Enhanced Prediction Model for Time Series Characterized by GARCH via Interval Type-2 Fuzzy Inference System

TL;DR

The paper tackles forecasting challenges in GARCH-type time series characterized by time-varying volatility and heteroskedasticity. It introduces IT2FIS-GARCH, a hybrid that dynamically embeds GARCH-estimated conditional variance into interval Type-2 fuzzy membership functions, turning IT2FIS into an adaptive, volatility-aware predictor. A mean–variance co-optimization mechanism and a volatility-driven defuzzification strategy are developed, enabling joint forecasting of the conditional mean and uncertainty. Empirical results on air quality, traffic, and energy datasets show superior predictive accuracy and robustness compared with GARCH-TSK, Fixed Variance IT2FIS, GARCH-GRU, and LSTM baselines, highlighting the practical utility of integrating interval fuzzy inference with econometric volatility models for real-world time series forecasting.

Abstract

GARCH-type time series (characterized by Generalized Autoregressive Conditional Heteroskedasticity) exhibit pronounced volatility, autocorrelation, and heteroskedasticity. To address these challenges and enhance predictive accuracy, this study introduces a hybrid forecasting framework that integrates the Interval Type-2 Fuzzy Inference System (IT2FIS) with the GARCH model. Leveraging the interval-based uncertainty representation of IT2FIS and the volatility-capturing capability of GARCH, the proposed model effectively mitigates the adverse impact of heteroskedasticity on prediction reliability. Specifically, the GARCH component estimates conditional variance, which is subsequently incorporated into the Gaussian membership functions of IT2FIS. This integration transforms IT2FIS into an adaptive variable-parameter system, dynamically aligning with the time-varying volatility of the target series. Through systematic parameter optimization, the framework not only captures intricate volatility patterns but also accounts for heteroskedasticity and epistemic uncertainties during modeling, thereby improving both prediction precision and model robustness. Experimental validation employs diverse datasets, including air quality concentration, urban traffic flow, and energy consumption. Comparative analyses are conducted against models: the GARCH-Takagi-Sugeno-Kang (GARCH-TSK) model, fixed-variance time series models, the GARCH-Gated Recurrent Unit (GARCH-GRU), and Long Short-Term Memory (LSTM) networks. The results indicate that the proposed model achieves superior predictive performance across the majority of test scenarios in error metrics. These findings underscore the effectiveness of hybrid approaches in forecasting uncertainty for GARCH-type time series, highlighting their practical utility in real-world time series forecasting applications.

Figures (8)

• Figure 1: GARCH(1,1) model. For a dataset with $n$ data points, the process is as follows: First, determine the initial conditional variance. Then, estimate the model parameters. Finally, use an iterative algorithm to optimize the parameters $w$, $\alpha_1$ and $\beta_{1}$, and implement the prediction for the GARCH(1,1) model.
• Figure 2: Variance interval estimation based on GARCH model. For a dataset with $n$ data points, obtain interval estimates of the variance for each data point. Firstly, estimate the initial variance value and model parameters. Then, refer to the chi square distribution table to obtain $\chi^2_{\alpha/2}$ and $\chi^2_{1 - \alpha/2}$. Finally, use the interval values of the random variables to calculate the variance interval values of the data. This leads to the final implementation of interval estimation of variance.
• Figure 3: Interval type-2 fuzzy reasoning system. For a dataset with $n$ data points, input it into the second layer of the system and use the membership function to calculate the membership interval $[\underline{\mu}_{nk},\overline{\mu}_{nk}]$. Taking Gaussian membership function as an example in the figure. Match rules based on the rule library and output the corresponding rule consequent $[\underline{y}_n,\overline{y}_n]$. Defuzzify the fuzzy output result to obtain a clear value $\hat{Y}$, which serves as the final output of the system.
• Figure 4: Standard interval Gaussian membership function.
• Figure 5: The IT2FIS-GARCH model. For a dataset with $n$ data points, the variance interval estimate $[\underline{\sigma}_{t}^{2},\overline{\sigma}_{t}^{2}] (t = 1,\cdots,n)$ is first obtained according to the chi-square distribution theory. At the same time, the mean value $C_t$ of the dataset is estimated using the 3$\sigma$ rule. Then, the membership degree intervals of the input data are obtained using the Gaussian membership function. A rule base is used to match the input data to rules, and the interval values of the fuzzy output are then transformed into specific values $\hat{Y}$. During the training process, the predicted value at the final layer is minimized against the true value using the mean squared error operation, thereby training the parameters of the IT2FIS-GARCH model. In the prediction process, the predicted value $\hat{Y}$ from the last layer of the IT2FIS-GARCH model is output.