A hybrid dynamical-stochastic model of maximum temperature time series of Imphal, Northeast India incorporating nonlinear feedback and noise diagnostics

TL;DR

The paper tackles regional climate variability by modeling Imphal's Tmax using a data-driven hybrid framework that blends a deterministic backbone with nonlinear feedback and stochastic forcing. It constructs the backbone Lambda from SSA-derived dominant components and FFT, adds nonlinear feedback of cubic or Lorenz origin, and couples a stochastic term with white, colored, or Lévy noise; Langevin and Fokker-Planck formulations are provided to ground the dynamics in statistical physics. At the monthly timescale, the model achieves high accuracy and reproduces the observed complexity, with the Lorenz-based feedback and Lévy or colored noise providing the closest match on complexity-entropy analyses; hindcast tests demonstrate practical predictive skill. The work offers a physically interpretable, region-specific framework that can be extended to other variables and regions, bridging deterministic cycles and stochastic fluctuations and enabling better understanding and forecasting of regional climate behavior and extremes.

Abstract

Climate variability is a complex phenomenon resulting from numerous interacting components of a climate system across a wide range of temporal and spatial scales. Although significant advances have been made in understanding global climate variability, there are relatively less studies on regional climate modeling, particularly in developing countries. In this work, we propose a framework of data driven hybrid dynamical stochastic modeling to investigate the variability of maximum temperature recorded for the capital city of Imphal in the state of Manipur, located in the Northeast India. In light of increasing concerns over global warming, studying maximum temperature variability over varying time scales is an important area of research. Analysis using publicly available climate data over the course of 73 years, our approach yields key insights into the temperature dynamics, such as a positive increase in temperature in the region during the period investigated. Our hybrid model, combining spectral analysis and Fourier decomposition methods with stochastic noise terms and nonlinear feedback mechanisms, is found to effectively reproduce the observed dynamics of maximum temperature variability with high accuracy. Our results are validated by robust statistical and qualitative tests. We further derive Langevin and Fokker-Planck equations for the maximum temperature dynamics, offering the theoretical ground and analytical interpretation of the model that links the temperature dynamics with underlying physical principles.

Figures (15)

• Figure 1: Flowchart of our hybrid modeling framework: The modeling begins with the filtering of the maximum temperature $T_{\textrm{max}}$ time series data and then applying singular spectrum analysis (SSA) after determining the optimal window length. Next, fast Fourier transform (FFT) is applied to the dominant and secondary components, after which we construct the deterministic part of the model. Noise diagnostics are then performed on the noise component. Lastly, we assemble the final model by adding the feedback and noise terms to the deterministic part.
• Figure 2: (A) The original time series of the monthly $T_{\mathrm{max}}$ data of Imphal for January for the period 1951-2024, obtained after Kalman filtering. The red dot dashed line indicates the linear regression fit to $T_{\mathrm{max}}$, where the slope indicates the gradual rise in the maximum temperature per year. (B) Dominant component, (C) Secondary component, and (D) noise component, obtained after applying singular spectrum analysis (SSA) to the $T_{\mathrm{max}}$ data.
• Figure 3: Blue curve indicates the dominant and secondary components extracted from the original $T_{\mathrm{max}}$ data. The red curve represents the deterministic model, as in the first term of $\hat{T}_{\mathrm{max}}$ in eq. \ref{['fin_model']}, obtained after performing the fast Fourier transform (FFT) on the dominant and secondary components.
• Figure 4: We compare the values of permutation entropy $H$ (using embedding dimension $d=3,4$) for the original time series (OTS) with deterministic model (DM), hybrid models incorporating different noise and feedback types (see $x$-axis labels). WN(C): White noise (Cubic), CN(C): Colored noise (Cubic), LN(C): Lévy noise (Cubic), WN(L): White noise (Lorenz), CN(L): Colored noise (Lorenz), LN(L): Lévy noise (Lorenz). The dotted blue and red lines indicate $H$ values for the OTS when $d=3$ and 4, respectively.
• Figure 5: Complexity-Entropy ($CH$)-plane showing the values of ($H,C$) for the original time series (OTS), the deterministic model (DM), and the hybrid models for different noise and feedback configurations, for the monthly data of January of the period 1951-2024 (using $d=4$). In the legend, WN(C): White noise (Cubic), CN(C): Colored noise (Cubic), LN(C): Lévy noise (Cubic), WN(L): White noise (Lorenz), CN(L): Colored noise (Lorenz), LN(L): Lévy noise (Lorenz).