# Domains

Domains are used to define the `param`

portion of a `PumasModel`

. This segment of the documentation describes the available `Domain`

types that can be used within the `param`

portions.

## Matching Parameter Types and Domains

A `param`

specification in a `PumasModel`

directly defines the types which are required to be given as the input to the model's API functions such as `simobs`

and `fit`

. For example, if the `param`

specification is given as:

```
@param begin
θ ∈ VectorDomain(2, lower=[0.0,0.0], upper=[20.0,20.0])
Ω ∈ PSDDomain(2)
Σ ∈ ConstDomain(0.1)
end
```

Then in `simobs(model, data, param)`

or `fit(model, data, param)`

, the parameters `param`

must be a `NamedTuple`

of values where the type fits in the domain. For example, `param`

must have `param.θ`

as a `Vector`

. Thus the following would be a valid definition of `param`

for this structure:

```
param = (
θ = [0.4, 7.0],
Ω = [0.04 0.0
0.0 0.01],
Σ = 0.1
)
```

Below are the specifications of the `Domain`

types and their matching value types.

## RealDomain and VectorDomain

`RealDomain`

and `VectorDomain`

are the core domain types of Pumas. A `RealDomain`

defines a scalar value which exists in the real line, while a `VectorDomain`

defines a real-valued vector which lives in a hypercube of $\mathbb{R}^n$. The length `n`

of a `VectorDomain`

is a required positional argument. Each of these allow keyword arguments for setting an `upper`

and `lower`

bound for the segment, where for the `RealDomain`

these are scalars and for the `VectorDomain`

it is a vector of upper and lower bound for each component of the vector. Thus the constructors are:

```
RealDomain(; upper, lower)
VectorDomain(n; upper, lower)
```

A `RealDomain`

requires that the matching parameters are an `AbstractFloat`

type. A `VectorDomain`

requires that the matching parameters are a `Vector`

of an `AbstactFloat`

type which has the correct size.

## ConstDomain

`ConstDomain`

is a special domain type which holds the parameter constant during estimation routines. A `ConstDomain`

is defined by its value:

`ConstDomain(val)`

A `ConstDomain`

requires the matching value in the parameters.

## Positive-Definite Matrix Domains

In many cases, one may wish to specify a positive-definite covariance matrix as a parameter in a model. A common use case for this functionality is for defining the domain of a random effect. There are two domains for positive-definite matrices: `PSDDomain`

and `PDiagDomain`

. Both of the constructors require the size `n`

of the `n x n`

postive-definite matrix:

```
PSDDomain(n)
PDiagDomain(n)
```

A `PSDDomain`

requires that the matching matrix input is positive definite but puts no further restrictions on the matrix. If a domain is specified as `Ω ∈ PSDDomain(2)`

, then

```
Ω = [0.04 0.0
0.0 0.01]
```

is a valid parameter specification. The `PDiagDomain`

restricts the parameter space to positive definite diagonal matricesis, i.e. diagonal matrices with positive diagonal elemenets. Thus, if a domain is specified as `Ω ∈ PDiagDomain(2)`

, then

`Ω = Diagonal([0.0, 1.0])`

is a valid parameter specification.

## Distributional Domains

Instead of using a `Domain`

type from Pumas, a `Distribution`

can be used to specify a domain. For example,

`Ω ~ Normal(0,1)`

is a valid domain specification. If this is done, then the probability distribution is used and interpreted as the prior distribution. Implicitly, the domain of a distribution is treated as the support of the distribution. A distributional domain requires that the matching parameter is of the same type as a sample from the domain. For example, if `Ω ~ Normal(0,1)`

, then `Ω`

should be given as a scalar.

## Constrained

The `Constrained`

domain is for defining a `Constrained`

probability distribution. `Constrained`

takes in a distribution and has keyword arguments for `upper`

and `lower`

bounds. For example, `ψ ~ Constrained(MVNormal(Ω),lower=[0.0,0.0])`

defines a `ψ`

to be from the distributional domain which corresponds to a multivariate normal distribution, but is constrained to be positive. Like with the distributional domains, `Constrained`

requires that the matching parameter is of the same type as a sample from the domain.