Count Time Series Following Generalised Linear Models

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Description

The function tsglm fits a generalised linear model (GLM) for time series of counts. The specification of the linear predictor allows for regressing on past observations, past values of the linear predictor and covariates as defined in the Details section. There is the so-called INGARCH model with the identity link (see for example Ferland et al., 2006, Fokianos et al., 2009) and another model with the logarithmic link (see for example Fokianos and Tjostheim, 2011), which also differ in the specification of the linear predictor. The conditional distribution can be chosen to be either Poisson or negative binomial.

Estimation is done by conditional maximum likelihood for the Poisson distribution or by a conditional quasi-likelihood approach based on the Poisson likelihood function for the negative binomial distribution.

There is a vignette available which introduces the functionality of tsglm and related functions of this package and its underlying statistical methods (vignette("tsglm", package="tscount")).

The function mean.fit is a lower level function to fit the mean specification of such a model assuming a Poisson distribution. It is called by tsglm. It has additional arguments allowing for a finer control of the fitting procedure, which can be handed over from the function tsglm by its ... argument. Note that it is usually not necessary for a user to call this lower level functions nor to worry about the additional arguments provided by this function. The defaults of these arguments have been chosen wisely by the authors of this package and should perform well in most applications.

Usage

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tsglm(ts, model = list(past_obs = NULL, past_mean = NULL,
            external = NULL), xreg = NULL, link = c("identity", "log"),
            distr = c("poisson", "nbinom"), ...)

mean.fit(ts, model, xreg, link, score = TRUE,
            info = c("score", "none", "hessian", "sandwich"),
            init.method=c("marginal", "iid", "firstobs", "zero"),
            init.drop = FALSE, epsilon = 1e-06, slackvar = 1e-06,
            start.control = list(), final.control = list(),
            inter.control = NULL)

Arguments

ts

a univariate time series.

model

a named list specifying the model for the linear predictor, which can be of the following elements:

past_obs

integer vector giving the previous observations to be regressed on (autoregression). This is a vector with the elements i[1],…,i[p] (see Details). If omitted, or of length zero, there will be no regression on previous observations.

past_mean

integer vector giving the previous conditional means to be regressed on. This is a vector with the elements j[1],…,j[q] (see Details). If omitted, or of length zero, there will be no regression on previous conditional means.

external

logical vector of length ncol(xreg) specifying for each covariate wether its effect should be external or not (see Details). If this is a scalar this choice will be used for all covariates. If omitted, all covariates will have an internal effect (i.e. external=FALSE).

xreg

matrix with covariates in the columns, i.e. its number of rows must be length(ts). This is the matrix X (see Details). If omitted no covariates will be included. For the identity link the covariates have to be non-negative.

link

character giving the link function. Default is "identity", fitting an INGARCH model. Another possible choice is "log", fitting a log-linear model.

distr

character giving the conditional distribution. Default is "poisson", i.e. a Poisson distribution.

...

additional arguments to be passed to the lower level fitting function mean.fit. See below.

score

logical value indicating whether the score vector should be computed.

info

character that determines if and how to compute the information matrix. Can be set to "score" (the default) for calculation via the outer product of the score vector, or to "hessian" for calculation via the Hessian matrix of second derivatives. For info="sandwich" the information matrix is estimated by a sandwich formula using both the outer score product and the Hessian matrix. If set to "none", no information matrix is computed. For distr="nbinom" one can only use info="score".

init.method

character that determines how the recursion of the conditional mean (and possibly of its derivatives) is initialised. If set to "marginal" (the default), the marginal mean of a model without covariates and its derivatives are used. If set to "iid", all values are initialised by the marginal mean under the assumption of i.i.d. data, which depends on the intercept only. If set to "firstobs" the first obersvation is used. If set to "zero", the recursions are initialised by the value zero.

init.drop

logical value that determines which observations are considered for computation of the log-likelihood, the score vector and, if applicable, the information matrix. If TRUE, the first max(model$past_obs) observations, which are needed for the autoregression, are not considered. If FALSE (the default), all observations are considered and pre-sample values determined by the method specified by the argument itit.method are used for the autoregression. Note that in the first case the effective number of observations used for maximum likelihood estimation is lower than the total number of observations of the original time series. Consequently only this lower number of observations is considered in the output. Note that for init.drop=TRUE the log-likelihood function for models of different orders might not be comparable if the effective number of observations is different.

epsilon

numeric positive but small value determining how close the parameters may come to the limits of the parameter space.

slackvar

numeric positive but small value determining how true inequalities among the parameter restrictions are treated; a true inequality x < y will be transformed to x + slackvar <= y.

start.control

named list with optional elements that determine how to make the start estimation. Possible list elements are:

use

integer vector of length one or two giving the number of observations from the beginning (if of length one) or the range of observations (if of length two) used for start estimation. For use = Inf all observations are used, which is the default.

method

character specifying how start estimators should be estimated. Possible values are "iid", "CSS", "CSS-ML", "ML", "MM", "GLM" and "fixed". If method is "iid" (the default), a moment estimator assuming an iid model without covariates is used. If method="MM", the start estimate is an ARMA(1,1) fit by moment estimators and parameters of higher order than one are set to zero. For this method the starting parameter values for the covariates are zero by default and can be set by the list element xreg. If method is "CSS", "CSS-ML" or "ML", the start estimate is based on an ARMA fit using the function arima, and list element method is passed to its argument of the same name. If method="GLM", the estimated parameters of a generalised linear model with regression on the specified past observations and covariates, but not on past conditional means, are used as start estimates. Initial estimates for the coefficients of past conditional means are set to zero. If method="fixed", parameters given in further named list elements of start.control are used when available, else the predefined values given in the following are used.

intercept

numeric value with the start value for the intercept parameter. Default value is 1.

past_obs

numeric vector with the start values for parameters for regression on previous observations. Default values are zero.

past_mean

numeric vector with the start values for parameters for regression on previous conditional means. Default values are zero.

xreg

numeric vector with the start values for the regression parameters. These values will also be used if method="MM". Default values are zero.

final.control

named list with optional elements that determine how to make the final maximum likelihood estimation. If final.control=NULL, only start estimates are computed and a list with fewer elements which has not the class "tsglm" is returned. Possible list elements of this argument are:

constrained

named list whose elements are passed to function constrOptim with possible elements mu, outer.iterations and outer.eps (see constrOptim for details). If constrained=NULL, an unconstrained optimisation is made with function optim. Note that this is likely to result in a fitted model which is non-stationary, which might cause further problems.

optim.method

character which is passed to functions constrOptim or optim as argument method. The default is "BFGS".

optim.control

named list which is passed to function constrOptim or optim as the argument control. Must not contain the list element fnscale. The default is list(maxit=20, reltol=1e-8).

inter.control

named list determining how to maximise the log-likelihood function in a first step. This intermediate optimisation will start from the start estimation and be followed by the final optimisation, which will in turn start from the intermediate optimisation result. This intermediate optimisation is intended to use a very quick but imprecise optimisation algorithm. Possible elements are the same as for final.control. The default is inter.control=NULL, which skips this intermediate optimisation step.

Details

The INGARCH model (argument link="identity") used here follows the definition

Z[t]|F[t-1] ~ Poi(ν[t]) or Z[t]|F[t-1] ~ NegBin(ν[t], φ),

where F[t-1] denotes the history of the process up to time t-1, Poi and NegBin is the Poisson respectively the negative binomial distribution with the parametrisation as specified below. For the model with covariates having an internal effect (the default) the linear predictor of the INGARCH model (which is in that case identical to the conditional mean) is given by

ν[t] = β[0] + β[1] Z[t-i[1]] + … + β[p] Z[t-i[p]] + α[1] ν[t-j[1]] + … + α[q] ν[t-j[q]] + η[1] X[t,1] + … + η[r] X[t,r].

The log-linear model (argument link="log") used here follows the definition

Z[t]|F[t-1] ~ Poi(λ[t]) or Z[t]|F[t-1] ~ NegBin(λ[t], φ),

with λ[t] = \exp(ν[t]) and F[t-1] as above. For the model with covariates having an internal effect (the default) the linear predictor ν[t] = \log(λ[t]) of the log-linear model is given by

ν[t] = β[0] + β[1] \log(Z[t-i[1]]+1) + … + β[p] \log(Z[t-i[p]]+1) + α[1] ν[t-j[1]] + … + α[q] ν[t-j[q]] + η[1] X[t,1] + … + η[r] X[t,r].

Note that because of the logarithmic link function the effect of single summands in the linear predictor on the conditional mean is multiplicative and hence the parameters play a different role than in the INGARCH model, although they are denoted by the same letters.

The Poisson distribution is parametrised by the mean lambda according to the definition in Poisson. The negative binomial distribution is parametrised by the mean mu with an additional dispersion parameter size according to the definition in NegBinomial. In the notation above its mean parameter mu is ν[t] and its dispersion parameter size is φ.

This function allows to include covariates in two different ways. A covariate can have a so-called internal effect as defined above, where its effect propagates via the regression on past values of the linear predictor and on past observations. Alternatively, it can have a so-called external effect, where its effect does not directly propagates via the feedback on past values of the linear predictor, but only via past observations. For external effects of the covariates, the linear predictor for the model with identity link is given by

ν[t] = μ[t] + η[1] X[t,1] + … + η[r] X[t,r],

μ[t] = β[0] + β[1] Z[t-i[1]] + … + β[p] Z[t-i[p] + α[1] μ[t-j[1]] + … + α[q] μ[t-j[q]],

and analoguesly for the model with logarithmic link by

ν[t] = μ[t] + η[1] X[t,1] + … + η[r] X[t,r],

μ[t] = β[0] + β[1] \log(Z[t-i[1]]+1) + … + β[p] \log(Z[t-i[p]+1) + α[1] μ[t-j[1]] + … + α[q] μ[t-j[q]].

This is described in more detail by Liboschik et al. (2014) for the case of deterministic covariates for modelling interventions. It is also possible to model a combination of external and internal covariates, which can be defined straightforwardly by adding each covariate either to the linear predictor ν[t] itself (for an internal effect) or to μ[t] defined above (for an external effect).

Value

An object of class "tsglm", which is a list with at least the following elements:

coefficients

a named vector of the maximum likelihood estimated coefficients, which can be extracted by the coef method.

start

a named vector of the start estimation for the coefficients.

residuals

a vector of residuals, which can be extracted by the residuals method.

fitted.values

the fitted values, which can be extracted by the fitted method.

linear.predictors

the linear fit on link scale.

response

a vector of the response values (this is usually the original time series but possibly without the first few observations used for initialization if argument init.drop=TRUE).

logLik

the log-likelihood of the fitted model, which can be extracted by the logLik method. This is the complete log-likelihood including all constant terms. It is based on n_eff observations (see below).

score

the score vector at the maximum likelihood estimation.

info.matrix

the information matrix at the maximum likelihood estimation assuming a Poisson distribution.

info.matrix_corrected

the information matrix at the maximum likelihood estimation assuming the distribution specified in distr.

call

the matched call.

n_obs

the number of observations.

n_eff

the effective number of observations used for maximum likelihood estimation (might be lower than n_obs if argument init.drop=TRUE).

ts

the original time series.

model

the model specification.

xreg

the given covariates.

distr

a character giving the fitted conditional distribution.

distrcoefs

a named vector of the estimated additional coefficients specifying the conditional distribution. Is NULL in case of a Poisson distribution.

sigmasq

the estimated overdispersion coefficient. Is zero in case of a Poisson distribution.

The functions ingarch.fit and loglin.fit have the same output except the elements distr, distrcoefs and sigmasq. In addition, they return the following list elements:

inter

some details on the intermediate estimation of the coefficients as returned by constrOptim or optim.

final

some details on the final estimation of the coefficients as returned by constrOptim or optim.

durations

named vector of the durations of the model fit (in seconds).

outerscoreprod

array of outer products of score vectors at each time point.

Author(s)

Tobias Liboschik, Philipp Probst, Konstantinos Fokianos and Roland Fried

References

Christou, V. and Fokianos, K. (2014) Quasi-likelihood inference for negative binomial time series models. Journal of Time Series Analysis 35(1), 55–78, http://dx.doi.org/10.1002/jtsa.12050.

Christou, V. and Fokianos, K. (2015) Estimation and testing linearity for non-linear mixed poisson autoregressions. Electronic Journal of Statistics 9, 1357–1377, http://dx.doi.org/10.1214/15-EJS1044.

Ferland, R., Latour, A. and Oraichi, D. (2006) Integer-valued GARCH process. Journal of Time Series Analysis 27(6), 923–942, http://dx.doi.org/10.1111/j.1467-9892.2006.00496.x.

Fokianos, K. and Fried, R. (2010) Interventions in INGARCH processes. Journal of Time Series Analysis 31(3), 210–225, http://dx.doi.org/10.1111/j.1467-9892.2010.00657.x.

Fokianos, K., and Fried, R. (2012) Interventions in log-linear Poisson autoregression. Statistical Modelling 12(4), 299–322. http://dx.doi.org/10.1177/1471082X1201200401.

Fokianos, K., Rahbek, A. and Tjostheim, D. (2009) Poisson autoregression. Journal of the American Statistical Association 104(488), 1430–1439, http://dx.doi.org/10.1198/jasa.2009.tm08270.

Fokianos, K. and Tjostheim, D. (2011) Log-linear Poisson autoregression. Journal of Multivariate Analysis 102(3), 563–578, http://dx.doi.org/10.1016/j.jmva.2010.11.002.

Liboschik, T., Kerschke, P., Fokianos, K. and Fried, R. (2014) Modelling interventions in INGARCH processes. International Journal of Computer Mathematics (published online), http://dx.doi.org/10.1080/00207160.2014.949250.

See Also

S3 methods print, summary, residuals, plot, fitted, coef, predict, logLik, vcov, AIC, BIC and QIC for the class "tsglm". The S3 method se computes the standard errors of the parameter estimates. Additionally, there are the S3 methods pit, marcal and scoring for predictive model assessment.

S3 methods interv_test, interv_detect and interv_multiple for tests and detection procedures for intervention effects. tsglm.sim for simulation from GLM-type model for time series of counts. ingarch.mean, ingarch.var and ingarch.acf for calculation of analytical mean, variance and autocorrelation function of an INGARCH model (i.e. with identity link) without covariates.

Example time series of counts are campy, ecoli, ehec, influenza, measles in this package, polio in package gamlss.data.

Examples

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###Campylobacter infections in Canada (see help("campy"))
interventions <- interv_covariate(n=length(campy), tau=c(84, 100),
  delta=c(1, 0)) #detected by Fokianos and Fried (2010, 2012)
#Linear link function with Negative Binomial distribution:
campyfit <- tsglm(campy, model=list(past_obs=1, past_mean=13),
  xreg=interventions, distr="nbinom")
campyfit
plot(campyfit)

###Road casualties in Great Britain (see help("Seatbelts"))
timeseries <- Seatbelts[, "VanKilled"]
regressors <- cbind(PetrolPrice=Seatbelts[, c("PetrolPrice")],
                    linearTrend=seq(along=timeseries)/12)
#Logarithmic link function with Poisson distribution:                    
seatbeltsfit <- tsglm(ts=timeseries, link="log",
  model=list(past_obs=c(1, 12)), xreg=regressors, distr="poisson")
summary(seatbeltsfit)

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