R: Constructor for monotone increasing P-splines in SCAMs

smooth.construct.mpi.smooth.spec {scam} | R Documentation

## Constructor for monotone increasing P-splines in SCAMs

### Description

This is a special method function for creating smooths subject to a monotone increasing constraint 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="mpi",m=2)`, where `k` denotes the basis dimension and `m+1` is the order of the B-spline basis.

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

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

### Usage

    ## S3 method for class 'mpi.smooth.spec'
    smooth.construct(object, data, knots)
    ## S3 method for class 'mpiBy.smooth.spec'
    smooth.construct(object, data, knots)

### 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.

### Details

The constructor is not called directly, but as with `gam(mgcv)` is used internally.

If the knots of the spline are not supplied, then they are placed evenly throughout the covariate values. If the knots are supplied, then the number of supplied knots should be `k+m+2`, and the range of the middle `k-m` knots must include all the covariate values.

### Value

An object of class `"mpi.smooth"`, `"mpiBy.smooth"`.

### Author(s)

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

### See Also

`smooth.construct.mpd.smooth.spec`, `smooth.construct.cv.smooth.spec`, `smooth.construct.cx.smooth.spec`, `smooth.construct.mdcv.smooth.spec`, `smooth.construct.mdcx.smooth.spec`, `smooth.construct.micv.smooth.spec`, `smooth.construct.micx.smooth.spec`

### Examples

    ## Monotone increasing P-splines example
      ## simulating data...
       require(scam)
       set.seed(12)
       n <- 100
       x <- runif(n)*4-1
       f <- 4*exp(4*x)/(1+exp(4*x))
       y <- rpois(n,exp(f))
       dat <- data.frame(x=x,y=y)
     ## fit model ...
       b <- scam(y~s(x,k=15,bs="mpi"),family=poisson(link="log"),
           data=dat)
     ## fit unconstrained model...
       b1 <- scam(y~s(x,k=15,bs="ps"),family=poisson(link="log"),
             data=dat)
     ## plot results ...
       plot(x,y,xlab="x",ylab="y")
       x1 <- sort(x,index=TRUE)
       lines(x1$x,exp(f)[x1$ix])      ## the true function
       lines(x1$x,b$fitted.values[x1$ix],col=2)  ## monotone fit
       lines(x1$x,b1$fitted.values[x1$ix],col=3) ## unconstrained fit

    ## example with supplied knots...
       knots <- list(x=c (-1.5,  -1.2, -.99, -.97, -.7, -.5, -.3, 0, 0.7,
               0.9,1.1, 1.22,1.5,2.2,2.77,2.93,2.99, 3.2,3.6))
       b2 <- scam(y~s(x,k=15,bs="mpi"),knots=knots,
              family=poisson(link="log"), data=dat)
       summary(b2)
       plot(b2,shade=TRUE)

    ## Not run:
    ## example with two terms...
       set.seed(0)
       n <- 200
       x1 <- runif(n)*6-3
       f1 <- 3*exp(-x1^2) # unconstrained term
       x2 <- runif(n)*4-1;
       f2 <- exp(4*x2)/(1+exp(4*x2)) # monotone increasing smooth
       f <- f1+f2
       y <- f+rnorm(n)*.7
       dat <- data.frame(x1=x1,x2=x2,y=y)
       knots <- list(x1=c(-4,-3.5,-2.99,-2.7,-2.5,-1.9,-1.1,-.9,-.3,0.3,.8,1.2,1.9,2.3,
    2.7,2.99,3.5,4.1,4.5), x2=c(-1.5,-1.2,-1.1, -.89,-.69,-.5,-.3,0,0.7,
    0.9,1.1,1.22,1.5,2.2,2.77,2.99,3.1, 3.2,3.6))
       b3 <- scam(y~s(x1,k=15)+s(x2,bs="mpi", k=15),
             knots=knots,data=dat)
       summary(b3)
       plot(b3,pages=1,shade=TRUE)
    ## setting knots for f(x2) only...
       knots <- list(x2=c(-1.5,-1.2,-1.1, -.89,-.69,-.5,-.3,
       0,0.7,0.9,1.1,1.22,1.5,2.2,2.77,2.99,3.1, 3.2,3.6))
       b4 <- scam(y~s(x1,k=15,bs="bs")+s(x2,bs="mpi",k=15),
           knots=knots,data=dat)
       summary(b4)
       plot(b4,pages=1,shade=TRUE)

    ## 'by' factor example...
     set.seed(2)
     n <- 400
     x <- runif(n, 0, 1)
     ## all three smooths are increasing...
     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)
     b5 <- scam(y ~ fac+s(x,by=fac,bs="mpi"),data=dat)
     plot(b5,pages=1,scale=0,shade=TRUE)
     summary(b5)
     vis.scam(b5,theta=50,color="terrain",cond=list(z=1))

     ## comparing with unconstrained fit...
     b6 <- scam(y ~ fac+s(x,by=fac),data=dat)
     x11()
     plot(b6,pages=1,scale=0,shade=TRUE)
     summary(b6)

     ## Note that 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.
     ## The difference with scam() is that here a 'zero intercept' constraint is
     ## applied in place of 'centering' (although scam's fitted smooths are centred for plotting).
     ## compare with a fit of the model without 'fac' parametric term...
     b7 <- scam(y ~ s(x,by=fac,bs="mpi"),data=dat)
     summary(b7)
     summary(b5)

    ## numeric 'by' variable example...
     set.seed(3)
     n <- 200
     x <- sort(runif(n,-1,2))
     z <- runif(n,-2,3)
     f <- exp(1.3*x)-5
     y <- f*z + rnorm(n)*2
     dat <- data.frame(x=x,y=y,z=z)
     b <- scam(y~s(x,by=z,bs="mpiBy"),data=dat)
     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)

    ## End(Not run)

[Package scam version 1.2-12 Index]