The function glarma is used to fit generalized linear
autoregressive moving average models with various distributions
(Poisson, binomial, negative binomial) using either Pearson residuals
or score residuals, and for the binomial distribution, identity
residuals. It also estimates the parameters of the GLARMA model with
various distributions by using Fisher scoring or Newton-Raphson
iteration.
For Poisson and negative binomial response distributions the log link is currently used. For binomial responses the logit link is currently used.
glarma(y, X, offset = NULL, type = "Poi", method = "FS", residuals = "Pearson",
phiLags, thetaLags, phiInit, thetaInit, beta, alphaInit,
alpha = 1, maxit = 30, grad = 2.22e-16)glarmaPoissonPearson(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaPoissonScore(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaBinomialIdentity(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaBinomialPearson(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaBinomialScore(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaNegBinPearson(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
glarmaNegBinScore(y, X, offset = NULL, delta, phiLags, thetaLags,
method = "FS")
Numeric vector; the response variable. If the response variable is for the model with the binomial distribution, it should be a n by 2 matrix, one column is the number of successes and another is the number of failures.
Matrix; the explanatory variables. A vector of ones should be
added to the data matrix as the first column for the beta of
the intercept.
Either NULL or a numeric vector of length equal
to the number of cases. Used to specify an a priori known
component to be included in the linear predictor during fitting.
Numeric vector; initial values of the regression coefficients.
Numeric vector; AR orders.
Numeric vector; initial values for the corresponding AR orders.
Numeric vector; MA orders.
Numeric vector; initial values for the corresponding MA orders.
Numeric vector; initial values of the parameters for the
GLARMA estimation procedure. It is a combination of the parameters of
beta, the AR terms and the MA terms.
Numeric; an optional initial shape parameter for
glm.nb.
Numeric; an initial shape parameter for
glarma for negative binomial counts.
Character; the count distribution. Possible values are
"Poi" (Poisson), "Bin" (binomial) and "NegBin"
(negative binomial). The default is the
Poisson distribution.
Character; method of iteration to be used. Possible
values are "FS" (Fisher scoring), and "NR"
(Newton-Raphson). The default is to use Fisher scoring to estimate
the parameters of a GLARMA model.
Character; the type of residuals to be used. Possible
values are "Pearson" and "Score", and for the binomial
distribution "Identity" is also allowed. The default is to
use Pearson residuals.
Numeric; the maximum number of iterations allowed.
Numeric; the tolerance for recognizing numbers, which are smaller than the specified tolerance, as zero.
The function summary (i.e., summary.glarma)
can be used to obtain or print a summary of the results.
The generic accessor functions coef (i.e.,
coef.glarma), logLik (i.e.,
logLik.glarma), fitted (i.e.,
fitted.glarma), residuals (i.e.,
residuals.glarma), nobs (i.e.,
nobs.glarma), model.frame (i.e.,
model.frame.glarma) and extractAIC (i.e.,
extractAIC.glarma) can be used to extract various useful
features of the value returned by glarma.
glarma returns an object of class "glarma" with components:
a vector of coefficients for beta, AR and
MA.
the loglikelihood of the specific distribution.
the derivative of the loglikelhood of the specified distribution.
the second derivative of the loglikelihood of the specified distribution.
the estimated linear predictor.
the GLARMA estimated mean.
the GLARMA fitted values.
the residuals of the type specified.
the estimated covariance matrix of the maximum likelihood estimators.
vector of AR orders.
vector of MA orders.
the number of columns in the model matrix.
the number of phiLags plus the number of
thetaLags.
the deviance from the initial GLM fit.
the degrees of freedom from the initial GLM fit.
the \(y\) vector used in the GLARMA model.
the model matrix.
the offset, NULL if there is no offset.
the distribution of the counts.
the method of iteration used.
the type of the residuals returned.
the matched call.
the number of iterations.
the error code; 0 indicating successful convergence of the iteration method, 1 indicating failure.
error code for finiteness of \(W\); 0 indicating all values of \(W\) are finite, 1 indicating at least one infinite value.
the minimum of the absolute value of the gradient.
A version of Akaike's An Information Criterion, minus twice the maximized log-likelihood plus twice the number of parameters, computed by the aic component of the family. For binomial and Poisson families the dispersion is fixed at one and the number of parameters is the number of coefficients.
Models for glarma are specified symbolically. A typical model
has the form y (response), X (terms) where y is
the count or factor reponse vector, X is a series of terms
which specifies a linear predictor for the response. It should be noted
that the first column of X should be a vector of 1s as the
intercept in the model. Four initial parameters that need to be
estimated are combined into \(\delta = (\beta, \phi, \theta,
\alpha)\), where \(\alpha\)
is an optional parameter to accomodate the negative binomial
model. Note that in the function glm.nb from the
package MASS, this parameter is called theta.
For Poisson and negative binomial response distributions the log link is currently used. For binomial responses the logit link is currently used.
The generalized linear autoregressive moving average models are computed as follows.
The linear predictor for the response is $$\log{\mu_t} = W_t = X_t^T\beta + \mbox{offset} + Z_t.$$
The infinite moving average from the linear predictor is $$Z_t = \sum_{i=1}^\infty \gamma_i e_{t-i}.$$
This infinite moving average, is computed using the autoregressive moving average recursions
$$Z_t = \phi_1 (Z_{t-1} + e_{t-1}) + ... +% \phi_p (Z_{t-p} + e_{t-p}) + \theta_1 e_{t-1}% + ... + \theta_q e_{t-q}$$
where \(p\) and \(q\) are the orders of \(\phi\)
and \(\theta\) respectively and the non-zero lags of the vectors
phi and theta may be specified by the user via the
arguments phiLag and thetaLag.
There are two types of residuals which may be used in each recursion, Pearson residuals or score residuals, and in addition, for the binomial distribution, identity residuals may be used. The infinite moving average, \(Z_t\), depends on the type of residuals used, as do the final parameters obtained from the filter. Standardisation of past observed counts is necessary to avoid instability, therefore the user should choose the appropriate type of residuals depending on the situation.
The method of estimation for parameters implemented in the function aims to maximise the log likelihood by an iterative method commencing from suitably chosen initial values for the parameters. Starting from initial values \(\hat\delta^{(0)}\) for the vector of parameters updates are obtained using the iterations
$$\hat{\delta}^{(k+1)}=\hat{\delta}^{(k)}+\Omega(\hat{\delta}^{(k)}% )\frac{\partial l(\hat{\delta}^{(k)})}{\partial\delta}$$
where \(\Omega (\hat\delta^{(k)})\) is some suitably chosen matrix.
Iterations continue for \(k\geq1\) until convergence is reached or the number of iterations \(k\) reaches a user specified upper limit on maximum iterations in which case they will stop. The convergence criterion used in our implementation is that based on \(\eta\), the maximum of absolute values of the first derivatives.
When \(\eta\) is less than a user specified value grad
the iterations stop. There are two methods of optimization of the
likelihood, Newton-Raphson and Fisher scoring. The method used is
specified by the argument method. It should be noticed that if
the initial value for parameters are not chosen well, the
optimization of the likelihood might fail to converge. Care is needed
when fitting mixed ARMA specifications because there is potential for
the AR and MA parameters to be non-identifiable if the orders \(p\) and
\(q\) are too large. Lack of identifiability manifests itself in the
algorithm to optimize the likelihood failing to converge and/or the
hessian being singular---check the warning messages and convergence
error codes.
Dunsmuir, William T. M. and Scott, David J. (2015) The glarma Package for Observation-Driven Time Series Regression of Counts. Journal of Statistical Software, 67(7), 1--36. http://dx.doi.org/10.18637/jss.v067.i07
Additional examples may be found in Asthma,
OxBoatRace, RobberyConvict, and
DriverDeaths.
# NOT RUN {
### Example from Davis, Dunsmuir Wang (1999)
## MA(1,2,5), Pearson Residuals, Fisher Scoring
data(Polio)
y <- Polio[, 2]
X <- as.matrix(Polio[, 3:8])
glarmamod <- glarma(y, X, thetaLags = c(1,2,5), type = "Poi", method = "FS",
residuals = "Pearson", maxit = 100, grad = 1e-6)
glarmamod
summary(glarmamod)
## Score Type (GAS) Residuals, Fisher Scoring
glarmamod <- glarma(y, X, thetaLags = c(1,2,5), type = "Poi", method = "FS",
residuals = "Score", maxit = 100, grad = 1e-6)
glarmamod
summary(glarmamod)
## Score Type (GAS) Residuals, Newton Raphson
## Note: Newton Raphson fails to converge from GLM initial estimates.
## Setting up the initial estimates by ourselves
init.delta <- glarmamod$delta
beta <- init.delta[1:6]
thetaInit <- init.delta[7:9]
glarmamod <- glarma(y, X, beta = beta, thetaLags = c(1, 2, 5),
thetaInit = thetaInit, type ="Poi", method = "NR",
residuals = "Score", maxit = 100, grad = 1e-6)
glarmamod
summary(glarmamod)
## AR(1,5), Pearson Residuals, Fisher Scoring
glarmamod <- glarma(y, X, phiLags = c(1, 5), type = "Poi", method = "FS",
residuals = "Pearson", maxit = 100, grad = 1e-6)
glarmamod
summary(glarmamod)
# }
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