mloglik.td(x, model,
KF.version = eval(formals(KFKSDS::KalmanFilter)$KF.version),
KF.args = list(), check.KF.args = TRUE,
barrier = list(type = c("1", "2"), mu = 0), inf = 99999)KFconvar(model, P0cov = FALSE, barrier = list(type = "1", mu = 0), debug = TRUE)
mloglik.td.deriv(model, gradient = TRUE, infomat = TRUE,
KF.args = list(), version = c("1", "2"), kfres = NULL,
convergence = c(0.001, length(model@y)))
mloglik.td.grad(x, model, KF.version, KF.args = list(),
convergence = c(0.001, length(model@y)),
check.KF.args, barrier, inf)
optim.stsm.TRUE, the elements passed in argument KF.args are
checked for consistency with KF.version.m@lower and m@upper.NA or non-finite.TRUE, values outside the diagonal of the covariance matrix
of the initial state vector are set equal to the values in the diagonal,
as done in StructTSTRUE, the correctness of the result is double-checked.TRUE, first order derivatives of the
negative of the spectral log-likelihood function are returned.TRUE, the information matrix
of the spectral log-likelihood are returned."2" or "2"
(the default) should be used.They yield the same result but are kept for debugging and
comparison of timings. This argument may be removed in future versions.KF.deriv.KF.model or at x if this argument is not NULL.
If the value happens to be NA or non-finite the value of argument
inf is returned.
This function is suited to be passed as the objective function to
optim.KFconvar returns a list containing the element mll,
the negative of the concentrated minus log-likelihood function and
the element cpar, the optimal value of the parameter that is
concentrated out of the likelihood.
mloglik.td.deriv returns a list containing a vector of the first order derivatives
of the negative of the time domain likelihood function and a matrix for the information
matrix. They are set to NULL if any of them are not requested.
mloglik.td.grad returns a numeric vector containing the gradient.
This function is suited to be passed as the gradient function to
optim.
for $t=1,\dots,n$ and $a[1] \sim N(a0, P0)$. $Z$ is a matrix of dimension $1\times m$; $H$ is $1\times 1$; $T$ is $m\times m$; $R$ is $m\times r$; $V$ is $r\times r$; $a0$ is $m\times 1$ and $P0$ is $m\times m$, where $r$ is the number of variance parameters in the state vector.
The Kalman filtering recursions for the model above are:
Prediction $$a[t] = T a[t-1]$$ $$P[t] = T P[t-1] T' + R V R'$$ $$v[t] = y[t] - Z a[t]$$ $$F[t] = Z P[t] Z' + H$$
Updating $$K[t] = P[t] Z' F[t]^{-1}$$ $$a[t] = a[t] + K[t] v[t]$$ $$P[t] = P[t] - K[t] Z P[t]'$$
for $t=2,\dots,n$, starting with $a[1]$ and $P[1]$ equal
to a0 and P0. $v[t]$ is the prediction error at observation
in time $t$ and $F[t]$ is the variance of $v[t]$.
The log-likelihood of the model for a given set of parameter values is:
$$logLik = -0.5 log(2\pi) - 0.5 \sum_{t=1}^n { log F[t] + v[t]^2 / F[t] }$$
For details about the options than can be passed through argument KF.args
see the documentation of the same argument of function
KalmanFilter in package mloglik.td.deriv, the only element that is used if provided in
KF.args is P0cov (a logical indicating whether the covariance matrix
of the initial state vector diagonal or not).
The argument x is an auxiliar vector that is necessary in some contexts.
For example, the input to function optim must contain
as first argument the vector of parameters where optimization is performed.
If it is not required or is redundant information contained in model@pars
it can be set to NULL.
For further information about the barrier term see
Bounds on parameters and barrier term
in the details section in maxlik.fd.scoring.
KFconvar evaluates the concentrated likelihood function.
The likelihood is concentrated with respect to the parameter defined in model@cpar.
The optimal value of the parameter that is concentrated out of the likelihood is:$$s2 = (1/n) \sum_{t=1}^n v[t]/ F[t]$$and the concentrated likelihood function is given by:$$clogLik = (n/2) log(2\pi + 1) + 0.5 \sum_{t=1}^n log(f[t]) + (n/2) log(s2).$$
The gradient and the information matrix are calculated upon their corresponding analytical expressions.
Arguments KF.version, check.KF.args, barrier and inf
are not used by mloglik.td.grad
but they are needed in maxlik.td.optim, where this function is passed
as the gradient to be used by optim
including the arguments inf and barrier.
Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
barrier.eval,
logLik,
maxlik.td,
stsm,
KalmanFilter,
KalmanLike.# local level plus seasonal model for a sample simulated series
data("llmseas")
m <- stsm.model(model = "llm+seas", y = llmseas,
pars = c("var1" = 300, "var2" = 10, "var3" = 100))
# evaluate the time domain likelihood function using
# excluding the contributions of the first 8 observations
mloglik.td(model = m, KF.version = "KFKSDS", KF.args = list(t0 = 9))
# compare analytical and numerical derivatives
# identical gradient up to a tolerance
a <- mloglik.td.deriv(m, infomat = TRUE)
g <- numDeriv::grad(func = mloglik.td, x = m@pars,
model = m, KF.version = "KFKSDS")
h <- numDeriv::hessian(func = mloglik.td, x = m@pars,
model = m, KF.version = "KFKSDS")
all.equal(a$gradient, g, check.attributes = FALSE)Run the code above in your browser using DataLab