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This library provides methods to construct transport maps that transform the uniform distribution on an interval to some distribution on the same interval that has mass concentrated in user-specified regions.

Install

Add (currently) unregistered public Julia package for dependency before installing IntervalMonoFuncs.jl

import Pkg
Pkg.add(path="https://github.com/RoyCCWang/IntervalMonoFuncs.jl")

To update this package once it is installed, do

Pkg.update("IntervalMonoFuncs")

Overview

The following summarizes the usage of the public API:

  • getpiecewiselines() constructs piecewise-linear functions from finite intervals on $\mathbb{R}$ to the same interval (an endomorphism in mathematics). The constructed function is a transport map that transforms the uniform distribution over the interval to a piecewise-uniform distribution over the interval. The construction method requires inputs that specify properties of the piecewise-uniform distribution and the endomorphism domain/range interval.

  • methods to evaluate piecewise-linear and a the composition of a logistic and a probit function, and their inverse. A composite function of this type is referred as a logistic-probit function throughout this package. It has domain and range [0,1].

  • createendopiewiselines1() is similar to getpiecewiselines(), but is designed specifically to generate a family of two-segment piecewise-linear functions. The generated functions are evenly "spaced/centered" over the user-specified subintervals in the domain and range. The domain and range is fixed to a subset of [-1,1] in the current version of this package. This creates a transport map that could drastically relocate the mass to a single interval.

  • getlogisticprobitparameters() is a routine to fit the parameters of a logistic-probit function to each family of two-segment piecewise-linear functions generated by createendopiewiselines1(), if the domain and range is set to the interval [0,1]. This creates a smooth version for each of the piecewise-uniform transport maps (each a two-segment piecewise-linear function).

Nomenclature

See our nomenclature page.

Usage examples

Julia examples: