Intertemporal choice data are usually summarized through scalar discount-rate parameters or fitted by predetermined parametric discount functions, although relevant information may lie in the shape of the whole discounting trajectory. This paper proposes a Functional Data Analysis framework for reconstructing and analyzing implicit subjective-time trajectories from discrete intertemporal equivalence judgments. Monetary equivalence responses from a multilingual questionnaire are transformed into individual discount curves, regularized by monotone smoothing, and used to recover normalized implicit subjective-time trajectories. The trajectories are examined through derivative summaries, Functional Principal Component Analysis, and clustering on standardized component scores. The empirical application, based on 107 participants, shows that heterogeneity in intertemporal choice is not fully captured by scalar discount-rate variation. The first two functional principal components explain 97.44% of the variability, indicating a low-dimensional structure. Functional clustering identifies three stable profiles of temporal deformation, supported by bootstrap stability analysis and sensitivity checks on components, algorithms, distances, smoothing specifications, and outlier treatment. Parametric benchmarks based on exponential, Weber-Fechner, and Stevens specifications provide accurate fits for many individuals, but do not fully recover the functional clustering structure. The comparison with explicit subjective-time perception measures reveals only partial alignment between implicit trajectories reconstructed from choices and directly reported temporal perception. Functional Data Analysis provides an applied statistical framework for representing intertemporal choice heterogeneity as variation in functional shape, complementing scalar discount-rate and parametric subjective-time models.

翻译：跨期选择数据通常通过标量折扣率参数汇总，或通过预设的参数化折扣函数拟合，然而相关信息可能潜藏于整个折扣轨迹的形态之中。本文提出一种函数型数据分析框架，用于从离散跨期等价判断中重建并分析隐式主观时间轨迹。来自多语言问卷的货币等价响应被转化为个体折扣曲线，经单调平滑正则化后，用于恢复归一化的隐式主观时间轨迹。通过导数摘要、函数型主成分分析以及基于标准化成分得分的聚类，对这些轨迹进行检验。基于107名参与者的实证应用表明，跨期选择的异质性无法完全通过标量折扣率变化来捕捉。前两个函数型主成分解释了97.44%的变异，显示出低维结构。函数型聚类识别出三种稳定的时间变形模式，并通过Bootstrap稳定性分析及对成分、算法、距离、平滑规范与异常值处理的敏感性检验加以支持。基于指数、韦伯-费希纳和史蒂文斯规范化的参数基准为许多个体提供了精确拟合，但未能完全恢复函数型聚类结构。与显式主观时间感知测量的比较显示，从选择中重建的隐式时间轨迹与直接报告的时间感知仅部分一致。函数型数据分析为将跨期选择异质性表示为函数形态变异提供了应用统计框架，补充了标量折扣率与参数化主观时间模型。