pcls {mgcv}R Documentation

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

M is the single list argument to pcls. It should have the following elements (last 3 are not in argument for mgcv) :
    y
    The response data vector.
    w
    A vector of weights for the data (often proportional to the reciprocal of the variance).
    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.
    C
    Matrix containing any linear equality constraints on the problem (i.e. C in Cp=0). If you have no equality constraints initialize this to a zero by zero matrix.
    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$df[i]*(j-1) on its ith row and jth column. The kth full penalty matrix is constructed by placing the matrix just derived into an appropriate matrix of zeroes with it's 1,1 element at M$off[k],M$off[k]
    off
    Offset values locating the elements of M$S[] in the correct location within each penalty coefficient matrix. (Zero offset implies starting in first location)

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

    sp
    An array of smoothing parameter estimates.
    p
    An array of feasible initial parameter estimates - these must satisfy the constraints, but should avoid satisfying the inequality constraints as equality constraints.
    Ain
    Matrix for the inequality constraints A_in p > b.
    bin
    vector in the inequality constraints.

Details

This solves the problem:

minimise || W^0.5 (Xp-y) ||^2 + lambda_1 p'S_1 p + lambda_1 p'S_2 p + . . .

subject to constraints Cp=0 and A_in p > b_in, w.r.t. p given the smoothing parameters lambda_i. X is a design matrix, p a parameter vector, y a data vector, W a diagonal weight matrix, S_i a positive semi-definite matrix of coefficients defining the ith penalty and C a matrix of coefficients defining the linear equality constraints on the problem. The smoothing parameters are the lambda_i. Note that X must be of full column rank, at least when projected into the null space of any equality constraints. A_in is a matrix of coefficients defining the inequality constraints, while 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. X'X is not formed explicitly. See Gill et al. 1981.

Value

The function returns an array containing the estimated parameter vector.

Author(s)

Simon N. Wood simon@stats.gla.ac.uk

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.stats.gla.ac.uk/~simon/

See Also

mgcv mono.con

Examples

# first an un-penalized example - fit E(y)=a+bx subject to a>0
set.seed(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.
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)
dat<-data.frame(x=x,y=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,dat,fit.method="mgcv")->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)

# now a tprs example of the same thing....

f.ug<-gam(y~s(x,k=10));lines(x,fitted(f.ug))
# Create Design matrix, constriants etc. for monotonic spline....
gam.setup(y~s(x,k=10),dat,fit.method="mgcv")->G;
GAMsetup(G)->G;
nc<-40         # number of constraints
xc<-0:nc/nc # points on [0,1]  
xc<-xc*4-1  # points at which to impose constraints
A0<-predict.gam(f.ug,data.frame(x=xc),type="lpmatrix") 
# ... A0
A1<-predict.gam(f.ug,data.frame(x=xc+1e-6),type="lpmatrix") 
A<-(A1-A0)/1e-6    
# ... approx. constraint matrix (A
G$Ain<-A;    # constraint matrix
G$bin<-rep(0,nc);  # constraint vector
G$sp<-f.ug$sp; # use smoothing parameters from un-constrained fit
G$p<-rep(0,11);G$p[11]<-0.1  
# ... monotonic start params, got by setting coefs of polynomial part
G$p[10]<- -mean(0.1*x) 
# ... must meet gam side conditions: sum of smooth over x's is zero 
G$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. 
f.ug$coefficients<-p; 
lines(x,predict.gam(f.ug,newdata=data.frame(x=x)),col=3)
[Package Contents]