Time Series Analysis of Discrete-Value Processes

Time series analysis of discrete-value processes focuses on sequences of observations that take values in a countable set, most commonly non-negative integers or finite categories. Such processes arise across disciplines—from epidemiology and ecology to finance and industrial quality control—and pose distinct challenges compared to real-valued series. Key features include integer support, possible zero-inflation or deflation, overdispersion or underdispersion, and complex autocorrelation structures. Methodological frameworks include thinning-based autoregressive constructions (INAR models), generalised count regression dynamics (INGARCH), hidden Markov formulations and score-driven updates. Recent advances have integrated Bayesian inference and efficient computational schemes, non-linear and heavy-tailed innovation distributions for extreme events, and machine-learning enhancements. Practical applications range from forecasting disease incidence and agricultural pests to modelling traffic accidents, financial transaction counts and categorical behavioural states, underscoring both global significance and diverse applicability.

Research from Nature Portfolio

Recent studies have introduced a flexible seasonal integer-valued autoregressive framework for modelling count data with recurring patterns and varying dispersion. The proposed seasonal INAR(p) model accommodates zero-inflation and deflation, overdispersion and underdispersion through a structured thinning operator. Multiple estimation strategies—including conditional likelihood, method of moments and Bayesian approaches—have been compared via simulation and applied to weekly influenza data, demonstrating improved fit and more accurate short-term forecasts than existing alternatives.

Time Series Analysis of Discrete-Value Processes publication trend

The graph below shows the total number of articles in time series analysis of discrete-value processes across all publications each year (not limited to Nature Index journals).

Technical terms

Discrete-value process: A time series whose observations belong to a countable set, such as non-negative integers or a finite set of categories.

Count time series: A sequence of non-negative integer observations indexed by time, often exhibiting autocorrelation and dispersion features.

Integer-valued autoregressive (INAR) model: A class of models for count data using a thinning operator to define lagged dependence.

Overdispersion: A condition in count data where the variance exceeds the mean, indicating extra variability beyond a Poisson assumption.

Zero-inflation: The presence of more zero observations in a count series than standard distributions predict.

Categorical time series: A sequence of observations taking values in a finite set of discrete categories, requiring specialised modelling of transitions.

Autocorrelation: The correlation between observations at different time lags, crucial for characterising temporal dependence structures.

References

1. Generalized Poisson difference autoregressive processes. International Journal of Forecasting (2024).
2. Some developments on seasonal INAR processes with application to influenza data. Scientific Reports (2023).
3. Analyzing categorical time series with the R package ctsfeatures. Journal of Computational Science (2024).
4. Beta–Negative Binomial Auto-Regressions for Modelling Integer-Valued Time Series with Extreme Observations. Journal of the Royal Statistical Society Series B Statistical Methodology (2020).