260 Handbook of Discrete-Valued Time Series
TABLE 11.2
Posterior statistics for ψ and γ for the model with covariates
Statistics ψ
CMHPI
ψ
COFI
ψ
FOR
ψ
Unemp
γ
25th 0.0063 0.7003 −1.5430 0.6252 0.2281
Mean 0.0160 0.8717 −1.3002 0.8191 0.2466
75th 0.0256 1.0510 −1.0550 1.0117 0.2643
St. Dev 0.0141 0.2663 0.3606 0.2826 0.0270
0
50
100
150
200
250
Default counts
Actual data
Model with covariates
0 20 40 60 80 100 120 140
t
FIGURE 11.5
Retrospective t of the model with covariates to data.
This approach would be of interest to grocery store managers who would like to be able
to differentiate the common effect from individual effects for store promotion purposes.
For example, inference about θ
t
may allow managers to carry out store-wide promotion
activities, whereas analysis on individual λ
j
s will allow managers to target specic house-
holds to promote store visits. For illustrative purposes, we have xed the discount factor
at γ = 0.5 and set the initial prior parameters as α
0
j
= α
0
= 0.001 and β
0
j
= β
0
= 0.001 with
j representing each individual household. As earlier, the behavior of the common arrival
rates, θ
t
s, can be described via their joint distribution, p(θ
1
, ..., θ
104
|D
104
). We present a
boxplot of θ
t
s in Figure 11.8 which provides insights into the temporal behavior of the com-
mon rate given the count data and the individual rate λ
j
s. Specically, we nd a drop in
the common rates in weeks 29–35 and in weeks 79–85, which indicates a possible seasonal
effect occurring over the calendar year. Note that, as previously discussed, such seasonal
effects can be easily incorporated into the model as covariates.
Furthermore, the posterior density plots of λ
j
s are shown in Figure 11.9. Clearly, house-
hold 1 can be characterized by a higher rate compared to household 2. Store managers may
261 Bayesian Modeling of Time Series of Counts with Business Applications
θ
400
300
200
100
0
1 7 14 22 30 38 46 54 62 70 78 86 94 103 113 123 133 143
t
FIGURE 11.6
Boxplots for the latent rates, θ
t
sfrom p(θ
1
, ..., θ
144
|D
144
).
15
10
0 20 40 60 80 100
Store visits
Household 1
Household 2
5
0
t
FIGURE 11.7
Time series plot of weekly grocery store visits.
262 Handbook of Discrete-Valued Time Series
0
1
2
3
4
5
6
7
θ
1 6 12 19 26 33 40 47 54 61 68 75 82 89 96 104
t
FIGURE 11.8
Boxplots for the common rates, θ
t
sfrom p(θ
1
, ..., θ
104
|D
104
).
(a)
λ
1
(b)
λ
2
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
0.0
0.5
1.0
1.5
FIGURE 11.9
Posterior density plots of (a) λ
1
and (b) λ
2
.
utilize our ndings in multiple ways. First, the manager may identify the season associated
with low common arrival rates and run a store-wide promotion strategy to lure customers.
In addition, store managers faced with a limited budget may also wish to focus their promo-
tion efforts on household 2 with a lower arrival rate rather than on household 1. Finally, one
may also extend the model to include covariates to identify the reasons behind differences
in individual household λ
j
s.
263 Bayesian Modeling of Time Series of Counts with Business Applications
11.6 Conclusion
In this chapter, we have introduced a general class of Poisson time series models. We
rst discussed univariate discrete-time state-space models with Poisson measurements and
their Bayesian inference via MCMC methods. Furthermore, we discussed issues of sequen-
tial updating, ltering, smoothing, and forecasting. We also introduced modeling strategies
for multivariate extensions. In order to show the implementation of the proposed models,
we have used real count data from different disciplines such as nance, operations, and
marketing.
We believe that several future directions can be pursued as a consequence of this study.
One such area is to treat γ as a time-varying or a series-specic discount factor which
could potentially create challenges in parameter estimation. Another possibility for estima-
tion purposes is to investigate the implementation of sequential particle ltering methods
instead of MCMC that are widely used for state-space applications.
References
Aktekin, T., Kim, B., and Soyer, R. (2014). Dynamic multivariate distributions for count data: A
Bayesian approach. Technical report, Institute for Integrating Statistics in Decision Sciences,
The George Washington University, Washington, DC.
Aktekin, T. and Soyer, R. (2011). Call center arrival modeling: A Bayesian state-space approach. Naval
Research Logistics, 58(1):28–42.
Aktekin, T., Soyer, R., and Xu, F. (2013). Assessment of mortgage default risk via Bayesian state space
models. Annals of Applied Statistics, 7(3):1450–1473.
Arbous, A. G. and Kerrich, J. (1951). Accident statistics and the concept of accident proneness.
Biometrics, 7:340–432.
Chib, S. and Greenberg, E. (1995). Understanding the Metropolis-Hasting algorithm. The American
Statistician, 49(4):327–335.
Chib, S., Greenberg, E., and Winkelmann, R. (1998). Posterior simulation and Bayes factors in panel
count data models. Journal of Econometrics, 86:33–54.
Chib, S. and Winkelmann, R. (2001). Markov chain Monte Carlo analysis of correlated count data.
Journal of Business and Economic Statistics, 19(4):428–435.
Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal
of Statistics, 8:93–115.
Davis, R. A., Dunsmuir, W. T. M., and Streett, S. B. (2003). Observation-driven models for Poisson
counts. Biometrika, 90(4):777–790.
Davis, R. A., Dunsmuir, W. T. M., and Wang, Y. (2000). On autocorrelation in a Poisson regression
model. Biometrika, 87(3):491–505.
Durbin, J. and Koopman, S. (2000). Time series analysis of non-Gaussian observations based on state
space models from both classical and Bayesian perspectives. JournaloftheRoyalStatisticalSociety.
Series B, 62(1):3–56.
Freeland, R. K. and McCabe, B. P. M. (2004). Analysis of low count time series data by Poisson
autocorrelation. Journal of Time Series Analysis, 25(5):701–722.
Fruhwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. Journal of Time Series
Analysis, 15:183–202.
Frühwirth-Schnatter, S. and Wagner, H. (2006). Auxiliary mixture sampling for parameter-driven
models of time series of counts with applications to state space modelling. Biometrika,
93(4):827–841.
264 Handbook of Discrete-Valued Time Series
Gamerman, D., Abanto-Valle, C. A., Silva, R. S., and Martins, T. G. (2015). Dynamic Bayesian models
for discrete-valued time series. In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds.,
Handbook of Discrete-Valued Time Series, pp. 165–186. Chapman & Hall, Boca Raton, FL.
Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo Stochastic Simulation for Bayesian
Inference. Chapman & Hall/CRC Press, Boca Raton, FL.
Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice.
Chapman & Hall/CRC Press, Boca Raton, FL.
Harvey, A. C. and Fernandes, C. (1989). Time series models for count or qualitative observations.
Journal of Business and Economic Statistics, 7(4):407–417.
Lindley, D. V. and Singpurwalla, N. D. (1986). Multivariate distributions for the life lengths of com-
ponents of a system sharing a common environment. Journal of Applied Probability, 23:418–431.
Ravishanker, N., Venkatesan, R., and Hu, S. (2015). Dynamic models for time series of counts with a
marketing application. In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook
of Discrete-Valued Time Series, pp. 425–446. Chapman & Hall, Boca Raton, FL.
Santos, T. R. D., Gamerman, D., and Franco, G. C. (2013). A non-Gaussian family of state-space models
with exact marginal likelihood. Journal of Time Series Analysis, 34(6):625–645.
Smith, A. F. M. and Gelman, A. E. (1992). Bayesian statistics without tears: A sampling perspective.
The American Statistician, 46(2):84–88.
Smith, R. and Miller, J. (1986). A non-Gaussian state space model and application to prediction of
records. Journal of the Royal Statistical Society. Series B, 48(1):79–88.
Zeger, S. L. (1988). A regression model for time series of counts. Biometrika, 75(4):621–629.
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset