First component

In order to maximize variance, the first weight vector w₍₁₎ thus has to satisfy

${\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}}$

Equivalently, writing this in matrix form gives

${\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}}$

Since w₍₁₎ has been defined to be a unit vector, it equivalently also satisfies

${\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}}$

The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as X^TX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector.

With w₍₁₎ found, the first principal component of a data vector x_(i) can then be given as a score t_1(i) = x_(i) ⋅ w₍₁₎ in the transformed co-ordinates, or as the corresponding vector in the original variables, {x_(i) ⋅ w₍₁₎} w₍₁₎.

Further components

The k-th component can be found by subtracting the first k − 1 principal components from X:

${\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\mathsf {T}}}$

and then finding the weight vector which extracts the maximum variance from this new data matrix

${\displaystyle \mathbf {w} _{(k)}=\mathop {\operatorname {arg\,max} } _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {\hat {X}} _{k}\mathbf {w} \right\|^{2}\right\}=\arg \max \left\{{\tfrac {\mathbf {w} ^{\mathsf {T}}\mathbf {\hat {X}} _{k}^{\mathsf {T}}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}$

It turns out that this gives the remaining eigenvectors of X^TX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of X^TX.

The k-th principal component of a data vector x_(i) can therefore be given as a score t_k(i) = x_(i) ⋅ w_(k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x_(i) ⋅ w_(k)} w_(k), where w_(k) is the kth eigenvector of X^TX.

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

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

where W is a p-by-p matrix of weights whose columns are the eigenvectors of X^TX. The transpose of W is sometimes called the whitening or sphering transformation. Columns of W multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called loadings in PCA or in Factor analysis.

Covariances

X^TX itself can be recognized as proportional to the empirical sample covariance matrix of the dataset X^T.^{[12]: 30–31}

The sample covariance Q between two of the different principal components over the dataset is given by:

${\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{\mathsf {T}}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {w} _{(k)}\end{aligned}}}$

where the eigenvalue property of w_(k) has been used to move from line 2 to line 3. However eigenvectors w_(j) and w_(k) corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.

Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.

In matrix form, the empirical covariance matrix for the original variables can be written

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

The empirical covariance matrix between the principal components becomes

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

where Λ is the diagonal matrix of eigenvalues λ_(k) of X^TX. λ_(k) is equal to the sum of the squares over the dataset associated with each component k, that is, λ_(k) = Σ_i t_k²_(i) = Σ_i (x_(i) ⋅ w_(k))².

Dimensionality reduction

The transformation T = X W maps a data vector x_(i) from an original space of p variables to a new space of p variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first L principal components, produced by using only the first L eigenvectors, gives the truncated transformation

${\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}}$

where the matrix T_L now has n rows but only L columns. In other words, PCA learns a linear transformation ${\displaystyle t=W_{L}^{\mathsf {T}}x,x\in \mathbb {R} ^{p},t\in \mathbb {R} ^{L},}$ where the columns of p × L matrix ${\displaystyle W_{L}}$ form an orthogonal basis for the L features (the components of representation t) that are decorrelated.^[13] By construction, of all the transformed data matrices with only L columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error ${\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}}$ or ${\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}}$.

A principal components analysis scatterplot of Y-STR haplotypes calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.
PCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent.

Such dimensionality reduction can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting L = 2 and keeping only the first two principal components finds the two-dimensional plane through the high-dimensional dataset in which the data is most spread out, so if the data contains clusters these too may be most spread out, and therefore most visible to be plotted out in a two-dimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more likely to substantially overlay each other, making them indistinguishable.

Similarly, in regression analysis, the larger the number of explanatory variables allowed, the greater is the chance of overfitting the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called principal component regression.

Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix W, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signal-to-noise ratio. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss. If the dataset is not too large, the significance of the principal components can be tested using parametric bootstrap, as an aid in determining how many principal components to retain.^[14]

Singular value decomposition

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

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

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

In terms of this factorization, the matrix X^TX can be written

${\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {U} ^{\mathsf {T}}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\hat {\Sigma }} ^{2}\mathbf {W} ^{\mathsf {T}}\end{aligned}}}$

where ${\displaystyle \mathbf {\hat {\Sigma }} }$ is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies ${\displaystyle \mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } }$. Comparison with the eigenvector factorization of X^TX establishes that the right singular vectors W of X are equivalent to the eigenvectors of X^TX, while the singular values σ_(k) of ${\displaystyle \mathbf {X} }$ are equal to the square-root of the eigenvalues λ_(k) of X^TX.

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

${\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}}$

so each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T.

Efficient algorithms exist to calculate the SVD of X without having to form the matrix X^TX, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix,^[15] unless only a handful of components are required.

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

${\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}}$

The truncation of a matrix M or T using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of rank L to the original matrix, in the sense of the difference between the two having the smallest possible Frobenius norm, a result known as the Eckart–Young theorem [1936].

Properties

Some properties of PCA include:^[12]

Property 1: For any integer q, 1 ≤ q ≤ p, consider the orthogonal linear transformation

${\displaystyle y=\mathbf {B'} x}$

where ${\displaystyle y}$ is a q-element vector and ${\displaystyle \mathbf {B'} }$ is a (q × p) matrix, and let ${\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} }$ be the variance-covariance matrix for ${\displaystyle y}$. Then the trace of ${\displaystyle \mathbf {\Sigma } _{y}}$, denoted ${\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}$, is maximized by taking ${\displaystyle \mathbf {B} =\mathbf {A} _{q}}$, where ${\displaystyle \mathbf {A} _{q}}$ consists of the first q columns of ${\displaystyle \mathbf {A} }$ ${\displaystyle (\mathbf {B'} }$ is the transpose of ${\displaystyle \mathbf {B} )}$. (${\displaystyle \mathbf {A} }$ is not defined here)

Property 2: Consider again the orthonormal transformation

${\displaystyle y=\mathbf {B'} x}$

with ${\displaystyle x,\mathbf {B} ,\mathbf {A} }$ and ${\displaystyle \mathbf {\Sigma } _{y}}$ defined as before. Then ${\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}$ is minimized by taking ${\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},}$ where ${\displaystyle \mathbf {A} _{q}^{*}}$ consists of the last q columns of ${\displaystyle \mathbf {A} }$.

The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of x, and they may also be useful in regression, in selecting a subset of variables from x, and in outlier detection.

Property 3: (Spectral decomposition of Σ)

${\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'}$

Before we look at its usage, we first look at diagonal elements,

${\displaystyle \operatorname {Var} (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}}$

Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of x into decreasing contributions due to each PC, but we can also decompose the whole covariance matrix into contributions ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ from each PC. Although not strictly decreasing, the elements of ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ will tend to become smaller as ${\displaystyle k}$ increases, as ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ is nonincreasing for increasing ${\displaystyle k}$, whereas the elements of ${\displaystyle \alpha _{k}}$ tend to stay about the same size because of the normalization constraints: ${\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p}$.

Limitations

As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.^[19]

The applicability of PCA as described above is limited by certain (tacit) assumptions^[20] made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see kernel PCA).

Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes,^[21] and forward modeling has to be performed to recover the true magnitude of the signals.^[22] As an alternative method, non-negative matrix factorization focusing only on the non-negative elements in the matrices, which is well-suited for astrophysical observations.^[23][24][25] See more at Relation between PCA and Non-negative Matrix Factorization.

PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were. They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss.^[26]

PCA relies on a linear model. If a dataset has a pattern hidden inside it that is nonlinear, then PCA can actually steer the analysis in the complete opposite direction of progress.^[27] Researchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results. "If the number of subjects or blocks is smaller than 30, and/or the researcher is interested in PC's beyond the first, it may be better to first correct for the serial correlation, before PCA is conducted".^[28] The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".^[28]

PCA and information theory

Dimensionality reduction results in a loss of information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.

Under the assumption that

${\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} ,}$

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

In particular, Linsker showed that if ${\displaystyle \mathbf {s} }$ is Gaussian and ${\displaystyle \mathbf {n} }$ is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information ${\displaystyle I(\mathbf {y}$ ;\mathbf {s} )} between the desired information ${\displaystyle \mathbf {s} }$ and the dimensionality-reduced output ${\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} }$.^[29]

If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector ${\displaystyle \mathbf {n} }$ are iid), but the information-bearing signal ${\displaystyle \mathbf {s} }$ is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as^[30][31]

${\displaystyle I(\mathbf {x}$ ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).}

The optimality of PCA is also preserved if the noise ${\displaystyle \mathbf {n} }$ is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal ${\displaystyle \mathbf {s} }$.^[32] In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise ${\displaystyle \mathbf {n} }$ becomes dependent.

Iterative computation

One way to compute the first principal component efficiently^[40] is shown in the following pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix.

r = a random vector of length p
r = r / norm(r)
do c times:
      s = 0 (a vector of length p)
      for each row x in X
            s = s + (x ⋅ r) x
      λ = rTs // λ is the eigenvalue
      error = |λ ⋅ r − s|
      r = s / norm(s)
      exit if error < tolerance
return λ, r

This power iteration algorithm simply calculates the vector X^T(X r), normalizes, and places the result back in r. The eigenvalue is approximated by r^T (X^TX) r, which is the Rayleigh quotient on the unit vector r for the covariance matrix X^TX . If the largest singular value is well separated from the next largest one, the vector r gets close to the first principal component of X within the number of iterations c, which is small relative to p, at the total cost 2cnp. The power iteration convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced matrix-free methods, such as the Lanczos algorithm or the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.

Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation. The latter approach in the block power method replaces single-vectors r and s with block-vectors, matrices R and S. Every column of R approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product X^T(X R). Implemented, for example, in LOBPCG, efficient blocking eliminates the accumulation of the errors, allows using high-level BLAS matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.

The NIPALS method

Non-linear iterative partial least squares (NIPALS) is a variant the classical power iteration with matrix deflation by subtraction implemented for computing the first few components in a principal component or partial least squares analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (for example, genomics, metabolomics) it is usually only necessary to compute the first few PCs. The non-linear iterative partial least squares (NIPALS) algorithm updates iterative approximations to the leading scores and loadings t₁ and r₁^T by the power iteration multiplying on every iteration by X on the left and on the right, that is, calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to X^TX, based on the function evaluating the product X^T(X r) = ((X r)^TX)^T.

The matrix deflation by subtraction is performed by subtracting the outer product, t₁r₁^T from X leaving the deflated residual matrix used to calculate the subsequent leading PCs.^[41] For large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality of PCs due to machine precision round-off errors accumulated in each iteration and matrix deflation by subtraction.^[42] A Gram–Schmidt re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.^[43] NIPALS reliance on single-vector multiplications cannot take advantage of high-level BLAS and results in slow convergence for clustered leading singular values—both these deficiencies are resolved in more sophisticated matrix-free block solvers, such as the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.

Online/sequential estimation

In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.^[44]

Intelligence

The earliest application of factor analysis was in locating and measuring components of human intelligence. it was believed that intelligence had various uncorrelated components such as spatial intelligence, verbal intelligence, induction, deduction etc and that scores on these could be adduced by factor analysis from results on various tests, to give a single index known as the Intelligence Quotient (IQ). The pioneering statistical psychologist Spearman actually developed factor analysis in 1904 for his two-factor theory of intelligence, adding a formal technique to the science of psychometrics. In 1924 Thurstone looked for 56 factors of intelligence, developing the notion of Mental Age. Standard IQ tests today are based on this early work.^[45]

Residential differentiation

In 1949, Shevky and Williams introduced the theory of factorial ecology, which dominated studies of residential differentiation from the 1950s to the 1970s.^[46] Neighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis. These were known as 'social rank' (an index of occupational status), 'familism' or family size, and 'ethnicity'; Cluster analysis could then be applied to divide the city into clusters or precincts according to values of the three key factor variables. An extensive literature developed around factorial ecology in urban geography, but the approach went out of fashion after 1980 as being methodologically primitive and having little place in postmodern geographical paradigms.

One of the problems with factor analysis has always been finding convincing names for the various artificial factors. In 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation. The principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities. The first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based. The next two components were 'disadvantage', which keeps people of similar status in separate neighbourhoods (mediated by planning), and ethnicity, where people of similar ethnic backgrounds try to co-locate.^[47]

About the same time, the Australian Bureau of Statistics defined distinct indexes of advantage and disadvantage taking the first principal component of sets of key variables that were thought to be important. These SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.^[48]

Development indexes

PCA has been the only formal method available for the development of indexes, which are otherwise a hit-or-miss ad hoc undertaking.

The City Development Index was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities. The first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for. The index ultimately used about 15 indicators but was a good predictor of many more variables. Its comparative value agreed very well with a subjective assessment of the condition of each city. The coefficients on items of infrastructure were roughly proportional to the average costs of providing the underlying services, suggesting the Index was actually a measure of effective physical and social investment in the city.

The country-level Human Development Index (HDI) from UNDP, which has been published since 1990 and is very extensively used in development studies,^[49] has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA.

Population genetics

In 1978 Cavalli-Sforza and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions. The components showed distinctive patterns, including gradients and sinusoidal waves. They interpreted these patterns as resulting from specific ancient migration events.

Since then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism. Genetics varies largely according to proximity, so the first two principal components actually show spatial distribution and may be used to map the relative geographical location of different population groups, thereby showing individuals who have wandered from their original locations.^[50]

PCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers. The lack of any measures of standard error in PCA are also an impediment to more consistent usage. In August 2022, the molecular biologist Eran Elhaik published a theoretical paper in Scientific Reports analyzing 12 PCA applications. He concluded that it was easy to manipulate the method, which, in his view, generated results that were 'erroneous, contradictory, and absurd.' Specifically, he argued, the results achieved in population genetics were characterized by cherry-picking and circular reasoning.^[51]

Market research and indexes of attitude

Market research has been an extensive user of PCA. It is used to develop customer satisfaction or customer loyalty scores for products, and with clustering, to develop market segments that may be targeted with advertising campaigns, in much the same way as factorial ecology will locate geographical areas with similar characteristics.^[52]

PCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed. In any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes. For example, the Oxford Internet Survey in 2013 asked 2000 people about their attitudes and beliefs, and from these analysts extracted four principal component dimensions, which they identified as 'escape', 'social networking', 'efficiency', and 'problem creating'.^[53]

Another example from Joe Flood in 2008 extracted an attitudinal index toward housing from 28 attitude questions in a national survey of 2697 households in Australia. The first principal component represented a general attitude toward property and home ownership. The index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice. The strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.^[54]

Quantitative finance

In quantitative finance, PCA is used^[55] in financial risk management, and has been applied to other problems such as portfolio optimization.

PCA is commonly used in problems involving fixed income securities and portfolios, and interest rate derivatives. Valuations here depend on the entire yield curve, comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements,^[56] thereby facilitating the modelling. One common risk management application is to calculating value at risk, VaR, applying PCA to the Monte Carlo simulation. ^[57] Here, for each simulation-sample, the components are stressed, and rates, and in turn option values, are then reconstructed; with VaR calculated, finally, over the entire run. PCA is also used in hedging exposure to interest rate risk, given partial durations and other sensitivities. ^[56] Under both, the first three, typically, principal components of the system are of interest (representing "shift", "twist", and "curvature"). These principal components are derived from an eigen-decomposition of the covariance matrix of yield at predefined maturities; ^[58] and where the variance of each component is its eigenvalue (and as the components are orthogonal, no correlation need be incorporated in subsequent modelling).

For equity, an optimal portfolio is one where the expected return is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see Markowitz model for discussion. Thus, one approach is to reduce portfolio risk, where allocation strategies are applied to the "principal portfolios" instead of the underlying stocks. A second approach is to enhance portfolio return, using the principal components to select stocks with upside potential. ^{[59] [60]} PCA has also been used to understand relationships ^[55] between international equity markets, and within markets between groups of companies in industries or sectors.

PCA may also be applied to stress testing, ^[61] essentially an analysis of a bank's ability to endure a hypothetical adverse economic scenario. Its utility is in "distilling the information contained in [several] macroeconomic variables into a more manageable data set, which can then [be used] for analysis."^[61] Here, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's eigenvector – and it is then observed how a "shock" to each of the factors affects the implied assets of each of the banks.

Neuroscience

A variant of principal components analysis is used in neuroscience to identify the specific properties of a stimulus that increases a neuron's probability of generating an action potential.^[62][63] This technique is known as spike-triggered covariance analysis. In a typical application an experimenter presents a white noise process as a stimulus (usually either as a sensory input to a test subject, or as a current injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the covariance matrix of the spike-triggered ensemble, the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The eigenvectors of the difference between the spike-triggered covariance matrix and the covariance matrix of the prior stimulus ensemble (the set of all stimuli, defined over the same length time window) then indicate the directions in the space of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.

In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs clustering analysis to associate specific action potentials with individual neurons.

PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, that is, order parameters, during phase transitions in the brain.^[64]

Correspondence analysis

Correspondence analysis (CA) was developed by Jean-Paul Benzécri^[65] and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. CA decomposes the chi-squared statistic associated to this table into orthogonal factors.^[66] Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not. Several variants of CA are available including detrended correspondence analysis and canonical correspondence analysis. One special extension is multiple correspondence analysis, which may be seen as the counterpart of principal component analysis for categorical data.^[67]

Factor analysis

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).

Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a pre-processing step before clustering. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (that is, translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.

Factor analysis is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance".^[68] In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (that is, shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.^[12]: 158 Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or causal modeling. If the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.^[69]

K-means clustering

It has been asserted that the relaxed solution of k-means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace.^[70][71] However, that PCA is a useful relaxation of k-means clustering was not a new result,^[72] and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.^[73]

Non-negative matrix factorization

Fractional residual variance (FRV) plots for PCA and NMF;^[25] for PCA, the theoretical values are the contribution from the residual eigenvalues. In comparison, the FRV curves for PCA reaches a flat plateau where no signal are captured effectively; while the NMF FRV curves decline continuously, indicating a better ability to capture signal. The FRV curves for NMF also converges to higher levels than PCA, indicating the less-overfitting property of NMF.

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,^[23][24][25] in the sense that astrophysical signals are non-negative. The PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis.

In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data.^[21] For NMF, its components are ranked based only on the empirical FRV curves.^[25] The residual fractional eigenvalue plots, that is, ${\displaystyle 1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}}$ as a function of component number ${\displaystyle k}$ given a total of ${\displaystyle n}$ 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).^[21] The FRV curves for NMF is decreasing continuously^[25] when the NMF components are constructed sequentially,^[24] indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA,^[25] indicating the less over-fitting property of NMF.

Iconography of correlations

It is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative. This leads the PCA user to a delicate elimination of several variables. If observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements. In addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane.

Iconography of correlations – Geochemistry of marine aerosols

The iconography of correlations, on the contrary, which is not a projection on a system of axes, does not have these drawbacks. We can therefore keep all the variables.

The principle of the diagram is to underline the "remarkable" correlations of the correlation matrix, by a solid line (positive correlation) or dotted line (negative correlation).

A strong correlation is not "remarkable" if it is not direct, but caused by the effect of a third variable. Conversely, weak correlations can be "remarkable". For example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet "remarkable".

Sparse PCA

A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables. Several approaches have been proposed, including

• a regression framework,^[74]
• a convex relaxation/semidefinite programming framework,^[75]
• a generalized power method framework^[76]
• an alternating maximization framework^[77]
• forward-backward greedy search and exact methods using branch-and-bound techniques,^[78]
• Bayesian formulation framework.^[79]

The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.^[80]

Nonlinear PCA

Linear PCA versus nonlinear Principal Manifolds^[81] 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.^[82] Software is available for free non-commercial use.^[83]

Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. Trevor Hastie expanded on this concept by proposing Principal curves^[84] as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data approximation followed by projecting the points onto it, as is illustrated by Fig. See also the elastic map algorithm and principal geodesic analysis.^[85] Another popular generalization is kernel PCA, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.

In multilinear subspace learning,^[86][87][88] PCA is generalized to multilinear PCA (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.

N-way principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.

Robust PCA

While PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to outliers in the data that produce large errors, something that the method tries to avoid in the first place. It is therefore common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in data mining algorithms like correlation clustering, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA^[89] based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.

Outlier-resistant variants of PCA have also been proposed, based on L1-norm formulations (L1-PCA).^[6][4]

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.^[90][91][92]

Independent component analysis

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

Network component analysis

Given a matrix ${\displaystyle E}$, it tries to decompose it into two matrices such that ${\displaystyle E=AP}$. A key difference from techniques such as PCA and ICA is that some of the entries of ${\displaystyle A}$ are constrained to be 0. Here ${\displaystyle P}$ is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :

1. ${\displaystyle A}$ has full column rank
2. Each column of ${\displaystyle A}$ must have at least ${\displaystyle L-1}$ zeroes where ${\displaystyle L}$ is the number of columns of ${\displaystyle A}$ (or alternatively the number of rows of ${\displaystyle P}$). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.
3. ${\displaystyle P}$ must have full row rank.

then the decomposition is unique up to multiplication by a scalar.^[93]

Discriminant analysis of principal components

Discriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals. Genetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former. Linear discriminants are linear combinations of alleles which best separate the clusters. Alleles that most contribute to this discrimination are therefore those that are the most markedly different across groups. The contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups^[94] In DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA).

A DAPC can be realized on R using the package Adegenet. (more info: adegenet on the web)

Directional component analysis

Directional component analysis (DCA) is a method used in the atmospheric sciences for analysing multivariate datasets.^[95] Like PCA, it allows for dimension reduction, improved visualization and improved interpretability of large data-sets. Also like PCA, it is based on a covariance matrix derived from the input dataset. The difference between PCA and DCA is that DCA additionally requires the input of a vector direction, referred to as the impact. Whereas PCA maximises explained variance, DCA maximises probability density given impact. The 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). DCA has been used to find the most likely and most serious heat-wave patterns in weather prediction ensembles ,^[96] and the most likely and most impactful changes in rainfall due to climate change .^[97]