[Paper Review] Spatial Functional Linear Model and its Estimation Method

Tingting Huang, Gilbert Saporta+2 more|arXiv (Cornell University)|Nov 1, 2018

Spatial and Panel Data Analysis19 references4 citations

TL;DR

This paper proposes a Spatial Functional Linear Model (SFLM) that extends the classical Functional Linear Model by incorporating spatial autoregressive dependence through a spatial weight matrix and spatial autoregressive parameter. The method combines functional principal component analysis (FPCA) and maximum likelihood estimation to handle high-dimensional functional predictors and network-structured data, demonstrating superior performance over standard FLM when spatial dependence is present, especially in fitting and prediction accuracy on real weather data.

ABSTRACT

The classical functional linear regression model (FLM) and its extensions, which are based on the assumption that all individuals are mutually independent, have been well studied and are used by many researchers. This independence assumption is sometimes violated in practice, especially when data with a network structure are collected in scientific disciplines including marketing, sociology and spatial economics. However, relatively few studies have examined the applications of FLM to data with network structures. We propose a novel spatial functional linear model (SFLM), that incorporates a spatial autoregressive parameter and a spatial weight matrix into FLM to accommodate spatial dependencies among individuals. The proposed model is relatively flexible as it takes advantage of FLM in handling high-dimensional covariates and spatial autoregressive (SAR) model in capturing network dependencies. We develop an estimation method based on functional principal component analysis (FPCA) and maximum likelihood estimation. Simulation studies show that our method performs as well as the FPCA-based method used with FLM when no network structure is present, and outperforms the latter when network structure is present. A real weather data is also employed to demonstrate the utility of the SFLM.

Motivation & Objective

• To address the limitation of classical Functional Linear Models (FLM) that assume independence among individuals, which is often violated in real-world data with network structures.
• To develop a flexible statistical model that integrates the strengths of functional linear models (for high-dimensional functional predictors) and spatial autoregressive (SAR) models (for capturing spatial dependence).
• To propose an estimation method based on functional principal component analysis (FPCA) and maximum likelihood estimation for the SFLM.
• To evaluate the performance of the SFLM in comparison to FLM under varying degrees of spatial autocorrelation.
• To demonstrate the model’s utility using real weather data from 34 Chinese cities, showing improved fit and prediction when spatial dependence is accounted for.

Proposed method

• The SFLM extends the classical FLM by introducing a spatial autoregressive parameter (ρ) and a spatial weight matrix (W) to model spatial dependence among responses.
• The model is formulated as: Y_i = α + ∫_Γ X_i(t)β(t)dt + ρ∑_j w_ij Y_j + ε_i, where Y_i is the scalar response, X_i(t) is the functional predictor, and w_ij represents spatial weights.
• Estimation is performed via functional principal component analysis (FPCA) to reduce the dimensionality of the functional covariates.
• Maximum likelihood estimation (MLE) is used to jointly estimate the spatial autoregressive parameter ρ and the slope function β(t).
• The method is validated through simulation studies under varying levels of spatial autocorrelation and compared to standard FPCA-based FLM estimation.
• The model is applied to real weather data (2005–2007) to predict annual precipitation using monthly temperature functions, with 2008 data used for out-of-sample prediction.

Experimental results

Research questions

• RQ1 Can a functional linear model be extended to account for spatial dependence among responses in network-structured data?
• RQ2 How does the proposed SFLM perform in comparison to the classical FLM when spatial autocorrelation is present in the data?
• RQ3 Does the inclusion of a spatial autoregressive parameter and spatial weight matrix improve model fit and predictive accuracy in functional data settings?
• RQ4 What is the impact of spatial dependence on residual structure, and can the SFLM effectively remove spatial autocorrelation in residuals?
• RQ5 How does the SFLM perform in real-world applications such as predicting annual precipitation from temperature functions across geographically connected cities?

Key findings

• The SFLM significantly reduces spatial autocorrelation in residuals compared to the classical FLM, as evidenced by a drop in Moran’s I statistic from 0.41 (FLM) to 0.15 (SFLM).
• The estimated spatial autoregressive parameter ρ is 0.58 with a p-value < 0.001, indicating strong and statistically significant spatial dependence in the weather data.
• The SFLM achieves a lower mean squared error (MSE) in fitting (0.24 vs. 0.33) and prediction (0.10 vs. 0.12) compared to the FLM, demonstrating improved accuracy.
• The estimated slope function β(t) under the SFLM is smoother than under the FLM, suggesting more stable and interpretable effects of temperature on precipitation over time.
• The SFLM captures stronger temperature effects on precipitation during winter months, with reduced overall sensitivity across the year compared to FLM.
• Simulation results confirm the consistency of the proposed estimators and show that the SFLM outperforms FLM when spatial dependence is present, while performing comparably when ρ = 0.