Poisson Autoregression Research Articles

TNPAR: Topological Neural Poisson Auto-Regressive Model for Learning Granger Causal Structure from Event Sequences

Learning Granger causality from event sequences is a challenging but essential task across various applications. Most existing methods rely on the assumption that event sequences are independent and identically distributed (i.i.d.). However, this i.i.d. assumption is often violated due to the inherent dependencies among the event sequences. Fortunately, in practice, we find these dependencies can be modeled by a topological network, suggesting a potential solution to the non-i.i.d. problem by introducing the prior topological network into Granger causal discovery. This observation prompts us to tackle two ensuing challenges: 1) how to model the event sequences while incorporating both the prior topological network and the latent Granger causal structure, and 2) how to learn the Granger causal structure. To this end, we devise a unified topological neural Poisson auto-regressive model with two processes. In the generation process, we employ a variant of the neural Poisson process to model the event sequences, considering influences from both the topological network and the Granger causal structure. In the inference process, we formulate an amortized inference algorithm to infer the latent Granger causal structure. We encapsulate these two processes within a unified likelihood function, providing an end-to-end framework for this task. Experiments on simulated and real-world data demonstrate the effectiveness of our approach.

Dynamic Network Poisson Autoregression with Application to COVID-19 Count Data

Dynamic Network Poisson Autoregression with Application to COVID-19 Count Data

Bayesian Forecasting of Bounded Poisson Distributed Time Series.

This research models and forecasts bounded ordinal time series data that can appear in various contexts, such as air quality index (AQI) levels, economic situations, and credit ratings. This class of time series data is characterized by being bounded and exhibiting a concentration of large probabilities on a few categories, such as states 0 and 1. We propose using Bayesian methods for modeling and forecasting in zero-one-inflated bounded Poisson autoregressive (ZOBPAR) models, which are specifically designed to capture the dynamic changes in such ordinal time series data. We innovatively extend models to incorporate exogenous variables, marking a new direction in Bayesian inferences and forecasting. Simulation studies demonstrate that the proposed methods accurately estimate all unknown parameters, and the posterior means of parameter estimates are robustly close to the actual values as the sample size increases. In the empirical study we investigate three datasets of daily AQI levels from three stations in Taiwan and consider five competing models for the real examples. The results exhibit that the proposed method reasonably predicts the AQI levels in the testing period, especially for the Miaoli station.

Nonlinear Poisson autoregression and nonlinear Hawkes processes

The nonlinear Hawkes process is a point process for which the occurrence of future events depends on its history, either by excitation or inhibition. This property made it popular in many fields, such as neuro-sciences and social dynamics. In this paper we propose a tractable nonlinear Poisson autoregression as a discrete-time Hawkes process. Our model allows for cross-excitation and inhibition between components, as well as for exogenous random noise on the intensity. We then prove a convergence theorem as the time step goes to zero. Finally, we suggest a parametric calibration method for the continuous-time Hawkes process based on the discrete-time approximation.

Estimation of Poisson Autoregressive Model for Multiple Time Series

A Poisson autoregressive (PAR) model accounting for discreteness and autocorrelation of count time series data is typically estimated in the state-space modeling framework through an extended Kalman filter. However, because of the complex dependencies in count time series, estimation becomes more challenging. PAR is viewed as an additive model and estimated using a hybrid of cubic smoothing splines and maximum likelihood estimation (MLE) in the backfitting framework. Simulation studies show that this estimation method is at the least comparable to PAR estimated in the state-space context, especially with larger counts. The flexibility of the additive model has two significant benefits: [1] robust estimation in the presence of temporary structural change and [2] viability to integrate the PAR model into a more complex model structure. We further generalized the PAR(p) model into multiple time series of counts and illustrated it with indicators in the financial markets.

A log linear Poisson autoregressive model to understand COVID-19 dynamics in Saudi Arabia

BackgroundOn March 2, 2020, the first case of COVID-19 infection in Saudi Arabia was identified and announced by the health authorities. From first week of March, the number of new confirmed COVID-cases has gradually increased, reaching 2932 confirmed cases on April 9, 2020. A period of increasing infection cases was noticed in June and July 2020. Many methods have been taken to model and predict the new confirmed cases of COVID-19, such as the traditional time series forecasting method and other several methods.ResultsWe present two statistical models, namely the log linear autoregressive Poisson model and the ARIMA model. The COVID-19 infectious dynamics were evaluated using models in Saudi Arabia, which can affect health, economics, finance, and other fields. We applied both models to daily confirmed cases of COVID-19 count time series data. Moreover, we compare the log linear Poisson autoregressive model with the automatic ARIMA model.ConclusionsThe result of this study showed that a log linear Poisson Autoregressive model gives better forecasting and the predicted results of the log linear Poisson Autoregressive model can be used as the baseline for additional interference to avoid future COVID-19 pandemic incidents. Moreover, the application of a log linear Poisson Autoregressive can be comprehensive to other cases in Saudi Arabia.

Model Selection for Time Series Count Data with Over-Dispersion

Time series of count with over-dispersion is the reality often encountered in many biomedical and public health applications. Statistical modelling of this type of series has been a great challenge. Rottenly, the Poisson and negative binomial distributions have been widely used in practice for discrete count time series data, their forms are too simplistic to accommodate features such as over-dispersion. Unable to account for these associated features while analyzing such data may result in incorrect and sometimes misleading inferences as well as detection of spurious associations. Therefore, the need for further investigation of count time series models suitable to fit count time series with over-dispersion of different level. The study therefore proposed a best model that can fit and forecast time series count data with different levels of over-dispersion and sample sizes Simulation studies were conducted using R statistical package, to investigate the performances of Autoregressiove Conditional Poisson (ACP) and Poisson Autoregressive (PAR) models. The predictive ability of the models were observed at different steps ahead. The relative performance of the models were examined using Akaike Information criteria (AIC) and Hannan-Quinn Information Criteria (HQIC). Conclusively, the best model to fit was ACP at different sample sizes. The predictive abilities of the four fitted models increased as sample size and number of steps ahead were increased

Monitoring COVID-19 contagion growth.

We present a statistical model that can be employed to monitor the time evolution of the COVID‐19 contagion curve and the associated reproduction rate. The model is a Poisson autoregression of the daily new observed cases and dynamically adapt its estimates to explain the evolution of contagion in terms of a short‐term and long‐term dependence of case counts, allowing for a comparative evaluation of health policy measures. We have applied the model to 2020 data from the countries most hit by the virus. Our empirical findings show that the proposed model describes the evolution of contagion dynamics and determines whether contagion growth can be affected by health policies. Based on our findings, we can draw two health policy conclusions that can be useful for all countries in the world. First, policy measures aimed at reducing contagion are very useful when contagion is at its peak to reduce the reproduction rate. Second, the contagion curve should be accurately monitored over time to apply policy measures that are cost‐effective.

Goodness–of–Fit Tests for Bivariate Time Series of Counts

This article considers goodness-of-fit tests for bivariate INAR and bivariate Poisson autoregression models. The test statistics are based on an L2-type distance between two estimators of the probability generating function of the observations: one being entirely nonparametric and the second one being semiparametric computed under the corresponding null hypothesis. The asymptotic distribution of the proposed tests statistics both under the null hypotheses as well as under alternatives is derived and consistency is proved. The case of testing bivariate generalized Poisson autoregression and extension of the methods to dimension higher than two are also discussed. The finite-sample performance of a parametric bootstrap version of the tests is illustrated via a series of Monte Carlo experiments. The article concludes with applications on real data sets and discussion.

Effects of air temperature on the number of ambulance calls for asthma during cold season in Nur-Sultan– the second coldest capital in the world

ABSTRACT Deleterious effect of cold on overall mortality is well-established. We studied associations between the air temperature and the number f ambulance calls for asthma in Nur-Sultan, Kazakhstan – the second coldest capital in the world. Daily counts of ambulance calls for asthma in Nur-Sultan for the cold seasons (October-March) 2006–2010 were obtained from the Municipal Ambulance Station. Associations between the number of calls and mean and minimum apparent temperatures (average for lags 0–15) were studied using first-order Poisson auto-regression models controlling for wind speed and effects of month, year, weekends and holidays. Altogether, there were 7373 ambulance calls for asthma during the study period. An inverse association between minimum apparent temperature and the number of calls was observed for the age-group 60 years and older. A decrease of the minimum apparent temperature by 1°C was associated with an increase in the number of calls by 1.7% (95% CI: 0.1%-3.3%) across the whole temperature spectrum. No associations in other age groups were found. Our results suggest an inverse association between the average 15-day lag minimum apparent temperature and the number of ambulance calls during the cold season in Nur-Sultan, but this is limited to the oldest age-group.

Pandemic risk: operational aspects

At the beginning of the COVID-19 pandemic, Intesa Sanpaolo has developed a contagion model aimed at calibrating the measures to be taken to safeguard its employees and the provision of banking services, according to the risk deriving from the external environment. The model is based on both external and internal views: the combination of such elements provides a holistic picture of the overall contagion risk level, enabling the opportunity to take informed decisions for each geographical area. One of the most useful external indicators to feed the model is the reproduction index (Rt) provided by the University of Pavia which is calculated using the Poisson autoregressive (PAR) model. Though more research and development are needed, the proposed model represents a useful tool for supporting the Top Managers in the decision process to continue business as usual safely.

Analysis of HIV/AIDS Epidemic and Socioeconomic Factors in Sub-Saharan Africa.

Sub-Saharan Africa has been the epicenter of the outbreak since the spread of acquired immunodeficiency syndrome (AIDS) began to be prevalent. This article proposes several regression models to investigate the relationships between the HIV/AIDS epidemic and socioeconomic factors (the gross domestic product per capita, and population density) in ten countries of Sub-Saharan Africa, for 2011–2016. The maximum likelihood method was used to estimate the unknown parameters of these models along with the Newton–Raphson procedure and Fisher scoring algorithm. Comparing these regression models, there exist significant spatiotemporal non-stationarity and auto-correlations between the HIV/AIDS epidemic and two socioeconomic factors. Based on the empirical results, we suggest that the geographically and temporally weighted Poisson autoregressive (GTWPAR) model is more suitable than other models, and has the better fitting results.

Necessary and sufficient conditions for the identifiability of observation‐driven models

In this contribution we are interested in proving that a given observation‐driven model is identifiable. In the case of a GARCH(p, q) model, a simple sufficient condition has been established in Berkes I, Horváth L, Kokoszka P. (2003). Bernoulli 9: 201–227 for showing the consistency of the quasi‐maximum likelihood estimator. It turns out that this condition applies for a much larger class of observation‐driven models, that we call the class of linearly observation‐driven models. This class includes standard integer valued observation‐driven time series such as the Poisson autoregression model and its numerous extensions. Our results also apply to vector‐valued time series such as the bivariate integer valued GARCH model, to nonlinear models such as the threshold Poisson autoregression or to observation‐driven models with exogenous covariates such as the PARX model.

Exploring the Application of the Linear Poisson Autoregressive Model for Analyzing the Dynamic Impact of Traffic Laws on Fatal Traffic Accident Frequency

Annual fatal traffic accident data often demonstrate time series characteristics. The existing traffic safety analysis approaches (e.g., negative binomial (NB) model) often cannot accommodate the dynamic impact of factors in fatal traffic accident data and may result in biased parameter estimation results. Thus, a linear Poisson autoregressive (PAR) model is proposed in this study. The objective of this study is to apply the PAR model to analyze the dynamic impact of traffic laws and climate on the frequency of fatal traffic accidents occurred in a large time span (from 1975 to 2016) in Illinois. Besides, the NB model, NB with a time trend, and autoregressive integrated moving average model with exogenous input variables (ARIMAX) are also developed to compare their performances. The important conclusions from the modelling results can be summarized as follows. (1) The PAR model is more appropriate for analyzing the dynamic impacts of traffic laws on annual fatal traffic accidents, especially the instantaneous impacts. (2) The law that allows motorcycles and bicycles to proceed on a red light following the rules applicable after a “reasonable period of time” leads to an increase in the frequency of annual fatal traffic accidents by 14.98% in the short term and 30.69% in the long term. The climate factors such as average temperature and precipitation concentration period have insignificant impacts on annual fatal traffic accidents in Illinois. Thus, the modelling results suggest that the PAR model is more appropriate for annual fatal traffic accident data and has an advantage in estimating the dynamic impact of traffic laws.

A proactive auto-scaling scheme with latency guarantees for multi-tenant NFV cloud

Network Functions Virtualization (NFV) is a promising technology to provide packet processing services. However, dynamic capacity provisioning to meet different tenants’ time-varying demands under service-level agreements is still a challenge for NFV service providers. The existing works generally perform the scaling-in/out actions for separate service chains and cannot promise a guarantee for the total processing time. Therefore, we propose Palm, a proactive auto-scaling framework to minimize the resource consumption while enforcing latency guarantees for multiple intersecting service chains. We first leverage the classed Jackson network model to analyze the packet processing latency. Then, a log-linear Poisson auto-regression method is employed to predict each tenant’s packet arrival rate. Based on the prediction result, we perform the capacity adjustment actions and update the flow forwarding policies. We formulate the capacity provisioning task as a nonlinear integer programming problem and propose an evolution based algorithm to tackle it. To simplify the auto-scaling problem, we develop an adaptive algorithm to divide the NFV cloud into persistent and temporary layers. Our comprehensive experiments show that Palm achieves a steady low-latency performance at a lower cost compared with the state-of-the-art dynamic scaling strategies.

A Poisson Autoregressive Model to Understand COVID-19 Contagion Dynamics

We present a statistical model which can be employed to understand the contagion dynamics of the COVID-19, which can heavily impact health, economics and finance. The model is a Poisson autoregression of the daily new observed cases, and can reveal whether contagion has a trend, and where is each country on that trend. Model results are exemplified from some observed series.

COVID-19 contagion and digital finance.

Digital finance is going to be heavily affected by the COVID-19 outbreak. We present a statistical model which can be employed to understand the contagion dynamics of the COVID-19, so that its impact on finance can possibly be anticipated, and digitally monitored. The model is a Poisson autoregression of the daily new observed cases, and considers both short-term and long-term dependence in the infections counts. Model results are presented for the observed time series of China, the first affected country, but can be easily reproduced for all countries.

A Poisson Autoregressive Model to Understand COVID-19 Contagion Dynamics

We present a statistical model which can be employed to understand the contagion dynamics of the COVID-19. The model is a Poisson autoregression, and can reveal whether contagion has a trend, and where is each country on that trend. Model results are presented from the observed series of China, Iran, Italy and South Korea.

A Statistical Model to Monitor COVID-19 Contagion Growth

We present a statistical model which can be employed to monitor the time evolution of the COVID-19 contagion curve, and of the associated reproduction rate R0. The model is a Poisson autoregression of the daily new observed cases and can, differently from classical exponential growth models, dynamically adapt its estimates to explain the evolution of contagion in different time periods and locations, allowing for the comparative evaluation of health policy measures. We have applied the model to the first two months of data from the countries most hit by the virus. Our empirical findings show that the proposed model is able to identify where a country lies on the contagion curve, at each point in time: behind a peak, on a peak, and after a peak. Based on this positioning, we draw three main health policy conclusions that can be useful for all countries in the world. First, countries that are still behind the peak (e.g. France and the United Kingdom) should maintain strong containment measures, such as diffuse testing and lockdown. Second, countries that are on the peak and have experienced a steep contagion growth (e.g. Italy, Spain and the United States), could partially relax containment measures but must couple them with continuous monitoring of the growth curve. Third, in countries that have experienced a less steep contagion growth (e.g. Germany) the approach to restrictive measures could be more cautious since there is a risk that social costs outweigh the benefits. Last, countries that have passed the peak (e.g. China) should relax containment measures, but continue statistical monitoring.

A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Integer‐valued time series data have an ever‐increasing presence in various applications (e.g., the number of purchases made in response to a marketing strategy, or the number of employees at a business) and need to be analyzed properly. While a Poisson autoregressive (PAR) model would seem like a natural choice to model such data, it is constrained by the equi‐dispersion assumption (i.e., that the variance and the mean equal). Hence, data that are over‐ or under‐dispersed (i.e., have the variance greater or less than the mean respectively) are improperly modeled, resulting in biased estimates and inaccurate forecasts. This work instead develops a flexible integer‐valued autoregressive model for count data that contain over‐ or under‐dispersion. Using the Conway–Maxwell–Poisson (CMP) distribution and related distributions as motivation, we develop a first‐order sum‐of‐CMP's autoregressive (SCMPAR(1)) model that will instead offer a generalizable construct that captures the PAR, and versions of what we refer to as a negative binomial AR model, and binomial AR model respectively as special cases, and serve as an overarching representation connecting these three special cases through the dispersion parameter. We illustrate the SCMPAR model's flexibility and ability to effectively model count time series data containing data dispersion through simulated and real data examples.