remove_all_constraints: Transform constrained and restricted parameter vector into...

In uGMAR: Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models

Transform constrained and restricted parameter vector into the regular form

Description

remove_all_constraints transforms constrained and restricted parameter vector into the regular form.

Usage

remove_all_constraints(
  p,
  M,
  params,
  model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE,
  constraints = NULL
)

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 \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2) \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M \nu=(\nu_{M1+1},...,\nu_{M}) M1 is the number of GMAR type regimes. In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0. If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints). For restricted models: Size (3M+M-M1+p-1x1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes. If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints). Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{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 \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu 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 \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.
constraints	specifies linear constraints imposed to each regime's autoregressive parameters separately. For non-restricted models: a list of size (pxq_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}). For restricted models: a size (pxq) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}. The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

Value

Returns such parameter vector corresponding to the input vector that is the form described in params for non-restricted and non-constrained models.

uGMAR documentation built on April 11, 2025, 5:50 p.m.