Random Processes

RandProc is a Haskell library, which defines and supports certain data structures that are convenient when working with random processes in a mathematically rigorous fashion.

The concepts embedded in RandProc come from the text I used in my graduate stochastic processes course:

Robert M. Gray, Lee D. Davisson; Random Processes - a Mathematical Approach for Engineers; Prentice-Hall Information and System Sciences Series, Thomas Kailath, Series Editor.

Authors: David Banas

Installation

Download the `randproc` package:

$ cabal install randproc

and import it:

import Data.RandProc

Description

Underlying concepts

As per the reference text, a probability space is taken to be composed of:

• an abstract space of samples,
• an event space, which must be a sigma field over the abstract sample space, and
• a probability measure over the event space.

In the code, the event space and probability measure are combined, in order to keep the code concise.

Public interface

New data types

The code defines an abstract data type, Sample, as follows:

data Sample  = Point Double
             | Range (Double, Double)
             | Empty
             | Full
   deriving (Eq, Show)

However, this type is not exported in the public interface of the `RandProc` module. Instead, helper functions point and range are provided, as alternatives, in order to protect against future implementation changes.

Note) There is, currently, a single exception to the above. The Empty value constructor is exported, as it is currently necessary. However, this is really just a hack designed to put off the task of making the support functions more intelligent, with regards to their handling of empty sample sets.

An event is defined in the code as simply an alias for a list of samples:

type Event = [Sample]

And a measure just attaches a probability of occurrence to an event:

data Measure = Measure {
   event :: Event
   ,prob :: Double
}
   deriving (Eq, Ord, Show)

Supporting functions

Several supporting functions are provided for working with the new data types described, above. A few of the more interesting examples are given, below. For a complete list, refer to the documentation.

checkProbMeas

Checks a value of type 'ProbSpace' for correctness, and returns a value of type 'TestResult'.

checkSigma

Checks whether event space is actually a Sigma field over the sample space.

smplSetUnion

Collapses a list of samples down to the maximally reduced set, which still composes a proper union of the input.

smplSetInt

Reduces a list of samples to a single sample representing their intersection.

Usage

For more extensive usage examples, see the `test/Test.hs` file.

Construction of a probability space representing a fair coin

fairCoin = ProbSpace
   [point 0.0, point 1.0]
   [Measure ( [Empty]) 0, Measure ( [point 0]) 0.5
   , Measure ( [point 1]) 0.5, Measure ( [point 0, point 1]) 1.0]

Testing the probability space defined, above

checkProbMeas fairCoin

Documentation

The documentation for this library is generated, via Haddock.

Related work

I suspect a potential symbiosis between this library and Probabilistic Functional Programming

Community

RandProcNotes

Blog dedicated to the `RandProc` Haskell library