| Copyright | (c) 2009 Bryan O'Sullivan |
|---|---|
| License | BSD3 |
| Maintainer | bos@serpentine.com |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell98 |
Statistics.Distribution
Description
Type classes for probability distributions
- class Distribution d where
- class Distribution d => DiscreteDistr d where
- class Distribution d => ContDistr d where
- class Distribution d => MaybeMean d where
- class MaybeMean d => Mean d where
- class MaybeMean d => MaybeVariance d where
- class (Mean d, MaybeVariance d) => Variance d where
- class Distribution d => MaybeEntropy d where
- class MaybeEntropy d => Entropy d where
- class Distribution d => ContGen d where
- class (DiscreteDistr d, ContGen d) => DiscreteGen d where
- genContinous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double
- findRoot :: ContDistr d => d -> Double -> Double -> Double -> Double -> Double
- sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double
Type classes
class Distribution d where #
Type class common to all distributions. Only c.d.f. could be defined for both discrete and continous distributions.
Minimal complete definition
Methods
cumulative :: d -> Double -> Double #
Cumulative distribution function. The probability that a random variable X is less or equal than x, i.e. P(X≤x). 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
class Distribution d => DiscreteDistr d where #
Discrete probability distribution.
Methods
probability :: d -> Int -> Double #
Probability of n-th outcome.
logProbability :: d -> Int -> Double #
Logarithm of probability of n-th outcome
class Distribution d => ContDistr d where #
Continuous probability distributuion.
Minimal complete definition is quantile and either density or
logDensity.
Minimal complete definition
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(X≤x) = p. If probability is outside
of [0,1] range function should call error
logDensity :: d -> Double -> Double #
Natural logarithm of density.
Instances
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
Instances
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
Instances
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
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.
Instances
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
Methods
maybeEntropy :: d -> Maybe Double #
Returns the entropy of a distribution, in nats, if such is defined.
Instances
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
Instances
Random number generation
class Distribution d => ContGen d where #
Generate discrete random variates which have given distribution.
Minimal complete definition
Instances
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
genContinous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double #
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 #
Sum probabilities in inclusive interval.