Math.NET Numerics provides a wide range of probability distributions. Given the distribution parameters they can be used to investigate their statistical properties or to sample non-uniform random numbers.
All the distributions implement a common set of operations such as evaluating the density (PDF) and the cumulative distribution (CDF) at a given point, or to compute the mean, standard deviation and other properties. Because it is often numerically more stable and faster to compute such statistical quantities in the logarithmic domain, we also provide a selection of them in the log domain with the "Ln" suffix, e.g. DensityLn for the logarithmic density.
Both probability functions and sampling are also available as static functions for simpler usage scenarios:
There are many ways to parametrize a distribution in the literature. When using the default constructor, read carefully which parameters it requires. For distributions where multiple ways are common there are also static methods, so you can use the one that fits best. For example, a normal distribution is usually parametrized with mean and standard deviation, but if you'd rather use mean and precision:
Since probability distributions can also be sampled to generate random numbers with the configured distribution, all constructors optionally accept a random generator as last argument.
A few more examples, this time in F#:
// some probability distributions let normal = Normal.WithMeanVariance(3.0, 1.5, a) let exponential = Exponential(2.4) let gamma = Gamma(2.0, 1.5, Random.crypto()) let cauchy = Cauchy(0.0, 1.0, Random.mrg32k3aWith 10 false) let poisson = Poisson(3.0) let geometric = Geometric(0.8, Random.system())
Some of the distributions also have routines for maximum-likelihood parameter estimation from a set of samples:
let estimation = LogNormal.Estimate([| 2.0; 1.5; 2.1; 1.2; 3.0; 2.4; 1.8 |]) let mean, variance = estimation.Mean, estimation.Variance let moreSamples = estimation.Samples() |> Seq.take 10 |> Seq.toArray
or in C#:
Each distribution provides methods to generate random numbers from that distribution.
These random variate generators work by accessing the distribution's member RandomSource
to provide uniform random numbers. By default, this member is an instance of System.Random
but one can easily replace this with more sophisticated random number generators from
MathNet.Numerics.Random (see Random Numbers for details).
// sample some random numbers from these distributions // continuous distributions sample to floating-point numbers: let continuous = [ yield normal.Sample() yield exponential.Sample() yield! gamma.Samples() |> Seq.take 10 ] // discrete distributions on the other hand sample to integers: let discrete = [ poisson.Sample() poisson.Sample() geometric.Sample() ]
Instead of creating a distribution object we can also sample directly with static functions. Note that no intermediate value caching is possible this way and parameters must be validated on each call.
// using the default number generator (SystemRandomSource.Default) let w = Rayleigh.Sample(1.5) let x = Hypergeometric.Sample(100, 20, 5) // or by manually providing the uniform random number generator let u = Normal.Sample(Random.system(), 2.0, 4.0) let v = Laplace.Samples(Random.mersenneTwister(), 1.0, 3.0) |> Seq.take 100 |> List.ofSeq
If you need to sample not just one or two values but a large number of them,
there are routines that either fill an existing array or return an enumerable.
The variant that fills an array is generally the fastest. Routines to sample
more than one value use the plural form
Samples instead of
Let's sample 100'000 values from a laplace distribution with mean 1.0 and scale 2.0 in C#:
Let's do some random walks in F# (TODO: Graph):
Seq.scan (+) 0.0 (Normal.Samples(0.0, 1.0)) |> Seq.take 10 |> Seq.toArray Seq.scan (+) 0.0 (Cauchy.Samples(0.0, 1.0)) |> Seq.take 10 |> Seq.toArray
Distributions can not just be used to generate non-uniform random samples.
Once parametrized they can compute a variety of distribution properties
or evaluate distribution functions. Because it is often numerically more stable
and faster to compute and work with such quantities in the logarithmic domain,
some of them are also available with the
// distribution properties of the gamma we've configured above let gammaStats = ( gamma.Mean, gamma.Variance, gamma.StdDev, gamma.Entropy, gamma.Skewness, gamma.Mode ) // probability distribution functions of the normal we've configured above. let nd = normal.Density(4.0) (* PDF *) let ndLn = normal.DensityLn(4.0) (* ln(PDF) *) let nc = normal.CumulativeDistribution(4.0) (* CDF *) let nic = normal.InverseCumulativeDistribution(0.7) (* CDF^(-1) *) // Distribution functions can also be evaluated without creating an object, // but then you have to pass in the distribution parameters as first arguments: let nd2 = Normal.PDF(3.0, sqrt 1.5, 4.0) let ndLn2 = Normal.PDFLn(3.0, sqrt 1.5, 4.0) let nc2 = Normal.CDF(3.0, sqrt 1.5, 4.0) let nic2 = Normal.InvCDF(3.0, sqrt 1.5, 0.7)
Specifically for F# there is also a
Sample module that allows a somewhat more functional
view on distribution sampling functions by having the random source passed in as last argument.
This way they can be composed and transformed arbitrarily if curried:
/// Transform a sample from a distribution let s1 rng = tanh (Sample.normal 2.0 0.5 rng) /// But we really want to transform the function, not the resulting sample: let s1f rng = Sample.map tanh (Sample.normal 2.0 0.5) rng /// Exactly the same also works with functions generating full sequences let s1s rng = Sample.mapSeq tanh (Sample.normalSeq 2.0 0.5) rng /// Now with multiple distributions, e.g. their product: let s2 rng = (Sample.normal 2.0 1.5 rng) * (Sample.cauchy 2.0 0.5 rng) let s2f rng = Sample.map2 (*) (Sample.normal 2.0 1.5) (Sample.cauchy 2.0 0.5) rng let s2s rng = Sample.mapSeq2 (*) (Sample.normalSeq 2.0 1.5) (Sample.cauchySeq 2.0 0.5) rng // Taking some samples from the composed function Seq.take 10 (s2s (Random.system())) |> Seq.toArray