g03aaf performs a principal component analysis on a data matrix; both the principal component loadings and the principal component scores are returned.
The routine may be called by the names g03aaf or nagf_mv_prin_comp.
3Description
Let $X$ be an $n\times p$ data matrix of $n$ observations on $p$ variables ${x}_{1},{x}_{2},\dots ,{x}_{p}$ and let the $p\times p$ variance-covariance matrix of ${x}_{1},{x}_{2},\dots ,{x}_{p}$ be $S$. A vector ${a}_{1}$ of length $p$ is found such that:
$${a}_{1}^{\mathrm{T}}S{a}_{1}\text{\hspace{1em} is maximized subject to \hspace{1em}}{a}_{1}^{\mathrm{T}}{a}_{1}=1\text{.}$$
The variable ${z}_{1}={\displaystyle \sum _{i=1}^{p}}{a}_{1i}{x}_{i}$ is known as the first principal component and gives the linear combination of the variables that gives the maximum variation. A second principal component, ${z}_{2}={\displaystyle \sum _{i=1}^{p}}{a}_{2i}{x}_{i}$, is found such that:
$${a}_{2}^{\mathrm{T}}S{a}_{2}\text{\hspace{1em} is maximized subject to}{a}_{2}^{\mathrm{T}}{a}_{2}=1\text{and}{a}_{2}^{\mathrm{T}}{a}_{1}=0\text{.}$$
This gives the linear combination of variables that is orthogonal to the first principal component that gives the maximum variation. Further principal components are derived in a similar way.
The vectors ${a}_{1},{a}_{2},\dots ,{a}_{p}$, are the eigenvectors of the matrix $S$ and associated with each eigenvector is the eigenvalue, ${\lambda}_{i}^{2}$. The value of ${\lambda}_{i}^{2}/\sum {\lambda}_{i}^{2}$ gives the proportion of variation explained by the $i$th principal component. Alternatively, the ${a}_{i}$'s can be considered as the right singular vectors in a singular value decomposition with singular values ${\lambda}_{i}$ of the data matrix centred about its mean and scaled by $1/\sqrt{(n-1)}$, ${X}_{s}$. This latter approach is used in g03aaf, with
$${X}_{s}=V\Lambda {P}^{\prime}$$
where $\Lambda $ is a diagonal matrix with elements ${\lambda}_{i}$, $P$ is the $p\times p$ matrix with columns ${a}_{i}$ and $V$ is an $n\times p$ matrix with ${V}^{\prime}V=I$, which gives the principal component scores.
Principal component analysis is often used to reduce the dimension of a dataset, replacing a large number of correlated variables with a smaller number of orthogonal variables that still contain most of the information in the original dataset.
The choice of the number of dimensions required is usually based on the amount of variation accounted for by the leading principal components. If $k$ principal components are selected, then a test of the equality of the remaining $p-k$ eigenvalues is
which has, asymptotically, a ${\chi}^{2}$-distribution with $\frac{1}{2}(p-k-1)(p-k+2)$ degrees of freedom.
Equality of the remaining eigenvalues indicates that if any more principal components are to be considered then they all should be considered.
Instead of the variance-covariance matrix the correlation matrix, the sums of squares and cross-products matrix or a standardized sums of squares and cross-products matrix may be used. In the last case $S$ is replaced by ${\sigma}^{-\frac{1}{2}}S{\sigma}^{-\frac{1}{2}}$ for a diagonal matrix $\sigma $ with positive elements. If the correlation matrix is used, the ${\chi}^{2}$ approximation for the statistic given above is not valid.
The principal component scores, $F$, are the values of the principal component variables for the observations. These can be standardized so that the variance of these scores for each principal component is $1.0$ or equal to the corresponding eigenvalue.
Weights can be used with the analysis, in which case the matrix $X$ is first centred about the weighted means then each row is scaled by an amount $\sqrt{{w}_{i}}$, where ${w}_{i}$ is the weight for the $i$th observation.
4References
Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall
Cooley W C and Lohnes P R (1971) Multivariate Data Analysis Wiley
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl.20(3) 2–25
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill
5Arguments
1: $\mathbf{matrix}$ – Character(1)Input
On entry: indicates for which type of matrix the principal component analysis is to be carried out.
${\mathbf{matrix}}=\text{'C'}$
It is for the correlation matrix.
${\mathbf{matrix}}=\text{'S'}$
It is for a standardized matrix, with standardizations given by s.
${\mathbf{matrix}}=\text{'U'}$
It is for the sums of squares and cross-products matrix.
${\mathbf{matrix}}=\text{'V'}$
It is for the variance-covariance matrix.
Constraint:
${\mathbf{matrix}}=\text{'C'}$, $\text{'S'}$, $\text{'U'}$ or $\text{'V'}$.
2: $\mathbf{std}$ – Character(1)Input
On entry: indicates if the principal component scores are to be standardized.
${\mathbf{std}}=\text{'S'}$
The principal component scores are standardized so that ${F}^{\prime}F=I$, i.e., $F={X}_{s}P{\Lambda}^{-1}=V$.
${\mathbf{std}}=\text{'U'}$
The principal component scores are unstandardized, i.e., $F={X}_{s}P=V\Lambda $.
${\mathbf{std}}=\text{'Z'}$
The principal component scores are standardized so that they have unit variance.
${\mathbf{std}}=\text{'E'}$
The principal component scores are standardized so that they have variance equal to the corresponding eigenvalue.
Constraint:
${\mathbf{std}}=\text{'E'}$, $\text{'S'}$, $\text{'U'}$ or $\text{'Z'}$.
Constraint:
${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
4: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint:
${\mathbf{n}}\ge 2$.
5: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables in the data matrix.
Constraint:
${\mathbf{m}}\ge 1$.
6: $\mathbf{x}({\mathbf{ldx}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}(\mathit{i},\mathit{j})$ must contain the $\mathit{i}$th observation for the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
7: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g03aaf is called.
On entry: ${\mathbf{isx}}\left(j\right)$ indicates whether or not the $j$th variable is to be included in the analysis.
If
${\mathbf{isx}}\left(\mathit{j}\right)>0$, the variable contained in the $\mathit{j}$th column of x is included in the principal component analysis, for $\mathit{j}=1,2,\dots ,m$.
Constraint:
${\mathbf{isx}}\left(j\right)>0$ for nvar values of $j$.
9: $\mathbf{s}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the standardizations to be used, if any.
If ${\mathbf{matrix}}=\text{'S'}$, the first $m$ elements of s must contain the standardization coefficients, the diagonal elements of $\sigma $.
Constraint:
if ${\mathbf{isx}}\left(j\right)>0$,
${\mathbf{s}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,m$.
On exit: if ${\mathbf{matrix}}=\text{'S'}$, s is unchanged on exit.
If ${\mathbf{matrix}}=\text{'C'}$, s contains the variances of the selected variables. ${\mathbf{s}}\left(j\right)$ contains the variance of the variable in the $j$th column of x if ${\mathbf{isx}}\left(j\right)>0$.
If ${\mathbf{matrix}}=\text{'U'}$ or $\text{'V'}$, s is not referenced.
10: $\mathbf{wt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array wt
must be at least
${\mathbf{n}}$ if ${\mathbf{weight}}=\text{'W'}$, and at least $1$ otherwise.
On entry: if ${\mathbf{weight}}=\text{'W'}$, the first $n$ elements of wt must contain the weights to be used in the principal component analysis.
If ${\mathbf{wt}}\left(i\right)=0.0$, the $i$th observation is not included in the analysis. The effective number of observations is the sum of the weights.
If ${\mathbf{weight}}=\text{'U'}$, wt is not referenced and the effective number of observations is $n$.
Constraints:
${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$;
the sum of weights $\text{}\ge {\mathbf{nvar}}+1$.
11: $\mathbf{nvar}$ – IntegerInput
On entry: $p$, the number of variables in the principal component analysis.
12: $\mathbf{e}({\mathbf{lde}},6)$ – Real (Kind=nag_wp) arrayOutput
On exit: the statistics of the principal component analysis.
${\mathbf{e}}(i,1)$
The eigenvalues associated with the
$\mathit{i}$th principal component, ${\lambda}_{\mathit{i}}^{2}$, for $\mathit{i}=1,2,\dots ,p$.
${\mathbf{e}}(i,2)$
The proportion of variation explained by the
$\mathit{i}$th principal component, for $\mathit{i}=1,2,\dots ,p$.
${\mathbf{e}}(\mathit{i},3)$
The cumulative proportion of variation explained by the first
$\mathit{i}$th principal components, for $\mathit{i}=1,2,\dots ,p$.
${\mathbf{e}}(\mathit{i},4)$
The ${\chi}^{2}$ statistics, for $i=1,2,\dots ,p$.
${\mathbf{e}}(i,5)$
The degrees of freedom for the ${\chi}^{2}$ statistics, for $i=1,2,\dots ,p$.
If ${\mathbf{matrix}}\ne \text{'C'}$,
${\mathbf{e}}(\mathit{i},6)$ contains significance level for the ${\chi}^{2}$ statistic, for $\mathit{i}=1,2,\dots ,p$.
If ${\mathbf{matrix}}=\text{'C'}$, ${\mathbf{e}}(i,6)$ is returned as zero.
13: $\mathbf{lde}$ – IntegerInput
On entry: the first dimension of the array e as declared in the (sub)program from which g03aaf is called.
Constraint:
${\mathbf{lde}}\ge {\mathbf{nvar}}$.
14: $\mathbf{p}({\mathbf{ldp}},{\mathbf{nvar}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the first nvar columns of p contain the principal component loadings, ${a}_{i}$. The $j$th column of p contains the nvar coefficients for the $j$th principal component.
15: $\mathbf{ldp}$ – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which g03aaf is called.
Constraint:
${\mathbf{ldp}}\ge {\mathbf{nvar}}$.
16: $\mathbf{v}({\mathbf{ldv}},{\mathbf{nvar}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the first nvar columns of v contain the principal component scores. The $j$th column of v contains the n scores for the $j$th principal component.
If ${\mathbf{weight}}=\text{'W'}$, any rows for which ${\mathbf{wt}}\left(i\right)$ is zero will be set to zero.
17: $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which g03aaf is called.
Constraint:
${\mathbf{ldv}}\ge {\mathbf{n}}$.
18: $\mathbf{wk}\left(1\right)$ – Real (Kind=nag_wp) arrayInput
This argument is no longer accessed by g03aaf. Workspace is provided internally by dynamic allocation instead.
19: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{lde}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lde}}\ge {\mathbf{nvar}}$.
On entry, ${\mathbf{ldp}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{nvar}}$.
On entry, ${\mathbf{ldv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{matrix}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{matrix}}=\text{'C'}$, $\text{'S'}$, $\text{'U'}$ or $\text{'V'}$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>{\mathbf{nvar}}$.
On entry, ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvar}}\ge 1$.
On entry, ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvar}}\le {\mathbf{m}}$.
On entry, ${\mathbf{std}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{std}}=\text{'E'}$, $\text{'S'}$, $\text{'U'}$ or $\text{'Z'}$.
On entry, ${\mathbf{weight}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
${\mathbf{ifail}}=2$
On entry, $i=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{wt}}\left(i\right)<0.0$.
Constraint: ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$.
${\mathbf{ifail}}=3$
Number of selected variables $\text{}\ge \text{}$ effective number of observations.
On entry, ${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$ and $\u27e8\mathit{\text{value}}\u27e9$ values of ${\mathbf{isx}}>0$.
Constraint: exactly nvar elements of ${\mathbf{isx}}>0$.
${\mathbf{ifail}}=4$
On entry, $j=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{s}}\left(j\right)\le 0.0$.
Constraint: ${\mathbf{s}}\left(j\right)>0.0$.
${\mathbf{ifail}}=5$
The singular value decomposition has failed to converge. This is an unlikely error exit.
${\mathbf{ifail}}=6$
All eigenvalues/singular values are zero. This will be caused by all the variables being constant.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
As g03aaf uses a singular value decomposition of the data matrix, it will be less affected by ill-conditioned problems than traditional methods using the eigenvalue decomposition of the variance-covariance matrix.
8Parallelism and Performance
g03aaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
None.
10Example
A dataset is taken from Cooley and Lohnes (1971), it consists of ten observations on three variables. The unweighted principal components based on the variance-covariance matrix are computed and the principal component scores requested. The principal component scores are standardized so that they have variance equal to the corresponding eigenvalue.