statistics-0.14.0.2: A library of statistical types, data, and functions

Copyright(c) 2009 Bryan O'Sullivan
LicenseBSD3
Maintainerbos@serpentine.com
Stabilityexperimental
Portabilityportable
Safe HaskellNone
LanguageHaskell98

Statistics.Distribution

Contents

Description

Type classes for probability distributions

Synopsis

Type classes

class Distribution d where #

Type class common to all distributions. Only c.d.f. could be defined for both discrete and continuous distributions.

Minimal complete definition

cumulative

Methods

cumulative :: d -> Double -> Double #

Cumulative distribution function. The probability that a random variable X is less or equal than x, i.e. P(Xx). Cumulative should be defined for infinities as well:

cumulative d +∞ = 1
cumulative d -∞ = 0

complCumulative :: d -> Double -> Double #

One's complement of cumulative distibution:

complCumulative d x = 1 - cumulative d x

It's useful when one is interested in P(X>x) and expression on the right side begin to lose precision. This function have default implementation but implementors are encouraged to provide more precise implementation.

Instances

Distribution ChiSquared # 
Distribution BetaDistribution # 
Distribution BinomialDistribution # 
Distribution PoissonDistribution # 
Distribution CauchyDistribution # 
Distribution DiscreteUniform # 
Distribution ExponentialDistribution # 
Distribution GammaDistribution # 
Distribution GeometricDistribution0 # 
Distribution GeometricDistribution # 
Distribution HypergeometricDistribution # 
Distribution NormalDistribution # 
Distribution StudentT # 
Distribution UniformDistribution # 
Distribution FDistribution # 
Distribution LaplaceDistribution # 
Distribution d => Distribution (LinearTransform d) # 

class Distribution d => ContDistr d where #

Continuous probability distributuion.

Minimal complete definition is quantile and either density or logDensity.

Minimal complete definition

quantile

Methods

density :: d -> Double -> Double #

Probability density function. Probability that random variable X lies in the infinitesimal interval [x,x+δx) equal to density(x)⋅δx

quantile :: d -> Double -> Double #

Inverse of the cumulative distribution function. The value x for which P(Xx) = p. If probability is outside of [0,1] range function should call error

complQuantile :: d -> Double -> Double #

1-complement of quantile:

complQuantile x ≡ quantile (1 - x)

logDensity :: d -> Double -> Double #

Natural logarithm of density.

Instances

ContDistr ChiSquared # 
ContDistr BetaDistribution # 
ContDistr CauchyDistribution # 
ContDistr ExponentialDistribution # 
ContDistr GammaDistribution # 
ContDistr NormalDistribution # 
ContDistr StudentT # 
ContDistr UniformDistribution # 
ContDistr FDistribution # 
ContDistr LaplaceDistribution # 
ContDistr d => ContDistr (LinearTransform d) # 

Distribution statistics

class Distribution d => MaybeMean d where #

Type class for distributions with mean. maybeMean should return Nothing if it's undefined for current value of data

Minimal complete definition

maybeMean

Methods

maybeMean :: d -> Maybe Double #

class MaybeMean d => Mean d where #

Type class for distributions with mean. If distribution have finite mean for all valid values of parameters it should be instance of this type class.

Minimal complete definition

mean

Methods

mean :: d -> Double #

class MaybeMean d => MaybeVariance d where #

Type class for distributions with variance. If variance is undefined for some parameter values both maybeVariance and maybeStdDev should return Nothing.

Minimal complete definition is maybeVariance or maybeStdDev

Instances

MaybeVariance ChiSquared # 
MaybeVariance BetaDistribution # 
MaybeVariance BinomialDistribution # 
MaybeVariance PoissonDistribution # 
MaybeVariance DiscreteUniform # 
MaybeVariance ExponentialDistribution # 
MaybeVariance GammaDistribution # 
MaybeVariance GeometricDistribution0 # 
MaybeVariance GeometricDistribution # 
MaybeVariance HypergeometricDistribution # 
MaybeVariance NormalDistribution # 
MaybeVariance StudentT # 
MaybeVariance UniformDistribution # 
MaybeVariance FDistribution # 
MaybeVariance LaplaceDistribution # 
MaybeVariance d => MaybeVariance (LinearTransform d) # 

class (Mean d, MaybeVariance d) => Variance d where #

Type class for distributions with variance. If distibution have finite variance for all valid parameter values it should be instance of this type class.

Minimal complete definition is variance or stdDev

Methods

variance :: d -> Double #

stdDev :: d -> Double #

Instances

Variance ChiSquared # 
Variance BetaDistribution # 
Variance BinomialDistribution # 
Variance PoissonDistribution # 
Variance DiscreteUniform # 
Variance ExponentialDistribution # 
Variance GammaDistribution # 
Variance GeometricDistribution0 # 
Variance GeometricDistribution # 
Variance HypergeometricDistribution # 
Variance NormalDistribution # 
Variance UniformDistribution # 
Variance LaplaceDistribution # 
Variance d => Variance (LinearTransform d) # 

class Distribution d => MaybeEntropy d where #

Type class for distributions with entropy, meaning Shannon entropy in the case of a discrete distribution, or differential entropy in the case of a continuous one. maybeEntropy should return Nothing if entropy is undefined for the chosen parameter values.

Minimal complete definition

maybeEntropy

Methods

maybeEntropy :: d -> Maybe Double #

Returns the entropy of a distribution, in nats, if such is defined.

Instances

MaybeEntropy ChiSquared # 
MaybeEntropy BetaDistribution # 
MaybeEntropy BinomialDistribution # 
MaybeEntropy PoissonDistribution # 
MaybeEntropy CauchyDistribution # 
MaybeEntropy DiscreteUniform # 
MaybeEntropy ExponentialDistribution # 
MaybeEntropy GammaDistribution # 
MaybeEntropy GeometricDistribution0 # 
MaybeEntropy GeometricDistribution # 
MaybeEntropy HypergeometricDistribution # 
MaybeEntropy NormalDistribution # 
MaybeEntropy StudentT # 
MaybeEntropy UniformDistribution # 
MaybeEntropy FDistribution # 
MaybeEntropy LaplaceDistribution # 
MaybeEntropy d => MaybeEntropy (LinearTransform d) # 

class MaybeEntropy d => Entropy d where #

Type class for distributions with entropy, meaning Shannon entropy in the case of a discrete distribution, or differential entropy in the case of a continuous one. If the distribution has well-defined entropy for all valid parameter values then it should be an instance of this type class.

Minimal complete definition

entropy

Methods

entropy :: d -> Double #

Returns the entropy of a distribution, in nats.

class FromSample d a where #

Estimate distribution from sample. First parameter in sample is distribution type and second is element type.

Minimal complete definition

fromSample

Methods

fromSample :: Vector v a => v a -> Maybe d #

Estimate distribution from sample. Returns nothing is there's not enough data to estimate or sample clearly doesn't come from distribution in question. For example if there's negative samples in exponential distribution.

Instances

FromSample ExponentialDistribution Double #

Create exponential distribution from sample. Returns Nothing if sample is empty or contains negative elements. No other tests are made to check whether it truly is exponential.

FromSample NormalDistribution Double #

Variance is estimated using maximum likelihood method (biased estimation).

Returns Nothing if sample contains less than one element or variance is zero (all elements are equal)

FromSample LaplaceDistribution Double #

Create Laplace distribution from sample. No tests are made to check whether it truly is Laplace. Location of distribution estimated as median of sample.

Random number generation

class Distribution d => ContGen d where #

Generate discrete random variates which have given distribution.

Minimal complete definition

genContVar

Methods

genContVar :: PrimMonad m => d -> Gen (PrimState m) -> m Double #

Instances

ContGen ChiSquared # 

Methods

genContVar :: PrimMonad m => ChiSquared -> Gen (PrimState m) -> m Double #

ContGen BetaDistribution # 
ContGen CauchyDistribution # 
ContGen ExponentialDistribution # 
ContGen GammaDistribution # 
ContGen GeometricDistribution0 # 
ContGen GeometricDistribution # 
ContGen NormalDistribution # 
ContGen StudentT # 

Methods

genContVar :: PrimMonad m => StudentT -> Gen (PrimState m) -> m Double #

ContGen UniformDistribution # 
ContGen FDistribution # 
ContGen LaplaceDistribution # 
ContGen d => ContGen (LinearTransform d) # 

class (DiscreteDistr d, ContGen d) => DiscreteGen d where #

Generate discrete random variates which have given distribution. ContGen is superclass because it's always possible to generate real-valued variates from integer values

Minimal complete definition

genDiscreteVar

Methods

genDiscreteVar :: PrimMonad m => d -> Gen (PrimState m) -> m Int #

genContinuous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double #

Generate variates from continuous distribution using inverse transform rule.

genContinous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double #

Deprecated: Use genContinuous

Backwards compatibility with genContinuous.

Helper functions

findRoot #

Arguments

:: ContDistr d 
=> d

Distribution

-> Double

Probability p

-> Double

Initial guess

-> Double

Lower bound on interval

-> Double

Upper bound on interval

-> Double 

Approximate the value of X for which P(x>X)=p.

This method uses a combination of Newton-Raphson iteration and bisection with the given guess as a starting point. The upper and lower bounds specify the interval in which the probability distribution reaches the value p.

sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double #

Sum probabilities in inclusive interval.