smooth.construct.mpd.smooth.spec: Constructor for monotone decreasing P-splines in SCAMs

Description

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

mpdBy.smooth.spec works similar to mpd.smooth.spec but without applying an identifiability constraint ('zero intercept' constraint). mpdBy.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="mpdBy"). See an example below.

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

Usage

# S3 method for mpd.smooth.spec
smooth.construct(object, data, knots)
# S3 method for mpdBy.smooth.spec
smooth.construct(object, data, knots)

Value

An object of class "mpd.smooth", "mpdBy.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) {
## Monotone decreasing SCOP-splines example... 
  ## simulating data...
   require(scam)
   set.seed(3)
   n <- 100
   x <- runif(n)*3-1
   f <- exp(-1.3*x)
   y <- rpois(n,exp(f))
   dat <- data.frame(x=x,y=y)
 ## fit model ...
   b <- scam(y~s(x,k=15,bs="mpd"),family=poisson(link="log"),
       data=dat)
 ## unconstrained model fit for comparison...
   b1 <- scam(y~s(x,k=15,bs="ps"),family=poisson(link="log"),
         data=dat)
 ## plot results ...
   plot(x,y,xlab="x",ylab="y",cex=.5)
   x1 <- sort(x,index=TRUE)
   lines(x1$x,exp(f)[x1$ix])      ## the true function
   lines(x1$x,b$fitted.values[x1$ix],col=2)  ## decreasing fit 
   lines(x1$x,b1$fitted.values[x1$ix],col=3) ## unconstrained fit 

 ## 'by' factor example... 
 set.seed(3)
 n <- 400
 x <- runif(n, 0, 1)
 ## all three smooths are decreasing...
 f1 <- -log(x *5) 
 f2 <-  -exp(2 * x) + 4
 f3 <-  -5* sin(x)
 e <- rnorm(n, 0, 2)
 fac <- as.factor(sample(1:3,n,replace=TRUE))
 fac.1 <- as.numeric(fac==1)
 fac.2 <- as.numeric(fac==2)
 fac.3 <- as.numeric(fac==3)
 y <- f1*fac.1 + f2*fac.2 + f3*fac.3 + e 
 dat <- data.frame(y=y,x=x,fac=fac,f1=f1,f2=f2,f3=f3)
 b2 <- scam(y ~ fac+s(x,by=fac,bs="mpd"),data=dat)  
 plot(b2,pages=1,scale=0,shade=TRUE)
 summary(b2)
 vis.scam(b2,theta=120,color="terrain")

 ## comparing with unconstrained fit...
 b3 <- scam(y ~ fac+s(x,by=fac),data=dat) 
 x11()
 plot(b3,pages=1,scale=0,shade=TRUE)
 summary(b3)

 ## Note that since in scam() as in mgcv::gam() when using factor 'by' variables, 'centering'
 ## constraints are applied to the smooths, which usually means that the 'by'
 ## factor variable should be included as a parametric term, as well. 


## numeric 'by' variable example...
set.seed(3)
n <- 100
x <- sort(runif(n,-1,2))
z <- runif(n,-2,3)
f <- exp(-1.3*x)
y <- f*z + rnorm(n)*0.4
dat <- data.frame(x=x,y=y,z=z)
b <- scam(y~s(x,k=15,by=z,bs="mpdBy"),data=dat,optimizer="efs")
plot(b,shade=TRUE)
summary(b)
## unconstrained fit...
b1 <- scam(y~s(x,k=15,by=z),data=dat)
plot(b1,shade=TRUE)
summary(b1)
  }

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