smooth.construct.cx.smooth.spec: Constructor for convex P-splines in SCAMs

Description

This is a special method function for creating smooths subject to convexity constraint which is built by the mgcv constructor function for smooth terms, smooth.construct. It is constructed using convex P-splines. This smooth is specified via model terms such as s(x,k,bs="cx",m=2), where k denotes the basis dimension and m+1 is the order of the B-spline basis.

cxBy.smooth.spec works similar to cx.smooth.spec but without applying an identifiability constraint ('zero intercept' constraint). cxBy.smooth.spec should be used when the smooth term has a numeric by variable that takes more than one value. In such cases, the smooth terms are fully identifiable without a 'zero intercept' constraint, so they are left unconstrained. This smooth is specified as s(x,by=z,bs="cxBy"). See an example below.

However a factor by variable requires identifiability constraints, so s(x,by=fac,bs="cx") is used in this case.

Usage

# S3 method for cx.smooth.spec
smooth.construct(object, data, knots)
# S3 method for cxBy.smooth.spec
smooth.construct(object, data, knots)

Value

An object of class "cx.smooth", "cxBy.smooth".

Arguments

object: A smooth specification object, generated by an s term in a GAM formula.
data: A data frame or list containing the data required by this term, with names given by object$term. The by variable is the last element.
knots: An optional list containing the knots supplied for basis setup. If it is NULL then the knot locations are generated automatically.

Author

Natalya Pya <nat.pya@gmail.com>

References

Pya, N. and Wood, S.N. (2015) Shape constrained additive models. Statistics and Computing, 25(3), 543-559

Pya, N. (2010) Additive models with shape constraints. PhD thesis. University of Bath. Department of Mathematical Sciences

Examples

Run this code

 if (FALSE) {
 
## Convex SCOP-splines example... 
  ## simulating data...
  require(scam)
  set.seed(16)
  n <- 100
  x <- sort(2*runif(n)-1)
  f <- 4*x^2
  y <- f + rnorm(n)*0.4
  dat <- data.frame(x=x,y=y)
  b <- scam(y~s(x,k=15,bs="cx"),family=gaussian,data=dat)
  ## unconstrained fit...
  b1 <- scam(y~s(x,k=15),family=gaussian, data=dat)
  ## plot results ...
  plot(x,y,xlab="x",ylab="y")
  lines(x,f)      ## the true function
  lines(x,b$fitted,col=2)  ## constrained fit 
  lines(x,b1$fitted,col=3) ## unconstrained fit 

## Poisson version...
  set.seed(18)
  y <- rpois(n,exp(f))
  dat <- data.frame(x=x,y=y)
  ## fit shape constrained model ...
  b <- scam(y~s(x,k=15,bs="cx"),family=poisson(link="log"),data=dat,optimizer="efs")
  ## unconstrained fit... 
  b1 <- scam(y~s(x,k=15),family=poisson(link="log"), data=dat,optimizer="efs")
  ## plot results ...
  plot(x,y,xlab="x",ylab="y")
  lines(x,exp(f))      ## the true function
  lines(x,b$fitted,col=2)  ## constrained fit 
  lines(x,b1$fitted,col=3) ## unconstrained fit 

## 'by' factor example... 
  set.seed(9)
  n <- 400
  x <- sort(runif(n,-.5,.5))
  f1 <- .7*x-cos(x)+3
  f2 <- 20*x^2 
  par(mfrow=c(1,2))
  plot(x,f1,type="l");plot(x,f2,type="l")
  e <- rnorm(n, 0, 1.5)
  fac <- as.factor(sample(1:2,n,replace=TRUE))
  fac.1 <- as.numeric(fac==1)
  fac.2 <- as.numeric(fac==2)
  y <- f1*fac.1 + f2*fac.2 + e 
  dat <- data.frame(y=y,x=x,fac=fac,f1=f1,f2=f2)
  b2 <- scam(y ~ fac+s(x,by=fac,bs="cx"),data=dat,optimizer="efs")  
  plot(b2,pages=1,scale=0)
  summary(b2)
  x11()
  vis.scam(b2,theta=50,color="terrain")

 ## numeric 'by' variable example... 
 set.seed(6)
 n <- 100
 x <- sort(2*runif(n)-1)
 z <- runif(n,-2,3)
 f <- 4*x^2
 y <- f*z + rnorm(n)*.6
 dat <- data.frame(x=x,z=z,y=y)
 b <- scam(y~s(x,k=15,by=z,bs="cxBy"),data=dat)
 summary(b)
 par(mfrow=c(1,2))
 plot(b,shade=TRUE)
 ## unconstrained fit...
 b1 <- scam(y~s(x,k=15,by=z),data=dat)
 plot(b1,shade=TRUE)
 summary(b1)
 }

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