predict.pgam {pgam}

Prediction

Description

Prediction and forecasting of the fitted model.

Usage

## S3 method for class 'pgam'
predict(object, forecast = FALSE, k = 1, x = NULL, ...)

Arguments

object

object of class pgam holding the fitted model

forecast

if TRUE the function tries to forecast

k

steps for forecasting

x

covariate values for forecasting if the model has covariates. Must have the k rows and p columns

...

further arguments passed to method

Details

It estimates predicted values, their variances, deviance components, generalized Pearson statistics components, local level, smoothed prediction and forecast.

Considering a Poisson process and a gamma priori, the predictive distribution of the model is negative binomial with parameters a_{t|t-1} and b_{t|t-1}. So, the conditional mean and variance are given by

E\left(y_{t}|Y_{t-1}\right)=a_{t|t-1}/b_{t|t-1}

and

Var\left(y_{t}|Y_{t-1}\right)=a_{t|t-1}\left(1+b_{t|t-1}\right)/b_{t|t-1}^{2}

Deviance components are estimated as follow

D\left(y;\hat\mu\right)=2\sum_{t=\tau+1}^{n}{a_{t|t-1}\log \left(\frac{a_{t|t-1}}{y_{t}b_{t|t-1}}\right)-\left(a_{t|t-1}+y_{t}\right)\log \frac{\left(y_{t}+a_{t|t-1}\right)}{\left(1+b_{t|t-1}\right)y_{t}}}

Generalized Pearson statistics has the form

X^{2}=\sum_{t=\tau+1}^{n}\frac{\left(y_{t}b_{t|t-1}-a_{t|t-1}\right)^{2}} {a_{t|t-1}\left(1+b_{t|t-1}\right)}

Approximate scale parameter is given by the expression

\hat\phi=frac{X^{2}}{edf}

where edf is the number o degrees of reedom of the fitted model.

Value

List with those described in Details

Author(s)

Washington Leite Junger [e-mail] and Antonio Ponce de Leon [e-mail]

References

Green, P. J., Silverman, B. W. (1994) Nonparametric Regression and Generalized Linear Models: a roughness penalty approach. Chapman and Hall, London

Harvey, A. C., Fernandes, C. (1989) Time series models for count data or qualitative observations. Journal of Business and Economic Statistics, 7(4):407–417

Junger, W. L. (2004) Semiparametric Poisson-Gamma models: a roughness penalty approach. MSc Dissertation. Rio de Janeiro, PUC-Rio, Department of Electrical Engineering.

Harvey, A. C. (1990) Forecasting, structural time series models and the Kalman Filter. Cambridge, New York

Hastie, T. J., Tibshirani, R. J.(1990) Generalized Additive Models. Chapman and Hall, London

McCullagh, P., Nelder, J. A. (1989). Generalized Linear Models. Chapman and Hall, 2nd edition, London

See Also

pgam, residuals.pgam

Examples

library(pgam)
data(aihrio)
attach(aihrio)
form <- ITRESP5~f(WEEK)+HOLIDAYS+rain+PM+g(tmpmax,7)+g(wet,3)
m <- pgam(form,aihrio,omega=.8,beta=.01,maxit=1e2,eps=1e-4,optim.method="BFGS")

p <- predict(m)$yhat
plot(ITRESP5)
lines(p)