Content deleted Content added
mNo edit summary Tag: Reverted |
m removed typo |
||
| (45 intermediate revisions by 23 users not shown) | |||
Line 1:
{{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.
The '''principal components''' of a collection of points in a [[real coordinate space]] are a sequence of <math>p</math> [[unit vector]]s, where the <math>i</math>-th vector is the direction of a line that best fits the data while being [[orthogonal]] to the first <math>i-1</math> vectors. Here, a best-fitting line is defined as one that minimizes the average squared [[perpendicular distance|perpendicular]] [[Distance from a point to a line|distance from the points to the line]]. These directions (i.e., principal components) constitute an [[orthonormal basis]] in which different individual dimensions of the data are [[Linear correlation|linearly uncorrelated]]. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points.<ref>{{
Principal component analysis has applications in many fields such as [[population genetics]], [[microbiome]] studies, and [[atmospheric science]].<ref>{{Cite journal |last1=Jolliffe |first1=Ian T. |last2=Cadima |first2=Jorge |date=2016-04-13 |title=Principal component analysis: a review and recent developments |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=374 |issue=2065 |pages=20150202 |bibcode=2016RSPTA.37450202J |doi=10.1098/rsta.2015.0202 |pmc=4792409 |pmid=26953178}}</ref>
== Overview ==
When performing PCA, the first principal component of a set of <math>p</math> variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through <math>p</math> iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an [[linear independence|independent set]].
The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The <math>i</math>-th principal component can be taken as a direction orthogonal to the first <math>i-1</math> principal components that maximizes the variance of the projected data.
For either objective, it can be shown that the principal components are [[eigenvectors]] of the data's [[covariance matrix]]. Thus, the principal components are often computed by [[Eigendecomposition of a matrix|eigendecomposition]] of the data covariance matrix or [[singular value decomposition]] of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to [[factor analysis]]. Factor analysis typically incorporates more ___domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to [[Canonical correlation|canonical correlation analysis (CCA)]]. CCA defines coordinate systems that optimally describe the [[cross-covariance]] between two datasets while PCA defines a new [[orthogonal coordinate system]] that optimally describes variance in a single dataset.<ref>{{Cite journal|author1=Barnett, T. P. |author2=R. Preisendorfer. |name-list-style=amp |title=Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis |journal=Monthly Weather Review |volume=115 |issue=9 |pages=1825 |year=1987 |doi=10.1175/1520-0493(1987)115<1825:oaloma>2.0.co;2|bibcode=1987MWRv..115.1825B|doi-access=free }}</ref><ref>{{Cite book |last1=Hsu|first1=Daniel |first2=Sham M.|last2=Kakade |first3=Tong|last3=Zhang |title=A spectral algorithm for learning hidden markov models |arxiv=0811.4413 |year=2008 |bibcode=2008arXiv0811.4413H}}</ref><ref name="mark2017">{{cite journal|last1=Markopoulos|first1=Panos P.|last2=Kundu|first2=Sandipan|last3=Chamadia|first3=Shubham |last4=Pados|first4=Dimitris A.|title=Efficient L1-Norm Principal-Component Analysis via Bit Flipping|journal=IEEE Transactions on Signal Processing|date=15 August 2017|volume=65|issue=16|pages=4252–4264|doi=10.1109/TSP.2017.2708023|arxiv=1610.01959|bibcode=2017ITSP...65.4252M|s2cid=7931130}}</ref><ref name="l1tucker">{{cite journal|last1=Chachlakis|first1=Dimitris G.|last2=Prater-Bennette|first2=Ashley|last3=Markopoulos|first3=Panos P.|title=L1-norm Tucker Tensor Decomposition|journal=IEEE Access|date=22 November 2019|volume=7|pages=178454–178465|doi=10.1109/ACCESS.2019.2955134|arxiv=1904.06455|doi-access=free|bibcode=2019IEEEA...7q8454C }}</ref> [[Robust
== History ==
Line 31:
== Details ==
PCA is defined as an [[orthogonal transformation|orthogonal]] [[linear transformation]] on a real [[inner product space]] that transforms the data to a new [[coordinate system]] such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.<ref name="Jolliffe2002
Consider an <math>n \times p</math> data [[Matrix (mathematics)|matrix]], '''X''', with column-wise zero [[empirical mean]] (the sample mean of each column has been shifted to zero), where each of the ''n'' rows represents a different repetition of the experiment, and each of the ''p'' columns gives a particular kind of feature (say, the results from a particular sensor).
Line 111:
=== 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.]]
Line 122 ⟶ 126:
Similarly, in [[regression analysis]], the larger the number of [[explanatory variable]]s 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 [[Bootstrapping (statistics)#Parametric bootstrap|parametric bootstrap]], as an aid in determining how many principal components to retain.<ref>{{Cite journal|author=Forkman J., Josse, J., Piepho, H. P. |year=2019 |title= Hypothesis tests for principal component analysis when variables are standardized |journal= Journal of Agricultural, Biological, and Environmental Statistics|volume=24 |issue=2 |pages= 289–308 |doi=10.1007/s13253-019-00355-5|doi-access=free |bibcode=2019JABES..24..289F }}</ref>
=== Singular value decomposition ===
Line 154 ⟶ 158:
:<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>
'''Theorem (Optimal k‑dimensional fit).'''
Let P be an n×m data matrix whose columns have been mean‑centered and scaled, and let
<math>P = U \,\Sigma\, V^{T}</math>
be its singular value decomposition. Then the best rank‑k approximation to P in the least‑squares (Frobenius‑norm) sense is
<math>P_{k} = U_{k}\,\Sigma_{k}\,V_{k}^{T}</math>,
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 ==
Line 159 ⟶ 173:
The singular values (in '''Σ''') are the square roots of the [[eigenvalue]]s of the matrix '''X'''<sup>T</sup>'''X'''. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see [[Principle Component Analysis#PCA and information theory|below]]). PCA is often used in this manner for [[dimensionality reduction]]. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the [[discrete cosine transform]], and in particular to the DCT-II which is simply known as the "DCT". [[Nonlinear dimensionality reduction]] techniques tend to be more computationally demanding than PCA.
PCA is sensitive to the scaling of the variables. Mathematically this sensitivity comes from the way a rescaling changes the sample‑covariance matrix that PCA diagonalises.<ref name="Holmes2023">
PCA is sensitive to the scaling of the variables. 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.▼
{{cite book
|last=Holmes
|first=Mark H.
|title=Introduction to Scientific Computing and Data Analysis
|series=Texts in Computational Science and Engineering
|edition=2nd
|year=2023
|publisher=Springer
|isbn=978-3-031-22429-4
|pages=475–490
}}
Let <math>\mathbf X_\text{c}</math> be the *centered* data matrix (''n'' rows, ''p'' columns) and define the covariance
<math>\Sigma = \frac{1}{n}\,\mathbf X_\text{c}^{\mathsf T}\mathbf X_\text{c}.</math>
If the <math>j</math>‑th variable is multiplied by a factor <math>\alpha_j</math> we obtain
<math>\mathbf X_\text{c}^{(\alpha)} = \mathbf X_\text{c}D,\qquad
D = \operatorname{diag}(\alpha_1,\ldots,\alpha_p).</math>
Hence the new covariance is
<math>\Sigma^{(\alpha)} = D^{\mathsf T}\,\Sigma\,D.</math>
Because the eigenvalues and eigenvectors of <math>\Sigma^{(\alpha)}</math> are those of <math>\Sigma</math> scaled by <math>D</math>, the principal axes rotate toward any column whose variance has been inflated, exactly as the 2‑D example below illustrates.
▲
Classical PCA assumes the cloud of points has already been translated so its centroid is at the origin.<ref name="Holmes2023" />
Write each observation as
<math>\mathbf q_i = \boldsymbol\mu + \mathbf z_i,\qquad
\boldsymbol\mu = \tfrac{1}{n}\sum_{i=1}^{n}\mathbf q_i.</math>
Without subtracting <math>\boldsymbol\mu</math> we are in effect diagonalising
<math>\Sigma_{\text{unc}} \;=\; n\,\boldsymbol\mu\boldsymbol\mu^{\mathsf T}
\;+\;\tfrac{1}{n}\,\mathbf Z^{\mathsf T}\mathbf Z,</math>
where <math>\mathbf Z</math> is the centered matrix.
The rank‑one term <math>n\,\boldsymbol\mu\boldsymbol\mu^{\mathsf T}</math> often dominates, forcing the leading eigenvector to point almost exactly toward the mean and obliterating any structure in the centred part <math>\mathbf Z</math>.
After mean subtraction that term vanishes and the principal axes align with the true directions of maximal variance.
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 ==
Line 282 ⟶ 334:
The applicability of PCA as described above is limited by certain (tacit) assumptions<ref>Jonathon Shlens, [https://arxiv.org/abs/1404.1100 A Tutorial on Principal Component Analysis.]</ref> 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 principal component analysis|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,<ref name="soummer12"/> and forward modeling has to be performed to recover the true magnitude of the signals.<ref name="pueyo16">{{Cite journal|arxiv= 1604.06097 |last1= Pueyo|first1= Laurent |title= Detection and Characterization of Exoplanets using Projections on Karhunen Loeve Eigenimages: Forward Modeling |journal= The Astrophysical Journal |volume= 824|issue= 2|pages= 117|year= 2016|doi= 10.3847/0004-637X/824/2/117|bibcode = 2016ApJ...824..117P|s2cid= 118349503|doi-access= free}}</ref> As an alternative method, [[non-negative matrix factorization]] focusing only on the non-negative elements in the matrices
PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms the 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.<ref>{{cite web | title=What are the Pros and cons of the PCA? | website=i2tutorials | date=September 1, 2019 | url=https://www.i2tutorials.com/what-are-the-pros-and-cons-of-the-pca/ | access-date=June 4, 2021}}</ref>
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.<ref name=abbott>{{cite book | title=Applied Predictive Analytics | last=Abbott | first=Dean | isbn=9781118727966 | date=May 2014 | publisher=Wiley}}</ref>{{Page needed|date=June 2021}} 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".<ref name=jiang /> The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".<ref name=jiang>{{cite journal| title=Bias in Principal Components Analysis Due to Correlated Observations| url=https://newprairiepress.org/agstatconference/2000/proceedings/13/ |last1=Jiang | first1=Hong| last2=Eskridge | first2=Kent M.| year=2000 | journal=Conference on Applied Statistics in Agriculture |issn=2475-7772| doi=10.4148/2475-7772.1247| doi-access=free}}</ref>
Line 358 ⟶ 410:
<li>
'''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>
<li>
Line 432 ⟶ 483:
| year = 1986
| doi = 10.1016/0003-2670(86)80028-9
| bibcode = 1986AcAC..185....1G
▲ }}</ref>
}}</ref>
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.<ref>{{cite book |last=Kramer |first=R. |year=1998 |title=Chemometric Techniques for Quantitative Analysis |publisher=CRC Press |___location=New York |isbn= 9780203909805|url=https://books.google.com/books?id=iBpOzwAOfHYC}}</ref> 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.<ref>{{cite journal |first=M. |last=Andrecut |title=Parallel GPU Implementation of Iterative PCA Algorithms |journal=Journal of Computational Biology |volume=16 |issue=11 |year=2009 |pages=1593–1599 |doi=10.1089/cmb.2008.0221 |pmid=19772385 |arxiv=0811.1081 |s2cid=1362603 }}</ref> 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.
Line 471 ⟶ 523:
=== Development indexes ===
PCA can be used as a formal method for the development of indexes. As an alternative [[confirmatory composite analysis]] has been proposed to develop and assess indexes.<ref>{{cite journal |last1=Schamberger |first1=Tamara |last2=Schuberth |first2=Florian |last3=Henseler |first3=Jörg |title=Confirmatory composite analysis in human development research |journal=International Journal of Behavioral Development |date=2023 |volume=47 |issue=1 |pages=
The
The country-level [[Human Development Index]] (HDI) from [[United Nations Development Programme|UNDP]], which has been published since 1990 and is very extensively used in development studies,<ref>{{Cite web |last=Human Development Reports |title=Human Development Index |url=https://hdr.undp.org/en/content/human-development-index-hdi |access-date=2022-05-06 |website=United Nations Development Programme}}</ref> has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA.
Line 485 ⟶ 537:
=== 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.<ref>{{Cite journal |last1=DeSarbo |first1=Wayne |last2=Hausmann |first2=Robert |last3=Kukitz |first3=Jeffrey |date=2007 |title=Restricted principal components analysis for marketing research |url=https://www.researchgate.net/publication/247623679 |journal=Journal of Marketing in Management |volume=2 |pages=305–328 |via=
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'.<ref>{{Cite book |last1=Dutton |first1=William H |url=http://oxis.oii.ox.ac.uk/wp-content/uploads/2014/11/OxIS-2013.pdf |title=Cultures of the Internet: The Internet in Britain |last2=Blank |first2=Grant |publisher=Oxford Internet Institute |year=2013 |pages=6}}</ref>
Another example from
=== Quantitative finance ===
Line 904 ⟶ 956:
* {{YouTube|BfTMmoDFXyE|A layman's introduction to principal component analysis}} (a video of less than 100 seconds.)
* {{YouTube|FgakZw6K1QQ|StatQuest: StatQuest: Principal Component Analysis (PCA), Step-by-Step}}
* [https://stats.stackexchange.com/a/140579 Layman's explanation in making sense of principal component analysis, eigenvectors & eigenvalues] on [[Stack Overflow]]
* See also the list of [[#Software/source code|Software implementations]]
| |||