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A Quantile-Based Kumaraswamy-Teissier autoregressive moving average models

This paper proposes a flexible quantile-based Kumaraswamy-Teissier autoregressive moving average (KTARMA) model for positive-valued time series, demonstrating through simulations and a case study on Northwest Himalayan rainfall that it outperforms existing KARMA and $\beta$ ARMA models in predictive accuracy.

Original authors: Kamana Mishra (School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India), Tanmay Kayal (School of Mathematical and Statistical Sciences, Indian In

Published 2026-09-07

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Original authors: Kamana Mishra (School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India), Tanmay Kayal (School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India), Sarita Azad (School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India)

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

Rainfall is a stubbornly unpredictable force. It does not fall in neat, bell-shaped curves, nor does it behave the same way in every storm. In many parts of the world, especially in mountainous regions, rain is skewed; it can be absent for long stretches and then arrive in violent, concentrated bursts. Traditional statistical tools, which were built on the assumption that nature follows a standard, symmetrical pattern, often struggle to make sense of this. They frequently predict rain that falls outside the natural limits of reality or miss the mark entirely when trying to forecast extreme events. This is a critical problem for regions like the Northwest Himalayas, where communities rely on accurate predictions for agriculture, water management, and safety. The challenge for scientists has been to build a mathematical model that respects the jagged, uneven reality of rain while still capturing the patterns that link one day's weather to the next.

A team of researchers at the Indian Institute of Technology Mandi has developed a new approach to solve this problem. They created a specialized model called the KTARMA model, designed specifically for positive-valued time series data, such as rainfall amounts that cannot be negative. Instead of trying to force the data into a standard shape, this model embraces the unique, skewed nature of rainfall. It does this by focusing on specific points within the distribution of rain, known as quantiles. Rather than just predicting the average amount of rain, which can be misleading when extreme events occur, the model allows scientists to ask questions about different parts of the rainfall spectrum. They can investigate what drives the light drizzles, the moderate showers, or the heavy downpours, treating each as a distinct part of the weather system.

To build this model, the researchers combined a flexible statistical distribution, known as the Kumaraswamy–Teissier, with a structure that accounts for how weather changes over time. Think of the distribution as a mold that can be stretched and shaped to fit the specific, irregular contours of rainfall data, while the time-based structure acts like a memory, remembering how past rain influences the present. This combination allows the model to learn from history without being trapped by it. The researchers tested their creation rigorously. First, they ran thousands of computer simulations with different settings to ensure the model could accurately recover the true patterns hidden within the data. These tests showed that as the amount of data increased, the model's estimates became sharper and more reliable, with fewer errors.

The true test, however, came when they applied the model to real-world data from the Northwest Himalayas. This region is vast and complex, covering parts of Jammu and Kashmir, Himachal Pradesh, and Uttarakhand. To make sense of the 525 individual grid squares that cover this area, the researchers used a technique to group them into four distinct zones based on how their rainfall behaved. They then fed the model not just the history of rain, but also a wide array of atmospheric clues, including temperature, humidity, wind patterns at different heights in the sky, and large-scale climate signals like the North Atlantic Oscillation. The model learned how these factors interacted differently in each zone. For instance, in some areas, rainfall was strongly linked to lagged climate indices, while in others, it was driven more by local wind patterns or humidity.

The results were striking. When the researchers used the model to forecast rainfall for the year 2025, it outperformed two other established models, the KARMA and the beta-ARMA models. While the other models sometimes struggled to capture the full intensity of the rain or produced larger errors during peak periods, the new model consistently delivered more accurate predictions. It was particularly effective at minimizing the size of its mistakes, ensuring that even when it missed the mark, the error was small. The model also proved flexible enough to handle the different needs of each zone; in some areas, the best forecasts came from looking at the median rainfall, while in others, focusing on higher or lower quantiles provided the clearest picture.

Ultimately, this work offers a more nuanced way to understand and predict rainfall in complex environments. By moving away from the idea that rain must follow a single, average path, the KTARMA model acknowledges that the weather is a multi-faceted phenomenon. It provides a tool that can be tuned to look at the light rain, the heavy rain, or anything in between, all while accounting for the memory of the past and the influence of the wider climate. For the Northwest Himalayas, and potentially other regions with similar weather patterns, this means a step forward in the ability to anticipate the future of the skies, turning a chaotic natural process into something that can be understood and prepared for with greater confidence.

Technical Summary: A Quantile-Based Kumaraswamy–Teissier Autoregressive Moving Average Model

Problem Statement
Time series data in environmental sciences, particularly rainfall, frequently exhibit seasonality, asymmetry, and serial dependence. Classical Autoregressive Integrated Moving Average (ARIMA) models rely on normality assumptions that are ill-suited for highly skewed, non-Gaussian variables. Furthermore, standard ARIMA approaches often produce out-of-sample forecasts that violate the natural bounded support of the observed process. While distribution-based frameworks like the Generalized ARMA (GARMA), $\beta$ ARMA (for unit interval data), and Kumaraswamy ARMA (KARMA) have addressed some of these issues, they are often restricted to modeling the conditional median ( $\rho = 0.5$ ). This limitation is insufficient for applications where specific regimes of the distribution (e.g., extreme low or high rainfall) are of practical interest. Additionally, existing models for positive-valued data often lack the flexibility to model the entire conditional distribution through specific quantile levels.

Methodology
The paper proposes the Kumaraswamy–Teissier Autoregressive Moving Average (KTARMA) model, an observation-driven time series framework designed for positive-valued processes.

Distributional Framework: The model utilizes the Kumaraswamy–Teissier distribution (KTD), a flexible distribution for positive-valued observations. To introduce temporal dependence, the authors employ a quantile-based parameterization. The shape parameter $b$ of the KTD is re-parameterized in terms of a conditional $\rho$ -th quantile ( $\mu_{\rho,t}$ ), where $\rho \in (0,1)$ is a fixed quantile level. This allows the time-varying dynamics to be modeled directly through the conditional quantile rather than the mean.
Dynamic Structure: The model extends the GARMA framework by linking the conditional quantile $\mu_{\rho,t}$ $μ_{ρ, t}$ to a linear predictor $\eta_{\rho,t}$ $η_{ρ, t}$ via a monotone link function $g(\cdot)$ $g (\cdot)$ . The linear predictor incorporates:
- An intercept ( $\alpha_\rho$ ).
- Exogenous covariates ( $x_t$ ) with regression coefficients ( $\beta_\rho$ ).
- Autoregressive (AR) terms acting on the transformed lagged observations.
- Moving Average (MA) terms acting on the transformed lagged residuals.
  The error term is defined as the deviation of the transformed observation from its conditional quantile.
Estimation: Parameters are estimated using the Conditional Maximum Likelihood Estimation (CMLE) method. The paper derives explicit analytical expressions for:
- The conditional log-likelihood function.
- The score vector (first derivatives).
- The conditional information matrix (second derivatives), utilizing specific lemmas to handle the expectations of the complex KTD derivatives.
Diagnostics and Forecasting: Model adequacy is assessed using randomized quantile residuals, which are expected to follow a standard normal distribution under correct specification. Serial independence is tested using the Ljung–Box test. Out-of-sample forecasting is performed recursively, requiring future covariate values for the prediction horizon.

Key Contributions
The paper outlines three primary contributions:

Novel Model Formulation: Introduction of the KTARMA model, which integrates the KTD with an ARMA structure specifically for positive-valued time series.
Generalized Quantile Framework: Extension of the conventional median-based dynamic modeling (as seen in KARMA) to a general conditional quantile formulation. This allows researchers to examine different regions of the conditional rainfall distribution by varying $\rho$ .
Empirical Application and Validation: Application of the model to spatially heterogeneous rainfall data in the Northwest Himalayas (NWH), including the development of diagnostic procedures and a comparative forecasting analysis.

Results

Simulation Study: A Monte Carlo simulation study was conducted with sample sizes ranging from $n=50$ to $n=500$ across two parameter scenarios and three quantile levels ( $\rho = 0.25, 0.50, 0.75$ ). The results demonstrated that as sample size increased, the empirical mean estimates converged to true parameter values, while bias and Mean Squared Error (MSE) decreased monotonically. This verifies the asymptotic consistency and efficiency of the CMLE estimators in finite samples.
Real-World Application (NWH Rainfall):
- Data: Monthly rainfall data (2001–2025) from 525 grids in the NWH were aggregated into four homogeneous zones using Self-Organizing Maps (SOM).
- Model Fit: The KTARMA model was fitted to each zone using various quantile levels. Diagnostic checks (Ljung–Box test on randomized quantile residuals) confirmed no significant serial correlation in the residuals, indicating adequate model fit.
- Covariate Effects: The analysis revealed distinct rainfall responses across zones. For instance, Zone 1 and Zone 4 showed inverse relationships with temperature and specific wind components, while Zone 3 (Ladakh) was significantly influenced by lagged North Atlantic Oscillation (NAO) and geopotential height.
- Forecasting Performance: In out-of-sample forecasting for 2025, the KTARMA model was compared against the median-based KARMA and $\beta$ $β$ ARMA models.
  - Zone 1 & 4: KTARMA achieved the lowest MSE.
  - Zone 2: KARMA had a marginally lower Mean Absolute Error (MAE), but KTARMA achieved the lowest MSE, indicating better control over large deviations.
  - Zone 3: KTARMA outperformed both competitors in both MAE and MSE, particularly in handling the low-volume, highly skewed rainfall regime where $\beta$ ARMA showed substantial errors.

Significance
The paper claims that the KTARMA model provides a flexible, observation-driven framework for modeling positive-valued time series with asymmetric distributions, temporal dependence, and seasonal effects. By moving beyond the conditional median to a general quantile formulation, the model allows for a more nuanced analysis of rainfall dynamics, particularly in regions with high spatial heterogeneity like the Northwest Himalayas. The study demonstrates that the KTARMA model delivers highly competitive predictive performance, consistently achieving lower mean squared errors across identified zones compared to existing KARMA and $\beta$ ARMA models, making it a robust tool for rainfall forecasting and risk assessment in varying climatic regimes.

Technical Summary: A Quantile-Based Kumaraswamy–Teissier Autoregressive Moving Average Model

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