Forecasting of Multiple Seasonal Categorical Time Series Using Fourier Series with Application to AQI Data of Kolkata

TL;DR

This paper introduces a Fourier series-based framework for modeling multiple seasonal patterns in categorical time series, specifically applied to AQI data, improving forecasting accuracy over traditional methods.

Contribution

It develops a novel approach combining Fourier series and indicator functions to capture multiple seasonalities in categorical data, extending TBATS methodology.

Findings

The model accurately captures complex seasonal dynamics.

It outperforms Markov and machine learning models in forecasting AQI.

Simulation confirms empirical consistency of parameter estimates.

Abstract

Multiple seasonalities have been widely studied in continuous time series using models such as TBATS, for instance in electricity demand forecasting. However, their treatment in categorical time series, such as air quality index (AQI) data, remains limited. Categorical AQI often exhibits distinct seasonal patterns at multiple frequencies, which are not captured by standard models. In this paper, we propose a framework that models multiple seasonalities using Fourier series and indicator functions, inspired by the TBATS methodology. The approach accommodates the ordinal nature of AQI categories while explicitly capturing daily, weekly and yearly seasonal cycles. Simulation studies demonstrate the empirical consistency of parameter estimates under the proposed model. We further illustrate its applicability using real categorical AQI data from Kolkata and compare forecasting performance with Markov models and machine learning methods. Results indicate that our approach effectively captures complex seasonal dynamics and provides improved predictive accuracy. The proposed methodology offers a flexible and interpretable framework for analyzing categorical time series exhibiting multiple seasonal patterns, with potential applications in air quality monitoring, energy consumption and other environmental domains.