深度神經網路因具有高度彈性的函數逼近能力，近年來已廣泛應用於複雜資料建模。然而，傳統空間自迴歸模型通常仰賴研究者預先指定空間權重矩陣，而一般深度神經網路亦不易提供變數層級的解釋。基於此，本研究探討具有非參數內生效果之高階非參數空間自迴歸模型中的變數選擇問題，並提出一個整合學習式空間權重機制、非參數內生空間效果與結構化變數選擇的深度神經網路架構。
在空間結構方面，所提模型透過多個由資料學習的空間權重機制及其對應之非參數內生效果，刻畫可能存在的多重與非線性空間相依關係。在主效應建構方面，將主效應函數分解為線性 skip component 與非線性神經網路分支，並結合 L1 懲罰與階層式限制，使變數在線性係數被壓縮為零時，其於非線性分支中的連結亦同步移除，從而在保留非線性擬合能力的同時進行變數選擇。
在理論方面，本研究基於固定點定理，推導所提模型之反應值解存在性與唯一性的充分條件，以說明模型在適當條件下的數學可行性。在計算方面，依參數所涉及的最佳化結構，將估計程序區分為一般梯度更新與階層式 proximal 更新，並結合路徑式稀疏化策略與驗證集模型選擇，以同步完成參數估計與重要變數辨識。
模擬研究與實際資料分析結果顯示，所提方法能有效辨識具有線性與非線性效果的重要變數，並在預測表現上優於比較方法。實證結果亦顯示，學習式空間權重機制與結構化變數選擇可共同提升模型對複雜空間資料的建模能力。整體而言，本研究所提出之方法能兼顧空間相依關係的建模彈性、預測表現與變數層級解釋性，顯示其應用於複雜空間資料分析的可行性。

This thesis proposes a higher-order nonparametric spatial autoregressive neural network model with variable selection. The proposed framework is motivated by two limitations in spatial data analysis: conventional spatial autoregressive models usually rely on pre-specified spatial weight matrices, while flexible deep neural networks often lack variable-level interpretability.
The proposed model integrates multiple learnable spatial weight mechanisms, nonparametric endogenous spatial effects, and a structured main-effect component. The spatial component is written as Σ_{m=1}^M W_{space,Φ_m}(X) h_{Γ_m}(Y), allowing the model to learn multiple spatial dependence structures and nonlinear spatial feedback effects from data. The main-effect component is decomposed as f_{θ,Ξ_G}(X) = Xθ + g_{Ξ_G}(X), where the linear skip component supports variable selection and the nonlinear neural network component preserves flexible function approximation. By combining an L1 penalty with a hierarchical constraint, the model can remove unimportant variables from both the linear and nonlinear main-effect branches.
Theoretical analysis establishes sufficient conditions for the existence and uniqueness of the response solution based on the Banach fixed-point theorem. Computationally, the model is estimated through gradient-based updates and hierarchical proximal updates along a regularization path.
Simulation studies show that the proposed method can recover important variables and improve predictive performance, especially when a covariate is involved in both spatial weight construction and the true main effect. Empirical analysis using the California Housing Prices data further demonstrates that the proposed method achieves better test performance than the fixed-W comparison method while retaining meaningful variables in the main-effect support.

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