# reform_parameters: Reform any parameter vector into standard form. In uGMAR: Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models

## Description

`reform_parameters` takes a parameter vector of any (non-constrained) GMAR, StMAR, or G-StMAR model and returns a list with the parameter vector in the standard form, parameter matrix containing AR coefficients and component variances, mixing weights alphas, and in case of StMAR or G-StMAR model also degrees of freedom parameters.

## Usage

 ```1 2 3 4 5 6 7``` ```reform_parameters( p, M, params, model = c("GMAR", "StMAR", "G-StMAR"), restricted = FALSE ) ```

## Arguments

 `p` a positive integer specifying the autoregressive order of the model. `M` For GMAR and StMAR models:a positive integer specifying the number of mixture components. For G-StMAR models:a size (2x1) integer vector specifying the number of GMAR type components `M1` in the first element and StMAR type components `M2` in the second element. The total number of mixture components is `M=M1+M2`. `params` a real valued parameter vector specifying the model. For non-restricted models: Size (M(p+3)+M-M1-1x1) vector θ=(υ_{1},...,υ_{M}, α_{1},...,α_{M-1},ν) where υ_{m}=(φ_{m,0},φ_{m},σ_{m}^2) φ_{m}=(φ_{m,1},...,φ_{m,p}), m=1,...,M ν=(ν_{M1+1},...,ν_{M}) M1 is the number of GMAR type regimes. In the GMAR model, M1=M and the parameter ν dropped. In the StMAR model, M1=0. If the model imposes linear constraints on the autoregressive parameters: Replace the vectors φ_{m} with the vectors ψ_{m} that satisfy φ_{m}=C_{m}ψ_{m} (see the argument `constraints`). For restricted models: Size (3M+M-M1+p-1x1) vector θ=(φ_{1,0},...,φ_{M,0},φ, σ_{1}^2,...,σ_{M}^2,α_{1},...,α_{M-1},ν), where φ=(φ_{1},...,φ_{p}) contains the AR coefficients, which are common for all regimes. If the model imposes linear constraints on the autoregressive parameters: Replace the vector φ with the vector ψ that satisfies φ=Cψ (see the argument `constraints`). Symbol φ denotes an AR coefficient, σ^2 a variance, α a mixing weight, and ν a degrees of freedom parameter. If `parametrization=="mean"`, just replace each intercept term φ_{m,0} with the regimewise mean μ_m = φ_{m,0}/(1-∑φ_{i,m}). In the G-StMAR model, the first `M1` components are GMAR type and the rest `M2` components are StMAR type. Note that in the case M=1, the mixing weight parameters α are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters ν have to be larger than 2. `model` is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first `M1` components are GMAR type and the rest `M2` components are StMAR type. `restricted` a logical argument stating whether the AR coefficients φ_{m,1},...,φ_{m,p} are restricted to be the same for all regimes.

## Details

This function does not support models imposing linear constraints. No argument checks in this function.

## Value

Returns a list with...

`\$params`

parameter vector in the standard form.

`\$pars`

corresponding parameter matrix containing AR coefficients and component variances. First row for phi0 or means depending on the parametrization. Column for each component.

`\$alphas`

numeric vector containing mixing weight parameters for all of the components (also for the last one).

`\$dfs`

numeric vector containing degrees of freedom parameters for all of components. Returned only if `model == "StMAR"` or `model == "G-StMAR"`.

@keywords internal

uGMAR documentation built on Jan. 24, 2022, 5:10 p.m.