In this study, a density-on-density regression model is introduced, where the association between densities is elucidated via a warping function. The proposed model has the advantage of a being straightforward demonstration of how one density transforms into another. Using the Riemannian representation of density functions, which is the square-root function (or half density), the model is defined in the correspondingly constructed Riemannian manifold. To estimate the warping function, it is proposed to minimize the average Hellinger distance, which is equivalent to minimizing the average Fisher-Rao distance between densities. An optimization algorithm is introduced by estimating the smooth monotone transformation of the warping function. Asymptotic properties of the proposed estimator are discussed. Simulation studies demonstrate the superior performance of the proposed approach over competing approaches in predicting outcome density functions. Applying to a proteomic-imaging study from the Alzheimer's Disease Neuroimaging Initiative, the proposed approach illustrates the connection between the distribution of protein abundance in the cerebrospinal fluid and the distribution of brain regional volume. Discrepancies among cognitive normal subjects, patients with mild cognitive impairment, and Alzheimer's disease (AD) are identified and the findings are in line with existing knowledge about AD.

翻译：在本研究中，提出了一种密度对密度回归模型，通过扭曲函数阐明密度之间的关联。该模型具有直观展示一种密度如何转变为另一种密度的优势。利用密度函数的黎曼表示——即平方根函数（或半密度），模型在相应构建的黎曼流形上定义。为估计扭曲函数，提出最小化平均Hellinger距离，这等价于最小化密度之间的平均Fisher-Rao距离。通过估计扭曲函数的光滑单调变换引入一种优化算法。讨论了所提估计量的渐近性质。模拟研究表明，所提方法在预测结果密度函数方面优于竞争方法。应用于阿尔茨海默病神经影像学倡议的蛋白质组学成像研究，所提方法揭示了脑脊液中蛋白质丰度分布与脑区域体积分布之间的联系。识别出认知正常受试者、轻度认知障碍患者和阿尔茨海默病（AD）患者之间的差异，且发现与现有AD知识一致。