uGMAR (version 1.0.1)

randomIndividual: Create somewhat random GMAR or StMAR model compatible parameter vector

Description

randomIndividual creates a somewhat random GMAR or StMAR model compatible parameter vector.

smartIndividual creates a somewhat random GMAR or StMAR model compatible parameter vector close to argument params.

Usage

randomIndividual(p, M, StMAR = FALSE, restricted = FALSE,
  constraints = FALSE, R, ar0scale, sigmascale)
smartIndividual(p, M, params, StMAR = FALSE, restricted = FALSE,
  constraints = FALSE, R, ar0scale, sigmascale, accuracy, whichRandom)

Arguments

p

a positive integer specifying the order of AR coefficients.

M

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

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.

ar0scale

a real valued vector of length two specifying the mean (the first element) and standard deviation (the second element) of the normal distribution from which the \(\phi_{m,0}\) parameters (for random regimes) should be generated.

sigmascale

a positive real number specifying the standard deviation of the (zero mean, positive only) normal distribution from which the component variance parameters (for random regimes) should be generated.

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 \(2\).

accuracy

a real number larger than zero specifying how close to params the generated parameter vector should be. Standard deviation of the normal distribution from which new parameter values are drawed from will be corresponding parameter value divided by accuracy.

whichRandom

an (optional) numeric vector of max length M specifying which regimes should be random instead of "smart" when using smartIndividual. Does not affect on mixing weight parameters. Default in none.

Value

Returns estimated parameter vector...

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:

Parameter vector as descripted above, but vectors \(\phi_{m}\) replaced with vectors \(\psi_{m}\) 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:

Parameter vector as descripted above, but vector \(\phi\) replaced with vector \(\psi\) 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 \(\nu\) a degrees of freedom parameter.

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.

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

Examples

# NOT RUN {
# GMAR model parameter vector
params22 <- randomIndividual(2, 2, ar0scale=c(0, 1), sigmascale=1)
smart22 <- smartIndividual(2, 2, params22, accuracy=10)
cbind(params22, smart22)


# Restricted GMAR parameter vector
params12r <- randomIndividual(1, 2, restricted=TRUE, ar0scale=c(-2, 2), sigmascale=2)
smart12r <- smartIndividual(1, 2, params12r, restricted=TRUE, accuracy=20)
cbind(params12r, smart12r)


# StMAR parameter vector: first regime is random in the "smart individual"
params13t <- randomIndividual(1, 3, StMAR=TRUE, ar0scale=c(3, 1), sigmascale=3)
smart13t <- smartIndividual(1, 3, params13t, StMAR=TRUE, accuracy=15,
                            ar0scale=c(3, 3), sigmascale=3, whichRandom=1)
cbind(params13t, smart13t)


# Restricted StMAR parameter vector
params22tr <- randomIndividual(2, 2, StMAR=TRUE, restricted=TRUE,
                               ar0scale=c(3, 2), sigmascale=0.5)
smart22tr <- smartIndividual(2, 2, params22tr, StMAR=TRUE, restricted=TRUE,
                            accuracy=30)
cbind(params22tr, smart22tr)


# 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 <- randomIndividual(2, 2, constraints=TRUE, R=R,
                              ar0scale=c(1, 1), sigmascale=1)
smart22c <- smartIndividual(2, 2, params22c, constraints=TRUE, R=R, accuracy=10)
cbind(params22c, smart22c)


# Such constrained StMAR(3, 2) model that the second order AR coefficients
# are constrained to zero.
R0 = matrix(c(1, 0, 0, 0, 0, 1), ncol=2)
R = list(R0, R0)
params32c <- randomIndividual(3, 2, StMAR=TRUE, constraints=TRUE, R=R,
                              ar0scale=c(1, 1), sigmascale=1)
smart32c <- smartIndividual(3, 2, params32c, StMAR=TRUE, constraints=TRUE, R=R, accuracy=10)
cbind(params32c, smart32c)


# 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. Second regime is random in the "smart individual".
params32trc <- randomIndividual(3, 2, StMAR=TRUE, restricted=TRUE, constraints=TRUE,
                                R=matrix(c(1, 0, 0, 0, 0, 1), ncol=2), ar0scale=c(-2, 0.5),
                                sigmascale=4)
smart32trc <- smartIndividual(3, 2, params32trc, StMAR=TRUE, restricted=TRUE,
                              constraints=TRUE, R=matrix(c(1, 0, 0, 0, 0, 1), ncol=2),
                              ar0scale=c(0, 0.1), sigmascale=0.1, whichRandom=2,
                              accuracy=20)
cbind(params32trc, smart32trc)
# }