Data Modeling With Polynomial Representations and Autoregressive Time-Series Representations, and Their Connections

Area of Science:

• Data Science
• Statistical Modeling
• Signal Processing

Background:

• Polynomial and time-series representations are distinct data modeling techniques.
• Understanding their relationship is crucial for advanced data analysis.

Purpose of the Study:

• To establish the connections and differences between polynomial and time-series data modeling.
• To prove that finite-degree polynomial data can be perfectly represented by autoregressive time-series models.

Main Methods:

• Theoretical analysis of uniformly sampled, noise-free data.
• Mathematical proof establishing the equivalence between polynomial and autoregressive time-series models.
• Numerical explorations using generated and real-world data (e.g., COVID-19 incidence).

Main Results:

• Data from finite-degree polynomial models are perfectly represented by autoregressive time-series models of a specific order.
• Polynomials of a given degree yield identical time-series coefficients, differing only in a constant term.
• Time-series with non-specific coefficients represent infinite-degree polynomials.
• Finite-order all-pole filters can represent polynomials of finite or infinite degree.

Conclusions:

• Autoregressive time-series models offer an exact representation for noise-free data modeled by finite-order all-pole filters.
• Time-series representations are a viable alternative for data modeling when specific polynomial coefficient values are not critical.