# Sampling Random Variables

`Fortuna.jl`

package allows to easily generate samples of both uncorrelated and correlated random variables using `rand()`

function using different sampling techniques. Current version of the package implements Inverse Transform Sampling (ITS) and Latin Hypercube Sampling (LHS) techniques.

## Sampling Random Variables

To generate samples of a random variable:

- Generate a random variable (
`X`

).

`X = randomvariable("Gamma", "M", [10, 1.5])`

- Sample the generated random variable using a sampling technique of your choice.

`XSamples = rand(X, 10000, LHS())`

## Sampling Random Vectors with Uncorrelated Marginals

To generate samples of a random vector with *uncorrelated* marginals:

- Generate random variables (
`X₁`

and`X₂`

).

```
X₁ = randomvariable("Gamma", "M", [10, 1.5])
X₂ = randomvariable("Gamma", "M", [15, 2.5])
```

- Define a random vector (
`X`

) with the generated random variables as marginals.

`X = [X₁, X₂]`

- Sample the defined random vector using a sampling technique of your choice.

`XSamples = rand(X, 10000, LHS())`

## Sampling Random Vectors with Correlated Marginals

When sampling random vectors with correlated marginals, the sampling technique determines how samples are generated in the space of uncorrelated standard normal random variables (referred to as $U$-space). These generated samples are then transformed into the target space of correlated non-normal random variables (referred to as $X$-space). For more information see Nataf Transformation and Rosenblatt Transformation.

To generate samples of a random vector with *correlated* marginals:

- Generate random variables (
`X₁`

and`X₂`

).

```
X₁ = randomvariable("Gamma", "M", [10, 1.5])
X₂ = randomvariable("Gamma", "M", [15, 2.5])
```

- Define a random vector (
`X`

) with the generated random variables as marginals.

`X = [X₁, X₂]`

- Define a correlated matrix (
`ρˣ`

) for the defined random vector.

`ρˣ = [1 -0.75; -0.75 1]`

- Define a transformation object that hold all information about the defined random vector.

`TransformationObject = NatafTransformation(X, ρˣ)`

- Sample the defined random vector using a sampling technique of your choice.

`XSamples, ZSamples, USamples = rand(TransformationObject, 10000, LHS())`

## API

`Fortuna.ITS`

— Type`struct ITS <: AbstractSamplingTechnique`

Type used to perform the Inverse Transform Sampling.

`Fortuna.LHS`

— Type`struct LHS <: AbstractSamplingTechnique`

Type used to perform the Latin Hypercube Sampling.

`Base.rand`

— Method`rand(RNG::Distributions.AbstractRNG, RandomVariable::Distributions.UnivariateDistribution, NumSamples::Int, SamplingTechnique::AbstractSamplingTechnique)`

Function used to generate samples from an *random variable*. If `SamplingTechnique`

is:

`ITS()`

samples are generated using Inverse Transform Sampling technique.`LHS()`

samples are generated using Latin Hypercube Sampling technique.

`Base.rand`

— Method`rand(RNG::Distributions.AbstractRNG, RandomVector::Vector{<:Distributions.UnivariateDistribution}, NumSamples::Int, SamplingTechnique::AbstractSamplingTechnique)`

Function used to generate samples from a *random vector with uncorrelated marginals*. If `SamplingTechnique`

is:

`ITS()`

samples are generated using Inverse Transform Sampling technique.`LHS()`

samples are generated using Latin Hypercube Sampling technique.

`Base.rand`

— Method`rand(RNG::Distributions.AbstractRNG, TransformationObject::NatafTransformation, NumSamples::Int, SamplingTechnique::AbstractSamplingTechnique)`

Function used to generate samples from a *random vector with correlated marginals* using Nataf Transformation object. If `SamplingTechnique`

is:

`ITS()`

samples are generated using Inverse Transform Sampling technique.`LHS()`

samples are generated using Latin Hypercube Sampling technique.