random-fu-0.1.4: Random number generation

Data.Random.Distribution

Synopsis

# Documentation

class Distribution d t whereSource

A `Distribution` is a data representation of a random variable's probability structure. For example, in Data.Random.Distribution.Normal, the `Normal` distribution is defined as:

``` data Normal a
= StdNormal
| Normal a a
```

Where the two parameters of the `Normal` data constructor are the mean and standard deviation of the random variable, respectively. To make use of the `Normal` type, one can convert it to an `rvar` and manipulate it or sample it directly:

``` x <- sample (rvar (Normal 10 2))
x <- sample (Normal 10 2)
```

A `Distribution` is typically more transparent than an `RVar` but less composable (precisely because of that transparency). There are several practical uses for types implementing `Distribution`:

• Typically, a `Distribution` will expose several parameters of a standard mathematical model of a probability distribution, such as mean and std deviation for the normal distribution. Thus, they can be manipulated analytically using mathematical insights about the distributions they represent. For example, a collection of bernoulli variables could be simplified into a (hopefully) smaller collection of binomial variables.
• Because they are generally just containers for parameters, they can be easily serialized to persistent storage or read from user-supplied configurations (eg, initialization data for a simulation).
• If a type additionally implements the `CDF` subclass, which extends `Distribution` with a cumulative density function, an arbitrary random variable `x` can be tested against the distribution by testing `fmap (cdf dist) x` for uniformity.

On the other hand, most `Distribution`s will not be closed under all the same operations as `RVar` (which, being a monad, has a fully turing-complete internal computational model). The sum of two uniformly-distributed variables, for example, is not uniformly distributed. To support general composition, the `Distribution` class defines a function `rvar` to construct the more-abstract and more-composable `RVar` representation of a random variable.

Methods

rvar :: d t -> RVar tSource

Return a random variable with this distribution.

rvarT :: d t -> RVarT n tSource

Return a random variable with the given distribution, pre-lifted to an arbitrary `RVarT`. Any arbitrary `RVar` can also be converted to an 'RVarT m' for an arbitrary `m`, using either `lift` or `sample`.

Instances

 Distribution StdUniform Bool Distribution StdUniform Char Distribution StdUniform Double Distribution StdUniform Float Distribution StdUniform Int Distribution StdUniform Int8 Distribution StdUniform Int16 Distribution StdUniform Int32 Distribution StdUniform Int64 Distribution StdUniform Ordering Distribution StdUniform Word Distribution StdUniform Word8 Distribution StdUniform Word16 Distribution StdUniform Word32 Distribution StdUniform Word64 Distribution StdUniform () Distribution Uniform Bool Distribution Uniform Char Distribution Uniform Double Distribution Uniform Float Distribution Uniform Int Distribution Uniform Int8 Distribution Uniform Int16 Distribution Uniform Int32 Distribution Uniform Int64 Distribution Uniform Integer Distribution Uniform Ordering Distribution Uniform Word Distribution Uniform Word8 Distribution Uniform Word16 Distribution Uniform Word32 Distribution Uniform Word64 Distribution Uniform () (Floating a, Distribution StdUniform a) => Distribution Weibull a (Floating a, Distribution StdUniform a) => Distribution Exponential a Distribution Normal Double Distribution Normal Float (Floating a, Ord a, Distribution Normal a, Distribution StdUniform a) => Distribution Gamma a Distribution Beta Double Distribution Beta Float (RealFloat a, Distribution StdUniform a) => Distribution Rayleigh a (RealFloat a, Ord a, Distribution StdUniform a) => Distribution Triangular a HasResolution r => Distribution StdUniform (Fixed r) HasResolution r => Distribution Uniform (Fixed r) (Fractional a, Distribution Gamma a) => Distribution Dirichlet [a] (Fractional b, Ord b, Distribution StdUniform b) => Distribution (Bernoulli b) Bool Distribution (Bernoulli b[a16gB]) Bool => Distribution (Bernoulli b[a16gB]) Word64 Distribution (Bernoulli b[a16gx]) Bool => Distribution (Bernoulli b[a16gx]) Word32 Distribution (Bernoulli b[a16gt]) Bool => Distribution (Bernoulli b[a16gt]) Word16 Distribution (Bernoulli b[a16gp]) Bool => Distribution (Bernoulli b[a16gp]) Word8 Distribution (Bernoulli b[a16gl]) Bool => Distribution (Bernoulli b[a16gl]) Word Distribution (Bernoulli b[a16gh]) Bool => Distribution (Bernoulli b[a16gh]) Int64 Distribution (Bernoulli b[a16gd]) Bool => Distribution (Bernoulli b[a16gd]) Int32 Distribution (Bernoulli b[a16g9]) Bool => Distribution (Bernoulli b[a16g9]) Int16 Distribution (Bernoulli b[a16g5]) Bool => Distribution (Bernoulli b[a16g5]) Int8 Distribution (Bernoulli b[a16g1]) Bool => Distribution (Bernoulli b[a16g1]) Int Distribution (Bernoulli b[a16fU]) Bool => Distribution (Bernoulli b[a16fU]) Integer Distribution (Bernoulli b[a16tQ]) Bool => Distribution (Bernoulli b[a16tQ]) Double Distribution (Bernoulli b[a16tM]) Bool => Distribution (Bernoulli b[a16tM]) Float (Fractional p, Ord p, Distribution StdUniform p) => Distribution (Categorical p) a (Num t, Ord t, Vector v t) => Distribution (Ziggurat v) t (Integral a, Floating b, Ord b, Distribution Normal b, Distribution StdUniform b) => Distribution (Erlang a) b (Floating b[a1ogO], Ord b[a1ogO], Distribution Beta b[a1ogO], Distribution StdUniform b[a1ogO]) => Distribution (Binomial b[a1ogO]) Word64 (Floating b[a1ogI], Ord b[a1ogI], Distribution Beta b[a1ogI], Distribution StdUniform b[a1ogI]) => Distribution (Binomial b[a1ogI]) Word32 (Floating b[a1ogC], Ord b[a1ogC], Distribution Beta b[a1ogC], Distribution StdUniform b[a1ogC]) => Distribution (Binomial b[a1ogC]) Word16 (Floating b[a1ogw], Ord b[a1ogw], Distribution Beta b[a1ogw], Distribution StdUniform b[a1ogw]) => Distribution (Binomial b[a1ogw]) Word8 (Floating b[a1ogq], Ord b[a1ogq], Distribution Beta b[a1ogq], Distribution StdUniform b[a1ogq]) => Distribution (Binomial b[a1ogq]) Word (Floating b[a1ogk], Ord b[a1ogk], Distribution Beta b[a1ogk], Distribution StdUniform b[a1ogk]) => Distribution (Binomial b[a1ogk]) Int64 (Floating b[a1oge], Ord b[a1oge], Distribution Beta b[a1oge], Distribution StdUniform b[a1oge]) => Distribution (Binomial b[a1oge]) Int32 (Floating b[a1og8], Ord b[a1og8], Distribution Beta b[a1og8], Distribution StdUniform b[a1og8]) => Distribution (Binomial b[a1og8]) Int16 (Floating b[a1og2], Ord b[a1og2], Distribution Beta b[a1og2], Distribution StdUniform b[a1og2]) => Distribution (Binomial b[a1og2]) Int8 (Floating b[a1ofW], Ord b[a1ofW], Distribution Beta b[a1ofW], Distribution StdUniform b[a1ofW]) => Distribution (Binomial b[a1ofW]) Int (Floating b[a1ofQ], Ord b[a1ofQ], Distribution Beta b[a1ofQ], Distribution StdUniform b[a1ofQ]) => Distribution (Binomial b[a1ofQ]) Integer Distribution (Binomial b[a1ous]) Integer => Distribution (Binomial b[a1ous]) Double Distribution (Binomial b[a1oum]) Integer => Distribution (Binomial b[a1oum]) Float (RealFloat b[a1u9B], Distribution StdUniform b[a1u9B], Distribution (Erlang Word64) b[a1u9B], Distribution (Binomial b[a1u9B]) Word64) => Distribution (Poisson b[a1u9B]) Word64 (RealFloat b[a1u9x], Distribution StdUniform b[a1u9x], Distribution (Erlang Word32) b[a1u9x], Distribution (Binomial b[a1u9x]) Word32) => Distribution (Poisson b[a1u9x]) Word32 (RealFloat b[a1u9t], Distribution StdUniform b[a1u9t], Distribution (Erlang Word16) b[a1u9t], Distribution (Binomial b[a1u9t]) Word16) => Distribution (Poisson b[a1u9t]) Word16 (RealFloat b[a1u9p], Distribution StdUniform b[a1u9p], Distribution (Erlang Word8) b[a1u9p], Distribution (Binomial b[a1u9p]) Word8) => Distribution (Poisson b[a1u9p]) Word8 (RealFloat b[a1u9l], Distribution StdUniform b[a1u9l], Distribution (Erlang Word) b[a1u9l], Distribution (Binomial b[a1u9l]) Word) => Distribution (Poisson b[a1u9l]) Word (RealFloat b[a1u9h], Distribution StdUniform b[a1u9h], Distribution (Erlang Int64) b[a1u9h], Distribution (Binomial b[a1u9h]) Int64) => Distribution (Poisson b[a1u9h]) Int64 (RealFloat b[a1u9d], Distribution StdUniform b[a1u9d], Distribution (Erlang Int32) b[a1u9d], Distribution (Binomial b[a1u9d]) Int32) => Distribution (Poisson b[a1u9d]) Int32 (RealFloat b[a1u99], Distribution StdUniform b[a1u99], Distribution (Erlang Int16) b[a1u99], Distribution (Binomial b[a1u99]) Int16) => Distribution (Poisson b[a1u99]) Int16 (RealFloat b[a1u95], Distribution StdUniform b[a1u95], Distribution (Erlang Int8) b[a1u95], Distribution (Binomial b[a1u95]) Int8) => Distribution (Poisson b[a1u95]) Int8 (RealFloat b[a1u91], Distribution StdUniform b[a1u91], Distribution (Erlang Int) b[a1u91], Distribution (Binomial b[a1u91]) Int) => Distribution (Poisson b[a1u91]) Int (RealFloat b[a1u8X], Distribution StdUniform b[a1u8X], Distribution (Erlang Integer) b[a1u8X], Distribution (Binomial b[a1u8X]) Integer) => Distribution (Poisson b[a1u8X]) Integer Distribution (Poisson b[a1umc]) Integer => Distribution (Poisson b[a1umc]) Double Distribution (Poisson b[a1um8]) Integer => Distribution (Poisson b[a1um8]) Float (Distribution (Bernoulli b) Bool, RealFloat a) => Distribution (Bernoulli b) (Complex a) (Distribution (Bernoulli b) Bool, Integral a) => Distribution (Bernoulli b) (Ratio a) (Num a, Fractional p, Distribution (Binomial p) a) => Distribution (Multinomial p) [a]

class Distribution d t => CDF d t whereSource

Methods

cdf :: d t -> t -> DoubleSource

Return the cumulative distribution function of this distribution. That is, a function taking `x :: t` to the probability that the next sample will return a value less than or equal to x, according to some order or partial order (not necessarily an obvious one).

In the case where `t` is an instance of Ord, `cdf` should correspond to the CDF with respect to that order.

In other cases, `cdf` is only required to satisfy the following law: `fmap (cdf d) (rvar d)` must be uniformly distributed over (0,1). Inclusion of either endpoint is optional, though the preferred range is (0,1].

Note that this definition requires that `cdf` for a product type should _not_ be a joint CDF as commonly defined, as that definition violates both conditions. Instead, it should be a univariate CDF over the product type. That is, it should represent the CDF with respect to the lexicographic order of the product.

The present specification is probably only really useful for testing conformance of a variable to its target distribution, and I am open to suggestions for more-useful specifications (especially with regard to the interaction with product types).

Instances

 CDF StdUniform Bool CDF StdUniform Char CDF StdUniform Double CDF StdUniform Float CDF StdUniform Int CDF StdUniform Int8 CDF StdUniform Int16 CDF StdUniform Int32 CDF StdUniform Int64 CDF StdUniform Ordering CDF StdUniform Word CDF StdUniform Word8 CDF StdUniform Word16 CDF StdUniform Word32 CDF StdUniform Word64 CDF StdUniform () CDF Uniform Bool CDF Uniform Char CDF Uniform Double CDF Uniform Float CDF Uniform Int CDF Uniform Int8 CDF Uniform Int16 CDF Uniform Int32 CDF Uniform Int64 CDF Uniform Integer CDF Uniform Ordering CDF Uniform Word CDF Uniform Word8 CDF Uniform Word16 CDF Uniform Word32 CDF Uniform Word64 CDF Uniform () (Real a, Distribution Weibull a) => CDF Weibull a (Real a, Distribution Exponential a) => CDF Exponential a (Real a, Distribution Normal a) => CDF Normal a (Real a, Distribution Rayleigh a) => CDF Rayleigh a (RealFrac a, Distribution Triangular a) => CDF Triangular a HasResolution r => CDF StdUniform (Fixed r) HasResolution r => CDF Uniform (Fixed r) (Distribution (Bernoulli b) Bool, Real b) => CDF (Bernoulli b) Bool CDF (Bernoulli b[a16gD]) Bool => CDF (Bernoulli b[a16gD]) Word64 CDF (Bernoulli b[a16gz]) Bool => CDF (Bernoulli b[a16gz]) Word32 CDF (Bernoulli b[a16gv]) Bool => CDF (Bernoulli b[a16gv]) Word16 CDF (Bernoulli b[a16gr]) Bool => CDF (Bernoulli b[a16gr]) Word8 CDF (Bernoulli b[a16gn]) Bool => CDF (Bernoulli b[a16gn]) Word CDF (Bernoulli b[a16gj]) Bool => CDF (Bernoulli b[a16gj]) Int64 CDF (Bernoulli b[a16gf]) Bool => CDF (Bernoulli b[a16gf]) Int32 CDF (Bernoulli b[a16gb]) Bool => CDF (Bernoulli b[a16gb]) Int16 CDF (Bernoulli b[a16g7]) Bool => CDF (Bernoulli b[a16g7]) Int8 CDF (Bernoulli b[a16g3]) Bool => CDF (Bernoulli b[a16g3]) Int CDF (Bernoulli b[a16fW]) Bool => CDF (Bernoulli b[a16fW]) Integer CDF (Bernoulli b[a16tS]) Bool => CDF (Bernoulli b[a16tS]) Double CDF (Bernoulli b[a16tO]) Bool => CDF (Bernoulli b[a16tO]) Float (Real b[a1ogR], Distribution (Binomial b[a1ogR]) Word64) => CDF (Binomial b[a1ogR]) Word64 (Real b[a1ogL], Distribution (Binomial b[a1ogL]) Word32) => CDF (Binomial b[a1ogL]) Word32 (Real b[a1ogF], Distribution (Binomial b[a1ogF]) Word16) => CDF (Binomial b[a1ogF]) Word16 (Real b[a1ogz], Distribution (Binomial b[a1ogz]) Word8) => CDF (Binomial b[a1ogz]) Word8 (Real b[a1ogt], Distribution (Binomial b[a1ogt]) Word) => CDF (Binomial b[a1ogt]) Word (Real b[a1ogn], Distribution (Binomial b[a1ogn]) Int64) => CDF (Binomial b[a1ogn]) Int64 (Real b[a1ogh], Distribution (Binomial b[a1ogh]) Int32) => CDF (Binomial b[a1ogh]) Int32 (Real b[a1ogb], Distribution (Binomial b[a1ogb]) Int16) => CDF (Binomial b[a1ogb]) Int16 (Real b[a1og5], Distribution (Binomial b[a1og5]) Int8) => CDF (Binomial b[a1og5]) Int8 (Real b[a1ofZ], Distribution (Binomial b[a1ofZ]) Int) => CDF (Binomial b[a1ofZ]) Int (Real b[a1ofT], Distribution (Binomial b[a1ofT]) Integer) => CDF (Binomial b[a1ofT]) Integer CDF (Binomial b[a1ouv]) Integer => CDF (Binomial b[a1ouv]) Double CDF (Binomial b[a1oup]) Integer => CDF (Binomial b[a1oup]) Float (Real b[a1u9D], Distribution (Poisson b[a1u9D]) Word64) => CDF (Poisson b[a1u9D]) Word64 (Real b[a1u9z], Distribution (Poisson b[a1u9z]) Word32) => CDF (Poisson b[a1u9z]) Word32 (Real b[a1u9v], Distribution (Poisson b[a1u9v]) Word16) => CDF (Poisson b[a1u9v]) Word16 (Real b[a1u9r], Distribution (Poisson b[a1u9r]) Word8) => CDF (Poisson b[a1u9r]) Word8 (Real b[a1u9n], Distribution (Poisson b[a1u9n]) Word) => CDF (Poisson b[a1u9n]) Word (Real b[a1u9j], Distribution (Poisson b[a1u9j]) Int64) => CDF (Poisson b[a1u9j]) Int64 (Real b[a1u9f], Distribution (Poisson b[a1u9f]) Int32) => CDF (Poisson b[a1u9f]) Int32 (Real b[a1u9b], Distribution (Poisson b[a1u9b]) Int16) => CDF (Poisson b[a1u9b]) Int16 (Real b[a1u97], Distribution (Poisson b[a1u97]) Int8) => CDF (Poisson b[a1u97]) Int8 (Real b[a1u93], Distribution (Poisson b[a1u93]) Int) => CDF (Poisson b[a1u93]) Int (Real b[a1u8Z], Distribution (Poisson b[a1u8Z]) Integer) => CDF (Poisson b[a1u8Z]) Integer CDF (Poisson b[a1ume]) Integer => CDF (Poisson b[a1ume]) Double CDF (Poisson b[a1uma]) Integer => CDF (Poisson b[a1uma]) Float (CDF (Bernoulli b) Bool, RealFloat a) => CDF (Bernoulli b) (Complex a) (CDF (Bernoulli b) Bool, Integral a) => CDF (Bernoulli b) (Ratio a)