statistics-0.13.3.0: A library of statistical types, data, and functions

Statistics.Distribution

Description

Type classes for probability distributions

Synopsis

# Type classes

class Distribution d where Source #

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

Minimal complete definition

cumulative

Methods

cumulative :: d -> Double -> Double Source #

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 Source #

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

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

class Distribution d => DiscreteDistr d where Source #

Discrete probability distribution.

Methods

probability :: d -> Int -> Double Source #

Probability of n-th outcome.

logProbability :: d -> Int -> Double Source #

Logarithm of probability of n-th outcome

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

class Distribution d => ContDistr d where Source #

Continuous probability distributuion.

Minimal complete definition is quantile and either density or logDensity.

Minimal complete definition

quantile

Methods

density :: d -> Double -> Double Source #

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 Source #

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

logDensity :: d -> Double -> Double Source #

Natural logarithm of density.

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

## Distribution statistics

class Distribution d => MaybeMean d where Source #

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 Source #

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

class MaybeMean d => Mean d where Source #

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 Source #

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Mean d => Mean (LinearTransform d) Source # Methods

class MaybeMean d => MaybeVariance d where Source #

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

Methods

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

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

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 Source #

stdDev :: d -> Double Source #

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

class Distribution d => MaybeEntropy d where Source #

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

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

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

class MaybeEntropy d => Entropy d where Source #

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 Source #

Returns the entropy of a distribution, in nats.

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods (Entropy d, DiscreteDistr d) => Entropy (LinearTransform d) Source # Methods

## Random number generation

class Distribution d => ContGen d where Source #

Generate discrete random variates which have given distribution.

Minimal complete definition

genContVar

Methods

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

Instances

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

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 Source #

Instances

 Source # Methods Source # Methods

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

Generate variates from continous distribution using inverse transform rule.

# Helper functions

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 Source #

Sum probabilities in inclusive interval.