pcls: Penalized Constrained Least Squares Fitting

Description

Solves least squares problems with quadratic penalties subject to linear equality and inequality constraints using quadratic programming.

Usage

pcls(M)

Arguments

is the single list argument to pcls. It should have the following elements (last 3 are not in argument for mgcv) :

M$y

The response data vector.

M$w

A vector of weights for the data (often proportional to the reciprocal of the variance).

M$X

The design matrix for the problem, note that ncol(M$X) must give the number of model parameters, while nrow(M$X) should give the number of data.

M$C

Matrix containing any linear equality constraints on the problem (i.e. $\bf C$ in ${\bf Cp}={\bf 0}$). If you have no equality constraints initialize this to a zero by zero matrix.

M$S

A one dimensional array containing the non-zero elements of the penalty matrices. Let start<-sum(M$df[1:(k-1)]^2) or start<-0 if k==1. Then penalty matrix k has M$S[start+i+M$df[i]*(j-1)

M$off

Offset values locating the elements of M$S[] in
   the correct location within each penalty coefficient matrix. (Zero
   offset implies starting in first location)

M$df

M$df[i] gives the number of rows and columns of
               M$S[i] that are non-zero.

M$sp

An array of smoothing parameter estimates.

M$p

An array of feasible initial parameter estimates - these must
satisfy the constraints, but should avoid satisfying the inequality
constraints as equality constraints.

M$Ain

Matrix for the inequality constraints ${\bf A}_{in}
    {\bf p} > {\bf b}_{in}$.

M$bin

vector in the inequality constraints.

`Value`

The function returns an array containing the estimated parameter
  vector.

`Details`

This solves the problem:
 
$$minimise~ \| { \bf W}^{1/2} ({ \bf Xp - y} ) \|^2  +  \sum_{i=1}^m
\lambda_i {\bf p^\prime S}_i{\bf p}$$
subject to constraints ${\bf Cp}={\bf 0}$ and ${\bf
    A}_{in}{\bf p}>{\bf b}_{in}$, w.r.t. $\bf p$ given the
smoothing parameters $\lambda_i$.
${\bf X}$ is a design matrix, $\bf p$ a parameter vector, 
$\bf y$ a data vector, $\bf W$ a diagonal weight matrix,
${\bf S}_i$ a positive semi-definite matrix  of coefficients
defining the ith penalty and $\bf C$ a matrix of coefficients 
defining the linear equality constraints on the problem. The smoothing
parameters are the $\lambda_i$. Note that ${\bf X}$
must be of full column rank, at least when projected  into the null space
of any equality constraints. ${\bf A}_{in}$ is a matrix of
coefficients defining the inequality constraints, while ${\bf
    b}_{in}$ is a vector involved in defining the inequality constraints.  
Quadratic programming is used to perform the solution. The method used
is designed for maximum stability with least squares problems:
i.e. ${\bf X}^\prime {\bf X}$ is not formed explicitly. See
Gill et al. 1981.

`References`

Gill, P.E., Murray, W. and Wright, M.H. (1981) Practical Optimization. Academic
Press, London. 
Wood, S.N. (1994) Monotonic smoothing splines fitted by cross validation SIAM
Journal on Scientific Computing 15(5):1126-1133
http://www.ruwpa.st-and.ac.uk/simon.html

`See Also`

mgcv mono.con

`Examples`

Run this code# first an un-penalized example - fit E(y)=a+bx subject to a>0
n<-100
x<-runif(n);y<-x-0.2+rnorm(n)*0.1
M<-list(X=matrix(0,n,2),p=c(0.1,0.5),off=array(0,0),df=array(0,0),S=0,
Ain=matrix(0,1,2),bin=0,C=matrix(0,0,0),sp<-0,y=y,w=y*0+1)
M$X[,1]<-1;M$X[,2]<-x;M$Ain[1,]<-c(1,0)
pcls(M)->M$p
plot(x,y);abline(M$p,col=2);abline(coef(lm(y~x)),col=3)

# and now a penalized example: a monotonic penalized regression spline .....

# Generate data from a monotonic truth.
set.seed(10);x<-runif(100)*4-1;x<-sort(x);
f<-exp(4*x)/(1+exp(4*x));y<-f+rnorm(100)*0.1;plot(x,y)
# Show regular spline fit (and save fitted object)
f.ug<-gam(y~s(x,k=10,bs="cr"));lines(x,fitted(f.ug))
# Create Design matrix, constriants etc. for monotonic spline....
gam.setup(y~s(x,k=10,bs="cr")-1)->G;GAMsetup(G)->G;F<-mono.con(G$xp);
G$Ain<-F$A;G$bin<-F$b;G$C<-matrix(0,0,0);G$sp<-f.ug$sp;
G$p<-G$xp;G$y<-y;G$w<-y*0+1

p<-pcls(G);  # fit spline (using s.p. from unconstrained fit)

# now modify the gam object from unconstrained fit a little, to use it
# for predicting and plotting constrained fit. 
p<-c(0,p);f.ug$coefficients<-p; 
lines(x,predict.gam(f.ug,newdata=data.frame(x=x)),col=2)
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