{"history":[],"article":{"id":"faedde6f-1ee3-40a5-828e-fa46cbd39a70","title":"Principal component analysis","slug":"principal-component-analysis","content":":::summary\n:::\n\n# \n**Principal component analysis** (**PCA**) is a [linear](https://bliptext.com/articles/linear-map) [dimensionality reduction](https://bliptext.com/articles/dimensionality-reduction) technique with applications in [exploratory data analysis](https://bliptext.com/articles/exploratory-data-analysis), visualization and [data preprocessing](https://bliptext.com/articles/data-preprocessing).\nThe data is linearly transformed onto a new [coordinate system](https://bliptext.com/articles/coordinate-system) such that the directions (principal components) capturing the largest variation in the data can be easily identified.\nThe **principal components** of a collection of points in a [real coordinate space](https://bliptext.com/articles/real-coordinate-space) are a sequence of p [unit vectors](https://bliptext.com/articles/unit-vector), where the i-th vector is the direction of a line that best fits the data while being [orthogonal](https://bliptext.com/articles/orthogonal) to the first i-1 vectors.\nHere, a best-fitting line is defined as one that minimizes the average squared [perpendicular](https://bliptext.com/articles/perpendicular-distance) [distance from the points to the line](https://bliptext.com/articles/distance-from-a-point-to-a-line).\nThese directions (i.e., principal components) constitute an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) in which different individual dimensions of the data are [linearly uncorrelated](https://bliptext.com/articles/linear-correlation).\nMany 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.\nPrincipal component analysis has applications in many fields such as [population genetics](https://bliptext.com/articles/population-genetics), [microbiome](https://bliptext.com/articles/microbiome) studies, and [atmospheric science](https://bliptext.com/articles/atmospheric-science).\n# [PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean. | upload.wikimedia.org/wikipedia/commons/f/f5/GaussianScatterPCA.svg]\n\n# Overview\nWhen performing PCA, the first principal component of a set of p variables is the derived variable formed as a linear combination of the original variables that explains the most variance.\nThe 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.\nPCA 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.\nThe first principal component can equivalently be defined as a direction that maximizes the variance of the projected data.\nThe 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.\nFor either objective, it can be shown that the principal components are [eigenvectors](https://bliptext.com/articles/eigenvectors) of the data's [covariance matrix](https://bliptext.com/articles/covariance-matrix).\nThus, the principal components are often computed by eigendecomposition of the data covariance matrix or [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) of the data matrix.\nPCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to [factor analysis](https://bliptext.com/articles/factor-analysis).\nFactor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.\nPCA is also related to [canonical correlation analysis (CCA)](https://bliptext.com/articles/canonical-correlation).\nCCA defines coordinate systems that optimally describe the [cross-covariance](https://bliptext.com/articles/cross-covariance) between two datasets while PCA defines a new [orthogonal coordinate system](https://bliptext.com/articles/orthogonal-coordinate-system) that optimally describes variance in a single dataset.\n[Robust](https://bliptext.com/articles/robust-statistics) and [L1-norm](https://bliptext.com/articles/lp-space)-based variants of standard PCA have also been proposed.\n\n# History\nPCA was invented in 1901 by [Karl Pearson](https://bliptext.com/articles/karl-pearson), as an analogue of the [principal axis theorem](https://bliptext.com/articles/principal-axis-theorem) in mechanics; it was later independently developed and named by [Harold Hotelling](https://bliptext.com/articles/harold-hotelling) in the 1930s.\nDepending on the field of application, it is also named the discrete [Karhunen–Loève](https://bliptext.com/articles/karhunen-lo-ve-theorem) transform (KLT) in [signal processing](https://bliptext.com/articles/signal-processing), the [Hotelling](https://bliptext.com/articles/harold-hotelling) transform in multivariate quality control, [proper orthogonal decomposition](https://bliptext.com/articles/proper-orthogonal-decomposition) (POD) in mechanical engineering, [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) (SVD) of ******X****** (invented in the last quarter of the 19th century ), [eigenvalue decomposition](https://bliptext.com/articles/eigendecomposition) (EVD) of ******X******T******X****** in linear algebra, [factor analysis](https://bliptext.com/articles/factor-analysis) (for a discussion of the differences between PCA and [factor analysis](https://bliptext.com/articles/factor-analysis) see Ch. 7 of Jolliffe's Principal Component Analysis), [Eckart–Young theorem](https://bliptext.com/articles/eckart-young-theorem) (Harman, 1960), or [empirical orthogonal functions](https://bliptext.com/articles/empirical-orthogonal-functions) (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), [spectral decomposition](https://bliptext.com/articles/spectral-theorem) in noise and vibration, and [empirical modal analysis](https://bliptext.com/articles/mode-shape) in structural dynamics.\n\n# Intuition\nPCA can be thought of as fitting a p-dimensional [ellipsoid](https://bliptext.com/articles/ellipsoid) to the data, where each axis of the [ellipsoid](https://bliptext.com/articles/ellipsoid) represents a principal component.\nIf some axis of the ellipsoid is small, then the variance along that axis is also small.\nTo 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.\nThese transformed values are used instead of the original observed values for each of the variables.\nThen, we compute the [covariance matrix](https://bliptext.com/articles/covariance-matrix) of the data and calculate the eigenvalues and corresponding eigenvectors of this [covariance matrix](https://bliptext.com/articles/covariance-matrix).\nThen we must [normalize](https://bliptext.com/articles/normalization-statistics) each of the orthogonal eigenvectors to turn them into unit vectors.\nOnce this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data.\nThis choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis.\nThe 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.\n[Biplots](https://bliptext.com/articles/biplot) and [scree plots](https://bliptext.com/articles/scree-plot) (degree of [explained variance](https://bliptext.com/articles/explained-variance)) are used to interpret findings of the PCA.\n# [A scree plot that is meant to help interpret the PCA and decide how many components to retain. The start of the bend in the line (point of inflexion or \"knee\") should indicate how many components are retained, hence in this example, three factors should be retained. | upload.wikimedia.org/wikipedia/commons/e/e8/SCREE///plot.jpg]\n\n# Details\nPCA is defined as an [orthogonal](https://bliptext.com/articles/orthogonal-transformation) [linear transformation](https://bliptext.com/articles/linear-transformation) on a real [inner product space](https://bliptext.com/articles/inner-product-space) that transforms the data to a new [coordinate system](https://bliptext.com/articles/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.\nConsider an n \\times p data [matrix](https://bliptext.com/articles/matrix-mathematics), **X**, with column-wise zero [empirical mean](https://bliptext.com/articles/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).\nMathematically, the transformation is defined by a set of size l 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\nin such a **w**ay **t**ha**t** **t**he individual variables of **t** considered over **t**he da**t**a se**t** successively inheri**t** **t**he maximum possible variance from **X**, **w**i**t**h each coefficien**t** vec**t**or **w** cons**t**rained **t**o be a uni**t** vec**t**or (**w**here l is usually selec**t**ed **t**o be s**t**ric**t**ly less **t**han p **t**o reduce dimensionali**t**y).\nThe above may equivalently be written in matrix form as where\n,, and\n.\n\n## First component\nIn order to maximize variance, the first **w**eight vector **w**(1) thus has to satisfy\nEquivalently, writing this in matrix form gives\nSince **w**(1) has been defined to be a unit vector, it equivalently also satisfies\nThe quantity to be maximised can be recognised as a [Rayleigh quotient](https://bliptext.com/articles/rayleigh-quotient).\nA standard result for a [positive semidefinite matrix](https://bliptext.com/articles/positive-semidefinite-matrix) such as ****X****T****X**** is that the quotient's maximum possible value is the largest [eigenvalue](https://bliptext.com/articles/eigenvalue) of the matrix, **w**hich occurs **w**hen **w** is the corresponding [eigenvector](https://bliptext.com/articles/eigenvector).\nWith ********w********(1) found, the first principal component of a data vector ******x******(i) can then be given as a score t1(i) = ******x******(i) ⋅ ********w********(1) in the transformed co-ordinates, or as the corresponding vector in the original variables, {******x******(i) ⋅ ********w********(1)} ********w********(1).\n\n## Further components\nThe k-th component can be found by subtracting the first k − 1 principal components from **X**:\nand then finding the weight vector which extracts the maximum variance from this new data matrix\nIt 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.\nThus the weight vectors are eigenvectors of ****X****T****X****.\nThe k-th principal component of a data vector ******x******(i) can therefore be given as a score tk(i) = ******x******(i) ⋅ ********w********(k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {******x******(i) ⋅ ********w********(k)} ********w********(k), ********w********here ********w********(k) is the kth eigenvector of ****X****T****X****.\nThe 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******.\nThe transpose of **W** is sometimes called the [whitening or sphering transformation](https://bliptext.com/articles/whitening-transformation).\nColumns 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.\n\n## Covariances\n****X****T****X**** itself can be recognized as proportional to the empirical sample [covariance matrix](https://bliptext.com/articles/covariance-matrix) of the dataset ****X****T.\nThe sample covariance Q between two of the different principal components over the dataset is given by:\n**w**here the eigenvalue property of **w**(k) has been used to move from line 2 to line 3.\nHo****w****ever 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).\nThe product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.\nAnother way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.\nIn matrix form, the empirical covariance matrix for the original variables can be written\nThe empirical covariance matrix between the principal components becomes\nwhere **Λ** is the diagonal matrix of eigenvalues λ(k) of ****X****T****X****.\nλ(k) is equal to the sum of the squares over the dataset associated **w**ith each component k, that is, λ(k) = Σi tk2(i) = Σi (**x**(i) ⋅ **w**(k))2.\n\n## Dimensionality reduction\n**T**he 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.\nHowever, not all the principal components need to be kept.\nKeeping only the first L principal components, produced by using only the first L eigenvectors, gives the truncated transformation\nwhere the matrix **T**L now has n rows but only L columns.\nIn other words, PCA learns a linear transformation where the columns of\np × L\nmatrix W_L form an orthogonal basis for the L features (the components of representation t) that are decorrelated.\nBy 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.\nSuch [dimensionality reduction](https://bliptext.com/articles/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.\nFor 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](https://bliptext.com/articles/cluster-analysis) 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](https://bliptext.com/articles/cluster-analysis) 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.\nSimilarly, in [regression analysis](https://bliptext.com/articles/regression-analysis), the larger the number of [explanatory variables](https://bliptext.com/articles/explanatory-variable) allowed, the greater is the chance of [overfitting](https://bliptext.com/articles/overfitting) the model, producing conclusions that fail to generalise to other datasets.\nOne 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](https://bliptext.com/articles/principal-component-regression).\nDimensionality reduction may also be appropriate when the variables in a dataset are noisy.\nIf 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).\nHowever, 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](https://bliptext.com/articles/signal-to-noise-ratio).\nPCA 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.\nIf the dataset is not too large, the significance of the principal components can be tested using [parametric bootstrap](https://bliptext.com/articles/bootstrapping-statistics), as an aid in determining how many principal components to retain.\n# [A principal components analysis scatterplot of Y-STR haplotypes calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.\nPCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent. | upload.wikimedia.org/wikipedia/commons/6/69/PCA///of///Haplogroup///J///using///37///STRs.png]\n\n## Singular value decomposition\nThe principal components transformation can also be associated with another matrix factorization, the [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) (SVD) of **X**,\nHere **Σ** is an n-by-p [rectangular diagonal matrix](https://bliptext.com/articles/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 matrix whose columns are orthogonal unit vectors of length p and called the right singular vectors of ******X******.\nIn terms of this factorization, the matrix ****X****T****X**** can be written\nwhere**** ******** ******** ****is**** ****the**** ****square**** ****diagonal**** ****matrix**** ****with**** ****the**** ****singular**** ****values**** ****of**** ****X**** ****and**** ****the**** ****excess**** ****zeros**** ****chopped**** ****off**** ****that**** ****satisfies**** ****.\nComparison** **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**************.\nUsing the singular value decomposition the score matrix **T** can be written\nso each column of **T** is given by one of the left singular vectors of **X** multiplied by the corresponding singular value.\n**T**his form is also the [polar decomposition](https://bliptext.com/articles/polar-decomposition) of **T**.\nEfficient 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, unless only a handful of components are required.\nAs with the eigen-decomposition, a truncated\nn × L\nscore matrix **T**L can be obtained by considering only the first L largest singular values and their singular vectors:\n**T**he 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](https://bliptext.com/articles/rank-linear-algebra) L to the original matrix, in the sense of the difference between the two having the smallest possible [Frobenius norm](https://bliptext.com/articles/frobenius-norm), a result known as the [Eckart–Young theorem](https://bliptext.com/articles/low-rank-approximation) [1936].\n\n# Further considerations\nThe singular values (in **Σ**) are the square roots of the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the matrix ****X****T****X****.\nEach 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.\nThe sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean.\nPCA essentially rotates the set of points around their mean in order to align with the principal components.\nThis moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions.\nThe values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see [below](https://bliptext.com/articles/principle-component-analysis)).\nPCA is often used in this manner for [dimensionality reduction](https://bliptext.com/articles/dimensionality-reduction).\nPCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest \"variance\" (as defined above).\nThis advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the [discrete cosine transform](https://bliptext.com/articles/discrete-cosine-transform), and in particular to the DCT-II which is simply known as the \"DCT\".\n[Nonlinear dimensionality reduction](https://bliptext.com/articles/nonlinear-dimensionality-reduction) techniques tend to be more computationally demanding than PCA.\nPCA is sensitive to the scaling of the variables.\nIf we have just two variables and they have the same [sample variance](https://bliptext.com/articles/sample-variance) 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.\nBut 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.\nThis means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis.\n(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.\nOne 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.\nHowever, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.\nMean 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.\nIf mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data.\nA mean of zero is needed for finding a basis that minimizes the [mean square error](https://bliptext.com/articles/minimum-mean-square-error) of the approximation of the data.\nMean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations.\nCorrelations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation).\nAlso see the article by Kromrey & Foster-Johnson (1998) on \"Mean-centering in Moderated Regression: Much Ado About Nothing\".\nSince [covariances are correlations of normalized variables](https://bliptext.com/articles/covariance-matrix) (**Z**- or standard-scores) a PCA based on the correlation matrix of ****X**** is [equal](https://bliptext.com/articles/equality-mathematics) to a PCA based on the covariance matrix of **Z**, the standardized version of ****X****.\nPCA is a popular primary technique in [pattern recognition](https://bliptext.com/articles/pattern-recognition).\nIt is not, however, optimized for class separability.\nHowever, 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.\nThe [linear discriminant analysis](https://bliptext.com/articles/linear-discriminant-analysis) is an alternative which is optimized for class separability.\n\n# Table of symbols and abbreviations\n\n# Properties and limitations\n\n## Properties\nSome properties of PCA include:\nThe statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs.\nBecause these last PCs have variances as small as possible they are useful in their own right.\nThey can help to detect unsuspected near-constant linear relationships between the elements of x, and they may also be useful in [regression](https://bliptext.com/articles/regression-analysis), in selecting a subset of variables from x, and in outlier detection.\nΣ\n)\nBefore we look at its usage, we first look at [diagonal](https://bliptext.com/articles/diagonal) elements,\nThen, 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](https://bliptext.com/articles/covariance-matrix) into contributions from each PC.\nAlthough not strictly decreasing, the elements of will tend to become smaller as k increases, as is nonincreasing for increasing k, whereas the elements of \\alpha_k tend to stay about the same size because of the normalization constraints:.\n\n## Limitations\nAs noted above, the results of PCA depend on the scaling of the variables.\nThis can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.\nThe applicability of PCA as described above is limited by certain (tacit) assumptions made in its derivation.\nIn particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference).\nIn some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see [kernel PCA](https://bliptext.com/articles/kernel-principal-component-analysis)).\nAnother limitation is the mean-removal process before constructing the covariance matrix for PCA.\nIn 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.\nAs an alternative method, [non-negative matrix factorization](https://bliptext.com/articles/non-negative-matrix-factorization) focusing only on the non-negative elements in the matrices, which is well-suited for astrophysical observations.\nSee more at [Relation between PCA and Non-negative Matrix Factorization](https://bliptext.com/articles/).\nPCA is at a disadvantage if the data has not been standardized before applying the algorithm to it.\nPCA 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.\nThey are linear interpretations of the original variables.\nAlso, if PCA is not performed properly, there is a high likelihood of information loss.\nPCA relies on a linear model.\nIf 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.\nResearchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results.\n\"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\".\nThe researchers at Kansas State also found that PCA could be \"seriously biased if the autocorrelation structure of the data is not correctly handled\".\n\n## PCA and information theory\nDimensionality reduction results in a loss of information, in general.\nPCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.\nUnder the assumption that\nthat is, that the data vector \\mathbf{x} is the sum of the desired information-bearing signal \\mathbf{s} and a noise signal \\mathbf{n} one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.\nIn particular, Linsker showed that if \\mathbf{s} is Gaussian and \\mathbf{n} is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the [mutual information](https://bliptext.com/articles/mutual-information) between the desired information \\mathbf{s} and the dimensionality-reduced output.\nIf the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector \\mathbf{n} are [iid](https://bliptext.com/articles/iid)), but the information-bearing signal \\mathbf{s} is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as\nThe optimality of PCA is also preserved if the noise \\mathbf{n} is iid and at least more Gaussian (in terms of the [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence)) than the information-bearing signal \\mathbf{s}.\nIn general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise \\mathbf{n} becomes dependent.\n\n# Computation using the covariance method\nThe following is a detailed description of PCA using the covariance method as opposed to the correlation method.\nThe goal is to transform a given data set **X** of dimension p to an alternative data set **Y** of smaller dimension L.\nEquivalently, we are seeking to find the matrix ****Y****, where ****Y**** is the [Karhunen–Loève](https://bliptext.com/articles/karhunen-lo-ve-theorem) transform (KLT) of matrix **X**:\n

\n**Organize the data set**\nSuppose 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.\nSuppose further, that the data are arranged as a set of n data vectors with each representing a single grouped observation of the p variables.\n
\n**Calculate the empirical mean**\n
\n**Calculate the deviations from the mean**\nMean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.\nHence we proceed by centering the data as follows:\nn × 1\ncolumn vector of all 1s:\nIn some applications, each variable (column of **B**) may also be scaled to have a variance equal to 1 (see [Z-score](https://bliptext.com/articles/z-score)).\nThis step affects the calculated principal components, but makes them independent of the units used to measure the different variables.\n
\n**Find the covariance matrix**\nn − 1\ninstead of n to calculate the covariance is [Bessel's correction](https://bliptext.com/articles/bessel-s-correction).\n
\n**Find the eigenvectors and eigenvalues of the covariance matrix**\n
\n**Rearrange the eigenvectors and eigenvalues**\n
\n**Compute the cumulative energy content for each eigenvector**\n
\n**Select a subset of the eigenvectors as basis vectors**\n
\n**Project the data onto the new basis**\nThat is, the first column of \\mathbf{T} is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc.

\n\n# Derivation using the covariance method\nLet **X** be a d-dimensional random vector expressed as column vector.\nWithout loss of generality, assume **X** has zero mean.\nWe want to find (\\ast) a\nd × d\n[orthonormal transformation matrix](https://bliptext.com/articles/orthonormal-basis) **P** so that **P**X has a diagonal covariance matrix (that is, **P**X is a random vector with all its distinct components pairwise uncorrelated).\nA quick computation assuming P were unitary yields:\nHence (\\ast) holds if and only if were diagonalisable by P.\nThis 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.\n\n# Covariance-free computation\nIn practical implementations, especially with [high dimensional data](https://bliptext.com/articles/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.\nThe covariance-free approach avoids the\nnp2\noperations of explicitly calculating and storing the covariance matrix\nXTX\n, instead utilizing one of [matrix-free methods](https://bliptext.com/articles/matrix-free-methods), for example, based on the function evaluating the product\nXT(X r)\nat the cost of\n2np\noperations.\n\n## Iterative computation\nOne way to compute the first principal component efficiently is shown in the following pseudo-code, for a data matrix\nX\nwith zero mean, without ever computing its covariance matrix.\nr\n= a ******r******andom vecto******r****** of length p ******r****** = ******r****** / no******r******m(******r******)\ndo c times:\ns = 0\n(a vector of length p) for each row x in X s = s + (x ⋅ r) x λ = rTs // λ is the eigenvalue\nλ ⋅ r − s r = s / norm(s)\nexit if error < tolerance\nreturn λ, r\nThis [power iteration](https://bliptext.com/articles/power-iteration) algorithm simply calculates the vector\nXT(X r)\n, normalizes, and places the result back in\nr\n. The eigenvalue is approximated by\nrT (XTX) r\n, which is the [Rayleigh quotient](https://bliptext.com/articles/rayleigh-quotient) on the unit vector\nr\nfor the covariance matrix\nXTX\n. If the largest singular value is well separated from the next largest one, the vector\nr\ngets close to the first principal component of\nX\nwithin the number of iterations c, which is small relative to p, at the total cost\n2cnp\n. The [power iteration](https://bliptext.com/articles/power-iteration) convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced [matrix-free methods](https://bliptext.com/articles/matrix-free-methods), such as the [Lanczos algorithm](https://bliptext.com/articles/lanczos-algorithm) or the Locally Optimal Block Preconditioned Conjugate Gradient ([LOBPCG](https://bliptext.com/articles/lobpcg)) method.\nSubsequent principal components can be computed one-by-one via deflation or simultaneously as a block.\nIn 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.\nThe latter approach in the block power method replaces single-vectors\nr\nand\ns\nwith block-vectors, matrices\nR\nand\nS\n. Every column of\nR\napproximates one of the leading principal components, while all columns are iterated simultaneously.\nThe main calculation is evaluation of the product\nXT(X R)\n. Implemented, for example, in [LOBPCG](https://bliptext.com/articles/lobpcg), efficient blocking eliminates the accumulation of the errors, allows using high-level [BLAS](https://bliptext.com/articles/blas) matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.\n\n## The NIPALS method\nNon-linear iterative [partial least squares](https://bliptext.com/articles/partial-least-squares) (NIPALS) is a variant the classical [power iteration](https://bliptext.com/articles/power-iteration) with matrix deflation by subtraction implemented for computing the first few components in a principal component or [partial least squares](https://bliptext.com/articles/partial-least-squares) analysis.\nFor very-high-dimensional datasets, such as those generated in the *omics sciences (for example, [genomics](https://bliptext.com/articles/genomics), [metabolomics](https://bliptext.com/articles/metabolomics)) it is usually only necessary to compute the first few PCs.\nThe non-linea**r** i**t**e**r**a**t**ive pa**r****t**ial leas**t** squa**r**es (NIPALS) algo**r**i**t**hm upda**t**es i**t**e**r**a**t**ive app**r**oxima**t**ions **t**o **t**he leading sco**r**es and loadings **t**1 and **r**1T by **t**he powe**r** i**t**e**r**a**t**ion mul**t**iplying on eve**r**y i**t**e**r**a**t**ion by **X** on **t**he lef**t** and on **t**he **r**igh**t**, **t**ha**t** is, calcula**t**ion of **t**he cova**r**iance ma**t****r**ix is avoided, jus**t** as in **t**he ma**t****r**ix-f**r**ee implemen**t**a**t**ion of **t**he powe**r** i**t**e**r**a**t**ions **t**o\nXTX\n, based on the function evaluating the product\nXT(X r) = ((X r)TX)T\n.\nThe ma**t****r**ix defla**t**ion by sub**t****r**ac**t**ion is pe**r**fo**r**med by sub**t****r**ac**t**ing **t**he ou**t**e**r** p**r**oduc**t**, **t**1**r**1T f**r**om **X** leaving **t**he defla**t**ed **r**esidual ma**t****r**ix used **t**o calcula**t**e **t**he subsequen**t** leading PCs.\nFor 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](https://bliptext.com/articles/round-off-errors) accumulated in each iteration and matrix deflation by subtraction.\nA [Gram–Schmidt](https://bliptext.com/articles/gram-schmidt) re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.\nNIPALS reliance on single-vector multiplications cannot take advantage of high-level [BLAS](https://bliptext.com/articles/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](https://bliptext.com/articles/lobpcg)) method.\n\n## Online/sequential estimation\nIn 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.\nThis can be done efficiently, but requires different algorithms.\n\n# Qualitative variables\nIn PCA, it is common that we want to introduce qualitative variables as supplementary elements.\nFor example, many quantitative variables have been measured on plants.\nFor these plants, some qualitative variables are available as, for example, the species to which the plant belongs.\nThese data were subjected to PCA for quantitative variables.\nWhen analyzing the results, it is natural to connect the principal components to the qualitative variable species.\nFor this, the following results are produced.\nThese results are what is called introducing a qualitative variable as supplementary element.\nThis procedure is detailed in and Husson, Lê, & Pagès (2009) and Pagès (2013).\nFew software offer this option in an \"automatic\" way.\nThis is the case of [SPAD](https://bliptext.com/articles/undefined) that historically, following the work of [Ludovic Lebart](https://bliptext.com/articles/ludovic-lebart), was the first to propose this option, and the R package [FactoMineR](https://bliptext.com/articles/undefined).\n\n# Applications\n\n## Intelligence\nThe earliest application of factor analysis was in locating and measuring components of human intelligence.\nIt 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](https://bliptext.com/articles/intelligence-quotient) (IQ).\nThe pioneering statistical psychologist [Spearman](https://bliptext.com/articles/charles-spearman) actually developed factor analysis in 1904 for his [two-factor theory](https://bliptext.com/articles/two-factor-theory-of-intelligence) of intelligence, adding a formal technique to the science of [psychometrics](https://bliptext.com/articles/psychometrics).\nIn 1924 [Thurstone](https://bliptext.com/articles/louis-leon-thurstone) looked for 56 factors of intelligence, developing the notion of Mental Age.\nStandard IQ tests today are based on this early work.\n\n## Residential differentiation\nIn 1949, Shevky and Williams introduced the theory of **factorial ecology**, which dominated studies of residential differentiation from the 1950s to the 1970s.\nNeighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis.\nThese 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.\nAn 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.\nOne of the problems with factor analysis has always been finding convincing names for the various artificial factors.\nIn 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation.\nThe principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities.\nThe first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based.\nThe 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.\nAbout 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.\nThese SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.\n\n## Development indexes\nPCA can be used as a formal method for the development of indexes.\nAs an alternative [confirmatory composite analysis](https://bliptext.com/articles/confirmatory-composite-analysis) has been proposed to develop and assess indexes.\nThe [City Development Index](https://bliptext.com/articles/city-development-index) was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities.\nThe first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for.\nThe index ultimately used about 15 indicators but was a good predictor of many more variables.\nIts comparative value agreed very well with a subjective assessment of the condition of each city.\nThe 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.\nThe country-level [Human Development Index](https://bliptext.com/articles/human-development-index) (HDI) from [UNDP](https://bliptext.com/articles/united-nations-development-programme), 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.\n\n## Population genetics\nIn 1978 [Cavalli-Sforza](https://bliptext.com/articles/luigi-luca-cavalli-sforza) and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions.\nThe components showed distinctive patterns, including gradients and sinusoidal waves.\nThey interpreted these patterns as resulting from specific ancient migration events.\nSince then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism.\nGenetics 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.\nPCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers.\nThe lack of any measures of standard error in PCA are also an impediment to more consistent usage.\nIn August 2022, the molecular biologist [Eran Elhaik](https://bliptext.com/articles/eran-elhaik) published a theoretical paper in [Scientific Reports](https://bliptext.com/articles/scientific-reports) analyzing 12 PCA applications.\nHe 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](https://bliptext.com/articles/circular-reasoning).\n\n## Market research and indexes of attitude\nMarket research has been an extensive user of PCA.\nIt 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.\nPCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed.\nIn any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes.\nFor 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'.\nAnother 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.\nThe first principal component represented a general attitude toward property and home ownership.\nThe index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice.\nThe strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.\n\n## Quantitative finance\nIn [quantitative finance](https://bliptext.com/articles/quantitative-finance), PCA is used\nin [financial risk management](https://bliptext.com/articles/financial-risk-management), and has been applied to [other problems](https://bliptext.com/articles/financial-modeling) such as [portfolio optimization](https://bliptext.com/articles/portfolio-optimization).\nPCA is commonly used in problems involving [fixed income](https://bliptext.com/articles/fixed-income) securities and [portfolios](https://bliptext.com/articles/bond-fund), and [interest rate derivatives](https://bliptext.com/articles/interest-rate-derivative).\nValuations here depend on the entire [yield curve](https://bliptext.com/articles/yield-curve), comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements,\nthereby facilitating the modelling.\nOne common risk management application is to [calculating value at risk](https://bliptext.com/articles/value-at-risk), VaR, applying PCA to the [Monte Carlo simulation](https://bliptext.com/articles/monte-carlo-methods-in-finance).\nHere, for each simulation-sample, the components are stressed, and rates, and [in turn option values](https://bliptext.com/articles/monte-carlo-methods-for-option-pricing), are then reconstructed;\nwith VaR calculated, finally, over the entire run.\nPCA is also used in [hedging](https://bliptext.com/articles/hedge-finance) exposure to [interest rate risk](https://bliptext.com/articles/interest-rate-risk), given [partial durations](https://bliptext.com/articles/key-rate-duration) and other sensitivities.\nUnder both, the first three, typically, principal components of the system are of interest ([representing](https://bliptext.com/articles/fixed-income-attribution) \"shift\", \"twist\", and \"curvature\").\nThese principal components are derived from an eigen-decomposition of the [covariance matrix](https://bliptext.com/articles/covariance-matrix) of [yield](https://bliptext.com/articles/yield-curve) at predefined maturities;\nand where the [variance](https://bliptext.com/articles/variance) of each component is its [eigenvalue](https://bliptext.com/articles/eigenvalue) (and as the components are [orthogonal](https://bliptext.com/articles/orthogonal), no correlation need be incorporated in subsequent modelling).\nFor [equity](https://bliptext.com/articles/equity-finance), an optimal portfolio is one where the [expected return](https://bliptext.com/articles/expected-return) is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see [Markowitz model](https://bliptext.com/articles/markowitz-model) for discussion.\nThus, one approach is to reduce portfolio risk, where [allocation strategies](https://bliptext.com/articles/asset-allocation) are applied to the \"principal portfolios\" instead of the underlying [stocks](https://bliptext.com/articles/capital-stock).\nA second approach is to enhance portfolio return, using the principal components to select companies' stocks with upside potential.\nPCA has also been used to understand relationships between international [equity markets](https://bliptext.com/articles/equity-market), and within markets between groups of companies in industries or [sectors](https://bliptext.com/articles/stock-market-index).\nPCA may also be applied to [stress testing](https://bliptext.com/articles/stress-test-financial), essentially an analysis of a bank's ability to endure [a hypothetical adverse economic scenario](https://bliptext.com/articles/list-of-bank-stress-tests).\nIts utility is in \"distilling the information contained in [several] [macroeconomic variables](https://bliptext.com/articles/macroeconomic-model) into a more manageable data set, which can then [be used] for analysis.\"\nHere, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's [eigenvector](https://bliptext.com/articles/eigenvector) – and it is then observed how a \"shock\" to each of the factors affects the implied assets of each of the banks.\n\n## Neuroscience\nA variant of principal components analysis is used in [neuroscience](https://bliptext.com/articles/neuroscience) to identify the specific properties of a stimulus that increases a [neuron](https://bliptext.com/articles/neuron)'s probability of generating an [action potential](https://bliptext.com/articles/action-potential).\nThis technique is known as [spike-triggered covariance analysis](https://bliptext.com/articles/spike-triggered-covariance).\nIn a typical application an experimenter presents a [white noise](https://bliptext.com/articles/white-noise) process as a stimulus (usually either as a sensory input to a test subject, or as a [current](https://bliptext.com/articles/electric-current) injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result.\nPresumably, certain features of the stimulus make the neuron more likely to spike.\nIn order to extract these features, the experimenter calculates the [covariance matrix](https://bliptext.com/articles/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.\nThe [eigenvectors](https://bliptext.com/articles/eigenvectors-and-eigenvalues) 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](https://bliptext.com/articles/vector-space) of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble.\nSpecifically, 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.\nSince 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.\nIn neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential.\n[Spike sorting](https://bliptext.com/articles/spike-sorting) is an important procedure because [extracellular](https://bliptext.com/articles/electrophysiology) recording techniques often pick up signals from more than one neuron.\nIn spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs [clustering analysis](https://bliptext.com/articles/cluster-analysis) to associate specific action potentials with individual neurons.\nPCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles.\nIt has been used in determining collective variables, that is, [order parameters](https://bliptext.com/articles/order-parameters), during [phase transitions](https://bliptext.com/articles/phase-transitions) in the brain.\n\n# Relation with other methods\n\n## Correspondence analysis\n[Correspondence analysis](https://bliptext.com/articles/correspondence-analysis) (CA)\nwas developed by [Jean-Paul Benzécri](https://bliptext.com/articles/jean-paul-benz-cri)\nand is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently.\nIt is traditionally applied to [contingency tables](https://bliptext.com/articles/contingency-tables).\nCA decomposes the [chi-squared statistic](https://bliptext.com/articles/chi-squared-statistic) associated to this table into orthogonal factors.\nBecause CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not.\nSeveral variants of CA are available including [detrended correspondence analysis](https://bliptext.com/articles/detrended-correspondence-analysis) and [canonical correspondence analysis](https://bliptext.com/articles/canonical-correspondence-analysis).\nOne special extension is [multiple correspondence analysis](https://bliptext.com/articles/multiple-correspondence-analysis), which may be seen as the counterpart of principal component analysis for categorical data.\n\n## Factor analysis\nPrincipal component analysis creates variables that are linear combinations of the original variables.\nThe new variables have the property that the variables are all orthogonal.\nThe PCA transformation can be helpful as a pre-processing step before clustering.\nPCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable.\nPCA 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.\n[Factor analysis](https://bliptext.com/articles/factor-analysis) is similar to principal component analysis, in that factor analysis also involves linear combinations of variables.\nDifferent 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\".\nIn 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.\nHowever, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.\nResults 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.\nFactor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or [causal modeling](https://bliptext.com/articles/causal-modeling).\nIf the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.\n# [The above picture is an example of the difference between PCA and Factor Analysis. In the top diagram the \"factor\" (e.g., career path) represents the three observed variables (e.g., doctor, lawyer, teacher) whereas in the bottom diagram the observed variables (e.g., pre-school teacher, middle school teacher, high school teacher) are reduced into the component of interest (e.g., teacher). | upload.wikimedia.org/wikipedia/commons/2/29/PCA///versus///Factor///Analysis.jpg]\n\n## K \n\n-means clustering\nIt has been asserted that the relaxed solution of [[k-means clustering|\nk\n-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.\nHowever, that PCA is a useful relaxation of\nk\n-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.\n\n## Non-negative matrix factorization\n[Non-negative matrix factorization](https://bliptext.com/articles/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.\nThe PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis.\nIn 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.\nFor NMF, its components are ranked based only on the empirical FRV curves.\nThe residual fractional eigenvalue plots, that is, as a function of component number k given a total of n components, for PCA have a flat plateau, where no data is captured to remove the quasi-static noise, then the curves drop quickly as an indication of over-fitting (random noise).\nThe FRV curves for NMF is decreasing continuously when the NMF components are constructed [sequentially](https://bliptext.com/articles/non-negative-matrix-factorization), indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA, indicating the less over-fitting property of NMF.\n\n## Iconography of correlations\nIt is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative.\nThis leads the PCA user to a delicate elimination of several variables.\nIf observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements.\nIn addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane.\nThe [iconography of correlations](https://bliptext.com/articles/iconography-of-correlations), on the contrary, which is not a projection on a system of axes, does not have these drawbacks.\nWe can therefore keep all the variables.\nThe 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).\nA strong correlation is not \"remarkable\" if it is not direct, but caused by the effect of a third variable.\nConversely, weak correlations can be \"remarkable\".\nFor example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet \"remarkable\".\n# [Iconography of correlations – Geochemistry of marine aerosols | upload.wikimedia.org/wikipedia/commons/b/b3/AirMerIconographyCorrelation.jpg]\n\n# Generalizations\n\n## Sparse PCA\nA particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables.\n[Sparse PCA](https://bliptext.com/articles/sparse-pca) overcomes this disadvantage by finding linear combinations that contain just a few input variables.\nIt extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables.\nSeveral approaches have been proposed, including\nThe methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.\n\n## Nonlinear PCA\nMost of the modern methods for [nonlinear dimensionality reduction](https://bliptext.com/articles/nonlinear-dimensionality-reduction) find their theoretical and algorithmic roots in PCA or K-means.\nPearson's original idea was to take a straight line (or plane) which will be \"the best fit\" to a set of data points.\n[Trevor Hastie](https://bliptext.com/articles/trevor-hastie) expanded on this concept by proposing **Principal [curves](https://bliptext.com/articles/curve)** as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data [approximation](https://bliptext.com/articles/approximation) followed by [projecting](https://bliptext.com/articles/projection-mathematics) the points onto it.\nSee also the [elastic map](https://bliptext.com/articles/elastic-map) algorithm and [principal geodesic analysis](https://bliptext.com/articles/principal-geodesic-analysis).\nAnother popular generalization is [kernel PCA](https://bliptext.com/articles/kernel-pca), which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.\nIn [multilinear subspace learning](https://bliptext.com/articles/multilinear-subspace-learning), PCA is generalized to [multilinear PCA](https://bliptext.com/articles/multilinear-principal-component-analysis) (MPCA) that extracts features directly from tensor representations.\nMPCA is solved by performing PCA in each mode of the tensor iteratively.\nMPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.\nN-way principal component analysis may be performed with models such as [Tucker decomposition](https://bliptext.com/articles/tucker-decomposition), [PARAFAC](https://bliptext.com/articles/parafac), multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.\n# [Linear PCA versus nonlinear Principal Manifolds for visualization of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The \"basal\" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition. Software is available for free non-commercial use. | upload.wikimedia.org/wikipedia/commons/8/84/Elmap///breastcancer///wiki.png]\n\n## Robust PCA\nWhile PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to [outliers](https://bliptext.com/articles/outlier) in the data that produce large errors, something that the method tries to avoid in the first place.\nIt is therefore common practice to remove outliers before computing PCA.\nHowever, in some contexts, outliers can be difficult to identify.\nFor example, in [data mining](https://bliptext.com/articles/data-mining) algorithms like [correlation clustering](https://bliptext.com/articles/correlation-clustering), the assignment of points to clusters and outliers is not known beforehand.\nA recently proposed generalization of PCA based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.\nOutlier-resistant variants of PCA have also been proposed, based on L1-norm formulations ([L1-PCA](https://bliptext.com/articles/l1-norm-principal-component-analysis)).\n[Robust principal component analysis](https://bliptext.com/articles/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.\n\n# Similar techniques\n\n## Independent component analysis\n[Independent component analysis](https://bliptext.com/articles/independent-component-analysis) (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.\n\n## Network component analysis\nGiven a matrix E, it tries to decompose it into two matrices such that E=AP.\nA key difference from techniques such as PCA and ICA is that some of the entries of A are constrained to be 0.\nHere P is termed the regulatory layer.\nWhile in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :\nthen the decomposition is unique up to multiplication by a scalar.\n\n## Discriminant analysis of principal components\nDiscriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals.\nGenetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former.\nLinear discriminants are linear combinations of alleles which best separate the clusters.\nAlleles that most contribute to this discrimination are therefore those that are the most markedly different across groups.\nThe contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups\nIn DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA).\nA DAPC can be realized on R using the package Adegenet.\n(more info: [adegenet on the web](https://bliptext.com/articles/undefined))\n\n## Directional component analysis\n[Directional component analysis](https://bliptext.com/articles/directional-component-analysis) (DCA) is a method used in the atmospheric sciences for analysing multivariate datasets.\nLike PCA, it allows for dimension reduction, improved visualization and improved interpretability of large data-sets.\nAlso like PCA, it is based on a covariance matrix derived from the input dataset.\nThe difference between PCA and DCA is that DCA additionally requires the input of a vector direction, referred to as the impact.\nWhereas PCA maximises explained variance, DCA maximises probability density given impact.\nThe motivation for DCA is to find components of a multivariate dataset that are both likely (measured using probability density) and important (measured using the impact).\nDCA has been used to find the most likely and most serious heat-wave patterns in weather prediction ensembles\n, and the most likely and most impactful changes in rainfall due to climate change\n.\n\n# Software/source code","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:46:47.623Z","updated_at":"2025-01-25T17:46:47.623Z","search_vector":"'analysi':3 'compon':2 'princip':1","search_content":" \n \n\n# \n**Principal component analysis** (**PCA**) is a [linear](https://bliptext.com/articles/linear-map) [dimensionality reduction](https://bliptext.com/articles/dimensionality-reduction) technique with applications in [exploratory data analysis](https://bliptext.com/articles/exploratory-data-analysis), visualization and [data preprocessing](https://bliptext.com/articles/data-preprocessing).\nThe data is linearly transformed onto a new [coordinate system](https://bliptext.com/articles/coordinate-system) such that the directions (principal components) capturing the largest variation in the data can be easily identified.\nThe **principal components** of a collection of points in a [real coordinate space](https://bliptext.com/articles/real-coordinate-space) are a sequence of p [unit vectors](https://bliptext.com/articles/unit-vector), where the i-th vector is the direction of a line that best fits the data while being [orthogonal](https://bliptext.com/articles/orthogonal) to the first i-1 vectors.\nHere, a best-fitting line is defined as one that minimizes the average squared [perpendicular](https://bliptext.com/articles/perpendicular-distance) [distance from the points to the line](https://bliptext.com/articles/distance-from-a-point-to-a-line).\nThese directions (i.e., principal components) constitute an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) in which different individual dimensions of the data are [linearly uncorrelated](https://bliptext.com/articles/linear-correlation).\nMany 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.\nPrincipal component analysis has applications in many fields such as [population genetics](https://bliptext.com/articles/population-genetics), [microbiome](https://bliptext.com/articles/microbiome) studies, and [atmospheric science](https://bliptext.com/articles/atmospheric-science).\n# [PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean. | upload.wikimedia.org/wikipedia/commons/f/f5/GaussianScatterPCA.svg]\n\n# Overview\nWhen performing PCA, the first principal component of a set of p variables is the derived variable formed as a linear combination of the original variables that explains the most variance.\nThe 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.\nPCA 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.\nThe first principal component can equivalently be defined as a direction that maximizes the variance of the projected data.\nThe 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.\nFor either objective, it can be shown that the principal components are [eigenvectors](https://bliptext.com/articles/eigenvectors) of the data's [covariance matrix](https://bliptext.com/articles/covariance-matrix).\nThus, the principal components are often computed by eigendecomposition of the data covariance matrix or [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) of the data matrix.\nPCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to [factor analysis](https://bliptext.com/articles/factor-analysis).\nFactor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.\nPCA is also related to [canonical correlation analysis (CCA)](https://bliptext.com/articles/canonical-correlation).\nCCA defines coordinate systems that optimally describe the [cross-covariance](https://bliptext.com/articles/cross-covariance) between two datasets while PCA defines a new [orthogonal coordinate system](https://bliptext.com/articles/orthogonal-coordinate-system) that optimally describes variance in a single dataset.\n[Robust](https://bliptext.com/articles/robust-statistics) and [L1-norm](https://bliptext.com/articles/lp-space)-based variants of standard PCA have also been proposed.\n\n# History\nPCA was invented in 1901 by [Karl Pearson](https://bliptext.com/articles/karl-pearson), as an analogue of the [principal axis theorem](https://bliptext.com/articles/principal-axis-theorem) in mechanics; it was later independently developed and named by [Harold Hotelling](https://bliptext.com/articles/harold-hotelling) in the 1930s.\nDepending on the field of application, it is also named the discrete [Karhunen–Loève](https://bliptext.com/articles/karhunen-lo-ve-theorem) transform (KLT) in [signal processing](https://bliptext.com/articles/signal-processing), the [Hotelling](https://bliptext.com/articles/harold-hotelling) transform in multivariate quality control, [proper orthogonal decomposition](https://bliptext.com/articles/proper-orthogonal-decomposition) (POD) in mechanical engineering, [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) (SVD) of ******X****** (invented in the last quarter of the 19th century ), [eigenvalue decomposition](https://bliptext.com/articles/eigendecomposition) (EVD) of ******X******T******X****** in linear algebra, [factor analysis](https://bliptext.com/articles/factor-analysis) (for a discussion of the differences between PCA and [factor analysis](https://bliptext.com/articles/factor-analysis) see Ch. 7 of Jolliffe's Principal Component Analysis), [Eckart–Young theorem](https://bliptext.com/articles/eckart-young-theorem) (Harman, 1960), or [empirical orthogonal functions](https://bliptext.com/articles/empirical-orthogonal-functions) (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), [spectral decomposition](https://bliptext.com/articles/spectral-theorem) in noise and vibration, and [empirical modal analysis](https://bliptext.com/articles/mode-shape) in structural dynamics.\n\n# Intuition\nPCA can be thought of as fitting a p-dimensional [ellipsoid](https://bliptext.com/articles/ellipsoid) to the data, where each axis of the [ellipsoid](https://bliptext.com/articles/ellipsoid) represents a principal component.\nIf some axis of the ellipsoid is small, then the variance along that axis is also small.\nTo 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.\nThese transformed values are used instead of the original observed values for each of the variables.\nThen, we compute the [covariance matrix](https://bliptext.com/articles/covariance-matrix) of the data and calculate the eigenvalues and corresponding eigenvectors of this [covariance matrix](https://bliptext.com/articles/covariance-matrix).\nThen we must [normalize](https://bliptext.com/articles/normalization-statistics) each of the orthogonal eigenvectors to turn them into unit vectors.\nOnce this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data.\nThis choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis.\nThe 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.\n[Biplots](https://bliptext.com/articles/biplot) and [scree plots](https://bliptext.com/articles/scree-plot) (degree of [explained variance](https://bliptext.com/articles/explained-variance)) are used to interpret findings of the PCA.\n# [A scree plot that is meant to help interpret the PCA and decide how many components to retain. The start of the bend in the line (point of inflexion or \"knee\") should indicate how many components are retained, hence in this example, three factors should be retained. | upload.wikimedia.org/wikipedia/commons/e/e8/SCREE///plot.jpg]\n\n# Details\nPCA is defined as an [orthogonal](https://bliptext.com/articles/orthogonal-transformation) [linear transformation](https://bliptext.com/articles/linear-transformation) on a real [inner product space](https://bliptext.com/articles/inner-product-space) that transforms the data to a new [coordinate system](https://bliptext.com/articles/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.\nConsider an n \\times p data [matrix](https://bliptext.com/articles/matrix-mathematics), **X**, with column-wise zero [empirical mean](https://bliptext.com/articles/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).\nMathematically, the transformation is defined by a set of size l 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\nin such a **w**ay **t**ha**t** **t**he individual variables of **t** considered over **t**he da**t**a se**t** successively inheri**t** **t**he maximum possible variance from **X**, **w**i**t**h each coefficien**t** vec**t**or **w** cons**t**rained **t**o be a uni**t** vec**t**or (**w**here l is usually selec**t**ed **t**o be s**t**ric**t**ly less **t**han p **t**o reduce dimensionali**t**y).\nThe above may equivalently be written in matrix form as where\n,, and\n.\n\n## First component\nIn order to maximize variance, the first **w**eight vector **w**(1) thus has to satisfy\nEquivalently, writing this in matrix form gives\nSince **w**(1) has been defined to be a unit vector, it equivalently also satisfies\nThe quantity to be maximised can be recognised as a [Rayleigh quotient](https://bliptext.com/articles/rayleigh-quotient).\nA standard result for a [positive semidefinite matrix](https://bliptext.com/articles/positive-semidefinite-matrix) such as ****X****T****X**** is that the quotient's maximum possible value is the largest [eigenvalue](https://bliptext.com/articles/eigenvalue) of the matrix, **w**hich occurs **w**hen **w** is the corresponding [eigenvector](https://bliptext.com/articles/eigenvector).\nWith ********w********(1) found, the first principal component of a data vector ******x******(i) can then be given as a score t1(i) = ******x******(i) ⋅ ********w********(1) in the transformed co-ordinates, or as the corresponding vector in the original variables, {******x******(i) ⋅ ********w********(1)} ********w********(1).\n\n## Further components\nThe k-th component can be found by subtracting the first k − 1 principal components from **X**:\nand then finding the weight vector which extracts the maximum variance from this new data matrix\nIt 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.\nThus the weight vectors are eigenvectors of ****X****T****X****.\nThe k-th principal component of a data vector ******x******(i) can therefore be given as a score tk(i) = ******x******(i) ⋅ ********w********(k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {******x******(i) ⋅ ********w********(k)} ********w********(k), ********w********here ********w********(k) is the kth eigenvector of ****X****T****X****.\nThe 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******.\nThe transpose of **W** is sometimes called the [whitening or sphering transformation](https://bliptext.com/articles/whitening-transformation).\nColumns 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.\n\n## Covariances\n****X****T****X**** itself can be recognized as proportional to the empirical sample [covariance matrix](https://bliptext.com/articles/covariance-matrix) of the dataset ****X****T.\nThe sample covariance Q between two of the different principal components over the dataset is given by:\n**w**here the eigenvalue property of **w**(k) has been used to move from line 2 to line 3.\nHo****w****ever 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).\nThe product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.\nAnother way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.\nIn matrix form, the empirical covariance matrix for the original variables can be written\nThe empirical covariance matrix between the principal components becomes\nwhere **Λ** is the diagonal matrix of eigenvalues λ(k) of ****X****T****X****.\nλ(k) is equal to the sum of the squares over the dataset associated **w**ith each component k, that is, λ(k) = Σi tk2(i) = Σi (**x**(i) ⋅ **w**(k))2.\n\n## Dimensionality reduction\n**T**he 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.\nHowever, not all the principal components need to be kept.\nKeeping only the first L principal components, produced by using only the first L eigenvectors, gives the truncated transformation\nwhere the matrix **T**L now has n rows but only L columns.\nIn other words, PCA learns a linear transformation where the columns of\np × L\nmatrix W_L form an orthogonal basis for the L features (the components of representation t) that are decorrelated.\nBy 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.\nSuch [dimensionality reduction](https://bliptext.com/articles/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.\nFor 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](https://bliptext.com/articles/cluster-analysis) 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](https://bliptext.com/articles/cluster-analysis) 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.\nSimilarly, in [regression analysis](https://bliptext.com/articles/regression-analysis), the larger the number of [explanatory variables](https://bliptext.com/articles/explanatory-variable) allowed, the greater is the chance of [overfitting](https://bliptext.com/articles/overfitting) the model, producing conclusions that fail to generalise to other datasets.\nOne 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](https://bliptext.com/articles/principal-component-regression).\nDimensionality reduction may also be appropriate when the variables in a dataset are noisy.\nIf 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).\nHowever, 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](https://bliptext.com/articles/signal-to-noise-ratio).\nPCA 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.\nIf the dataset is not too large, the significance of the principal components can be tested using [parametric bootstrap](https://bliptext.com/articles/bootstrapping-statistics), as an aid in determining how many principal components to retain.\n# [A principal components analysis scatterplot of Y-STR haplotypes calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.\nPCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent. | upload.wikimedia.org/wikipedia/commons/6/69/PCA///of///Haplogroup///J///using///37///STRs.png]\n\n## Singular value decomposition\nThe principal components transformation can also be associated with another matrix factorization, the [singular value decomposition](https://bliptext.com/articles/singular-value-decomposition) (SVD) of **X**,\nHere **Σ** is an n-by-p [rectangular diagonal matrix](https://bliptext.com/articles/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 matrix whose columns are orthogonal unit vectors of length p and called the right singular vectors of ******X******.\nIn terms of this factorization, the matrix ****X****T****X**** can be written\nwhere**** ******** ******** ****is**** ****the**** ****square**** ****diagonal**** ****matrix**** ****with**** ****the**** ****singular**** ****values**** ****of**** ****X**** ****and**** ****the**** ****excess**** ****zeros**** ****chopped**** ****off**** ****that**** ****satisfies**** ****.\nComparison** **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**************.\nUsing the singular value decomposition the score matrix **T** can be written\nso each column of **T** is given by one of the left singular vectors of **X** multiplied by the corresponding singular value.\n**T**his form is also the [polar decomposition](https://bliptext.com/articles/polar-decomposition) of **T**.\nEfficient 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, unless only a handful of components are required.\nAs with the eigen-decomposition, a truncated\nn × L\nscore matrix **T**L can be obtained by considering only the first L largest singular values and their singular vectors:\n**T**he 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](https://bliptext.com/articles/rank-linear-algebra) L to the original matrix, in the sense of the difference between the two having the smallest possible [Frobenius norm](https://bliptext.com/articles/frobenius-norm), a result known as the [Eckart–Young theorem](https://bliptext.com/articles/low-rank-approximation) [1936].\n\n# Further considerations\nThe singular values (in **Σ**) are the square roots of the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the matrix ****X****T****X****.\nEach 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.\nThe sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean.\nPCA essentially rotates the set of points around their mean in order to align with the principal components.\nThis moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions.\nThe values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see [below](https://bliptext.com/articles/principle-component-analysis)).\nPCA is often used in this manner for [dimensionality reduction](https://bliptext.com/articles/dimensionality-reduction).\nPCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest \"variance\" (as defined above).\nThis advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the [discrete cosine transform](https://bliptext.com/articles/discrete-cosine-transform), and in particular to the DCT-II which is simply known as the \"DCT\".\n[Nonlinear dimensionality reduction](https://bliptext.com/articles/nonlinear-dimensionality-reduction) techniques tend to be more computationally demanding than PCA.\nPCA is sensitive to the scaling of the variables.\nIf we have just two variables and they have the same [sample variance](https://bliptext.com/articles/sample-variance) 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.\nBut 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.\nThis means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis.\n(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.\nOne 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.\nHowever, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.\nMean 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.\nIf mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data.\nA mean of zero is needed for finding a basis that minimizes the [mean square error](https://bliptext.com/articles/minimum-mean-square-error) of the approximation of the data.\nMean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations.\nCorrelations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation).\nAlso see the article by Kromrey & Foster-Johnson (1998) on \"Mean-centering in Moderated Regression: Much Ado About Nothing\".\nSince [covariances are correlations of normalized variables](https://bliptext.com/articles/covariance-matrix) (**Z**- or standard-scores) a PCA based on the correlation matrix of ****X**** is [equal](https://bliptext.com/articles/equality-mathematics) to a PCA based on the covariance matrix of **Z**, the standardized version of ****X****.\nPCA is a popular primary technique in [pattern recognition](https://bliptext.com/articles/pattern-recognition).\nIt is not, however, optimized for class separability.\nHowever, 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.\nThe [linear discriminant analysis](https://bliptext.com/articles/linear-discriminant-analysis) is an alternative which is optimized for class separability.\n\n# Table of symbols and abbreviations\n\n# Properties and limitations\n\n## Properties\nSome properties of PCA include:\nThe statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs.\nBecause these last PCs have variances as small as possible they are useful in their own right.\nThey can help to detect unsuspected near-constant linear relationships between the elements of x, and they may also be useful in [regression](https://bliptext.com/articles/regression-analysis), in selecting a subset of variables from x, and in outlier detection.\nΣ\n)\nBefore we look at its usage, we first look at [diagonal](https://bliptext.com/articles/diagonal) elements,\nThen, 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](https://bliptext.com/articles/covariance-matrix) into contributions from each PC.\nAlthough not strictly decreasing, the elements of will tend to become smaller as k increases, as is nonincreasing for increasing k, whereas the elements of \\alpha_k tend to stay about the same size because of the normalization constraints:.\n\n## Limitations\nAs noted above, the results of PCA depend on the scaling of the variables.\nThis can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.\nThe applicability of PCA as described above is limited by certain (tacit) assumptions made in its derivation.\nIn particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference).\nIn some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see [kernel PCA](https://bliptext.com/articles/kernel-principal-component-analysis)).\nAnother limitation is the mean-removal process before constructing the covariance matrix for PCA.\nIn 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.\nAs an alternative method, [non-negative matrix factorization](https://bliptext.com/articles/non-negative-matrix-factorization) focusing only on the non-negative elements in the matrices, which is well-suited for astrophysical observations.\nSee more at [Relation between PCA and Non-negative Matrix Factorization](https://bliptext.com/articles/).\nPCA is at a disadvantage if the data has not been standardized before applying the algorithm to it.\nPCA 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.\nThey are linear interpretations of the original variables.\nAlso, if PCA is not performed properly, there is a high likelihood of information loss.\nPCA relies on a linear model.\nIf 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.\nResearchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results.\n\"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\".\nThe researchers at Kansas State also found that PCA could be \"seriously biased if the autocorrelation structure of the data is not correctly handled\".\n\n## PCA and information theory\nDimensionality reduction results in a loss of information, in general.\nPCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.\nUnder the assumption that\nthat is, that the data vector \\mathbf{x} is the sum of the desired information-bearing signal \\mathbf{s} and a noise signal \\mathbf{n} one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.\nIn particular, Linsker showed that if \\mathbf{s} is Gaussian and \\mathbf{n} is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the [mutual information](https://bliptext.com/articles/mutual-information) between the desired information \\mathbf{s} and the dimensionality-reduced output.\nIf the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector \\mathbf{n} are [iid](https://bliptext.com/articles/iid)), but the information-bearing signal \\mathbf{s} is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as\nThe optimality of PCA is also preserved if the noise \\mathbf{n} is iid and at least more Gaussian (in terms of the [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence)) than the information-bearing signal \\mathbf{s}.\nIn general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise \\mathbf{n} becomes dependent.\n\n# Computation using the covariance method\nThe following is a detailed description of PCA using the covariance method as opposed to the correlation method.\nThe goal is to transform a given data set **X** of dimension p to an alternative data set **Y** of smaller dimension L.\nEquivalently, we are seeking to find the matrix ****Y****, where ****Y**** is the [Karhunen–Loève](https://bliptext.com/articles/karhunen-lo-ve-theorem) transform (KLT) of matrix **X**:\n \n \n**Organize the data set**\nSuppose 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 \n \n**Calculate the empirical mean**\n \n \n**Calculate the deviations from the mean**\nMean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.\nHence we proceed by centering the data as follows:\nn × 1\ncolumn vector of all 1s:\nIn some applications, each variable (column of **B**) may also be scaled to have a variance equal to 1 (see [Z-score](https://bliptext.com/articles/z-score)).\nThis step affects the calculated principal components, but makes them independent of the units used to measure the different variables.\n \n \n**Find the covariance matrix**\nn − 1\ninstead of n to calculate the covariance is [Bessel's correction](https://bliptext.com/articles/bessel-s-correction).\n \n \n**Find the eigenvectors and eigenvalues of the covariance matrix**\n \n \n**Rearrange the eigenvectors and eigenvalues**\n \n \n**Compute the cumulative energy content for each eigenvector**\n \n \n**Select a subset of the eigenvectors as basis vectors**\n \n \n**Project the data onto the new basis**\nThat is, the first column of \\mathbf{T} is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc. \n \n\n# Derivation using the covariance method\nLet **X** be a d-dimensional random vector expressed as column vector.\nWithout loss of generality, assume **X** has zero mean.\nWe want to find (\\ast) a\nd × d\n[orthonormal transformation matrix](https://bliptext.com/articles/orthonormal-basis) **P** so that **P**X has a diagonal covariance matrix (that is, **P**X is a random vector with all its distinct components pairwise uncorrelated).\nA quick computation assuming P were unitary yields:\nHence (\\ast) holds if and only if were diagonalisable by P.\nThis 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.\n\n# Covariance-free computation\nIn practical implementations, especially with [high dimensional data](https://bliptext.com/articles/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.\nThe covariance-free approach avoids the\nnp2\noperations of explicitly calculating and storing the covariance matrix\nXTX\n, instead utilizing one of [matrix-free methods](https://bliptext.com/articles/matrix-free-methods), for example, based on the function evaluating the product\nXT(X r)\nat the cost of\n2np\noperations.\n\n## Iterative computation\nOne way to compute the first principal component efficiently is shown in the following pseudo-code, for a data matrix\nX\nwith zero mean, without ever computing its covariance matrix.\nr\n= a ******r******andom vecto******r****** of length p ******r****** = ******r****** / no******r******m(******r******)\ndo c times:\ns = 0\n(a vector of length p) for each row x in X s = s + (x ⋅ r) x λ = rTs // λ is the eigenvalue\nλ ⋅ r − s r = s / norm(s)\nexit if error < tolerance\nreturn λ, r\nThis [power iteration](https://bliptext.com/articles/power-iteration) algorithm simply calculates the vector\nXT(X r)\n, normalizes, and places the result back in\nr\n. The eigenvalue is approximated by\nrT (XTX) r\n, which is the [Rayleigh quotient](https://bliptext.com/articles/rayleigh-quotient) on the unit vector\nr\nfor the covariance matrix\nXTX\n. If the largest singular value is well separated from the next largest one, the vector\nr\ngets close to the first principal component of\nX\nwithin the number of iterations c, which is small relative to p, at the total cost\n2cnp\n. The [power iteration](https://bliptext.com/articles/power-iteration) convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced [matrix-free methods](https://bliptext.com/articles/matrix-free-methods), such as the [Lanczos algorithm](https://bliptext.com/articles/lanczos-algorithm) or the Locally Optimal Block Preconditioned Conjugate Gradient ([LOBPCG](https://bliptext.com/articles/lobpcg)) method.\nSubsequent principal components can be computed one-by-one via deflation or simultaneously as a block.\nIn 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.\nThe latter approach in the block power method replaces single-vectors\nr\nand\ns\nwith block-vectors, matrices\nR\nand\nS\n. Every column of\nR\napproximates one of the leading principal components, while all columns are iterated simultaneously.\nThe main calculation is evaluation of the product\nXT(X R)\n. Implemented, for example, in [LOBPCG](https://bliptext.com/articles/lobpcg), efficient blocking eliminates the accumulation of the errors, allows using high-level [BLAS](https://bliptext.com/articles/blas) matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.\n\n## The NIPALS method\nNon-linear iterative [partial least squares](https://bliptext.com/articles/partial-least-squares) (NIPALS) is a variant the classical [power iteration](https://bliptext.com/articles/power-iteration) with matrix deflation by subtraction implemented for computing the first few components in a principal component or [partial least squares](https://bliptext.com/articles/partial-least-squares) analysis.\nFor very-high-dimensional datasets, such as those generated in the *omics sciences (for example, [genomics](https://bliptext.com/articles/genomics), [metabolomics](https://bliptext.com/articles/metabolomics)) it is usually only necessary to compute the first few PCs.\nThe non-linea**r** i**t**e**r**a**t**ive pa**r****t**ial leas**t** squa**r**es (NIPALS) algo**r**i**t**hm upda**t**es i**t**e**r**a**t**ive app**r**oxima**t**ions **t**o **t**he leading sco**r**es and loadings **t**1 and **r**1T by **t**he powe**r** i**t**e**r**a**t**ion mul**t**iplying on eve**r**y i**t**e**r**a**t**ion by **X** on **t**he lef**t** and on **t**he **r**igh**t**, **t**ha**t** is, calcula**t**ion of **t**he cova**r**iance ma**t****r**ix is avoided, jus**t** as in **t**he ma**t****r**ix-f**r**ee implemen**t**a**t**ion of **t**he powe**r** i**t**e**r**a**t**ions **t**o\nXTX\n, based on the function evaluating the product\nXT(X r) = ((X r)TX)T\n.\nThe ma**t****r**ix defla**t**ion by sub**t****r**ac**t**ion is pe**r**fo**r**med by sub**t****r**ac**t**ing **t**he ou**t**e**r** p**r**oduc**t**, **t**1**r**1T f**r**om **X** leaving **t**he defla**t**ed **r**esidual ma**t****r**ix used **t**o calcula**t**e **t**he subsequen**t** leading PCs.\nFor 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](https://bliptext.com/articles/round-off-errors) accumulated in each iteration and matrix deflation by subtraction.\nA [Gram–Schmidt](https://bliptext.com/articles/gram-schmidt) re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.\nNIPALS reliance on single-vector multiplications cannot take advantage of high-level [BLAS](https://bliptext.com/articles/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](https://bliptext.com/articles/lobpcg)) method.\n\n## Online/sequential estimation\nIn 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.\nThis can be done efficiently, but requires different algorithms.\n\n# Qualitative variables\nIn PCA, it is common that we want to introduce qualitative variables as supplementary elements.\nFor example, many quantitative variables have been measured on plants.\nFor these plants, some qualitative variables are available as, for example, the species to which the plant belongs.\nThese data were subjected to PCA for quantitative variables.\nWhen analyzing the results, it is natural to connect the principal components to the qualitative variable species.\nFor this, the following results are produced.\nThese results are what is called introducing a qualitative variable as supplementary element.\nThis procedure is detailed in and Husson, Lê, & Pagès (2009) and Pagès (2013).\nFew software offer this option in an \"automatic\" way.\nThis is the case of [SPAD](https://bliptext.com/articles/undefined) that historically, following the work of [Ludovic Lebart](https://bliptext.com/articles/ludovic-lebart), was the first to propose this option, and the R package [FactoMineR](https://bliptext.com/articles/undefined).\n\n# Applications\n\n## Intelligence\nThe earliest application of factor analysis was in locating and measuring components of human intelligence.\nIt 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](https://bliptext.com/articles/intelligence-quotient) (IQ).\nThe pioneering statistical psychologist [Spearman](https://bliptext.com/articles/charles-spearman) actually developed factor analysis in 1904 for his [two-factor theory](https://bliptext.com/articles/two-factor-theory-of-intelligence) of intelligence, adding a formal technique to the science of [psychometrics](https://bliptext.com/articles/psychometrics).\nIn 1924 [Thurstone](https://bliptext.com/articles/louis-leon-thurstone) looked for 56 factors of intelligence, developing the notion of Mental Age.\nStandard IQ tests today are based on this early work.\n\n## Residential differentiation\nIn 1949, Shevky and Williams introduced the theory of **factorial ecology**, which dominated studies of residential differentiation from the 1950s to the 1970s.\nNeighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis.\nThese 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.\nAn 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.\nOne of the problems with factor analysis has always been finding convincing names for the various artificial factors.\nIn 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation.\nThe principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities.\nThe first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based.\nThe 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.\nAbout 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.\nThese SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.\n\n## Development indexes\nPCA can be used as a formal method for the development of indexes.\nAs an alternative [confirmatory composite analysis](https://bliptext.com/articles/confirmatory-composite-analysis) has been proposed to develop and assess indexes.\nThe [City Development Index](https://bliptext.com/articles/city-development-index) was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities.\nThe first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for.\nThe index ultimately used about 15 indicators but was a good predictor of many more variables.\nIts comparative value agreed very well with a subjective assessment of the condition of each city.\nThe 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.\nThe country-level [Human Development Index](https://bliptext.com/articles/human-development-index) (HDI) from [UNDP](https://bliptext.com/articles/united-nations-development-programme), 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.\n\n## Population genetics\nIn 1978 [Cavalli-Sforza](https://bliptext.com/articles/luigi-luca-cavalli-sforza) and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions.\nThe components showed distinctive patterns, including gradients and sinusoidal waves.\nThey interpreted these patterns as resulting from specific ancient migration events.\nSince then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism.\nGenetics 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.\nPCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers.\nThe lack of any measures of standard error in PCA are also an impediment to more consistent usage.\nIn August 2022, the molecular biologist [Eran Elhaik](https://bliptext.com/articles/eran-elhaik) published a theoretical paper in [Scientific Reports](https://bliptext.com/articles/scientific-reports) analyzing 12 PCA applications.\nHe 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](https://bliptext.com/articles/circular-reasoning).\n\n## Market research and indexes of attitude\nMarket research has been an extensive user of PCA.\nIt 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.\nPCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed.\nIn any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes.\nFor 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'.\nAnother 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.\nThe first principal component represented a general attitude toward property and home ownership.\nThe index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice.\nThe strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.\n\n## Quantitative finance\nIn [quantitative finance](https://bliptext.com/articles/quantitative-finance), PCA is used\nin [financial risk management](https://bliptext.com/articles/financial-risk-management), and has been applied to [other problems](https://bliptext.com/articles/financial-modeling) such as [portfolio optimization](https://bliptext.com/articles/portfolio-optimization).\nPCA is commonly used in problems involving [fixed income](https://bliptext.com/articles/fixed-income) securities and [portfolios](https://bliptext.com/articles/bond-fund), and [interest rate derivatives](https://bliptext.com/articles/interest-rate-derivative).\nValuations here depend on the entire [yield curve](https://bliptext.com/articles/yield-curve), comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements,\nthereby facilitating the modelling.\nOne common risk management application is to [calculating value at risk](https://bliptext.com/articles/value-at-risk), VaR, applying PCA to the [Monte Carlo simulation](https://bliptext.com/articles/monte-carlo-methods-in-finance).\nHere, for each simulation-sample, the components are stressed, and rates, and [in turn option values](https://bliptext.com/articles/monte-carlo-methods-for-option-pricing), are then reconstructed;\nwith VaR calculated, finally, over the entire run.\nPCA is also used in [hedging](https://bliptext.com/articles/hedge-finance) exposure to [interest rate risk](https://bliptext.com/articles/interest-rate-risk), given [partial durations](https://bliptext.com/articles/key-rate-duration) and other sensitivities.\nUnder both, the first three, typically, principal components of the system are of interest ([representing](https://bliptext.com/articles/fixed-income-attribution) \"shift\", \"twist\", and \"curvature\").\nThese principal components are derived from an eigen-decomposition of the [covariance matrix](https://bliptext.com/articles/covariance-matrix) of [yield](https://bliptext.com/articles/yield-curve) at predefined maturities;\nand where the [variance](https://bliptext.com/articles/variance) of each component is its [eigenvalue](https://bliptext.com/articles/eigenvalue) (and as the components are [orthogonal](https://bliptext.com/articles/orthogonal), no correlation need be incorporated in subsequent modelling).\nFor [equity](https://bliptext.com/articles/equity-finance), an optimal portfolio is one where the [expected return](https://bliptext.com/articles/expected-return) is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see [Markowitz model](https://bliptext.com/articles/markowitz-model) for discussion.\nThus, one approach is to reduce portfolio risk, where [allocation strategies](https://bliptext.com/articles/asset-allocation) are applied to the \"principal portfolios\" instead of the underlying [stocks](https://bliptext.com/articles/capital-stock).\nA second approach is to enhance portfolio return, using the principal components to select companies' stocks with upside potential.\nPCA has also been used to understand relationships between international [equity markets](https://bliptext.com/articles/equity-market), and within markets between groups of companies in industries or [sectors](https://bliptext.com/articles/stock-market-index).\nPCA may also be applied to [stress testing](https://bliptext.com/articles/stress-test-financial), essentially an analysis of a bank's ability to endure [a hypothetical adverse economic scenario](https://bliptext.com/articles/list-of-bank-stress-tests).\nIts utility is in \"distilling the information contained in [several] [macroeconomic variables](https://bliptext.com/articles/macroeconomic-model) into a more manageable data set, which can then [be used] for analysis.\"\nHere, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's [eigenvector](https://bliptext.com/articles/eigenvector) – and it is then observed how a \"shock\" to each of the factors affects the implied assets of each of the banks.\n\n## Neuroscience\nA variant of principal components analysis is used in [neuroscience](https://bliptext.com/articles/neuroscience) to identify the specific properties of a stimulus that increases a [neuron](https://bliptext.com/articles/neuron)'s probability of generating an [action potential](https://bliptext.com/articles/action-potential).\nThis technique is known as [spike-triggered covariance analysis](https://bliptext.com/articles/spike-triggered-covariance).\nIn a typical application an experimenter presents a [white noise](https://bliptext.com/articles/white-noise) process as a stimulus (usually either as a sensory input to a test subject, or as a [current](https://bliptext.com/articles/electric-current) injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result.\nPresumably, certain features of the stimulus make the neuron more likely to spike.\nIn order to extract these features, the experimenter calculates the [covariance matrix](https://bliptext.com/articles/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.\nThe [eigenvectors](https://bliptext.com/articles/eigenvectors-and-eigenvalues) 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](https://bliptext.com/articles/vector-space) of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble.\nSpecifically, 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.\nSince 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.\nIn neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential.\n[Spike sorting](https://bliptext.com/articles/spike-sorting) is an important procedure because [extracellular](https://bliptext.com/articles/electrophysiology) recording techniques often pick up signals from more than one neuron.\nIn spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs [clustering analysis](https://bliptext.com/articles/cluster-analysis) to associate specific action potentials with individual neurons.\nPCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles.\nIt has been used in determining collective variables, that is, [order parameters](https://bliptext.com/articles/order-parameters), during [phase transitions](https://bliptext.com/articles/phase-transitions) in the brain.\n\n# Relation with other methods\n\n## Correspondence analysis\n[Correspondence analysis](https://bliptext.com/articles/correspondence-analysis) (CA)\nwas developed by [Jean-Paul Benzécri](https://bliptext.com/articles/jean-paul-benz-cri)\nand is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently.\nIt is traditionally applied to [contingency tables](https://bliptext.com/articles/contingency-tables).\nCA decomposes the [chi-squared statistic](https://bliptext.com/articles/chi-squared-statistic) associated to this table into orthogonal factors.\nBecause CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not.\nSeveral variants of CA are available including [detrended correspondence analysis](https://bliptext.com/articles/detrended-correspondence-analysis) and [canonical correspondence analysis](https://bliptext.com/articles/canonical-correspondence-analysis).\nOne special extension is [multiple correspondence analysis](https://bliptext.com/articles/multiple-correspondence-analysis), which may be seen as the counterpart of principal component analysis for categorical data.\n\n## Factor analysis\nPrincipal component analysis creates variables that are linear combinations of the original variables.\nThe new variables have the property that the variables are all orthogonal.\nThe PCA transformation can be helpful as a pre-processing step before clustering.\nPCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable.\nPCA 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.\n[Factor analysis](https://bliptext.com/articles/factor-analysis) is similar to principal component analysis, in that factor analysis also involves linear combinations of variables.\nDifferent 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\".\nIn 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.\nHowever, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.\nResults 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.\nFactor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or [causal modeling](https://bliptext.com/articles/causal-modeling).\nIf the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.\n# [The above picture is an example of the difference between PCA and Factor Analysis. In the top diagram the \"factor\" (e.g., career path) represents the three observed variables (e.g., doctor, lawyer, teacher) whereas in the bottom diagram the observed variables (e.g., pre-school teacher, middle school teacher, high school teacher) are reduced into the component of interest (e.g., teacher). | upload.wikimedia.org/wikipedia/commons/2/29/PCA///versus///Factor///Analysis.jpg]\n\n## K \n\n-means clustering\nIt has been asserted that the relaxed solution of [[k-means clustering|\nk\n-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.\nHowever, that PCA is a useful relaxation of\nk\n-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.\n\n## Non-negative matrix factorization\n[Non-negative matrix factorization](https://bliptext.com/articles/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.\nThe PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis.\nIn 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.\nFor NMF, its components are ranked based only on the empirical FRV curves.\nThe residual fractional eigenvalue plots, that is, as a function of component number k given a total of n components, for PCA have a flat plateau, where no data is captured to remove the quasi-static noise, then the curves drop quickly as an indication of over-fitting (random noise).\nThe FRV curves for NMF is decreasing continuously when the NMF components are constructed [sequentially](https://bliptext.com/articles/non-negative-matrix-factorization), indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA, indicating the less over-fitting property of NMF.\n\n## Iconography of correlations\nIt is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative.\nThis leads the PCA user to a delicate elimination of several variables.\nIf observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements.\nIn addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane.\nThe [iconography of correlations](https://bliptext.com/articles/iconography-of-correlations), on the contrary, which is not a projection on a system of axes, does not have these drawbacks.\nWe can therefore keep all the variables.\nThe 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).\nA strong correlation is not \"remarkable\" if it is not direct, but caused by the effect of a third variable.\nConversely, weak correlations can be \"remarkable\".\nFor example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet \"remarkable\".\n# [Iconography of correlations – Geochemistry of marine aerosols | upload.wikimedia.org/wikipedia/commons/b/b3/AirMerIconographyCorrelation.jpg]\n\n# Generalizations\n\n## Sparse PCA\nA particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables.\n[Sparse PCA](https://bliptext.com/articles/sparse-pca) overcomes this disadvantage by finding linear combinations that contain just a few input variables.\nIt extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables.\nSeveral approaches have been proposed, including\nThe methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.\n\n## Nonlinear PCA\nMost of the modern methods for [nonlinear dimensionality reduction](https://bliptext.com/articles/nonlinear-dimensionality-reduction) find their theoretical and algorithmic roots in PCA or K-means.\nPearson's original idea was to take a straight line (or plane) which will be \"the best fit\" to a set of data points.\n[Trevor Hastie](https://bliptext.com/articles/trevor-hastie) expanded on this concept by proposing **Principal [curves](https://bliptext.com/articles/curve)** as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data [approximation](https://bliptext.com/articles/approximation) followed by [projecting](https://bliptext.com/articles/projection-mathematics) the points onto it.\nSee also the [elastic map](https://bliptext.com/articles/elastic-map) algorithm and [principal geodesic analysis](https://bliptext.com/articles/principal-geodesic-analysis).\nAnother popular generalization is [kernel PCA](https://bliptext.com/articles/kernel-pca), which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.\nIn [multilinear subspace learning](https://bliptext.com/articles/multilinear-subspace-learning), PCA is generalized to [multilinear PCA](https://bliptext.com/articles/multilinear-principal-component-analysis) (MPCA) that extracts features directly from tensor representations.\nMPCA is solved by performing PCA in each mode of the tensor iteratively.\nMPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.\nN-way principal component analysis may be performed with models such as [Tucker decomposition](https://bliptext.com/articles/tucker-decomposition), [PARAFAC](https://bliptext.com/articles/parafac), multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.\n# [Linear PCA versus nonlinear Principal Manifolds for visualization of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The \"basal\" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition. Software is available for free non-commercial use. | upload.wikimedia.org/wikipedia/commons/8/84/Elmap///breastcancer///wiki.png]\n\n## Robust PCA\nWhile PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to [outliers](https://bliptext.com/articles/outlier) in the data that produce large errors, something that the method tries to avoid in the first place.\nIt is therefore common practice to remove outliers before computing PCA.\nHowever, in some contexts, outliers can be difficult to identify.\nFor example, in [data mining](https://bliptext.com/articles/data-mining) algorithms like [correlation clustering](https://bliptext.com/articles/correlation-clustering), the assignment of points to clusters and outliers is not known beforehand.\nA recently proposed generalization of PCA based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.\nOutlier-resistant variants of PCA have also been proposed, based on L1-norm formulations ([L1-PCA](https://bliptext.com/articles/l1-norm-principal-component-analysis)).\n[Robust principal component analysis](https://bliptext.com/articles/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.\n\n# Similar techniques\n\n## Independent component analysis\n[Independent component analysis](https://bliptext.com/articles/independent-component-analysis) (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.\n\n## Network component analysis\nGiven a matrix E, it tries to decompose it into two matrices such that E=AP.\nA key difference from techniques such as PCA and ICA is that some of the entries of A are constrained to be 0.\nHere P is termed the regulatory layer.\nWhile in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :\nthen the decomposition is unique up to multiplication by a scalar.\n\n## Discriminant analysis of principal components\nDiscriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals.\nGenetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former.\nLinear discriminants are linear combinations of alleles which best separate the clusters.\nAlleles that most contribute to this discrimination are therefore those that are the most markedly different across groups.\nThe contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups\nIn DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA).\nA DAPC can be realized on R using the package Adegenet.\n(more info: [adegenet on the web](https://bliptext.com/articles/undefined))\n\n## Directional component analysis\n[Directional component analysis](https://bliptext.com/articles/directional-component-analysis) (DCA) is a method used in the atmospheric sciences for analysing multivariate datasets.\nLike PCA, it allows for dimension reduction, improved visualization and improved interpretability of large data-sets.\nAlso like PCA, it is based on a covariance matrix derived from the input dataset.\nThe difference between PCA and DCA is that DCA additionally requires the input of a vector direction, referred to as the impact.\nWhereas PCA maximises explained variance, DCA maximises probability density given impact.\nThe motivation for DCA is to find components of a multivariate dataset that are both likely (measured using probability density) and important (measured using the impact).\nDCA has been used to find the most likely and most serious heat-wave patterns in weather prediction ensembles\n, and the most likely and most impactful changes in rainfall due to climate change\n.\n\n# Software/source code","revisionCount":0},"hasMore":false,"nextCursor":null}