Creating a Kernel

2024-07-08

Kernel Class

Similarly to implemented probability distributions, implemented kernels all inherit from a single class ‘kernel’, which in turn inherits from Distribution (see the uml diagram). But this is a much simpler class with reduced methods and many of these return the same result and are therefore defined in the kernel parent class. For every kernel the following is true

• kernel$mean() == 0
• kernel$median() == 0
• kernel$mode() == 0
• kernel$type() == Reals$new()
• kernel$valueSupport() == continuous
• kernel$variateForm() == univariate

And for all kernels the sampling method uses inverse transform sampling, so this is also defined in the parent class. Hence the only methods required to add are

• pdfSquared2Norm: This is new for kernels (i.e. not given for probability distributions)
• variance

And then the d/p/q functions are given in the constructor just like probability distributions.

Creating a Kernel

Kernel Variables

These are identical to the SDistribution public variables:

• name - Full (unique) name of kernel
• short_name - Short name (unique) id for kernel
• description - Short description, usually just the name
• package - The package in which the d/p/q functions are written

For the Epanechnikov kernel, the above all looks like

Epanechnikov <- R6::R6Class("Epanechnikov", inherit = Kernel, lock_objects = F)
Epanechnikov$set("public","name","Epanechnikov")
Epanechnikov$set("public","short_name","Epan")
Epanechnikov$set("public","description","Epanechnikov Kernel")

Note:

1. Again note the use of lock_objects = F which ensures correct usage with decorators.
2. The package variable is omitted as it defaults to ‘distr6’

Kernel Methods

As stated above, there are fewer methods that need to be implemented in kernels than probability distributions. These include

• variance
• pdfSquared2Norm: the squared 2-norm of the kernel’s pdf over the full support

Again the d/p/q methods are implemented in the constructor. So for the Epanechnikov kernel,

Epanechnikov$set("public","pdfSquared2Norm",function(){
  return(3/5)
})
Epanechnikov$set("public","variance",function(){
  return(1/5)
})

Note:

1. These methods will always return constants, they are defined as methods not variables for consistency
2. If no constant numeric is available, omit the method so a numeric one can be added after decoration
3. As well as the methods listen above, the following methods are also included by default and don’t need to be defined
   • prec
   • stdev
   • iqr

The Constructor

The constructor for kernels is much more simple than that of probability distributions and only need include the d/p/q methods and properties. For the Epanechnikov kernel:

Epanechnikov$set("public","initialize",function(decorators = NULL){
  pdf <- function(x1){
    return(0.75 * (1-x1^2))
  }
  cdf <- function(x1){
    return(3/4*x1 - 1/4*x1^3 + 1/2)
  }

  super$initialize(decorators = decorators, pdf = pdf, cdf = cdf,
                   support = set6::Interval$new(-1, 1),  symmetric = TRUE)
  invisible(self)
})

Note:

1. We omit the quantile method as no closed form analytic expression was found
2. The support will generally either be Reals$new() or Interval$new(-1,1)
3. Kernels are all symmetric
4. The constructor always takes one argument only, decorators which is passed to the parent-class constructor

Summary

Kernels are much simpler to extend than SDistributions. Just remember the following

1. The 4 public variables: name, short_name, description, package
2. The 2 public methods: variance, pdfSquared2Norm
3. The constructor: Includes d/p/q, the only argument is decorators, and the properties are generally identical

Extension Guidelines

• SDistribution
• Wrapper
• Decorator