This paper proposes a multivariate nonlinear function-on-function regression model, which allows both the response and the covariates can be multi-dimensional functions. The model is built upon the multivariate functional reproducing kernel Hilbert space (RKHS) theory. It predicts the response function by linearly combining each covariate function in their respective functional RKHS, and extends the representation theorem to accommodate model estimation. Further variable selection is proposed by adding the lasso penalty to the coefficients of the kernel functions. A block coordinate descent algorithm is proposed for model estimation, and several theoretical properties are discussed. Finally, we evaluate the efficacy of our proposed model using simulation data and a real-case dataset in meteorology.

翻译：本文提出了一种多元非线性函数对函数回归模型，该模型允许响应变量和协变量均为多维函数。该模型建立在多元函数再生核希尔伯特空间理论基础上，通过在各自函数再生核希尔伯特空间中对各协变量函数进行线性组合来预测响应函数，并扩展了表示定理以适应模型估计。通过向核函数系数添加lasso惩罚项，进一步提出了变量选择方法。本文提出了一种块坐标下降算法进行模型估计，并讨论了若干理论性质。最后，我们通过模拟数据和气象学实际案例数据集评估了所提出模型的有效性。