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: 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 (e.g. C in Cp=c). If you have no equality constraints initialize this to a zero by zero matrix. Note that there is no need to supply the vector c, it is defined implicitly by the initial parameter estimates p. S A list of penalty matrices. `S[[i]]` is the smallest contiguous matrix including all the non-zero elements of the ith penalty matrix. The first parameter it penalizes is given by `off[i]+1` (starting counting at 1). 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) 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=c 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.wood@r-project.org

### 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.maths.bath.ac.uk/~sw283/

### 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),S=list(),
    Ain=matrix(0,1,2),bin=0,C=matrix(0,0,0),sp=array(0,0),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)

    # Penalized example: 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, constraints etc. for monotonic spline....
    sm<-smoothCon(s(x,k=10,bs="cr"),dat,knots=NULL)
    F<-mono.con(sm$xp);   # get constraints
    G<-list(X=sm$X,C=matrix(0,0,0),sp=f.ug$sp,p=sm$xp,y=y,w=y*0+1)
    G$Ain<-F$A;G$bin<-F$b;G$S<-sm$S;G$off<-0

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

    fv<-Predict.matrix(sm,data.frame(x=x))%*%p
    lines(x,fv,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....
    sm<-smoothCon(s(x,k=10,bs="tp"),dat,knots=NULL)
    xc<-0:39/39 # points on [0,1]
    nc<-length(xc)  # number of constraints
    xc<-xc*4-1  # points at which to impose constraints
    A0<-Predict.matrix(sm,data.frame(x=xc))
    # ... A0
    A1<-Predict.matrix(sm,data.frame(x=xc+1e-6))
    A<-(A1-A0)/1e-6
    # ... approx. constraint matrix (A%*%p is -ve spline gradient at points xc)
    G<-list(X=sm$X,C=matrix(0,0,0),sp=f.ug$sp,y=y,w=y*0+1,S=sm$S,off=0)
    G$Ain<-A;    # constraint matrix
    G$bin<-rep(0,nc);  # constraint vector
    G$p<-rep(0,10);G$p[10]<-0.1
    # ... monotonic start params, got by setting coefs of polynomial part
    p<-pcls(G);  # fit spline (using s.p. from unconstrained fit)

    fv2<-Predict.matrix(sm,data.frame(x=x))%*%p
    lines(x,fv2,col=3)

[Package mgcv version 1.3-17 Index]