Probabilistic Crop Yields Forecasts With Spatio-Temporal Conditional Copula Using Extreme Weather Covariates

Abstract: We introduce a novel forecasting model for crop yields that explicitly accounts for spatio-temporal dependence and the influence of extreme weather and climatic events. Our approach combines Bayesian Structural Time Series for modeling marginal crop yields, ensuring a more robust quantification of uncertainty given the typically short historical records. To capture dynamic dependencies between regions, we develop a time-varying conditional copula model, where the copula parameter evolves over time as a function of its previous lag and extreme weather covariates. Unlike traditional approaches that treat climatic factors as fixed inputs, we incorporate dynamic Generalized Extreme Value models to characterize extreme weather events, enabling a more accurate reflection of their impact on crop yields. Furthermore, to ensure scalability for large-scale applications, we build on the existing Partitioning Around Medoids clustering algorithm and introduce a novel dissimilarity measure that integrates both spatial and copula-based dependence, enabling an effective reduction of the dimensionality in the dependence structure.

• The paper presents a novel framework that integrates BSTS and dynamic conditional copulas to forecast crop yields probabilistically.
• It uses extreme weather covariates and autoregressive processes to capture both temporal trends and tail dependencies in yield data.
• The approach improves forecast accuracy and offers a robust framework for extending predictions amid climate variability.

Probabilistic Crop Yield Forecasts Using Spatio-Temporal Conditional Copula

Introduction

The paper explores a probabilistic framework for forecasting crop yields by employing spatio-temporal conditional copula models, which integrate extreme weather covariates. This methodology aims to enhance predictive accuracy for crop yield forecasts by leveraging complex dependencies between regional yields and weather conditions beyond traditional linear models.

Marginal Distribution Modeling

In this study, the marginal distributions for regional crop yields are modeled using Bayesian Structural Time Series (BSTS) with a local linear trend component. Each region, represented as $D=2$, considers the maximum yearly precipitation as the primary covariate. The BSTS model incorporates a dynamic observation equation and an auto-regressive (AR) component to capture temporal correlations:

$$y_{d,t} = l_{d,t} + \psi_d e_{d,t-1} + \beta_{d,t} Z_{d,t} + \epsilon_{d,t}.$$

where $l_{d,t}$ denotes the local level, and $$\epsilon_{d,t} \sim \mathcal{N}(0,\sigma^2_{\epsilon_d})$$ represents the noise. The latent states, $l_{d,t}$ and $\tau_{d,t}$, follow random walk processes that account for potential trend fluctuations over time, with priors selected for variance terms using Inverse-Gamma distributions, ensuring robust Bayesian regression. AR(1) coefficients, $\psi_d$, are drawn from a uniform distribution ensuring stationarity.

Conditional Copula Approach

To effectively model joint distributions of crop yields across regions, the study employs the Clayton copula. This copula function, parameterized dynamically by a temporal function of precipitation, captures tail dependencies, crucial for extreme event analysis. The copula parameter $\theta_t(X_t)$ evolves according to an autoregressive process modulated by observed covariates:

$$\theta_t(X_t) = \exp(\omega + \alpha_t \theta_{t-1} + \gamma_t X_t + \epsilon_t).$$

This structure enables dynamic adaptation of dependency strengths, adjusted by extreme weather, thus incorporating larger spatio-temporal variability in forecast generation.

Dynamic GEV Distributions

For weather covariates, the study employs dynamic Generalized Extreme Value (GEV) distributions, providing flexibility for extreme precipitation modeling. The location parameter $\mu_{d,t}$ is expressed dynamically:

$\mu_{d,t} = \phi_d \mu_{d,t-1} + \epsilon_{d,t}.$

This dynamic adaptation allows for real-time updates in response to observed data, thus enhancing model responsiveness to environmental shifts.

Combined Model and Likelihood

The combined model, integrating dynamics from both the BSTS and copula approaches, yields a comprehensive pseudo-likelihood function. This function accounts for simultaneous fitting of the marginal distributions of yields and the copula dependencies:

$$\mathcal{L}_{\text{full}}(\Theta) = \mathcal{L}_{\text{copula}}(\alpha, \gamma, \sigma_\theta) \times \mathcal{L}_{\text{GEV}}(\phi, \sigma_\mu, \sigma, \xi).$$

The likelihood integrates copula density and GEV distribution with Bayesian inference to optimize the parameter set $\Theta$, crucial for encapsulating the copula-driven dependencies influenced by climatic extremes.

Conclusion

This paper presents a nuanced approach to crop yield forecasting by incorporating spatio-temporal interactions through copulas conditioned on extreme weather events. The Bayesian integration of structural time series with dynamic copula models fosters improved forecasting accuracy, especially under weather-induced stress conditions. Future research can extend this framework to more regions and diverse covariates, further refining crop yield predictions amid climate change scenarios.