PCA.txt

Principal component analysis

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Conversion of observations of possibly-correlated variables into values of
fewer, uncorrelated variables
[300px-GaussianScatterPCA]
 
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.

The principal components of a collection of points in a real coordinate space
are a sequence of p {\displaystyle p} p unit vectors, where the i {\
displaystyle i} i-th vector is the direction of a line that best fits the data
while being orthogonal to the first i − 1 {\displaystyle i-1} i-1 vectors.
Here, a best-fitting line is defined as one that minimizes the average squared
distance from the points to the line. These directions constitute an
orthonormal basis in which different individual dimensions of the data are
linearly uncorrelated. Principal component analysis (PCA) is the process of
computing the principal components and using them to perform a change of basis
on the data, sometimes using only the first few principal components and
ignoring the rest.

PCA is used in exploratory data analysis and for making predictive models. It
is commonly used for dimensionality reduction by projecting each data point
onto only the first few principal components to obtain lower-dimensional data
while preserving as much of the data's variation as possible. The first
principal component can equivalently be defined as a direction that maximizes
the variance of the projected data. The i {\displaystyle i} i-th principal
component can be taken as a direction orthogonal to the first i − 1 {\
displaystyle i-1} i-1 principal components that maximizes the variance of the
projected data.

From 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 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
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.^[1]^
[2]^[3]^[4] Robust and L1-norm-based variants of standard PCA have also been
proposed.^[5]^[6]^[4]

[ ]

Contents

  • 1 History
  • 2 Intuition
  • 3 Details
      □ 3.1 First component
      □ 3.2 Further components
      □ 3.3 Covariances
      □ 3.4 Dimensionality reduction
      □ 3.5 Singular value decomposition
  • 4 Further considerations
  • 5 Table of symbols and abbreviations
  • 6 Properties and limitations of PCA
      □ 6.1 Properties
      □ 6.2 Limitations
      □ 6.3 PCA and information theory
  • 7 Computing PCA using the covariance method
  • 8 Derivation of PCA using the covariance method
  • 9 Covariance-free computation
      □ 9.1 Iterative computation
      □ 9.2 The NIPALS method
      □ 9.3 Online/sequential estimation
  • 10 PCA and qualitative variables
  • 11 Applications
      □ 11.1 Quantitative finance
      □ 11.2 Neuroscience
  • 12 Relation with other methods
      □ 12.1 Correspondence analysis
      □ 12.2 Factor analysis
      □ 12.3 K-means clustering
      □ 12.4 Non-negative matrix factorization
      □ 12.5 Iconography of correlations
  • 13 Generalizations
      □ 13.1 Sparse PCA
      □ 13.2 Nonlinear PCA
      □ 13.3 Robust PCA
  • 14 Similar techniques
      □ 14.1 Independent component analysis
      □ 14.2 Network component analysis
      □ 14.3 Discriminant analysis of principal components
  • 15 Software/source code
  • 16 See also
  • 17 References
  • 18 Further reading
  • 19 External links

History[edit]

PCA was invented in 1901 by Karl Pearson,^[7] as an analogue of the principal
axis theorem in mechanics; it was later independently developed and named by
Harold Hotelling in the 1930s.^[8] Depending on the field of application, it is
also named the discrete Karhunen–Loève transform (KLT) in signal processing,
the Hotelling transform in multivariate quality control, proper orthogonal
decomposition (POD) in mechanical engineering, singular value decomposition
(SVD) of X (invented in the last quarter of the 19th century^[9]), eigenvalue
decomposition (EVD) of X^TX in linear algebra, factor analysis (for a
discussion of the differences between PCA and factor analysis see Ch. 7 of
Jolliffe's Principal Component Analysis),^[10] Eckart–Young theorem (Harman,
1960), or empirical orthogonal functions (EOF) in meteorological science,
empirical eigenfunction decomposition (Sirovich, 1987), empirical component
analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral
decomposition in noise and vibration, and empirical modal analysis in
structural dynamics.

Intuition[edit]

PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where
each axis of the ellipsoid represents a principal component. If some axis of
the ellipsoid is small, then the variance along that axis is also small.

To find the axes of the ellipsoid, we must first center the values of each
variable in the dataset on 0 by subtracting the mean of the variable's observed
values from each of those values. These transformed values are used instead of
the original observed values for each of the variables. Then, we compute the
covariance matrix of the data and calculate the eigenvalues and corresponding
eigenvectors of this covariance matrix. Then we must normalize each of the
orthogonal eigenvectors to turn them into unit vectors. Once this is done, each
of the mutually orthogonal unit eigenvectors can be interpreted as an axis of
the ellipsoid fitted to the data. This choice of basis will transform our
covariance matrix into a diagonalised form with the diagonal elements
representing the variance of each axis. The proportion of the variance that
each eigenvector represents can be calculated by dividing the eigenvalue
corresponding to that eigenvector by the sum of all eigenvalues.

Biplots and scree plots (degree of explained variance) are used to explain
findings of the PCA.

Details[edit]

PCA is defined as an orthogonal linear transformation 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.^[10]^[page needed]

Consider an n × p {\displaystyle n\times p} n\times p data 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).

Mathematically, the transformation is defined by a set of size l {\displaystyle
l} l of p-dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … ,
w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} \
mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)} that map each row vector x ( i )
{\displaystyle \mathbf {x} _{(i)}} \mathbf{x}_{(i)} of X to a new vector of
principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \
mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} {\displaystyle \mathbf {t} _{(i)}
=(t_{1},\dots ,t_{l})_{(i)}}, given by

    t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\
    displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad
    \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} {\displaystyle {t_
    {k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \
    qquad i=1,\dots ,n\qquad k=1,\dots ,l}

in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},
\dots ,t_{l}} {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data
set successively inherit the maximum possible variance from X, with each
coefficient vector w constrained to be a unit vector (where l {\displaystyle l}
l is usually selected to be strictly less than p {\displaystyle p} p to reduce
dimensionality).

First component[edit]

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

    w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1
    { ∑ i ( x ( i ) ⋅ w ) 2 } {\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\}} {\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

    w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T
    X w } {\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\}} {\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

    w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\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\}} {\
    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[edit]

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

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

    w ( k ) = a r g m a x ‖ w ‖ = 1 ⁡ { ‖ X ^ k w ‖ 2 } = arg ⁡ max { w T X ^ k
    T X ^ k w w T w } {\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\}} {\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

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

where W is a p-by-p matrix of weights whose columns are the eigenvectors of X^T
X. 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[edit]

X^TX itself can be recognized as proportional to the empirical sample
covariance matrix of the dataset X^T.^[10]^: 30–31 

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

    Q ( P C ( j ) , P C ( k ) ) ∝ ( X w ( j ) ) T ( X w ( k ) ) = w ( j ) T X T
    X w ( k ) = w ( j ) T λ ( k ) w ( k ) = λ ( k ) w ( j ) T w ( k ) {\
    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}}} {\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

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

    W T Q W ∝ W T W Λ W T W = Λ {\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
    } } {\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[k]^2[(i)] = Σ[i] (x[(i)] ⋅ w[(k)])^2.

Dimensionality reduction[edit]

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

    T L = X W L {\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}} \
    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 t = W L T x , x ∈ R p , t ∈ R L , {\displaystyle
t=W_{L}^{\mathsf {T}}x,x\in R^{p},t\in R^{L},} {\displaystyle t=W_{L}^{\mathsf
{T}}x,x\in R^{p},t\in R^{L},} where the columns of p × L matrix W L {\
displaystyle W_{L}} {\displaystyle W_{L}} form an orthogonal basis for the L
features (the components of representation t) that are decorrelated.^[11] 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 ‖ T W T − T
L W L T ‖ 2 2 {\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\
mathbf {W} _{L}^{T}\|_{2}^{2}} \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\
mathbf {W} _{L}^{T}\|_{2}^{2} or ‖ X − X L ‖ 2 2 {\displaystyle \|\mathbf {X} -
\mathbf {X} _{L}\|_{2}^{2}} \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}.

[220px-PCA_of_Haplogroup_J_]
 
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.^[12]

Singular value decomposition[edit]

Main article: Singular value decomposition

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

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

    X T X = W Σ T U T U Σ W T = W Σ T Σ W T = W Σ ^ 2 W T {\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}}} {\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 Σ ^ {\displaystyle \mathbf {\hat {\Sigma }} } {\displaystyle \mathbf {\
hat {\Sigma }} } is the square diagonal matrix with the singular values of X 
and the excess zeros chopped off that satisfies Σ ^ 2 = Σ T Σ {\displaystyle \
mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma
} } {\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 X {\displaystyle \
mathbf {X} } {\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

    T = X W = U Σ W T W = U Σ {\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}}} {\
    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^[citation needed], 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:

    T L = U L Σ L = X W L {\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\
    mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}} \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].

Further considerations[edit]

Given a set of points in Euclidean space, the first principal component
corresponds to a line that passes through the multidimensional mean and
minimizes the sum of squares of the distances of the points from the line^[
disputed – discuss]. The second principal component corresponds to the same
concept after all correlation with the first principal component has been
subtracted from the points. The singular values (in Σ) are the square roots of
the eigenvalues of the matrix X^TX. Each eigenvalue is proportional to the
portion of the "variance" (more correctly of the sum of the squared distances
of the points from their multidimensional mean) that is associated with each
eigenvector. The sum of all the eigenvalues is equal to the sum of the squared
distances of the points from their multidimensional mean. PCA essentially
rotates the set of points around their mean in order to align with the
principal components. This moves as much of the variance as possible (using an
orthogonal transformation) into the first few dimensions. The values in the
remaining dimensions, therefore, tend to be small and may be dropped with
minimal loss of information (see below). 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. If we have just two variables
and they have the same sample variance and are positively correlated, then the
PCA will entail a rotation by 45° and the "weights" (they are the cosines of
rotation) for the two variables with respect to the principal component will be
equal. But if we multiply all values of the first variable by 100, then the
first principal component will be almost the same as that variable, with a
small contribution from the other variable, whereas the second component will
be almost aligned with the second original variable. This means that whenever
the different variables have different units (like temperature and mass), PCA
is a somewhat arbitrary method of analysis. (Different results would be
obtained if one used Fahrenheit rather than Celsius for example.) Pearson's
original paper was entitled "On Lines and Planes of Closest Fit to Systems of
Points in Space" – "in space" implies physical Euclidean space where such
concerns do not arise. One way of making the PCA less arbitrary is to use
variables scaled so as to have unit variance, by standardizing the data and
hence use the autocorrelation matrix instead of the autocovariance matrix as a
basis for PCA. However, this compresses (or expands) the fluctuations in all
dimensions of the signal space to unit variance.

Mean subtraction (a.k.a. "mean centering") is necessary for performing
classical PCA to ensure that the first principal component describes the
direction of maximum variance. If mean subtraction is not performed, the first
principal component might instead correspond more or less to the mean of the
data. A mean of zero is needed for finding a basis that minimizes the mean
square error of the approximation of the data.^[13]

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".

PCA is a popular primary technique in pattern recognition. It is not, however,
optimized for class separability.^[14] 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.^[15] The linear discriminant analysis is
an alternative which is optimized for class separability.

Table of symbols and abbreviations[edit]

      Symbol                    Meaning              Dimensions     Indices
X = { X i j } {\                                                 i = 1 … n {\
displaystyle \                                                   displaystyle i
mathbf {X} =\{X_    data matrix, consisting of the  n × p {\     =1\ldots n} i
{ij}\}} {\          set of all data vectors, one    displaystyle = 1 \ldots n
displaystyle \      vector per row                  n\times p} n j = 1 … p {\
mathbf {X} =\{X_                                    \times p     displaystyle j
{ij}\}}                                                          =1\ldots p} j
                                                                 = 1 \ldots p
                                                    1 × 1 {\
n {\displaystyle n} the number of row vectors in    displaystyle scalar
n                   the data set                    1\times 1} 1
                                                    \times 1
                                                    1 × 1 {\
p {\displaystyle p} the number of elements in each  displaystyle scalar
p                   row vector (dimension)          1\times 1} 1
                                                    \times 1
                    the number of dimensions in the 1 × 1 {\
L {\displaystyle L} dimensionally reduced subspace, displaystyle scalar
L                   1 ≤ L ≤ p {\displaystyle 1\leq  1\times 1} 1
                    L\leq p} 1 \le L \le p          \times 1
u = { u j } {\
displaystyle \      vector of empirical means, one  p × 1 {\     j = 1 … p {\
mathbf {u} =\{u_{j} mean for each column j of the   displaystyle displaystyle j
\}} {\displaystyle  data matrix                     p\times 1} p =1\ldots p} j
\mathbf {u} =\{u_                                   \times 1     = 1 \ldots p
{j}\}}
s = { s j } {\
displaystyle \      vector of empirical standard    p × 1 {\     j = 1 … p {\
mathbf {s} =\{s_{j} deviations, one standard        displaystyle displaystyle j
\}} {\displaystyle  deviation for each column j of  p\times 1} p =1\ldots p} j
\mathbf {s} =\{s_   the data matrix                 \times 1     = 1 \ldots p
{j}\}}
h = { h i } {\
displaystyle \                                      1 × n {\     i = 1 … n {\
mathbf {h} =\{h_{i} vector of all 1's               displaystyle displaystyle i
\}} {\displaystyle                                  1\times n} 1 =1\ldots n} i
\mathbf {h} =\{h_                                   \times n     = 1 \ldots n
{i}\}}
B = { B i j } {\                                                 i = 1 … n {\
displaystyle \                                                   displaystyle i
mathbf {B} =\{B_    deviations from the mean of     n × p {\     =1\ldots n} i
{ij}\}} {\          each column j of the data       displaystyle = 1 \ldots n
displaystyle \      matrix                          n\times p} n j = 1 … p {\
mathbf {B} =\{B_                                    \times p     displaystyle j
{ij}\}}                                                          =1\ldots p} j
                                                                 = 1 \ldots p
Z = { Z i j } {\                                                 i = 1 … n {\
displaystyle \                                                   displaystyle i
mathbf {Z} =\{Z_    z-scores, computed using the    n × p {\     =1\ldots n} i
{ij}\}} {\          mean and standard deviation for displaystyle = 1 \ldots n
displaystyle \      each row m of the data matrix   n\times p} n j = 1 … p {\
mathbf {Z} =\{Z_                                    \times p     displaystyle j
{ij}\}}                                                          =1\ldots p} j
                                                                 = 1 \ldots p
                                                                 j = 1 … p {\
C = { C j j ′ } {\                                               displaystyle j
displaystyle \                                      p × p {\     =1\ldots p} j
mathbf {C} =\{C_                                    displaystyle = 1 \ldots p
{jj'}\}} {\         covariance matrix               p\times p} p j ′ = 1 … p {\
displaystyle \                                      \times p     displaystyle
mathbf {C} =\{C_                                                 j'=1\ldots p} 
{jj'}\}}                                                         {\displaystyle
                                                                 j'=1\ldots p}
                                                                 j = 1 … p {\
R = { R j j ′ } {\                                               displaystyle j
displaystyle \                                      p × p {\     =1\ldots p} j
mathbf {R} =\{R_                                    displaystyle = 1 \ldots p
{jj'}\}} {\         correlation matrix              p\times p} p j ′ = 1 … p {\
displaystyle \                                      \times p     displaystyle
mathbf {R} =\{R_                                                 j'=1\ldots p} 
{jj'}\}}                                                         {\displaystyle
                                                                 j'=1\ldots p}
                                                                 j = 1 … p {\
V = { V j j ′ } {\                                               displaystyle j
displaystyle \                                      p × p {\     =1\ldots p} j
mathbf {V} =\{V_    matrix consisting of the set of displaystyle = 1 \ldots p
{jj'}\}} {\         all eigenvectors of C, one      p\times p} p j ′ = 1 … p {\
displaystyle \      eigenvector per column          \times p     displaystyle
mathbf {V} =\{V_                                                 j'=1\ldots p} 
{jj'}\}}                                                         {\displaystyle
                                                                 j'=1\ldots p}
                    diagonal matrix consisting of                j = 1 … p {\
D = { D j j ′ } {\  the set of all eigenvalues of C              displaystyle j
displaystyle \      along its principal diagonal,   p × p {\     =1\ldots p} j
mathbf {D} =\{D_    and 0 for all other elements (  displaystyle = 1 \ldots p
{jj'}\}} {\         note Λ {\displaystyle \mathbf   p\times p} p j ′ = 1 … p {\
displaystyle \      {\Lambda } } {\displaystyle \   \times p     displaystyle
mathbf {D} =\{D_    mathbf {\Lambda } } used above               j'=1\ldots p} 
{jj'}\}}            )                                            {\displaystyle
                                                                 j'=1\ldots p}
                                                                 j = 1 … p {\
W = { W j l } {\    matrix of basis vectors, one                 displaystyle j
displaystyle \      vector per column, where each   p × L {\     =1\ldots p} j
mathbf {W} =\{W_    basis vector is one of the      displaystyle = 1 \ldots p
{jl}\}} {\          eigenvectors of C, and where    p\times L} p l = 1 … L {\
displaystyle \      the vectors in W are a sub-set  \times L     displaystyle l
mathbf {W} =\{W_    of those in V                                =1\ldots L} {\
{jl}\}}                                                          displaystyle l
                                                                 =1\ldots L}
                                                                 i = 1 … n {\
T = { T i l } {\    matrix consisting of n row                   displaystyle i
displaystyle \      vectors, where each vector is   n × L {\     =1\ldots n} i
mathbf {T} =\{T_    the projection of the           displaystyle = 1 \ldots n
{il}\}} {\          corresponding data vector from  n\times L} n l = 1 … L {\
displaystyle \      matrix X onto the basis vectors \times L     displaystyle l
mathbf {T} =\{T_    contained in the columns of                  =1\ldots L} {\
{il}\}}             matrix W.                                    displaystyle l
                                                                 =1\ldots L}

Properties and limitations of PCA[edit]

Properties[edit]

Some properties of PCA include:^[10]^[page needed]

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

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

    where y {\displaystyle y} y is a q-element vector and B ′ {\displaystyle \
    mathbf {B'} } \mathbf{B'} is a (q × p) matrix, and let Σ y = B ′ Σ B {\
    displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf
    {B} } {\mathbf {{\Sigma }}}_{y}={\mathbf {B'}}{\mathbf {\Sigma }}{\mathbf
    {B}} be the variance-covariance matrix for y {\displaystyle y} y. Then the
    trace of Σ y {\displaystyle \mathbf {\Sigma } _{y}} {\mathbf {\Sigma }}_{y}
    , denoted tr ⁡ ( Σ y ) {\displaystyle \operatorname {tr} (\mathbf {\Sigma }
    _{y})} {\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}, is
    maximized by taking B = A q {\displaystyle \mathbf {B} =\mathbf {A} _{q}} \
    mathbf{B} = \mathbf{A}_q, where A q {\displaystyle \mathbf {A} _{q}} \
    mathbf{A}_q consists of the first q columns of A {\displaystyle \mathbf {A}
    } \mathbf {A} ( B ′ {\displaystyle (\mathbf {B'} } (\mathbf{B'} is the
    transposition of B ) {\displaystyle \mathbf {B} )} \mathbf{B}).

    Property 2: Consider again the orthonormal transformation

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

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

        Σ = λ 1 α 1 α 1 ′ + ⋯ + λ p α p α p ′ {\displaystyle \mathbf {\Sigma }
        =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\
        alpha _{p}'} {\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,

    Var ⁡ ( x j ) = ∑ k = 1 P λ k α k j 2 {\displaystyle \operatorname {Var}
    (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}} {\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 λ k α k α k ′ {\displaystyle \lambda _{k}\
alpha _{k}\alpha _{k}'} {\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}
from each PC. Although not strictly decreasing, the elements of λ k α k α k ′
{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} {\displaystyle \lambda _{k}
\alpha _{k}\alpha _{k}'} will tend to become smaller as k {\displaystyle k} k
increases, as λ k α k α k ′ {\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}
{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} is nonincreasing for
increasing k {\displaystyle k} k, whereas the elements of α k {\displaystyle \
alpha _{k}} \alpha _{k} tend to stay about the same size because of the
normalization constraints: α k ′ α k = 1 , k = 1 , … , p {\displaystyle \alpha
_{k}'\alpha _{k}=1,k=1,\dots ,p} {\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\
dots ,p}.

Limitations[edit]

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.^[16]

The applicability of PCA as described above is limited by certain (tacit)
assumptions^[17] 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,^[18] and forward modeling has to be performed to recover the
true magnitude of the signals.^[19] 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.^[20]^[21]^[22]
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.^[23]

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.^[24]^[page needed] 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".^[25] The
researchers at Kansas State also found that PCA could be "seriously biased if
the autocorrelation structure of the data is not correctly handled".^[25]

PCA and information theory[edit]

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

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

that is, that the data vector x {\displaystyle \mathbf {x} } \mathbf {x} is the
sum of the desired information-bearing signal s {\displaystyle \mathbf {s} } \
mathbf {s} and a noise signal n {\displaystyle \mathbf {n} } \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 s {\displaystyle \mathbf {s} } \mathbf
{s} is Gaussian and n {\displaystyle \mathbf {n} } \mathbf {n} is Gaussian
noise with a covariance matrix proportional to the identity matrix, the PCA
maximizes the mutual information I ( y ; s ) {\displaystyle I(\mathbf {y} ;\
mathbf {s} )} I(\mathbf{y};\mathbf{s}) between the desired information s {\
displaystyle \mathbf {s} } \mathbf {s} and the dimensionality-reduced output y
= W L T x {\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} } \mathbf
{y}=\mathbf{W}_L^T\mathbf{x}.^[26]

If the noise is still Gaussian and has a covariance matrix proportional to the
identity matrix (that is, the components of the vector n {\displaystyle \mathbf
{n} } \mathbf {n} are iid), but the information-bearing signal s {\displaystyle
\mathbf {s} } \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^
[27]^[28]

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

The optimality of PCA is also preserved if the noise n {\displaystyle \mathbf
{n} } \mathbf {n} is iid and at least more Gaussian (in terms of the
Kullback–Leibler divergence) than the information-bearing signal s {\
displaystyle \mathbf {s} } \mathbf {s} .^[29] In general, even if the above
signal model holds, PCA loses its information-theoretic optimality as soon as
the noise n {\displaystyle \mathbf {n} } \mathbf {n} becomes dependent.

Computing PCA using the covariance method[edit]

The following is a detailed description of PCA using the covariance method (see
also here) as opposed to the correlation method.^[30]

The goal is to transform a given data set X of dimension p to an alternative
data set Y of smaller dimension L. Equivalently, we are seeking to find the
matrix Y, where Y is the Karhunen–Loève transform (KLT) of matrix X:

    Y = K L T { X } {\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}}
    \mathbf{Y} = \mathbb{KLT} \{ \mathbf{X} \}

Organize the data set

Suppose you have data comprising a set of observations of p variables, and you
want to reduce the data so that each observation can be described with only L
variables, L < p. Suppose further, that the data are arranged as a set of n
data vectors x 1 … x n {\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}} 
\mathbf{x}_1 \ldots \mathbf{x}_n with each x i {\displaystyle \mathbf {x} _{i}}
\mathbf {x} _{i} representing a single grouped observation of the p variables.

  • Write x 1 … x n {\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}} \
    mathbf{x}_1 \ldots \mathbf{x}_n as row vectors, each with p elements.
  • Place the row vectors into a single matrix X of dimensions n × p.

Calculate the empirical mean

  • Find the empirical mean along each column j = 1, ..., p.
  • Place the calculated mean values into an empirical mean vector u of
    dimensions p × 1.

        u j = 1 n ∑ i = 1 n X i j {\displaystyle u_{j}={\frac {1}{n}}\sum _{i=
        1}^{n}X_{ij}} {\displaystyle u_{j}={\frac {1}{n}}\sum _{i=1}^{n}X_{ij}}

Calculate the deviations from the mean

Mean subtraction is an integral part of the solution towards finding a
principal component basis that minimizes the mean square error of approximating
the data.^[31] Hence we proceed by centering the data as follows:

  • Subtract the empirical mean vector u T {\displaystyle \mathbf {u} ^{T}} {\
    displaystyle \mathbf {u} ^{T}} from each row of the data matrix X.
  • Store mean-subtracted data in the n × p matrix B.

        B = X − h u T {\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \
        mathbf {u} ^{T}} {\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \
        mathbf {u} ^{T}}
        where h is an n × 1 column vector of all 1s:

            h i = 1 for  i = 1 , … , n {\displaystyle h_{i}=1\,\qquad \qquad {\
            text{for }}i=1,\ldots ,n} {\displaystyle h_{i}=1\,\qquad \qquad {\
            text{for }}i=1,\ldots ,n}

In some applications, each variable (column of B) may also be scaled to have a
variance equal to 1 (see Z-score).^[32] This step affects the calculated
principal components, but makes them independent of the units used to measure
the different variables.

Find the covariance matrix

  • Find the p × p empirical covariance matrix C from matrix B:
    C = 1 n − 1 B ∗ B {\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^
    {*}\mathbf {B} }
    {\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\mathbf {B} }
    where ∗ {\displaystyle *} * is the conjugate transpose operator. If B
    consists entirely of real numbers, which is the case in many applications,
    the "conjugate transpose" is the same as the regular transpose.
  • The reasoning behind using n − 1 instead of n to calculate the covariance
    is Bessel's correction.

Find the eigenvectors and eigenvalues of the covariance matrix

  • Compute the matrix V of eigenvectors which diagonalizes the covariance
    matrix C:
    V − 1 C V = D {\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\
    mathbf {D} }
    {\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} }
    where D is the diagonal matrix of eigenvalues of C. This step will
    typically involve the use of a computer-based algorithm for computing
    eigenvectors and eigenvalues. These algorithms are readily available as
    sub-components of most matrix algebra systems, such as SAS,^[33] R, MATLAB,
    ^[34]^[35] Mathematica,^[36] SciPy, IDL (Interactive Data Language), or GNU
    Octave as well as OpenCV.
  • Matrix D will take the form of an p × p diagonal matrix, where
    D k ℓ = λ k for  k = ℓ {\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text
    {for }}k=\ell }
    {\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text{for }}k=\ell }
    is the jth eigenvalue of the covariance matrix C, and
    D k ℓ = 0 for  k ≠ ℓ . {\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq
    \ell .}
    {\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq \ell .}
  • Matrix V, also of dimension p × p, contains p column vectors, each of
    length p, which represent the p eigenvectors of the covariance matrix C.
  • The eigenvalues and eigenvectors are ordered and paired. The jth eigenvalue
    corresponds to the jth eigenvector.
  • Matrix V denotes the matrix of right eigenvectors (as opposed to left
    eigenvectors). In general, the matrix of right eigenvectors need not be the
    (conjugate) transpose of the matrix of left eigenvectors.

Rearrange the eigenvectors and eigenvalues

  • Sort the columns of the eigenvector matrix V and eigenvalue matrix D in
    order of decreasing eigenvalue.
  • Make sure to maintain the correct pairings between the columns in each
    matrix.

Compute the cumulative energy content for each eigenvector

  • The eigenvalues represent the distribution of the source data's energy^[
    clarification needed] among each of the eigenvectors, where the
    eigenvectors form a basis for the data. The cumulative energy content g for
    the jth eigenvector is the sum of the energy content across all of the
    eigenvalues from 1 through j:

        g j = ∑ k = 1 j D k k for  j = 1 , … , p {\displaystyle g_{j}=\sum _{k=
        1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p} {\displaystyle g_{j}=\sum
        _{k=1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p}^[citation needed]

Select a subset of the eigenvectors as basis vectors

  • Save the first L columns of V as the p × L matrix W:
    W k l = V k ℓ for  k = 1 , … , p ℓ = 1 , … , L {\displaystyle W_{kl}=V_{k\
    ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell =1,\dots ,L}
    {\displaystyle W_{kl}=V_{k\ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell
    =1,\dots ,L}
    where
    1 ≤ L ≤ p . {\displaystyle 1\leq L\leq p.}
    {\displaystyle 1\leq L\leq p.}
  • Use the vector g as a guide in choosing an appropriate value for L. The
    goal is to choose a value of L as small as possible while achieving a
    reasonably high value of g on a percentage basis. For example, you may want
    to choose L so that the cumulative energy g is above a certain threshold,
    like 90 percent. In this case, choose the smallest value of L such that
    g L g p ≥ 0.9 {\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9}
    {\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9}

Project the data onto the new basis

  • The projected data points are the rows of the matrix
    T = B ⋅ W {\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} }
    {\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} }

That is, the first column of T {\displaystyle \mathbf {T} } \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.

Derivation of PCA using the covariance method[edit]

Let X be a d-dimensional random vector expressed as column vector. Without loss
of generality, assume X has zero mean.

We want to find ( ∗ ) {\displaystyle (\ast )} {\displaystyle (\ast )} a d × d
orthonormal transformation matrix P so that PX has a diagonal covariance matrix
(that is, PX is a random vector with all its distinct components pairwise
uncorrelated).

A quick computation assuming P {\displaystyle P} P were unitary yields:

    cov ⁡ ( P X ) = E ⁡ [ P X   ( P X ) ∗ ] = E ⁡ [ P X   X ∗ P ∗ ] = P E ⁡ [ X
    X ∗ ] P ∗ = P cov ⁡ ( X ) P − 1 {\displaystyle {\begin{aligned}\
    operatorname {cov} (PX)&=\operatorname {E} [PX~(PX)^{*}]\\&=\operatorname
    {E} [PX~X^{*}P^{*}]\\&=P\operatorname {E} [XX^{*}]P^{*}\\&=P\operatorname
    {cov} (X)P^{-1}\\\end{aligned}}} {\displaystyle {\begin{aligned}\
    operatorname {cov} (PX)&=\operatorname {E} [PX~(PX)^{*}]\\&=\operatorname
    {E} [PX~X^{*}P^{*}]\\&=P\operatorname {E} [XX^{*}]P^{*}\\&=P\operatorname
    {cov} (X)P^{-1}\\\end{aligned}}}

Hence ( ∗ ) {\displaystyle (\ast )} {\displaystyle (\ast )} holds if and only
if cov ⁡ ( X ) {\displaystyle \operatorname {cov} (X)} \operatorname {cov}(X)
were diagonalisable by P {\displaystyle P} P.

This is very constructive, as cov(X) is guaranteed to be a non-negative
definite matrix and thus is guaranteed to be diagonalisable by some unitary
matrix.

Covariance-free computation[edit]

In practical implementations, especially with 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. The covariance-free approach avoids the np^2 operations of explicitly
calculating and storing the covariance matrix X^TX, instead utilizing one of
matrix-free methods, for example, based on the function evaluating the product 
X^T(X r) at the cost of 2np operations.

Iterative computation[edit]

One way to compute the first principal component efficiently^[37] 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 = r^Ts // 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[edit]

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 t[1] and r[1]^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, t[1]r[1]^T from X leaving the deflated residual matrix used to
calculate the subsequent leading PCs.^[38] 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.^[39] 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.^[40] 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[edit]

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.^[41]

PCA and qualitative variables[edit]

In PCA, it is common that we want to introduce qualitative variables as
supplementary elements. For example, many quantitative variables have been
measured on plants. For these plants, some qualitative variables are available
as, for example, the species to which the plant belongs. These data were
subjected to PCA for quantitative variables. When analyzing the results, it is
natural to connect the principal components to the qualitative variable species
. For this, the following results are produced.

  • Identification, on the factorial planes, of the different species, for
    example, using different colors.
  • Representation, on the factorial planes, of the centers of gravity of
    plants belonging to the same species.
  • For each center of gravity and each axis, p-value to judge the significance
    of the difference between the center of gravity and origin.

These results are what is called introducing a qualitative variable as
supplementary element. This procedure is detailed in and Husson, Lê & Pagès
2009 and Pagès 2013. Few software offer this option in an "automatic" way. This
is the case of SPAD that historically, following the work of Ludovic Lebart,
was the first to propose this option, and the R package FactoMineR.

Applications[edit]

Quantitative finance[edit]

See also: Portfolio optimization

In quantitative finance, principal component analysis can be directly applied
to the risk management of interest rate derivative portfolios.^[42] Trading
multiple swap instruments which are usually a function of 30–500 other market
quotable swap instruments is sought to be reduced to usually 3 or 4 principal
components, representing the path of interest rates on a macro basis.
Converting risks to be represented as those to factor loadings (or multipliers)
provides assessments and understanding beyond that available to simply
collectively viewing risks to individual 30–500 buckets.

PCA has also been applied to equity portfolios in a similar fashion,^[43] both
to portfolio risk and to risk return. One application is to reduce portfolio
risk, where allocation strategies are applied to the "principal portfolios"
instead of the underlying stocks.^[44] A second is to enhance portfolio return,
using the principal components to select stocks with upside potential.^[
citation needed]

Neuroscience[edit]

A variant of principal components analysis is used in neuroscience to identify
the specific properties of a stimulus that increase a neuron's probability of
generating an action potential.^[45] 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.^[46]

Relation with other methods[edit]

Correspondence analysis[edit]

Correspondence analysis (CA) was developed by Jean-Paul Benzécri^[47] 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.^[48] 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.^[49]

Factor analysis[edit]

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".^[50] 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.^[10]^: 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.^[51]

K-means clustering[edit]

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.^[52]^[53] However, that PCA is a useful relaxation of k
-means clustering was not a new result,^[54] and it is straightforward to
uncover counterexamples to the statement that the cluster centroid subspace is
spanned by the principal directions.^[55]

Non-negative matrix factorization[edit]

[page1-500px-Fractional_Residual_Variances_comparison]
 
Fractional residual variance (FRV) plots for PCA and NMF;^[22] 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 are declining 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,^[20]^[21]^[22] 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.^[18] For NMF, its components are
ranked based only on the empirical FRV curves.^[22] The residual fractional
eigenvalue plots, that is, 1 − ∑ i = 1 k λ i / ∑ j = 1 n λ j {\displaystyle 1-\
sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}} {\displaystyle
1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}} as a function
of component number k {\displaystyle k} k given a total of n {\displaystyle n} 
n components, for PCA has a flat plateau, where no data is captured to remove
the quasi-static noise, then the curves dropped quickly as an indication of
over-fitting and captures random noise.^[18] The FRV curves for NMF is
decreasing continuously^[22] when the NMF components are constructed
sequentially,^[21] indicating the continuous capturing of quasi-static noise;
then converge to higher levels than PCA,^[22] indicating the less over-fitting
property of NMF.

Iconography of correlations[edit]

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.

[220px-AirMerIconographyCor]
 
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".

Generalizations[edit]

Sparse PCA[edit]

Main article: 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,^[56]
  • a convex relaxation/semidefinite programming framework,^[57]
  • a generalized power method framework^[58]
  • an alternating maximization framework^[59]
  • forward-backward greedy search and exact methods using branch-and-bound
    techniques,^[60]
  • Bayesian formulation framework.^[61]

The methodological and theoretical developments of Sparse PCA as well as its
applications in scientific studies were recently reviewed in a survey paper.^
[62]

Nonlinear PCA[edit]

[300px-Elmap_breastcancer_wiki]
 
Linear PCA versus nonlinear Principal Manifolds^[63] 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.^[64]
Software is available for free non-commercial use.^[65]

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^[66] 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.^[67] 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,^[68] 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[edit]

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^[69] 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).^[5]^[3]

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.^[70]^[71]^[72]

Similar techniques[edit]

Independent component analysis[edit]

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[edit]

Given a matrix E {\displaystyle E} E, it tries to decompose it into two
matrices such that E = A P {\displaystyle E=AP} E=AP . A key difference from
techniques such as PCA and ICA is that some of the entries of A {\displaystyle
A} A are constrained to be 0. Here P {\displaystyle P} 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. A {\displaystyle A} A has full column rank
 2. Each column of A {\displaystyle A} A must have at least L − 1 {\
    displaystyle L-1} L-1 zeroes where L {\displaystyle L} L is the number of
    columns of A {\displaystyle A} A (or alternatively the number of rows of P
    {\displaystyle P} 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. P {\displaystyle P} P must have full row rank.

then the decomposition is unique up to multiplication by a scalar.^[73]

Discriminant analysis of principal components[edit]

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^[74] 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: https://
adegenet.r-forge.r-project.org/)

Software/source code[edit]

  • ALGLIB - a C++ and C# library that implements PCA and truncated PCA
  • Analytica – The built-in EigenDecomp function computes principal
    components.
  • ELKI – includes PCA for projection, including robust variants of PCA, as
    well as PCA-based clustering algorithms.
  • Gretl – principal component analysis can be performed either via the pca
    command or via the princomp() function.
  • Julia – Supports PCA with the pca function in the MultivariateStats package
  • KNIME – A java based nodal arranging software for Analysis, in this the
    nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily.
  • Mathematica – Implements principal component analysis with the
    PrincipalComponents command using both covariance and correlation methods.
  • MathPHP – PHP mathematics library with support for PCA.
  • MATLAB Statistics Toolbox – The functions princomp and pca (R2012b) give
    the principal components, while the function pcares gives the residuals and
    reconstructed matrix for a low-rank PCA approximation.
  • Matplotlib – Python library have a PCA package in the .mlab module.
  • mlpack – Provides an implementation of principal component analysis in C++.
  • NAG Library – Principal components analysis is implemented via the g03aa
    routine (available in both the Fortran versions of the Library).
  • NMath – Proprietary numerical library containing PCA for the .NET Framework
    .
  • GNU Octave – Free software computational environment mostly compatible with
    MATLAB, the function princomp gives the principal component.
  • OpenCV
  • Oracle Database 12c – Implemented via DBMS_DATA_MINING.SVDS_SCORING_MODE by
    specifying setting value SVDS_SCORING_PCA
  • Orange (software) – Integrates PCA in its visual programming environment.
    PCA displays a scree plot (degree of explained variance) where user can
    interactively select the number of principal components.
  • Origin – Contains PCA in its Pro version.
  • Qlucore – Commercial software for analyzing multivariate data with instant
    response using PCA.
  • R – Free statistical package, the functions princomp and prcomp can be used
    for principal component analysis; prcomp uses singular value decomposition
    which generally gives better numerical accuracy. Some packages that
    implement PCA in R, include, but are not limited to: ade4, vegan,
    ExPosition, dimRed, and FactoMineR.
  • SAS – Proprietary software; for example, see^[75]
  • Scikit-learn – Python library for machine learning which contains PCA,
    Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the
    decomposition module.
  • Weka – Java library for machine learning which contains modules for
    computing principal components.

See also[edit]

  • Correspondence analysis (for contingency tables)
  • Multiple correspondence analysis (for qualitative variables)
  • Factor analysis of mixed data (for quantitative and qualitative variables)
  • Canonical correlation
  • CUR matrix approximation (can replace of low-rank SVD approximation)
  • Detrended correspondence analysis
  • Dynamic mode decomposition
  • Eigenface
  • Exploratory factor analysis (Wikiversity)
  • Factorial code
  • Functional principal component analysis
  • Geometric data analysis
  • Independent component analysis
  • Kernel PCA
  • L1-norm principal component analysis
  • Low-rank approximation
  • Matrix decomposition
  • Non-negative matrix factorization
  • Nonlinear dimensionality reduction
  • Oja's rule
  • Point distribution model (PCA applied to morphometry and computer vision)
  • Principal component analysis (Wikibooks)
  • Principal component regression
  • Singular spectrum analysis
  • Singular value decomposition
  • Sparse PCA
  • Transform coding
  • Weighted least squares

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Further reading[edit]

  • Jackson, J.E. (1991). A User's Guide to Principal Components (Wiley).
  • Jolliffe, I. T. (1986). Principal Component Analysis. Springer Series in
    Statistics. Springer-Verlag. pp. 487. CiteSeerX 10.1.1.149.8828. doi:
    10.1007/b98835. ISBN 978-0-387-95442-4.
  • Jolliffe, I. T. (2002). Principal Component Analysis. Springer Series in
    Statistics. New York: Springer-Verlag. doi:10.1007/b98835. ISBN 
    978-0-387-95442-4.
  • Husson François, Lê Sébastien & Pagès Jérôme (2009). Exploratory
    Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series,
    London. 224p. ISBN 978-2-7535-0938-2
  • Pagès Jérôme (2014). Multiple Factor Analysis by Example Using R. Chapman &
    Hall/CRC The R Series London 272 p

External links[edit]

● Wikimedia Commons has media related to Principal component analysis.

  • University of Copenhagen video by Rasmus Bro on YouTube
  • Stanford University video by Andrew Ng on YouTube
  • A Tutorial on Principal Component Analysis
  • A layman's introduction to principal component analysis on YouTube (a video
    of less than 100 seconds.)
  • StatQuest: Principal Component Analysis (PCA) clearly explained on YouTube
  • See also the list of Software implementations

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                                          • Likelihood-ratio
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                     Goodness of     • Jarque–Bera
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                                     • Model selection
  Specific tests                         □ Cross validation
                                         □ AIC
                                         □ BIC

                                     • Sign
                                         □ Sample median
                                     • Signed rank (Wilcoxon)
                                         □ Hodges–Lehmann estimator
                         Rank        • Rank sum (Mann–Whitney)
                      statistics     • Nonparametric anova
                                         □ 1-way (Kruskal–Wallis)
                                         □ 2-way (Friedman)
                                         □ Ordered alternative 
                                           (Jonckheere–Terpstra)

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Bayesian inference    • Credible interval
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                                  • Correlation
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