uGMAR (version 1.0.1)

mixingWeights: Calculate mixing weights of GMAR or StMAR model

Description

mixingWeights calculates the mixing weights of the specified GMAR or StMAR model and returns them as a matrix.

Usage

mixingWeights(data, p, M, params, StMAR = FALSE, restricted = FALSE,
  constraints = FALSE, R)

Arguments

data

a numeric vector or column matrix containing the data. NA values are not supported.

p

a positive integer specifying the order of AR coefficients.

M

a positive integer specifying the number of mixture components or regimes.

params

a real valued parameter vector specifying the model.

For non-restricted models:

For GMAR model:

Size \((M(p+3)-1x1)\) vector \(\theta\)\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), where \(\upsilon_{m}\)\(=(\phi_{m,0},\)\(\phi_{m}\)\(, \sigma_{m}^2)\) and \(\phi_{m}\)=\((\phi_{m,1},...,\phi_{m,p}), m=1,...,M\).

For StMAR model:

Size \((M(p+4)-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}\)).

With linear constraints:

Replace the vectors \(\phi_{m}\) with vectors \(\psi_{m}\) and provide a list of constraint matrices R that satisfy \(\phi_{m}\)\(=\)\(R_{m}\psi_{m}\) for all \(m=1,...,M\), where \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).

For restricted models:

For GMAR model:

Size \((3M+p-1x1)\) vector \(\theta\)\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1})\), where \(\phi\)=\((\phi_{1},...,\phi_{M})\).

For StMAR model:

Size \((4M+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})\).

With linear constraints:

Replace the vector \(\phi\) with vector \(\psi\) and provide a constraint matrix \(R\) that satisfies \(\phi\)\(=\)\(R\psi\), where \(\psi\)\(=(\psi_{1},...,\psi_{q})\).

Symbol \(\phi\) denotes an AR coefficient, \(\sigma^2\) a variance, \(\alpha\) a mixing weight and \(v\) a degrees of freedom parameter. Note that in the case M=1 the parameter \(\alpha\) is dropped, and in the case of StMAR model the degrees of freedom parameters \(\nu_{m}\) have to be larger than \(2\).

StMAR

an (optional) logical argument stating whether StMAR model should be considered instead of GMAR model. Default is FALSE.

restricted

an (optional) logical argument stating whether the AR coefficients \(\phi_{m,1},...,\phi_{m,p}\) are restricted to be the same for all regimes. Default is FALSE.

constraints

an (optional) logical argument stating whether general linear constraints should be applied to the model. Default is FALSE.

R

Specifies the linear constraints.

For non-restricted models:

a list of size \((pxq_{m})\) constraint matrices \(R_{m}\) of full column rank satisfying \(\phi_{m}\)\(=\)\(R_{m}\psi_{m}\) 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 \(R\) of full column rank satisfying \(\phi\)\(=\)\(R\psi\), where \(\phi\)\(=(\phi_{1},...,\phi_{p})\) and \(\psi\)\(=\psi_{1},...,\psi_{q}\).

Symbol \(\phi\) denotes an AR coefficient. Note that regardless of any constraints, the nominal order of AR coefficients is alway p for all regimes. This argument is ignored if constraints==FALSE.

Value

Returns size \((TxM)\) matrix containing the mixing weights of the specified GMAR or StMAR model so that \(i\):th column corresponds to \(i\):th mixing component or regime.

References

• Kalliovirta L., Meitz M. and Saikkonen P. (2015) Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36, 247-266.

• Lutkepohl H. New Introduction to Multiple Time Series Analysis, Springer, 2005.

• Galbraith, R., Galbraith, J., (1974). On the inverses of some patterned matrices arising in the theory of stationary time series. Journal of Applied Probability 11, 63-71.

• References regarding the StMAR model and general linear constraints will be updated after they are published.

Examples

# NOT RUN {
# GMAR model
params12 <- c(1.1, 0.9, 0.3, 4.5, 0.7, 3.2, 0.8)
mw12 <- mixingWeights(VIX, 1, 2, params12)

# Restricted GMAR model
params12r <- c(1.4, 1.8, 0.9, 0.3, 3.2, 0.8)
mw12r <- mixingWeights(VIX, 1, 2, params12r, restricted=TRUE)

# StMAR model
params12t <- c(1.1, 0.9, 0.3, 4.5, 0.7, 3.2, 0.8, 5, 8)
mw12t <- mixingWeights(VIX, 1, 2, params12t, StMAR=TRUE)

# Non-mixture version of StMAR model
params11t <- c(0.76, 0.93, 1.35, 2.4)
mw11t <- mixingWeights(VIX, 1, 1, params11t, StMAR=TRUE)

# GMAR model as a mixture of AR(2) and AR(1) models
R <- list(diag(1, ncol=2, nrow=2), as.matrix(c(1, 0)))
params22c <- c(1.2, 0.8, 0.1, 0.3, 3.3, 0.8, 2.8, 0.8)
mw22c <- mixingWeights(VIX, 2, 2, params22c, constraints=TRUE, R=R)

# Such StMAR(3,2) that the AR coefficients are restricted to be
# the same for both regimes and that the second AR coefficients are
# constrained to zero.
params32trc <- c(2.2, 1.8, 0.88, -0.03, 2.4, 0.27, 0.40, 3.9, 1000)
mw32trc <- mixingWeights(VIX, 3, 2, params32trc, StMAR=TRUE,
                         restricted=TRUE, constraints=TRUE,
                         R=matrix(c(1, 0, 0, 0, 0, 1), ncol=2))
# }