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: yThe response data vector. wA vector of weights for the data (often proportional to the reciprocal of the variance). XThe 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. CMatrix 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. pAn array of feasible initial parameter estimates - these must satisfy the constraints, but should avoid satisfying the inequality constraints as equality constraints. AinMatrix for the inequality constraints A_in p > b. binvector in the inequality constraints.

### Details

This solves the problem:

min || 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

`magic`, `mono.con`

### Examples

```require(mgcv)
# 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%*%p evaluates spline at xc points
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 <- 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)

######################################
######################################

## First simulate data...

set.seed(10)
f1 <- function(x) 5*exp(4*x)/(1+exp(4*x));
f2 <- function(x) {
ind <- x > .5
f <- x*0
f[ind] <- (x[ind] - .5)^2*10
f
}
f3 <- function(x) 0.2 * x^11 * (10 * (1 - x))^6 +
10 * (10 * x)^3 * (1 - x)^10
n <- 200
x <- runif(n); z <- runif(n); v <- runif(n)
mu <- f1(x) + f2(z) + f3(v)
y <- mu + rnorm(n)

## Preliminary unconstrained gam fit...
G <- gam(y~s(x)+s(z)+s(v,k=20),fit=FALSE)
b <- gam(G=G)

## generate constraints, by finite differencing
## using predict.gam ....
eps <- 1e-7
pd0 <- data.frame(x=seq(0,1,length=100),z=rep(.5,100),
v=rep(.5,100))
pd1 <- data.frame(x=seq(0,1,length=100)+eps,z=rep(.5,100),
v=rep(.5,100))
X0 <- predict(b,newdata=pd0,type="lpmatrix")
X1 <- predict(b,newdata=pd1,type="lpmatrix")
Xx <- (X1 - X0)/eps ## Xx %*% coef(b) must be positive
pd0 <- data.frame(z=seq(0,1,length=100),x=rep(.5,100),
v=rep(.5,100))
pd1 <- data.frame(z=seq(0,1,length=100)+eps,x=rep(.5,100),
v=rep(.5,100))
X0 <- predict(b,newdata=pd0,type="lpmatrix")
X1 <- predict(b,newdata=pd1,type="lpmatrix")
Xz <- (X1-X0)/eps
G\$Ain <- rbind(Xx,Xz) ## inequality constraint matrix
G\$bin <- rep(0,nrow(G\$Ain))
G\$C = matrix(0,0,ncol(G\$X))
G\$sp <- b\$sp
G\$p <- coef(b)
G\$off <- G\$off-1 ## to match what pcls is expecting
## force inital parameters to meet constraint
G\$p[11:18] <- G\$p[2:9]<- 0
p <- pcls(G) ## constrained fit
par(mfrow=c(2,3))
plot(b) ## original fit
b\$coefficients <- p
plot(b) ## constrained fit
## note that standard errors in preceding plot are obtained from
## unconstrained fit

```

[Package mgcv version 1.8-28 Index]