RandProc.hs - a Haskell library for working with random processes in a mathematically rigorous way

(Concepts taken from Random Processes - a Mathematical Approach for Engineers by:

  • Robert M. Gray
  • Lee D. Davisson

Prentice-Hall Information and System Sciences Series, Thomas Kailath, Series Editor)

$Id: RandProc.hs 31 2011-06-22 13:49:48Z dbanas $

David Banas

12 March 2011

Copyright (c) 2011 by David Banas; All rights reserved World wide.

Revision History:

Date SVN #
2011-03-13 3
Data structures stabilized. isSigma working under minimal, discrete sample testing.
2011-03-18 4
Added isProbMeas, as well as monadic debugging versions of both it and isSigma. Added an example probability space representing a fair die.
2011-03-29 7
Custom intersection functions added and briefly tested.
2011-04-02 8
Custom union functions added and briefly tested. Solution is crude: it is O(N^2), and requires 2 passes over the sample list every time a join is successful. Perhaps, a pre-sort?
2011-06-06 9
Attempted fix of getCompEvent Added smplComp function, as helper to revised getCompEvent. Changed Point to accept Double. Moved all sample spaces to new file, Main.hs. Added input sorting to range. Changed Ranges to be open intervals, in order to allow for complementing out a Point from them.
2011-06-11 10
Major re-write. getCompEvent fixed. All 5 test spaces checking out ok.
2011-06-18 21
Removed sample set order dependency from checkSigma. All 7 test spaces checking out ok.
2011-06-19 22
Added 'union of events is an event' test to checkSigma.
2011-06-20 23
Changed Event from data constructor to type alias, in order to eliminate many instances of 'Event . f . getSamps' code.
2011-06-20 25
Modified smpsSetInt to use a fold.
2011-06-20 26
Defined public interface.
2011-06-21 27
Modified comments for Haddock, and generated HTML docs.
2011-06-22 31
Moved into Data directory.
End of Subversion revision history
This source has been moved to darcs.
Made smplSetUnion more efficient, and tuned remaining performance bottlenecks.

To Do:



data ProbSpace Source

We take a probability space to consist of the following:

  • an 'abstract space' composed of either discrete or continuous (or a mix) samples
  • an 'event space', which must be a Sigma field defined over the abstract space
  • a 'probability measure' defined over the event space
For the sake of efficient coding, the event space and the probability measure are combined in the Haskell data structure, below. This is permissable, because there has to be a 1:1 correspondance between them anyway. And it is preferable, because it:
  • keeps the probabilities more closely associated w/ the events, and
  • avoids duplication of code (i.e. - the list of events).


ProbSpace [Sample] [Measure] 


data Measure Source

Measure has 2 fields:

  • event - a list of samples from the space, and
  • prob - a number between 0 and 1 giving the events probability of occurence.


Measure Event Double 

data Sample Source

This is our abstract data type, which represents a sample in the abstract space.

It has a constructor representing every possible element in the abstract space we're modeling. (Currently, just points and ranges of Doubles.)

Normally, none of the constructors of this type will be called directly. Instead, helper functions are provided, such as point and range, which hide the implementation details from the user, and present a stable interface.

Currently, the sole exception to the above is the Empty constructor, which is really just a hack intended to put off the job of making the functions in this library more intelligent, with regard to their handling of empty lists.




data TestResult Source

Custom data type used for test results and error reporting.




err :: ErrType

checkProbMeas :: ProbSpace -> TestResultSource

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

point :: Double -> SampleSource

This is the helper function intended to be used for constructing a point sample.

range :: (Double, Double) -> SampleSource

This is the helper function intended to be used for constructing a range sample. The range is considered open. That is, its end points are not included.

makeProbSpace :: [(Sample, Double)] -> ProbSpaceSource

This helper function generates a complete and valid probability space, given a discrete sample space and set of probabilities.

rangeBegin :: Sample -> DoubleSource

Gets the beginning point of a range, which is not included in the range, since ranges are considered to be open.

rangeEnd :: Sample -> DoubleSource

Gets the ending point of a range, which is not included in the range,

getProb :: Measure -> DoubleSource

Extracts the probability from a Measure.

getEvent :: Measure -> EventSource

Extracts the Event from a Measure.

getCompEvent :: [Sample] -> Event -> EventSource

Get the complement of an event from the sample space.

eventInt :: Event -> Event -> EventSource

Calculates the intersection of 2 events (i.e. - list of samples).

smplComp :: Sample -> Sample -> [Sample]Source

Returns that portion of the first sample that is disjoint from the second.

isElem :: [Sample] -> Sample -> BoolSource

Determine if a sample is an element of a space.

(Need this, as opposed to just using elem, in order to accomodate ranges.)

noDupEvents :: [Measure] -> BoolSource

Checks a list of measures against duplicate events.

smplInt :: Sample -> Sample -> SampleSource

Returns the intersection between 2 samples.

smplSetInt :: [Sample] -> SampleSource

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

smplUnion :: Sample -> Sample -> [Sample]Source

Returns the union of 2 samples.

Unlike smplInt, smplUnion must return a list since, if the 2 input samples aren't adjacent or overlapping, the union of them is a list containing both.

smplSetUnion :: [Sample] -> [Sample]Source

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

checkSigma :: ProbSpace -> TestResultSource

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

getRsltStr :: TestResult -> StringSource

Turns a value of type TestResult into a human readable string.