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

This paper proposes a novel framework that adapts Fourier series and indicator functions to model multiple seasonalities in categorical time series, demonstrating improved forecasting accuracy for Kolkata's Air Quality Index data compared to traditional Markov and machine learning methods.

Original authors: Anirban Ghosh, Raju Maiti

Published 2026-03-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anirban Ghosh, Raju Maiti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather, but instead of "sunny" or "rainy," you are predicting the Air Quality Index (AQI). In cities like Kolkata, the air quality isn't just a number; it's a category: Good, Satisfactory, Moderate, Poor, Very Poor, or Severe.

The problem is that air quality doesn't just change randomly. It has rhythms, like a heartbeat.

The Daily Beat: It's often worse on weekdays when traffic is heavy and better on weekends.
The Seasonal Beat: It's terrible in winter (due to fog and lack of wind) but great during the monsoon (rain washes the dust away).
The Festival Beat: It spikes dramatically during festivals like Diwali because of fireworks.

This paper is about building a better "crystal ball" to predict these air quality categories by understanding these complex rhythms.

The Old Way vs. The New Way

The Old Way (The "Step Ladder" Approach):
Traditionally, statisticians used models like Markov Chains. Imagine a ladder where you can only step to the rung right above or below you. These models look at yesterday's air quality and guess today's.

The Flaw: They are like a rigid robot. They don't understand why the air is bad. They don't know that "Winter" or "Diwali" is coming up. They just see a pattern of "bad air yesterday = bad air today." They miss the big picture of the seasons and festivals.

The New Way (The "Music and Calendar" Approach):
The authors, Anirban Ghosh and Raju Maiti, propose two new models inspired by a famous method used for electricity forecasting called TBATS. They adapted it for air quality categories. Think of it as teaching the computer to "hear" the music of the seasons and "read" the calendar.

They created two versions of this new model:

1. TSOLR: The "Smooth Music" Model

The Analogy: Imagine the seasons as a smooth, flowing song. Winter is a low, deep note; Summer is a high, bright note.
How it works: This model uses Fourier Series (a fancy math term for combining sine and cosine waves) to draw smooth curves that represent these rhythms. It's like drawing a smooth wave on a graph that goes up and down to match the yearly cycle of pollution.
Why it's cool: It's great at capturing gradual changes. It understands that pollution slowly gets worse as winter approaches and slowly gets better as spring arrives. It can also handle "high notes" (sudden spikes) like Diwali fireworks.

2. ISOLR: The "Calendar Switch" Model

The Analogy: Imagine a calendar with big, bold switches. One switch says "Winter," another says "Diwali," another says "Weekend."
How it works: This model uses Indicator Functions. It's like flipping a light switch. If today is Diwali, the "Diwali Switch" is ON (1), and everything else is OFF (0).
Why it's cool: It's very direct. It doesn't try to smooth things out; it just says, "Hey, it's a festival day, so pollution will be high!" It's great for sudden, sharp changes.

The Experiment: Who Wins?

The authors tested these models using real data from Kolkata (2019–2024). They compared their new models against:

Old School: Markov Chains (the rigid ladder).
Machine Learning: Random Forests and Neural Networks (the "black box" computers that guess based on patterns but don't explain why).

The Results:

The Winner: The TSOLR (Smooth Music) model performed the best. It predicted the air quality category correctly about 74% of the time.
The Runner-Up: The ISOLR (Calendar Switch) model was a very close second.
The Surprise: The "Black Box" machine learning models (like Random Forests) actually did worse when they tried to use the smooth music (Fourier) approach. They seemed confused by the smooth waves, whereas the new statistical models thrived on them.

Why Does This Matter?

It's Interpretable: Unlike a "black box" AI that just gives an answer, these models tell you why. You can look at the math and say, "Ah, the model predicts 'Poor' air quality because the 'Winter Wave' is peaking and the 'Diwali Switch' is on."
It Handles Complexity: Real life has multiple rhythms happening at once (daily traffic + yearly seasons + one-time festivals). These models can juggle all those rhythms at the same time.
It's Practical: Better predictions mean better public health warnings. If we know the air will be "Severe" next week, cities can prepare by stopping construction or advising people to stay indoors.

The Bottom Line

Think of air quality forecasting like trying to predict the tides.

Old models just looked at the water level right now and guessed the next level.
The new models understand the moon's pull (seasons), the wind (weekdays), and the storm (festivals).

By combining the logic of a calendar with the smoothness of music waves, the authors built a tool that is not only accurate but also easy for humans to understand, helping us breathe a little easier (literally and figuratively) in the future.

1. Problem Statement

The paper addresses the challenge of modeling and forecasting categorical time series that exhibit multiple seasonalities.

Context: While models like TBATS (Trigonometric seasonality, Box–Cox transformation, ARMA errors, Trend, and Seasonal components) are well-established for continuous time series (e.g., electricity demand), they are not directly applicable to categorical data.
Specific Application: The study focuses on the Air Quality Index (AQI), which is reported in ordinal categories (Good, Satisfactory, Moderate, Poor, Very Poor, Severe) rather than continuous values.
The Gap: AQI data in urban environments (specifically Kolkata, India) displays complex seasonal patterns at multiple frequencies:
- Weekly: Differences between weekdays and weekends.
- Yearly/Seasonal: Variations between Summer, Monsoon, Autumn, and Winter.
- Event-based: Sharp spikes during festivals (e.g., Diwali).
Limitations of Existing Methods:
- Markov Chains: Struggle with parameter explosion as state space increases and often fail to capture multiple seasonal cycles explicitly.
- Machine Learning (RF, SVM, LSTM): Often lack interpretability and do not explicitly model seasonal structures in a statistically grounded way.
- Standard Logistic Regression: Often lacks the flexibility to model complex, multi-frequency seasonality without ad-hoc covariate selection.

2. Methodology

The authors propose two novel frameworks extending the TBATS philosophy to Ordinal Logistic Regression:

A. The Proposed Models

Both models utilize a Proportional Odds Ordinal Logistic Regression framework to handle the ordered nature of AQI categories. The cumulative probability is modeled as:
$\text{logit}(P(Y_t \le j)) = \theta_j - \eta_t$
Where $\eta_t$ is the linear predictor containing seasonal components.

TSOLR (Trigonometric Seasonality in Ordinal Logistic Regression):
- Mechanism: Uses Fourier series (sine and cosine terms) as covariates to model seasonality.
- Formula: $\eta_t = \sum [\beta_{1i} \cos(2\pi t/P_i) + \beta_{2i} \sin(2\pi t/P_i)] + X_t^T \gamma$ .
- Advantage: Captures smooth, continuous seasonal cycles (daily, weekly, yearly) and allows for the modeling of higher-frequency events (like festivals) using absolute sine/cosine terms. It is parsimonious and interpretable.
ISOLR (Indicator Seasonality in Ordinal Logistic Regression):
- Mechanism: Uses dummy (indicator) variables to model seasonality.
- Formula: $\eta_t = \beta_1 \cdot \text{Summer}_t + \beta_2 \cdot \text{Monsoon}_t + \dots + \beta_5 \cdot \text{Weekend}_t + \beta_6 \cdot Y_{t-1}$ .
- Advantage: Captures abrupt, discrete shifts in air quality (e.g., sudden spikes during Diwali or specific winter months) without assuming smooth transitions.

B. Estimation and Selection

Estimation: Parameters are estimated using Maximum Likelihood Estimation (MLE). The asymptotic normality of the estimators is established, and confidence intervals are derived from the observed Fisher information matrix (Hessian).
Order Selection: Since standard ACF is not applicable to categorical data, the authors use auto-association measures (Cohen's $\kappa$ , Cramér's $\nu$ , Goodman-Kruskal's $\tau$ , etc.) to determine the appropriate lag order for the models.

C. Benchmarking

The proposed models are compared against:

Traditional Time Series: MTD (Mixture Transition Distribution) and PAR (Pegram's Autoregressive) models.
Machine Learning: Random Forests, Support Vector Machines (SVM), and LSTM (Long Short-Term Memory) networks.
Feature Variations: All ML models were tested with both Indicator-based and Fourier-based seasonal features to ensure a fair comparison.

3. Key Contributions

Extension of TBATS to Categorical Data: Successfully adapts the multi-seasonal modeling capability of TBATS to ordinal categorical time series via logistic regression.
Dual Framework (TSOLR & ISOLR): Provides two complementary approaches: one for smooth seasonal trends (Fourier) and one for discrete/abrupt shifts (Indicators), offering flexibility based on the data's nature.
Handling Multiple Seasonalities: Explicitly models co-existing seasonal cycles (weekly, yearly, and event-based) in AQI data, a feature often missed by standard Markov models.
Interpretability: Unlike "black-box" deep learning models, the proposed logistic regression models offer clear statistical interpretation of how specific seasons or festivals impact the probability of AQI categories.
Empirical Validation: Demonstrates the consistency of MLE estimates through simulation studies and validates the approach on real-world Kolkata AQI data.

4. Results

Simulation Studies

Consistency: As sample sizes increased (500 to 10,000), the standard errors of parameter estimates decreased, confirming the empirical consistency of the TSOLR model.
Forecasting Performance: In simulated data, the ISOLR model slightly outperformed TSOLR, but the difference was negligible and decreased with larger sample sizes. This suggests TSOLR is robust enough for forecasting ordinal data with seasonal effects.

Real-World Application (Kolkata AQI Data, 2019–2024)

Data Characteristics: The analysis revealed clear seasonal patterns:
- Winter: High frequency of "Poor" and "Very Poor" categories.
- Monsoon: Dominance of "Good" and "Satisfactory" categories.
- Diwali: Sharp, short-term deterioration in air quality.
- Trend: A general improvement in air quality over the years (fewer "Severe" days).
Forecasting Accuracy (Test Set 2024):
- TSOLR achieved the highest performance: 74.04% Accuracy and 74.07% Weighted F1 Score.
- ISOLR followed closely (72.95% Accuracy).
- Machine Learning: Fourier-based versions of SVM (73.50%) and LSTM (73.70%) outperformed their indicator-based counterparts. Notably, Random Forest performance dropped significantly when using Fourier features (65.85%), suggesting it handles discrete indicators better than smooth trigonometric terms.
- Traditional Models: MTD and PAR models performed similarly to the indicator-based logistic models but lacked the ability to incorporate exogenous seasonal covariates effectively.

5. Significance and Implications

Methodological Advancement: The paper fills a critical gap in time series literature by providing a rigorous, interpretable framework for multi-seasonal categorical data.
Public Health Utility: The models provide actionable insights for urban planners and health officials. By quantifying the impact of specific seasons and festivals (like Diwali) on AQI categories, the models can support better pollution control strategies and public health warnings.
Flexibility: The framework is not limited to AQI; it is applicable to any ordinal time series with complex seasonality, such as energy consumption categories, traffic congestion levels, or weather severity indices.
Future Directions: The authors suggest extending the model to include spatial dependencies (multiple monitoring stations), incorporating exogenous meteorological variables (wind, humidity), and exploring Bayesian estimation for sparse data.

In conclusion, the paper demonstrates that TSOLR (Fourier-based) and ISOLR (Indicator-based) models offer a superior balance of predictive accuracy and interpretability for forecasting complex seasonal patterns in categorical time series, outperforming both traditional Markov models and several machine learning benchmarks in the context of Air Quality Index forecasting.