Data.RandProc

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 #

Description

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.

2011-06-27

Made smplSetUnion more efficient, and tuned remaining performance bottlenecks.

To Do:

Documentation

data ProbSpace

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

Note:

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).

Constructors

ProbSpace [Sample] [Measure]

Instances

Show ProbSpace

data Measure

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.

Constructors

Measure Event Double

Instances

Eq Measure

Ord Measure

Show Measure

data Sample

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.

Constructors

Empty

Instances

Eq Sample

Ord Sample

Show Sample

data TestResult

Custom data type used for test results and error reporting.

Constructors

Fail

Fields

err :: ErrType

Pass

Instances

Eq TestResult

Show TestResult

data ErrType

Custom data type for reporting different errors

Constructors

UnknownErr

EmptySampleSpace

EmptyEventSpace

MissingNullEvent

MissingCertainEvent

BadEventSamples

MissingCompEvent

MissingUnionEvent

EventMeasLenMismatch

DupEventsInMeas

MissingEventsInMeas

NullEventNonZeroProb

CertainEventNonUnityProb

EventAndCompNoSumOne

Instances

Eq ErrType

Show ErrType

checkProbMeas :: ProbSpace -> TestResult

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

point :: Double -> Sample

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

range :: (Double, Double) -> Sample

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)] -> ProbSpace

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

rangeBegin :: Sample -> Double

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

rangeEnd :: Sample -> Double

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

getProb :: Measure -> Double

Extracts the probability from a Measure.

getEvent :: Measure -> Event

Extracts the Event from a Measure.

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

Get the complement of an event from the sample space.

eventInt :: Event -> Event -> Event

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

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

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

isElem :: [Sample] -> Sample -> Bool

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] -> Bool

Checks a list of measures against duplicate events.

smplInt :: Sample -> Sample -> Sample

Returns the intersection between 2 samples.

smplSetInt :: [Sample] -> Sample

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

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

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]

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

checkSigma :: ProbSpace -> TestResult

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

getRsltStr :: TestResult -> String

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