naginterfaces.library.correg.pls_​wold

naginterfaces.library.correg.pls_wold(x, isx, y, iscale, xstd, ystd, maxfac, maxit=200, tau=0.0001, io_manager=None)

pls_wold fits an orthogonal scores partial least squares (PLS) regression by using Wold’s iterative method.

For full information please refer to the NAG Library document for g02lb

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02lbf.html

Parameters

xfloat, array-like, shape $(n,\text{mx})$

$x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}$th observation on the $\text{j}$th predictor variable, for $\text{j}=1,2,\dots ,\text{mx}$, for $\text{i}=1,2,\dots ,n$.

isxint, array-like, shape $\left(\text{mx}\right)$

Indicates which predictor variables are to be included in the model.

$isx[j-1]=1$

The $j$th predictor variable (with variates in the $j$th column of $X$) is included in the model.

$isx[j-1]=0$

Otherwise.

yfloat, array-like, shape $(n,\text{my})$

$y[\text{i}-1,\text{j}-1]$ must contain the $\text{i}$th observation for the $\text{j}$th response variable, for $\text{j}=1,2,\dots ,\text{my}$, for $\text{i}=1,2,\dots ,n$.

iscaleint

Indicates how predictor variables are scaled.

$iscale=1$

Data are scaled by the standard deviation of variables.

$iscale=2$

Data are scaled by user-supplied scalings.

$iscale=-1$

No scaling.

xstdfloat, array-like, shape $\left(\text{ip}\right)$

If $iscale=2$, $xstd[\text{j}-1]$ must contain the user-supplied scaling for the $\text{j}$th predictor variable in the model, for $\text{j}=1,2,\dots ,\text{ip}$. Otherwise $xstd$ need not be set.

ystdfloat, array-like, shape $\left(\text{my}\right)$

If $iscale=2$, $ystd[\text{j}-1]$ must contain the user-supplied scaling for the $\text{j}$th response variable in the model, for $\text{j}=1,2,\dots ,\text{my}$. Otherwise $ystd$ need not be set.

maxfacint

$k$, the number of latent variables to calculate.

maxitint, optional

If $\text{my}=1$, $maxit$ is not referenced; otherwise the maximum number of iterations used to calculate the $x$-weights.

taufloat, optional

If $\text{my}=1$, $tau$ is not referenced; otherwise the iterative procedure used to calculate the $x$-weights will halt if the Euclidean distance between two subsequent estimates is less than or equal to $tau$.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

xbarfloat, ndarray, shape $\left(\text{ip}\right)$

Mean values of predictor variables in the model.

ybarfloat, ndarray, shape $\left(\text{my}\right)$

The mean value of each response variable.

xstdfloat, ndarray, shape $\left(\text{ip}\right)$

If $iscale=1$, standard deviations of predictor variables in the model. Otherwise $xstd$ is not changed.

ystdfloat, ndarray, shape $\left(\text{my}\right)$

If $iscale=1$, the standard deviation of each response variable. Otherwise $ystd$ is not changed.

xresfloat, ndarray, shape $(n,\text{ip})$

The predictor variables’ residual matrix $X_k$.

yresfloat, ndarray, shape $(n,\text{my})$

The residuals for each response variable, $Y_k$.

wfloat, ndarray, shape $(\text{ip},maxfac)$

The $\text{j}$th column of $W$ contains the $x$-weights $w_{\text{j}}$, for $\text{j}=1,2,\dots ,maxfac$.

pfloat, ndarray, shape $(\text{ip},maxfac)$

The $\text{j}$th column of $P$ contains the $x$-loadings $p_{\text{j}}$, for $\text{j}=1,2,\dots ,maxfac$.

tfloat, ndarray, shape $(n,maxfac)$

The $\text{j}$th column of $T$ contains the $x$-scores $t_{\text{j}}$, for $\text{j}=1,2,\dots ,maxfac$.

cfloat, ndarray, shape $(\text{my},maxfac)$

The $\text{j}$th column of $C$ contains the $y$-loadings $c_{\text{j}}$, for $\text{j}=1,2,\dots ,maxfac$.

ufloat, ndarray, shape $(n,maxfac)$

The $\text{j}$th column of $U$ contains the $y$-scores $u_{\text{j}}$, for $\text{j}=1,2,\dots ,maxfac$.

xcvfloat, ndarray, shape $\left(maxfac\right)$

$xcv[\text{j}-1]$ contains the cumulative percentage of variance in the predictor variables explained by the first $\text{j}$ factors, for $\text{j}=1,2,\dots ,maxfac$.

ycvfloat, ndarray, shape $(maxfac,\text{my})$

$ycv[\text{i}-1,\text{j}-1]$ is the cumulative percentage of variance of the $\text{j}$th response variable explained by the first $\text{i}$ factors, for $\text{j}=1,2,\dots ,\text{my}$, for $\text{i}=1,2,\dots ,maxfac$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $1$)

On entry, $\text{mx}=⟨value⟩$.

Constraint: $\text{mx}>1$.

(errno $1$)

On entry, $isx[⟨value⟩]$ is invalid.

Constraint: $isx[j-1]=0$ or $1$, for all $j$.

(errno $1$)

On entry, $\text{my}=⟨value⟩$.

Constraint: $\text{my}\ge 1$.

(errno $1$)

On entry, $iscale=⟨value⟩$.

Constraint: $iscale=-1$ or $1$.

(errno $2$)

On entry, $\text{ip}=⟨value⟩$ and $\text{mx}=⟨value⟩$.

Constraint: $1<\text{ip}\le \text{mx}$.

(errno $2$)

On entry, $maxfac=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Constraint: $1\le maxfac\le \text{ip}$.

(errno $2$)

On entry, $\text{my}=⟨value⟩$ and $maxit=⟨value⟩$.

Constraint: if $\text{my}>1$, $maxit>1$.

(errno $2$)

On entry, $tau=⟨value⟩$.

Constraint: if $\text{my}>1$, $tau>0.0$.

(errno $3$)

On entry, $\text{ip}=⟨value⟩$ and $sum\left(isx\right)=⟨value⟩$.

Constraint: the sum of elements in $isx$ must equal $\text{ip}$.

Notes

Let $X_1$ be the mean-centred $n\times m$ data matrix $X$ of $n$ observations on $m$ predictor variables. Let $Y_1$ be the mean-centred $n\times r$ data matrix $Y$ of $n$ observations on $r$ response variables.

The first of the $k$ factors PLS methods extract from the data predicts both $X_1$ and $Y_1$ by regressing on a $t_1$ column vector of $n$ scores:

$$\begin{array}{c}\begin{array}{cc}{\hat{X}}_1=t_1{p}_1^T& \\ {\hat{Y}}_1=t_1{c}_1^T\text{,}& \text{with}{t}_1^T{t}_1=1\text{,}\end{array}\end{array}$$

where the column vectors of $m$ $x$-loadings $p_1$ and $r$ $y$-loadings $c_1$ are calculated in the least squares sense:

$$\begin{array}{c}\begin{array}{c}{p}_1^T=t_1^T{X}_1\\ c_1^T=t_1^T{Y}_1\text{.}\end{array}\end{array}$$

The $x$-score vector $t_1=X_1{w}_1$ is the linear combination of predictor data $X_1$ that has maximum covariance with the $y$-scores $u_1=Y_1{c}_1$, where the $x$-weights vector $w_1$ is the normalised first left singular vector of $X_1^T{Y}_1$.

The method extracts subsequent PLS factors by repeating the above process with the residual matrices:

$$\begin{array}{c}\begin{array}{c}{X}_i=X_{i-1}-{\hat{X}}_{i-1}\\ Y_i=Y_{i-1}-{\hat{Y}}_{i-1}\text{,}i=2,3,\dots ,k\text{,}\end{array}\end{array}$$

and with orthogonal scores:

$$t_i^T{t}_j=0\text{,}j=1,2,\dots ,i-1\text{.}$$

Optionally, in addition to being mean-centred, the data matrices $X_1$ and $Y_1$ may be scaled by standard deviations of the variables. If data are supplied mean-centred, the calculations are not affected within numerical accuracy.

References

Wold, H, 1966, Estimation of principal components and related models by iterative least squares, In: Multivariate Analysis, (ed P R Krishnaiah), 391–420, Academic Press NY