| Copyright | (c) 2009 Bryan O'Sullivan |
|---|---|
| License | BSD3 |
| Maintainer | bos@serpentine.com |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell2010 |
Statistics.Distribution
Description
Type classes for probability distributions
Synopsis
-
class
Distribution
d
where
- cumulative :: d -> Double -> Double
- complCumulative :: d -> Double -> Double
-
class
Distribution
d =>
DiscreteDistr
d
where
- probability :: d -> Int -> Double
- logProbability :: d -> Int -> Double
- class Distribution d => ContDistr d where
- class Distribution d => MaybeMean d where
- class MaybeMean d => Mean d where
-
class
MaybeMean
d =>
MaybeVariance
d
where
- maybeVariance :: d -> Maybe Double
- maybeStdDev :: d -> Maybe Double
- class ( Mean d, MaybeVariance d) => Variance d where
-
class
Distribution
d =>
MaybeEntropy
d
where
- maybeEntropy :: d -> Maybe Double
- class MaybeEntropy d => Entropy d where
-
class
FromSample
d a
where
- fromSample :: Vector v a => v a -> Maybe d
-
class
Distribution
d =>
ContGen
d
where
- genContVar :: StatefulGen g m => d -> g -> m Double
-
class
(
DiscreteDistr
d,
ContGen
d) =>
DiscreteGen
d
where
- genDiscreteVar :: StatefulGen g m => d -> g -> m Int
- genContinuous :: ( ContDistr d, StatefulGen g m) => d -> g -> m Double
- findRoot :: ContDistr d => d -> Double -> Double -> Double -> Double -> Double
- sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double
Type classes
class Distribution d where Source #
Type class common to all distributions. Only c.d.f. could be defined for both discrete and continuous distributions.
Minimal complete definition
( cumulative | complCumulative )
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( X ≤ x ). Cumulative should be defined for infinities as well:
cumulative d +∞ = 1 cumulative d -∞ = 0
complCumulative :: d -> Double -> Double Source #
One's complement of cumulative distribution:
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 Source #
Discrete probability distribution.
Minimal complete definition
( probability | logProbability )
Methods
probability :: d -> Int -> Double Source #
Probability of n-th outcome.
logProbability :: d -> Int -> Double Source #
Logarithm of probability of n-th outcome
Instances
class Distribution d => ContDistr d where Source #
Continuous probability distribution.
Minimal complete definition is
quantile
and either
density
or
logDensity
.
Minimal complete definition
( density | logDensity ), ( quantile | complQuantile )
Methods
density :: d -> Double -> Double Source #
Probability density function. Probability that random variable X lies in the infinitesimal interval [ x , x+ δ x ) equal to density(x) ⋅δ x
logDensity :: d -> Double -> Double Source #
Natural logarithm of density.
quantile :: d -> Double -> Double Source #
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
complQuantile :: d -> Double -> Double Source #
1-complement of
quantile
:
complQuantile x ≡ quantile (1 - x)
Instances
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
Instances
class MaybeMean d => Mean d where Source #
Type class for distributions with mean. If a distribution has finite mean for all valid values of parameters it should be instance of this type class.
Instances
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
Minimal complete definition
( maybeVariance | maybeStdDev )
Instances
class ( Mean d, MaybeVariance d) => Variance d where Source #
Type class for distributions with variance. If distribution have finite variance for all valid parameter values it should be instance of this type class.
Instances
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.
Methods
maybeEntropy :: d -> Maybe Double Source #
Returns the entropy of a distribution, in nats, if such is defined.
Instances
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.
Instances
class FromSample d a where Source #
Estimate distribution from sample. First parameter in sample is distribution type and second is element type.
Methods
fromSample :: Vector v a => v a -> Maybe d Source #
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 WeibullDistribution Double Source # |
Uses an approximation based on the mean and standard deviation in
Returns
|
|
Defined in Statistics.Distribution.Weibull Methods fromSample :: Vector v Double => v Double -> Maybe WeibullDistribution Source # |
|
| FromSample NormalDistribution Double Source # |
Variance is estimated using maximum likelihood method (biased estimation).
Returns
|
|
Defined in Statistics.Distribution.Normal Methods fromSample :: Vector v Double => v Double -> Maybe NormalDistribution Source # |
|
| FromSample LognormalDistribution Double Source # |
Variance is estimated using maximum likelihood method (biased estimation) over the log of the data.
Returns
|
|
Defined in Statistics.Distribution.Lognormal Methods fromSample :: Vector v Double => v Double -> Maybe LognormalDistribution Source # |
|
| FromSample LaplaceDistribution Double Source # |
Create Laplace distribution from sample. No tests are made to check whether it truly is Laplace. Location of distribution estimated as median of sample. |
|
Defined in Statistics.Distribution.Laplace Methods fromSample :: Vector v Double => v Double -> Maybe LaplaceDistribution Source # |
|
| FromSample ExponentialDistribution Double Source # |
Create exponential distribution from sample. Returns
|
|
Defined in Statistics.Distribution.Exponential Methods fromSample :: Vector v Double => v Double -> Maybe ExponentialDistribution Source # |
|
Random number generation
class Distribution d => ContGen d where Source #
Generate discrete random variates which have given distribution.
Methods
genContVar :: StatefulGen g m => d -> g -> 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
Methods
genDiscreteVar :: StatefulGen g m => d -> g -> m Int Source #
Instances
| DiscreteGen GeometricDistribution0 Source # | |
|
Defined in Statistics.Distribution.Geometric Methods genDiscreteVar :: StatefulGen g m => GeometricDistribution0 -> g -> m Int Source # |
|
| DiscreteGen GeometricDistribution Source # | |
|
Defined in Statistics.Distribution.Geometric Methods genDiscreteVar :: StatefulGen g m => GeometricDistribution -> g -> m Int Source # |
|
| DiscreteGen DiscreteUniform Source # | |
|
Defined in Statistics.Distribution.DiscreteUniform Methods genDiscreteVar :: StatefulGen g m => DiscreteUniform -> g -> m Int Source # |
|
genContinuous :: ( ContDistr d, StatefulGen g m) => d -> g -> m Double Source #
Generate variates from continuous 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.