PumasEMModel-Error models

The error models in a PumasEMModel implicitly define the dispersion parameters based on the provided distribution. The general syntax is

@error begin
    dv ~ ProportionalNormal(μ)
end

Where $dv$ is the observed variable, and $μ$ the expectation, specified in an earlier block. All currently supported error models follow the $observed ~ Distribution(expectation)$ syntax. When specifyiing inits, add a $σ$ field to the named tuple with one element per observed variable. Each element should itself be a tuple with a Float64 per parameter in the error model. For example, given

@error begin
    dv1 ~ ProportionalNormal(cp1) # parameterized by 1 dispersion parameter
    dv2 ~ CombinedNormal(cp2)    # parameterized by 2 dispersion parameters
end

one may, specify inits of the form $σ = ((2.0,),(0.9,0.5))$, where $2.0$ is the standard deviation parameter for dv1 and $0.9, 0.5$ are the parameters for $dv2$, in this case corresponding to the additive and proportional standard deviations of the combined normal error model. It is not necessary to initialize σ when fitting; if unspecified, each element is initialized to 1.0.

Gaussian models

Additive Normal

@error begin
   Y ~ Normal(μ)
end

Indicates that Y ~ Normal(μ, σ).

Proportional Normal

@error begin
   Y ~ ProportionalNormal(μ)
end

Indicates that Y ~ Normal(μ, abs(μ)*σ).

Combined Normal

@error begin
   Y ~ CombinedNormal(μ)
end

Indicates that Y ~ Normal(μ, √(σₐ² + μ²*σₚ²)).

Log Normal

@error begin
   Y ~ LogNormal(μ)
end

Indicates that Y ~ LogNormal(μ, σ).

0-Dispersion-Parameter models

Bernoulli

@error begin
   Y ~ Bernoulli(μ)
end

Indicates that Y ~ Bernoulli(μ).

Bernoulli Logit

@error begin
   Y ~ BernoulliLogit(μ)
end

Indicates that Y ~ Bernoulli( 1 /(1 + exp(-μ)) ).

Exponential

@error begin
   Y ~ Exponential(μ)
end

Indicates that Y ~ Exponential(μ).

Poisson

@error begin
   Y ~ Poisson(μ)
end

Indicates that Y ~ Poisson(μ).

1-Dispersion-Parameter models

Beta

@error begin
   Y ~ Beta(μ)
end

Indicates that Y ~ Beta(μ * 10/σ, (1 - μ) * 10/σ).

Gamma

@error begin
   Y ~ Gamma(μ)
end

Indicates that Y ~ Gamma( 1 / σ², μ * σ² ).

Summary of Error Models