# NAG CL Interfaceg02kac (ridge_​opt)

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## 1Purpose

g02kac calculates a ridge regression, optimizing the ridge parameter according to one of four prediction error criteria.

## 2Specification

 #include
 void g02kac (Nag_OrderType order, Integer n, Integer m, const double x[], Integer pdx, const Integer isx[], Integer ip, double tau, const double y[], double *h, Nag_PredictError opt, Integer *niter, double tol, double *nep, Nag_EstimatesOption orig, double b[], double vif[], double res[], double *rss, Integer *df, Nag_OptionLOO optloo, double perr[], NagError *fail)
The function may be called by the names: g02kac, nag_correg_ridge_opt or nag_regsn_ridge_opt.

## 3Description

A linear model has the form:
 $y = c+Xβ+ε ,$
where
• $y$ is an $n×1$ matrix of values of a dependent variable;
• $c$ is a scalar intercept term;
• $X$ is an $n×m$ matrix of values of independent variables;
• $\beta$ is an $m×1$ matrix of unknown values of parameters;
• $\epsilon$ is an $n×1$ matrix of unknown random errors such that variance of $\epsilon ={\sigma }^{2}I$.
Let $\stackrel{~}{X}$ be the mean-centred $X$ and $\stackrel{~}{y}$ the mean-centred $y$. Furthermore, $\stackrel{~}{X}$ is scaled such that the diagonal elements of the cross product matrix ${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}$ are one. The linear model now takes the form:
 $y~ = X~ β~ + ε .$
Ridge regression estimates the parameters $\stackrel{~}{\beta }$ in a penalised least squares sense by finding the $\stackrel{~}{b}$ that minimizes
 $‖X~b~-y~‖ 2 + h ‖b~‖ 2 , h>0 ,$
where $‖·‖$ denotes the ${\ell }_{2}$-norm and $h$ is a scalar regularization or ridge parameter. For a given value of $h$, the parameter estimates $\stackrel{~}{b}$ are found by evaluating
 $b~ = (X~TX~+hI)-1 X~T y~ .$
Note that if $h=0$ the ridge regression solution is equivalent to the ordinary least squares solution.
Rather than calculate the inverse of (${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}+hI$) directly, g02kac uses the singular value decomposition (SVD) of $\stackrel{~}{X}$. After decomposing $\stackrel{~}{X}$ into $UD{V}^{\mathrm{T}}$ where $U$ and $V$ are orthogonal matrices and $D$ is a diagonal matrix, the parameter estimates become
 $b~ = V (DTD+hI)-1 D UT y~ .$
A consequence of introducing the ridge parameter is that the effective number of parameters, $\gamma$, in the model is given by the sum of diagonal elements of
 $DT D (DTD+hI)-1 ,$
see Moody (1992) for details.
Any multi-collinearity in the design matrix $X$ may be highlighted by calculating the variance inflation factors for the fitted model. The $j$th variance inflation factor, ${v}_{j}$, is a scaled version of the multiple correlation coefficient between independent variable $j$ and the other independent variables, ${R}_{j}$, and is given by
 $vj = 1 1-Rj , j=1,2,…,m .$
The $m$ variance inflation factors are calculated as the diagonal elements of the matrix:
 $(X~TX~+hI)-1 X~T X~ (X~TX~+hI)-1 ,$
which, using the SVD of $\stackrel{~}{X}$, is equivalent to the diagonal elements of the matrix:
 $V (DTD+hI)-1 DT D (DTD+hI)-1 VT .$
Although parameter estimates $\stackrel{~}{b}$ are calculated by using $\stackrel{~}{X}$, it is usual to report the parameter estimates $b$ associated with $X$. These are calculated from $\stackrel{~}{b}$, and the means and scalings of $X$. Optionally, either $\stackrel{~}{b}$ or $b$ may be calculated.
The method can adopt one of four criteria to minimize while calculating a suitable value for $h$:
1. (a)Generalized cross-validation (GCV):
 $ns (n-γ) 2 ;$
2. (b)Unbiased estimate of variance (UEV):
 $s n-γ ;$
3. (c)Future prediction error (FPE):
 $1n (s+2γs n-γ ) ;$
4. (d)Bayesian information criterion (BIC):
 $1n (s+ log(n)γs n-γ ) ;$
where $s$ is the sum of squares of residuals. However, the function returns all four of the above prediction errors regardless of the one selected to minimize the ridge parameter, $h$. Furthermore, the function will optionally return the leave-one-out cross-validation (LOOCV) prediction error.

## 4References

Hastie T, Tibshirani R and Friedman J (2003) The Elements of Statistical Learning: Data Mining, Inference and Prediction Springer Series in Statistics
Moody J.E. (1992) The effective number of parameters: An analysis of generalisation and regularisation in nonlinear learning systems In: Neural Information Processing Systems (eds J E Moody, S J Hanson, and R P Lippmann) 4 847–854 Morgan Kaufmann San Mateo CA

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
3: $\mathbf{m}$Integer Input
On entry: the number of independent variables available in the data matrix $X$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
4: $\mathbf{x}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array x must be at least
• ${\mathbf{pdx}}×{\mathbf{m}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{pdx}}×{\mathbf{n}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the values of independent variables in the data matrix $X$.
5: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• otherwise ${\mathbf{pdx}}\ge {\mathbf{m}}$.
6: $\mathbf{isx}\left[{\mathbf{m}}\right]$const Integer Input
On entry: indicates which $m$ independent variables are included in the model.
${\mathbf{isx}}\left[j-1\right]=1$
The $j$th variable in x will be included in the model.
${\mathbf{isx}}\left[j-1\right]=0$
Variable $j$ is excluded.
Constraint: ${\mathbf{isx}}\left[\mathit{j}-1\right]=0$ or $1$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
7: $\mathbf{ip}$Integer Input
On entry: $m$, the number of independent variables in the model.
Constraints:
• $1\le {\mathbf{ip}}\le {\mathbf{m}}$;
• Exactly ip elements of isx must be equal to $1$.
8: $\mathbf{tau}$double Input
On entry: singular values less than tau of the SVD of the data matrix $X$ will be set equal to zero.
Suggested value: ${\mathbf{tau}}=0.0$.
Constraint: ${\mathbf{tau}}\ge 0.0$.
9: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input
On entry: the $n$ values of the dependent variable $y$.
10: $\mathbf{h}$double * Input/Output
On entry: an initial value for the ridge regression parameter $h$; used as a starting point for the optimization.
Constraint: ${\mathbf{h}}>0.0$.
On exit: h is the optimized value of the ridge regression parameter $h$.
11: $\mathbf{opt}$Nag_PredictError Input
On entry: the measure of prediction error used to optimize the ridge regression parameter $h$. The value of opt must be set equal to one of:
${\mathbf{opt}}=\mathrm{Nag_GCV}$
Generalized cross-validation (GCV);
${\mathbf{opt}}=\mathrm{Nag_UEV}$
Unbiased estimate of variance (UEV)
${\mathbf{opt}}=\mathrm{Nag_FPE}$
Future prediction error (FPE)
${\mathbf{opt}}=\mathrm{Nag_BIC}$
Bayesian information criteron (BIC).
Constraint: ${\mathbf{opt}}=\mathrm{Nag_GCV}$, $\mathrm{Nag_UEV}$, $\mathrm{Nag_FPE}$ or $\mathrm{Nag_BIC}$.
12: $\mathbf{niter}$Integer * Input/Output
On entry: the maximum number of iterations allowed to optimize the ridge regression parameter $h$.
Constraint: ${\mathbf{niter}}\ge 1$.
On exit: the number of iterations used to optimize the ridge regression parameter $h$ within tol.
13: $\mathbf{tol}$double Input
On entry: iterations of the ridge regression parameter $h$ will halt when consecutive values of $h$ lie within tol.
Constraint: ${\mathbf{tol}}>0.0$.
14: $\mathbf{nep}$double * Output
On exit: the number of effective parameters, $\gamma$, in the model.
15: $\mathbf{orig}$Nag_EstimatesOption Input
On entry: if ${\mathbf{orig}}=\mathrm{Nag_EstimatesOrig}$, the parameter estimates $b$ are calculated for the original data; otherwise ${\mathbf{orig}}=\mathrm{Nag_EstimatesStand}$ and the parameter estimates $\stackrel{~}{b}$ are calculated for the standardized data.
Constraint: ${\mathbf{orig}}=\mathrm{Nag_EstimatesOrig}$ or $\mathrm{Nag_EstimatesStand}$.
16: $\mathbf{b}\left[{\mathbf{ip}}+1\right]$double Output
On exit: contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by isx. The first element of b contains the estimate for the intercept; ${\mathbf{b}}\left[\mathit{j}\right]$ contains the parameter estimate for the $\mathit{j}$th independent variable in the model, for $\mathit{j}=1,2,\dots ,{\mathbf{ip}}$.
17: $\mathbf{vif}\left[{\mathbf{ip}}\right]$double Output
On exit: the variance inflation factors in the order indicated by isx. For the $\mathit{j}$th independent variable in the model, ${\mathbf{vif}}\left[\mathit{j}-1\right]$ is the value of ${v}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ip}}$.
18: $\mathbf{res}\left[{\mathbf{n}}\right]$double Output
On exit: ${\mathbf{res}}\left[\mathit{i}-1\right]$ is the value of the $\mathit{i}$th residual for the fitted ridge regression model, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
19: $\mathbf{rss}$double * Output
On exit: the sum of squares of residual values.
20: $\mathbf{df}$Integer * Output
On exit: the degrees of freedom for the residual sum of squares rss.
21: $\mathbf{optloo}$Nag_OptionLOO Input
On entry: if ${\mathbf{optloo}}=\mathrm{Nag_WantLOO}$, the leave-one-out cross-validation estimate of prediction error is calculated; otherwise no such estimate is calculated and ${\mathbf{optloo}}=\mathrm{Nag_NoLOO}$.
Constraint: ${\mathbf{optloo}}=\mathrm{Nag_NoLOO}$ or $\mathrm{Nag_WantLOO}$.
22: $\mathbf{perr}\left[5\right]$double Output
On exit: the first four elements contain, in this order, the measures of prediction error: GCV, UEV, FPE and BIC.
If ${\mathbf{optloo}}=\mathrm{Nag_WantLOO}$, ${\mathbf{perr}}\left[4\right]$ is the LOOCV estimate of prediction error; otherwise ${\mathbf{perr}}\left[4\right]$ is not referenced.
23: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_2_INT_ARG_CONS
On entry, ${\mathbf{ip}}=⟨\mathit{\text{value}}⟩$; ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{ip}}\le {\mathbf{m}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>1$.
On entry, ${\mathbf{niter}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{niter}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$; ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$; ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
NE_INT_ARG_CONS
On entry, ${\mathbf{ip}}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathrm{sum}\left({\mathbf{isx}}\right)={\mathbf{ip}}$.
NE_INT_ARRAY_VAL_1_OR_2
On entry, ${\mathbf{isx}}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{isx}}\left[j-1\right]=0$ or $1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{h}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{h}}>0.0$.
On entry, ${\mathbf{tau}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tau}}\ge 0.0$.
On entry, ${\mathbf{tol}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tol}}>0.0$.
NE_SVD_FAIL
SVD failed to converge.
NW_TOO_MANY_ITER
Maximum number of iterations used.

Not applicable.

## 8Parallelism and Performance

g02kac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02kac 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

g02kac allocates internally $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(5×\left({\mathbf{n}}-1\right),2×{\mathbf{ip}}×{\mathbf{ip}}\right)+\left({\mathbf{n}}+3\right)×{\mathbf{ip}}+{\mathbf{n}}$ elements of double precision storage.

## 10Example

This example reads in data from an experiment to model body fat, and a ridge regression is calculated that optimizes GCV prediction error.

### 10.1Program Text

Program Text (g02kace.c)

### 10.2Program Data

Program Data (g02kace.d)

### 10.3Program Results

Program Results (g02kace.r)