \mathbb \ ; Organize the data set 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_1 \ldots \mathbf_n with each \mathbf_i representing a single grouped observation of the ''p'' variables. * Write \mathbf_1 \ldots \mathbf_n as row vectors, each with ''p'' elements. * Place the row vectors into a single matrix X of dimensions ''n'' × ''p''. ; Calculate the empirical mean * 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 = \frac \sum_^n X_ ; Calculate the deviations from the mean Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data. Hence we proceed by centering the data as follows: * Subtract the empirical mean vector \mathbf^ from each row of the data matrix X. * Store mean-subtracted data in the ''n'' × ''p'' matrix B. *:\mathbf = \mathbf - \mathbf\mathbf^T *: where h is an column vector of all 1s: *::h_i = 1 \, \qquad \qquad \text i = 1, \ldots, n In some applications, each variable (column of B) may also be scaled to have a variance equal to 1 (see Z-score). This step affects the calculated principal components, but makes them independent of the units used to measure the different variables. ; Find the covariance matrix * Find the ''p'' × ''p'' empirical
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
C from matrix B: \mathbf = \mathbf^ \mathbf where * is the conjugate transpose operator. 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 instead of ''n'' to calculate the covariance is Bessel's correction. ; Find the eigenvectors and eigenvalues of the covariance matrix * Compute the matrix V of eigenvectors which diagonalizes the covariance matrix C: \mathbf^ \mathbf \mathbf = \mathbf where D is the diagonal matrix of
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s 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 In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, '' ...
systems, such as SAS, R,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
, Mathematica,
SciPy SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, ...
, IDL ( Interactive Data Language), or
GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lan ...
as well as OpenCV. * Matrix D will take the form of an ''p'' × ''p'' diagonal matrix, where D_ = \lambda_k \qquad \text k = \ell is the ''j''th eigenvalue of the covariance matrix C, and D_ = 0 \qquad \text k \ne \ell. * 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 ''j''th eigenvalue corresponds to the ''j''th 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 * Sort the columns of the eigenvector matrix V and eigenvalue matrix D in order of ''decreasing'' eigenvalue. * Make sure to maintain the correct pairings between the columns in each matrix. ; Compute the cumulative energy content for each eigenvector * The eigenvalues represent the distribution of the source data's energy among each of the eigenvectors, where the eigenvectors form a basis for the data. The cumulative energy content ''g'' for the ''j''th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through ''j'': *:g_j = \sum_^j D_ \qquad \text j = 1,\dots,p ; Select a subset of the eigenvectors as basis vectors * Save the first ''L'' columns of V as the ''p'' × ''L'' matrix W: W_ = V_ \qquad \text k = 1,\dots,p \qquad \ell = 1,\dots,L where 1 \leq L \leq 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 \frac \ge 0.9 ; Project the data onto the new basis * The projected data points are the rows of the matrix \mathbf = \mathbf \cdot \mathbf That is, the first column of \mathbf is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc.


Derivation of PCA using the covariance method

Let X be a ''d''-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean. We want to find (\ast) a 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 P were unitary yields: :\begin \operatorname(PX) &= \operatorname X~(PX)^\ &= \operatorname X~X^P^\ &= P\operatorname X^^\\ &= P\operatorname(X)P^\\ \end Hence (\ast) holds if and only if \operatorname(X) were diagonalisable by P. This is very constructive, as cov(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.


Covariance-free computation

In practical implementations, especially with
high dimensional data In statistical theory, the field of high-dimensional statistics studies data whose dimension is larger than typically considered in classical multivariate analysis. The area arose owing to the emergence of many modern data sets in which the dimen ...
(large ), 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 operations of explicitly calculating and storing the covariance matrix , instead utilizing one of matrix-free methods, for example, based on the function evaluating the product at the cost of operations.


Iterative computation

One way to compute the first principal component efficientlyRoweis, Sam. "EM Algorithms for PCA and SPCA." Advances in Neural Information Processing Systems. Ed. Michael I. Jordan, Michael J. Kearns, and
Sara A. Solla Sara A. Solla is an Argentine-American physicist and neuroscientist whose research applies ideas from statistical mechanics to problems involving neural networks, machine learning, and neuroscience. She is a professor of physics and of physiology ...
The MIT Press, 1998.
is shown in the following pseudo-code, for a data matrix with zero mean, without ever computing its covariance matrix. = a random vector of length r = r / norm(r) do times: (a vector of length ) return This power iteration algorithm simply calculates the vector , normalizes, and places the result back in . The eigenvalue is approximated by , which is the
Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the ...
on the unit vector for the covariance matrix . If the largest singular value is well separated from the next largest one, the vector gets close to the first principal component of within the number of iterations , which is small relative to , at the total cost . 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 The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n Hermitian matr ...
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 and with block-vectors, matrices and . Every column of approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product . Implemented, for example, in LOBPCG, efficient blocking eliminates the accumulation of the errors, allows using high-level
BLAS Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and mat ...
matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.


The NIPALS method

''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 Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes as well as its hierarchical, three-dim ...
, 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 t1 and r1T 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 , based on the function evaluating the product . The matrix deflation by subtraction is performed by subtracting the outer product, t1r1T from X leaving the deflated residual matrix used to calculate the subsequent leading PCs. 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 A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
accumulated in each iteration and matrix deflation by subtraction. 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. NIPALS reliance on single-vector multiplications cannot take advantage of high-level
BLAS Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and mat ...
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.


Online/sequential estimation

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.


PCA and qualitative variables

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. * Identification, on the factorial planes, of the different species, for example, 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 o
SPAD
that historically, following the work of Ludovic Lebart, was the first to propose this option, and the R packag
FactoMineR


Applications


Intelligence

The earliest application of factor analysis was in locating and measuring components of human intelligence. it was believed that intelligence had various uncorrelated components such as spatial intelligence, verbal intelligence, induction, deduction etc and that scores on these could be adduced by factor analysis from results on various tests, to give a single index known as the
Intelligence Quotient An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. The abbreviation "IQ" was coined by the psychologist William Stern for the German term ''Intelligen ...
(IQ). The pioneering statistical psychologist
Spearman A spear is a pole weapon consisting of a shaft, usually of wood, with a pointed head. The head may be simply the sharpened end of the shaft itself, as is the case with fire hardened spears, or it may be made of a more durable material fastene ...
actually developed factor analysis in 1904 for his two-factor theory of intelligence, adding a formal technique to the science of psychometrics. In 1924 Thurstone looked for 56 factors of intelligence, developing the notion of Mental Age. Standard IQ tests today are based on this early work.Kaplan, R.M., & Saccuzzo, D.P. (2010). ''Psychological Testing: Principles, Applications, and Issues.'' (8th ed.). Belmont, CA: Wadsworth, Cengage Learning.


Residential differentiation

In 1949, Shevky and Williams introduced the theory of factorial ecology, which dominated studies of residential differentiation from the 1950s to the 1970s. Neighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis. These were known as 'social rank' (an index of occupational status), 'familism' or family size, and 'ethnicity'; Cluster analysis could then be applied to divide the city into clusters or precincts according to values of the three key factor variables. An extensive literature developed around factorial ecology in urban geography, but the approach went out of fashion after 1980 as being methodologically primitive and having little place in postmodern geographical paradigms. One of the problems with factor analysis has always been finding convincing names for the various artificial factors. In 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation. The principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities. The first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based. The next two components were 'disadvantage', which keeps people of similar status in separate neighbourhoods (mediated by planning), and ethnicity, where people of similar ethnic backgrounds try to co-locate. About the same time, the Australian Bureau of Statistics defined distinct indexes of advantage and disadvantage taking the first principal component of sets of key variables that were thought to be important. These SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.


Development indexes

PCA has been the only formal method available for the development of indexes, which are otherwise a hit-or-miss ''ad hoc'' undertaking. The
City Development Index The City Development Index (CDI) was developed for the Second United Nations Conference on Human Settlements (Habitat II) in 1996 and measures the level of development in cities. The United Nations Human Settlements Programme (UN-Habitat The Un ...
was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities. The first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for. The index ultimately used about 15 indicators but was a good predictor of many more variables. Its comparative value agreed very well with a subjective assessment of the condition of each city. The coefficients on items of infrastructure were roughly proportional to the average costs of providing the underlying services, suggesting the Index was actually a measure of effective physical and social investment in the city. The country-level
Human Development Index The Human Development Index (HDI) is a statistic composite index of life expectancy, education (mean years of schooling completed and expected years of schooling upon entering the education system), and per capita income indicators, w ...
(HDI) from UNDP, which has been published since 1990 and is very extensively used in development studies, has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA.


Population genetics

In 1978
Cavalli-Sforza Luigi Luca Cavalli-Sforza (; 25 January 1922 – 31 August 2018) was an Italian geneticist. He was a population geneticist who taught at the University of Parma, the University of Pavia and then at Stanford University. Works Schooling and p ...
and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions. The components showed distinctive patterns, including gradients and sinusoidal waves. They interpreted these patterns as resulting from specific ancient migration events. Since then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism. Genetics varies largely according to proximity, so the first two principal components actually show spatial distribution and may be used to map the relative geographical location of different population groups, thereby showing individuals who have wandered from their original locations. PCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers. The lack of any measures of standard error in PCA are also an impediment to more consistent usage. In August 2022, the molecular biologist Eran Elhaik published a theoretical paper in
Scientific Reports ''Scientific Reports'' is a peer-reviewed open-access scientific mega journal published by Nature Portfolio, covering all areas of the natural sciences. The journal was established in 2011. The journal states that their aim is to assess sole ...
analyzing 12 PCA applications. He concluded that it was easy to manipulate the method, which, in his view, generated results that were 'erroneous, contradictory, and absurd.' Specifically, he argued, the results achieved in population genetics were characterized by cherry-picking and circular reasoning.


Market research and indexes of attitude

Market research has been an extensive user of PCA. It is used to develop customer satisfaction or customer loyalty scores for products, and with clustering, to develop market segments that may be targeted with advertising campaigns, in much the same way as factorial ecology will locate geographical areas with similar characteristics. PCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed. In any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes. For example, the Oxford Internet Survey in 2013 asked 2000 people about their attitudes and beliefs, and from these analysts extracted four principal component dimensions, which they identified as 'escape', 'social networking', 'efficiency', and 'problem creating'. Another example from Joe Flood in 2008 extracted an attitudinal index toward housing from 28 attitude questions in a national survey of 2697 households in Australia. The first principal component represented a general attitude toward property and home ownership. The index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice. The strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.


Quantitative finance

In quantitative finance, principal component analysis can be directly applied to the risk management of interest rate derivative portfolios.The Pricing and Hedging of Interest Rate Derivatives: A Practical Guide to Swaps
J H M Darbyshire, 2016,
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, 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. A second is to enhance portfolio return, using the principal components to select stocks with upside potential.


Neuroscience

A variant of principal components analysis is used in
neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...
to identify the specific properties of a stimulus that increases a
neuron A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa ...
's probability of generating an
action potential An action potential occurs when the membrane potential of a specific cell location rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of animal cells ...
.Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000). This technique is known as spike-triggered covariance analysis. In a typical application an experimenter presents a
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
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 In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
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 Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
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 varince 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 This glossary of biology terms is a list of definitions of fundamental terms and concepts used in biology, the study of life and of living organisms. It is intended as introductory material for novices; for more specific and technical definitions ...
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 In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, during phase transitions in the brain.


Relation with other methods


Correspondence analysis

Correspondence analysis (CA) was developed by
Jean-Paul Benzécri Jean-Paul Benzécri was a French mathematician and statistician. He studied at École Normale Supérieure and was professor at Université de Rennes and later for most of his career at the Paris Institute of Statistics (l'Institut de Statistiqu ...
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 In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business in ...
. CA decomposes the chi-squared statistic associated to this table into orthogonal factors. 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.


Factor analysis

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". 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. 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.


-means clustering

It has been asserted that the relaxed solution of -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. However, that PCA is a useful relaxation of -means clustering was not a new result, and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.


Non-negative matrix factorization

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, 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. For NMF, its components are ranked based only on the empirical FRV curves. The residual fractional eigenvalue plots, that is, 1-\sum_^k \lambda_i\Big/\sum_^n \lambda_j as a function of component number k given a total of 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. The FRV curves for NMF is decreasing continuously when the NMF components are constructed sequentially, indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA, indicating the less over-fitting property of NMF.


Iconography of correlations

It is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative. This leads the PCA user to a delicate elimination of several variables. If observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements. In addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane. The iconography of correlations, on the contrary, which is not a projection on a system of axes, does not have these drawbacks. We can therefore keep all the variables. The principle of the diagram is to underline the "remarkable" correlations of the correlation matrix, by a solid line (positive correlation) or dotted line (negative correlation). A strong correlation is not "remarkable" if it is not direct, but caused by the effect of a third variable. Conversely, weak correlations can be "remarkable". For example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet "remarkable".


Generalizations


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 * a regression framework, * a convex relaxation/semidefinite programming framework, * a generalized power method framework * an alternating maximization framework * forward-backward greedy search and exact methods using branch-and-bound techniques, * Bayesian formulation framework. The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.


Nonlinear PCA

Most of the modern methods for
nonlinear dimensionality reduction Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...
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. Trevor Hastie expanded on this concept by proposing Principal curves as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
followed by projecting the points onto it, as is illustrated by Fig. See also the elastic map algorithm and
principal geodesic analysis In geometric data analysis Geometric data analysis comprises geometric aspects of image analysis, pattern analysis, and shape analysis, and the approach of multivariate statistics, which treat arbitrary data sets as ''clouds of points'' in a space ...
. 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, 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 In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor an ...
,
PARAFAC In multilinear algebra, the tensor rank decomposition or the rank-R decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum R rank-1 tensors. This is an open problem. Canonical polyadic decomposition (CPD) is a var ...
, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.


Robust PCA

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 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).
Robust principal component analysis Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to ''grossly'' corrupted observations. A number of different approaches exist f ...
(RPCA) via decomposition in low-rank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations.


Similar techniques


Independent component analysis

Independent component analysis (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.


Network component analysis

Given a matrix E, it tries to decompose it into two matrices such that E=AP . A key difference from techniques such as PCA and ICA is that some of the entries of A are constrained to be 0. Here 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 has full column rank # Each column of A must have at least L-1 zeroes where L is the number of columns of A (or alternatively the number of rows of 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 must have full row rank. then the decomposition is unique up to multiplication by a scalar.


Discriminant analysis of principal components

Discriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals. Genetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former. Linear discriminants are linear combinations of alleles which best separate the clusters. Alleles that most contribute to this discrimination are therefore those that are the most markedly different across groups. The contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups In DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA). A DAPC can be realized on R using the package Adegenet. (more info
adegenet on the web


Software/source code

*
ALGLIB ALGLIB is a cross-platform open source numerical analysis and data processing library. It can be used from several programming languages (C++, C#, VB.NET, Python, Delphi). ALGLIB started in 1999 and has a long history of steady development wi ...
- a C++ and C# library that implements PCA and truncated PCA * Analytica – The built-in EigenDecomp function computes principal components. *
ELKI ELKI (for ''Environment for DeveLoping KDD-Applications Supported by Index-Structures'') is a data mining (KDD, knowledge discovery in databases) software framework developed for use in research and teaching. It was originally at the database s ...
– includes PCA for projection, including robust variants of PCA, as well as PCA-based clustering algorithms. *
Gretl gretl is an open-source statistical package, mainly for econometrics. The name is an acronym for ''G''nu ''R''egression, ''E''conometrics and ''T''ime-series ''L''ibrary. It has both a graphical user interface (GUI) and a command-line inter ...
– principal component analysis can be performed either via the pca command or via the princomp() function. * Julia – Supports PCA with the pca function in the MultivariateStats package *
KNIME KNIME (), the Konstanz Information Miner, is a free and open-source data analytics, reporting and integration platform. KNIME integrates various components for machine learning and data mining through its modular data pipelining "Building Blocks ...
– A java based nodal arranging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily. * Mathematica – Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods. * MathPHP – PHP mathematics library with support for PCA. *
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
- The SVD function is part of the basic system. In the 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. * MatplotlibPython library have a PCA package in the .mlab module. * mlpack – Provides an implementation of principal component analysis in C++. * NAG Library – Principal components analysis is implemented via the g03aa routine (available in both the Fortran versions of the Library). *
NMath NMath is a numerical package for the Microsoft .NET Framework. It is developed by CenterSpace Software. Version 1.0 was released in March, 2003 as NMath Core. The current version is called NMath 7.1, released in December, 2019. NMath is built o ...
– Proprietary numerical library containing PCA for the
.NET Framework The .NET Framework (pronounced as "''dot net"'') is a proprietary software framework developed by Microsoft that runs primarily on Microsoft Windows. It was the predominant implementation of the Common Language Infrastructure (CLI) until bein ...
. *
GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lan ...
– Free software computational environment mostly compatible with MATLAB, the function princomp gives the principal component. * OpenCV *
Oracle Database Oracle Database (commonly referred to as Oracle DBMS, Oracle Autonomous Database, or simply as Oracle) is a multi-model database management system produced and marketed by Oracle Corporation. It is a database commonly used for running online ...
12c – Implemented via DBMS_DATA_MINING.SVDS_SCORING_MODE by specifying setting value SVDS_SCORING_PCA *
Orange (software) Orange is an open-source data visualization, machine learning and data mining toolkit. It features a visual programming front-end for explorative rapid qualitative data analysis and interactive data visualization. Description Orange is a c ...
– Integrates PCA in its visual programming environment. PCA displays a scree plot (degree of explained variance) where user can interactively select the number of principal components. *
Origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
– Contains PCA in its Pro version. *
Qlucore Qlucore was founded in early 2007. Qlucore started as a collaborative research project at Lund University , motto = Ad utrumque , mottoeng = Prepared for both , established = , type ...
– Commercial software for analyzing multivariate data with instant response using PCA. * RFree statistical package, the functions princomp and prcomp can be used for principal component analysis; prcomp uses
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
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. * SAS – Proprietary software; for example, see * scikit-learn – Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module. *
Scilab Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simula ...
– Free and open-source, cross-platform numerical computational package, the function princomp computes principal component analysis, the function pca computes principal component analysis with standardized variables. *
SPSS SPSS Statistics is a statistical software suite developed by IBM for data management, advanced analytics, multivariate analysis, business intelligence, and criminal investigation. Long produced by SPSS Inc., it was acquired by IBM in 2009. C ...
– Proprietary software most commonly used by social scientists for PCA, factor analysis and associated cluster analysis. * Weka – Java library for machine learning which contains modules for computing principal components.


See also

* Correspondence analysis (for contingency tables) * Multiple correspondence analysis (for qualitative variables) *
Factor analysis of mixed data In statistics, factor analysis of mixed data or factorial analysis of mixed data (FAMD, in the French original: ''AFDM'' or ''Analyse Factorielle de Données Mixtes''), is the factorial method devoted to data tables in which a group of individuals ...
(for quantitative and qualitative variables) *
Canonical correlation In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors ''X'' = (''X''1, ..., ''X'n'') and ''Y' ...
*
CUR matrix approximation A CUR matrix approximation is a set of three matrices that, when multiplied together, closely approximate a given matrix. A CUR approximation can be used in the same way as the low-rank approximation of the singular value decomposition (SVD). CU ...
(can replace of low-rank SVD approximation) * Detrended correspondence analysis * Dynamic mode decomposition * Eigenface * Expectation–maximization algorithm * Exploratory factor analysis (Wikiversity) *
Factorial code {{Short description, Data representation for machine learning Most real world data sets consist of data vectors whose individual components are not statistically independent. In other words, knowing the value of an element will provide information ...
*
Functional principal component analysis Functional principal component analysis (FPCA) is a statistical method for investigating the dominant modes of variation of functional data. Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of t ...
* Geometric data analysis * Independent component analysis * Kernel PCA * L1-norm principal component analysis * Low-rank approximation * Matrix decomposition * Non-negative matrix factorization *
Nonlinear dimensionality reduction Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...
* Oja's rule * Point distribution model (PCA applied to morphometry and computer vision) * Principal component analysis (Wikibooks) * Principal component regression * Singular spectrum analysis *
Singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
* Sparse PCA *
Transform coding Transform coding is a type of data compression for "natural" data like audio signals or photographic images. The transformation is typically lossless (perfectly reversible) on its own but is used to enable better (more targeted) quantization, ...
* Weighted least squares


References


Further reading

* Jackson, J.E. (1991). ''A User's Guide to Principal Components'' (Wiley). * * * Husson François, Lê Sébastien & Pagès Jérôme (2009). ''Exploratory Multivariate Analysis by Example Using R''. Chapman & Hall/CRC The R Series, London. 224p. * Pagès Jérôme (2014).
Multiple Factor Analysis by Example Using R
'. Chapman & Hall/CRC The R Series London 272 p


External links

* *
A Tutorial on Principal Component Analysis
* (a video of less than 100 seconds.) * * See also the list of Software implementations {{Authority control Matrix decompositions Dimension reduction