Generalized Linear Autoregressive Moving Average Models with Various Distributions

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Description

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.

Usage

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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")

Arguments

y

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.

X

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.

offset

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.

beta

Numeric vector; initial values of the regression coefficients.

phiLags

Numeric vector; AR orders.

phiInit

Numeric vector; initial values for the corresponding AR orders.

thetaLags

Numeric vector; MA orders.

thetaInit

Numeric vector; initial values for the corresponding MA orders.

delta

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.

alpha

Numeric; an optional initial shape parameter for glm.nb.

alphaInit

Numeric; an initial shape parameter for glarma for negative binomial counts.

type

Character; the count distribution. Possible values are "Poi" (Poisson), "Bin" (binomial) and "NegBin" (negative binomial). The default is the Poisson distribution.

method

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.

residuals

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.

maxit

Numeric; the maximum number of iterations allowed.

grad

Numeric; the tolerance for recognizing numbers, which are smaller than the specified tolerance, as zero.

Details

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 = transpose(X_t) * beta + offset + Z_t.

The infinite moving average from the linear predictor is

Z_t = sum(gamma_i * residuals_(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_{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 delta hat^(0) for the vector of parameters updates are obtained using the iterations

delta_(k+1) = delta_(k) + Omega(delta_k) * first derivative of log(delta_k)

where Omega(delta hat ^(k)) is some suitably chosen matrix.

Iterations continue for k >= 1 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.

Value

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:

delta

a vector of coefficients for beta, AR and MA.

logLik

the loglikelihood of the specific distribution.

logLikDeriv

the derivative of the loglikelhood of the specified distribution.

logLikDeriv2

the second derivative of the loglikelihood of the specified distribution.

eta

the estimated linear predictor.

mu

the GLARMA estimated mean.

fitted.values

the GLARMA fitted values.

residuals

the residuals of the type specified.

cov

the estimated covariance matrix of the maximum likelihood estimators.

phiLags

vector of AR orders.

thetaLags

vector of MA orders.

r

the number of columns in the model matrix.

pq

the number of phiLags plus the number of thetaLags.

null.deviance

the deviance from the initial GLM fit.

df.null

the degrees of freedom from the initial GLM fit.

y

the y vector used in the GLARMA model.

X

the model matrix.

offset

the offset, NULL if there is no offset.

type

the distribution of the counts.

method

the method of iteration used.

residType

the type of the residuals returned.

call

the matched call.

iter

the number of iterations.

errCode

the error code; 0 indicating successful convergence of the iteration method, 1 indicating failure.

WError

error code for finiteness of W; 0 indicating all values of W are finite, 1 indicating at least one infinite value.

min

the minimum of the absolute value of the gradient.

aic

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.

Author(s)

The original GLARMA routine for Poisson responses was developed in collaboration with Richard A. Davis and Ying Wang. The binomial response version was developed with the assistance of Haolan Lu. The extension to negative binomial response was carried out by Bo Wang. Daniel Drescher contributed to the initial structure of the software used as the basis of the package.

The main author of the package is "William T.M. Dunsmuir" <w.dunsmuir@unsw.edu.au>. Package development was carried out by Cenanning Li supervised by David J. Scott.

References

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

See Also

Additional examples may be found in Asthma, OxBoatRace, RobberyConvict, and DriverDeaths.

Examples

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### 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)