Motivated by renal imaging studies that combine renogram curves with pharmacokinetic and demographic covariates, we propose Hybrid partial least squares (Hybrid PLS) for simultaneous supervised dimension reduction and regression in the presence of cross-modality correlations. The proposed approach embeds multiple functional and scalar predictors into a unified hybrid Hilbert space and rigorously extends the nonlinear iterative PLS (NIPALS) algorithm. This theoretical development is complemented by a sample-level algorithm that incorporates roughness penalties to control smoothness. By exploiting the rank-one structure of the resulting optimization problem, the algorithm admits a computationally efficient closed-form solution that requires solving only linear systems at each iteration. We establish fundamental geometric properties of the proposed framework, including orthogonality of the latent scores and PLS directions. Extensive numerical studies on synthetic data, together with an application to a renal imaging study, validate these theoretical results and demonstrate the method's ability to recover predictive structure under intermodal multicollinearity, yielding parsimonious low-dimensional representations.

翻译：受结合肾图曲线与药代动力学及人口统计学协变量的肾脏影像研究启发，我们提出混合偏最小二乘回归（Hybrid PLS），用于在存在跨模态相关性的场景中同时进行监督降维与回归分析。该方法将多类函数型与标量预测因子嵌入统一的混合希尔伯特空间，并严格推广了非线性迭代偏最小二乘（NIPALS）算法。理论构建辅以引入粗糙度惩罚以控制平滑性的样本级算法。通过利用所得优化问题的秩一结构，该算法获得计算高效的闭式解，每次迭代仅需求解线性方程组。我们建立了该框架的基本几何性质，包括潜在得分与PLS方向的正交性。基于合成数据的广泛数值研究，结合肾脏影像研究的实际应用，验证了理论结果，并证明该方法能在模态间多重共线性下恢复预测结构，生成简约的低维表示。