smooth.construct.mifo.smooth.spec: Constructor for monotone increasing SCOP-splines with an additional 'finish at zero' constraint

Description

This is a special method function for creating smooths subject to a monotone increasing constraint plus the smooths should pass through zero at the right-end point of the covariate range. This is similar to the pc argument to s in mgcv(gam) when pc=max(x), where x is a covariate. The smooth is built by the mgcv constructor function for smooth terms, smooth.construct. 'Zero intercept' identifiability constraints used for univariate SCOP-splines are substituted with a 'finish at zero' constraint here. This smooth is specified via model terms such as s(x,k,bs="mifo",m=2), where k denotes the basis dimension and m+1 is the order of the B-spline basis.

Usage

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

Value

An object of class "mifo.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>

Details

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

A 'finish at zero' constraint is achieved by setting the last (m+1) spline coefficients to zero. According to the B-spline basis functions properties, the value of the spline, f(x), is determined by m+2 non-zero basis functions, and only m+1 B-splines are non-zero at knots. Only m+2 B-splines are non-zero on any [k_i, k_{i+1}), and the sum of these m+2 basis functions is 1.

If the knots of the spline are not supplied, then they are placed evenly throughout the covariate values with an exception of the m inner knots preceeding the last inner knot that are joined with that last knot. This is done in order to avoid an otherwise plateau fit at the right-end region. 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.

Note: when a plateau region is expected at the righ-end covariate region, the smooth might result in some decrease when approaching to zero.

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

 
  ## Monotone increasing SCOP-spline examples with a finish at zero constraint...
  set.seed(53)
  n <- 100;x <- runif(n);z <- runif(n)
  pc <- max(x)
  y <- exp(3*x)/10-exp(3*pc)/10 + z*(1-z)*5 + rnorm(100)*.4
  m1 <- scam(y~s(x,bs='mifo')+s(z)) #,knots=knots) 
  plot(m1,pages=1,scale=0)
  summary(m1)
  newd<- data.frame(x=pc,z=0)
  predict(m1,newd, type='terms')

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