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{{Short description|Method of data analysis}}
[[File:GaussianScatterPCA.svg|thumb|upright=1.3|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 [[Eigenvalues and eigenvectors|eigenvectors]] of the [[covariance matrix]] scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.]]
{{Machine learning bar}}
'''Principal component analysis''' ('''PCA''') is a [[Linear map|linear]] [[dimensionality reduction]] technique with applications in [[exploratory data analysis]], visualization and [[
The data is [[linear map|linearly transformed]] onto a new [[coordinate system]] such that the directions (principal components) capturing the largest variation in the data can be easily identified.
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=== Dimensionality reduction ===
The transformation '''
To non-dimensionalize the centered data, let ''X<sub>c</sub>'' represent the characteristic values of data vectors ''X<sub>i</sub>'', given by:
* <math>\frac{1}{n} \|X\|_1</math> (mean absolute value), or
* <math>\frac{1}{\sqrt{n}} \|X\|_2</math> (normalized Euclidean norm),
for a dataset of size ''n''. These norms are used to transform the original space of variables ''x, y'' to a new space of uncorrelated variables ''p, q'' (given ''Y<sub>c</sub>'' with same meaning), such that <math>p_i = \frac{X_i}{X_c}, \quad q_i = \frac{Y_i}{Y_c}</math>;
and the new variables are linearly related as: <math>q = \alpha p</math>.
To find the optimal linear relationship, we minimize the total squared reconstruction error:
<math>E(\alpha) = \frac{1}{1 - \alpha^2} \sum_{i=1}^{n} (\alpha p_i - q_i)^2</math>; such that setting the derivative of the error function to zero <math>(E'(\alpha) = 0)</math> yields:<math>\alpha = \frac{1}{2} \left( -\lambda \pm \sqrt{\lambda^2 + 4} \right)</math> where<math>\lambda = \frac{p \cdot p - q \cdot q}{p \cdot q}</math>.<ref name="Holmes2023" />
[[File:PCA of Haplogroup J using 37 STRs.png|thumb|right|A principal components analysis scatterplot of [[Y-STR]] [[haplotype]]s calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.<br /> PCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent.]]
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:<math>\mathbf{T}_L = \mathbf{U}_L\mathbf{\Sigma}_L = \mathbf{X} \mathbf{W}_L </math>
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 (linear algebra)|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 [[Low-rank approximation#Proof of Eckart–Young–Mirsky theorem (for Frobenius norm)|Eckart–Young theorem]] [1936].
<blockquote>
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where V<sub>k</sub> consists of the first k columns of V. Moreover, the relative residual variance is
<math>R(k)=\frac{\sum_{j=k+1}^{m}\sigma_{j}^{2}}{\sum_{j=1}^{m}\sigma_{j}^{2}}</math>.
</blockquote><ref name="Holmes2023" />
== Further considerations ==
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If we have just two variables and they have the same [[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. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.
Classical PCA assumes the cloud of points has already been translated so its centroid is at the origin.<ref name="Holmes2023" />
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Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: ''Pearson Product-Moment Correlation''). Also see the article by Kromrey & Foster-Johnson (1998) on ''"Mean-centering in Moderated Regression: Much Ado About Nothing"''. Since [[Covariance matrix#Relation to the correlation matrix|covariances are correlations of normalized variables]] ([[Standard score#Calculation|Z- or standard-scores]]) a PCA based on the correlation matrix of '''X''' is [[Equality (mathematics)|equal]] to a PCA based on the covariance matrix of '''Z''', the standardized version of '''X'''.
PCA is a popular primary technique in [[pattern recognition]]. It is not, however, optimized for class separability.<ref>{{Cite book| last=Fukunaga|first=Keinosuke|author-link=Keinosuke Fukunaga | title = Introduction to Statistical Pattern Recognition |publisher=Elsevier | year = 1990 | url=https://dl.acm.org/doi/book/10.5555/92131| isbn=978-0-12-269851-4}}</ref> However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes.<ref>{{cite journal|last1=Alizadeh|first1=Elaheh|last2=Lyons|first2=Samanthe M|last3=Castle|first3=Jordan M|last4=Prasad|first4=Ashok|title=Measuring systematic changes in invasive cancer cell shape using Zernike moments|journal=Integrative Biology|date=2016|volume=8|issue=11|pages=1183–1193|doi=10.1039/C6IB00100A|pmid=27735002|url=https://pubs.rsc.org/en/Content/ArticleLanding/2016/IB/C6IB00100A|url-access=subscription}}</ref> The [[linear discriminant analysis]] is an alternative which is optimized for class separability.
== Table of symbols and abbreviations ==
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'''Compute the cumulative energy content for each eigenvector'''
* The eigenvalues represent the distribution of the source data's energy{{Clarify|date=March 2011}} among each of the eigenvectors, where the eigenvectors form a [[basis (linear algebra)|basis]] for the data. The cumulative energy content ''g'' for the ''j''th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through ''j'' divided by the sum of energy content across all eigenvalues (shown in step 8):{{Citation needed|date=March 2011}} <math display="block">g_j = \sum_{k=1}^j D_{kk} \qquad \text{for } j = 1,\dots,p </math>▼
▲* The eigenvalues represent the distribution of the source data's energy{{Clarify|date=March 2011}} among each of the eigenvectors, where the eigenvectors form a [[basis (linear algebra)|basis]] for the data. The cumulative energy content ''g'' for the ''j''th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through ''j'':{{Citation needed|date=March 2011}} <math display="block">g_j = \sum_{k=1}^j D_{kk} \qquad \text{for } j = 1,\dots,p </math>
</li>
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