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arxiv: 2605.31265 · v1 · pith:V76I7J4Anew · submitted 2026-05-29 · 📊 stat.AP

Subjective Time Deformation in Intertemporal Choice: A Functional Data Analysis Approach

Fabrizio Maturo , Salvador Cruz Rambaud , Vincenzo Li Calzi , Andrea Mazzitelli , Annamaria Porreca This is my paper

Pith reviewed 2026-06-28 20:23 UTC · model grok-4.3

classification 📊 stat.AP
keywords intertemporal choicefunctional data analysissubjective timediscount curvesfunctional principal component analysisfunctional clusteringtemporal deformation
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The pith

Heterogeneity in intertemporal choice shows up as distinct functional shapes of subjective time deformation rather than scalar discount-rate differences alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a functional data analysis approach that turns discrete monetary equivalence judgments into continuous individual discount curves via monotone smoothing, then recovers normalized implicit subjective-time trajectories from those curves. It applies functional principal component analysis and clustering to data from 107 participants and reports that the first two components explain 97.44 percent of the variability, pointing to a low-dimensional structure. Clustering on the component scores identifies three stable profiles of temporal deformation that remain consistent under bootstrap checks and sensitivity tests. Standard parametric discount functions fit many individuals yet fail to reproduce the same clustering partition. The reconstructed trajectories align only partially with separate explicit reports of subjective time perception.

Core claim

Heterogeneity in intertemporal choice is not fully captured by scalar discount-rate variation; instead, the full shape of each person's discounting trajectory can be reconstructed as a normalized implicit subjective-time function, and functional principal component analysis plus clustering on those functions reveals a low-dimensional structure consisting of three stable profiles of temporal deformation.

What carries the argument

Monotone smoothing of discrete equivalence judgments to produce individual discount curves, followed by functional principal component analysis and clustering on the resulting normalized subjective-time trajectories.

If this is right

  • The first two functional principal components explain 97.44 percent of variability in the implicit trajectories.
  • Functional clustering identifies three stable profiles of temporal deformation that survive bootstrap stability analysis.
  • Parametric models based on exponential, Weber-Fechner, and Stevens specifications fit many individuals accurately but do not recover the functional clustering partition.
  • Reconstructed implicit trajectories show only partial alignment with directly reported explicit subjective-time perception measures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The low-dimensional structure suggests that a small set of basis functions could approximate most individual differences in discounting behavior for modeling purposes.
  • If the three profiles correlate with observable traits such as age or cognitive measures, they could be used to stratify participants in future choice experiments.
  • The partial mismatch between implicit and explicit time measures indicates that choice-based reconstruction may capture decision-specific deformations not reported in direct perception tasks.

Load-bearing premise

Discrete intertemporal equivalence judgments from the questionnaire can be turned into individual discount curves by monotone smoothing without introducing systematic distortion to the recovered subjective-time trajectories.

What would settle it

Repeating the full pipeline on an independent sample of similar size and finding that the first two functional principal components explain far less than 97 percent of variability or that the clustering yields unstable or non-replicable profiles would falsify the low-dimensional structure claim.

Figures

Figures reproduced from arXiv: 2605.31265 by Andrea Mazzitelli, Annamaria Porreca, Fabrizio Maturo, Salvador Cruz Rambaud, Vincenzo Li Calzi.

Figure 1
Figure 1. Figure 1: Functional reconstruction of discount curves and subjective time trajectories. The upper-left panel reports the smoothed discount curves fi(t); the upper-right panel shows the recovered subjective time functions τi(t); the lower panels display the first derivative τ ′ i (t) and the second derivative τ ′′ i (t), respectively. The final analytical sample consists of N = 107 participants. 21 [PITH_FULL_IMAGE… view at source ↗
Figure 2
Figure 2. Figure 2: Diagnostic comparison between observed discount factors, parametric benchmark fits, and FDA monotone reconstruction for six representative subjects. Subjects are selected according to tertiles of the average parametric RMSE [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Functional Principal Component Analysis of centered subjective time trajectories. The figure reports the first functional principal components obtained from the centered functions τ c i (t), together with the percentage of variability explained by each component. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Functional clustering of centered subjective time trajectories. Thin lines represent individual centered trajectories τ c i (t), whereas thick lines represent the functional centroids of the clusters selected through the average silhouette criterion. a peak around t = 300–400 days and then declining progressively in the long run. These individuals show a relative expansion of time in the short-to-medium ho… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between implicit subjective time profiles and explicit STP profiles. The left panel reports the functional centroids of the τ (t) clusters. The right panel compares the standardized shape of the τ (t) centroids with the corresponding mean STP profiles within the same clusters. and mean STP is r = −0.253, indicating a weak-to-moderate association. The correlation between TSS and mean STP is r = 0… view at source ↗
Figure 6
Figure 6. Figure 6: Average silhouette index as a function of the number of clusters k on standardized FPCA scores. The maximum is attained at k = 3, supporting the three-cluster solution adopted in the main analysis. The distribution of subjects in the space of the first FPCA scores further shows that the three groups occupy distinct regions of the reduced space, indicating that the observed structure does not arise from iso… view at source ↗
Figure 7
Figure 7. Figure 7: Scatterplot of the first two FPCA scores, with subjects colored according to the three-cluster solution. The plot provides a visualization of the clustering structure in the reduced-dimensional FPCA space. resamples. The mean values of the Jaccard index are high for all three clusters, with values equal to 0.963 for Cluster 1, 0.928 for Cluster 2, and 0.969 for Cluster 3. These results indi￾cate that the g… view at source ↗
Figure 8
Figure 8. Figure 8: Graphical representation of the bootstrap stability of the three-cluster solution. The figure reports the mean Jaccard similarity for each functional cluster across bootstrap resamples. Higher values indicate greater cluster stability. Similar results are also obtained using the Euclidean distance on standardized FPCA scores, with values equal to 1 for PAM and 0.955 for Ward.D2. A different situation emerg… view at source ↗
Figure 9
Figure 9. Figure 9: reports the distribution of the discount factor fi(t) = A0,i(t)/100 for each of the thirteen time horizons considered. Median values show a progressive reduction as delay increases, consistent with the presence of positive time preference in the sample. Individual heterogeneity is marked across all horizons, with dispersion tending to increase for longer delays [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of STP responses by time horizon, expressed in subjective months normalized on the individual 7-day anchor. N = 107 [PITH_FULL_IMAGE:figures/full_fig_p041_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of core Temporal Sense Scale items by temporal window. The scale ranges from 1 (very slowly) to 7 (very fast). N = 107. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: reports the distribution of individual discount rates ki in the analytical sam￾ple. The distribution is right-skewed, with the majority of participants concentrated at low values (mean = 0.000212, median = 0.000021), and a small number of subjects showing substantially higher discount rates. This pattern is consistent with the presence of marked individual heterogeneity in discounting intensity [PITH_FUL… view at source ↗
Figure 13
Figure 13. Figure 13: Pearson correlation matrix among ki, FPCA scores, TSS core index, and mean STP. The heatmap summarizes the association structure between implicit subjective time measures, explicit time￾perception measures, and the scalar discount rate [PITH_FULL_IMAGE:figures/full_fig_p043_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scatterplot between the standardized first FPCA score of the centered subjective time trajec￾tories and the individual mean STP measure. The fitted line provides a visual summary of the association between implicit and explicit subjective time measures [PITH_FULL_IMAGE:figures/full_fig_p044_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Descriptive distributions of age, TSS core index, financial literacy, and the individual discount rate ki by functional cluster. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Robustness of the clustering solution with respect to the number of FPCs used for clustering. The Adjusted Rand Index compares each alternative partition with the reference solution based on the first two standardized FPCA scores [PITH_FULL_IMAGE:figures/full_fig_p046_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Robustness of the three-cluster solution under alternative clustering specifications. Values close to one indicate high agreement with the reference partition. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Robustness of the clustering solution under alternative functional distances and derivative￾based semi-metrics. Lower agreement for derivative-based semi-metrics indicates that local dynamic features capture complementary information relative to the centered trajectories [PITH_FULL_IMAGE:figures/full_fig_p047_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Sensitivity of the clustering solution to alternative smoothing-basis specifications for fi(t) and τi(t). Each cell reports the Adjusted Rand Index relative to the reference solution. 48 [PITH_FULL_IMAGE:figures/full_fig_p048_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Summary of robustness checks. The figure reports the Adjusted Rand Index between each alternative solution and the reference clustering partition. 49 [PITH_FULL_IMAGE:figures/full_fig_p049_20.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a Functional Data Analysis framework to reconstruct normalized implicit subjective-time trajectories from discrete intertemporal equivalence judgments in a multilingual questionnaire (n=107). Monetary responses are converted to discount curves via monotone smoothing, then analyzed with derivative summaries, FPCA (first two components explain 97.44% variance), and functional clustering into three profiles. The central claim is that scalar discount-rate variation does not fully capture heterogeneity in intertemporal choice; parametric benchmarks (exponential, Weber-Fechner, Stevens) fit many individuals but miss the functional clustering structure, while explicit time-perception measures show only partial alignment. Bootstrap stability and sensitivity checks on smoothing, components, algorithms, distances, and outliers are reported.

Significance. If the monotone-smoothing reconstruction is shown to be faithful, the work supplies a useful nonparametric complement to scalar and parametric models by demonstrating low-dimensional functional structure in discounting trajectories. The reported bootstrap stability, sensitivity analyses, and direct comparison to three parametric families are concrete strengths that make the empirical findings falsifiable and reproducible within the manuscript's scope.

major comments (2)
  1. [§3] §3 (monotone smoothing and trajectory recovery): The central claim that FPCA and clustering reveal genuine heterogeneity beyond scalar rates depends on the preprocessing step that converts discrete judgments into continuous normalized trajectories. Although the abstract and results mention sensitivity checks on smoothing specifications, no simulation study is described that generates discrete equivalence judgments from known parametric forms (e.g., exponential or hyperbolic) and verifies faithful recovery of the original trajectories after smoothing and normalization. Without this direct fidelity test, it remains possible that regularization artifacts contribute to the reported 97.44% variance in the first two components or the three-cluster partition.
  2. [§4.2] §4.2 (functional clustering): The number of clusters is listed as a free parameter in the analysis pipeline. The manuscript reports bootstrap stability for the three-profile solution but does not describe an objective selection procedure (e.g., silhouette, gap statistic, or cross-validated prediction error) or show results for neighboring values of k. Because the low-dimensional claim is tied to the existence of these specific stable profiles, the justification for k=3 versus other values needs to be made explicit and load-bearing.
minor comments (2)
  1. [§3] Notation for the normalized implicit subjective-time function (e.g., how the normalization to [0,1] interval is performed after smoothing) should be stated once in a single equation early in §3 rather than re-described in multiple places.
  2. [§2] Table 1 (participant demographics) and the multilingual questionnaire description would benefit from an explicit statement of the exact number of equivalence judgments per participant and the range of delays used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (monotone smoothing and trajectory recovery): The central claim that FPCA and clustering reveal genuine heterogeneity beyond scalar rates depends on the preprocessing step that converts discrete judgments into continuous normalized trajectories. Although the abstract and results mention sensitivity checks on smoothing specifications, no simulation study is described that generates discrete equivalence judgments from known parametric forms (e.g., exponential or hyperbolic) and verifies faithful recovery of the original trajectories after smoothing and normalization. Without this direct fidelity test, it remains possible that regularization artifacts contribute to the reported 97.44% variance in the first two components or the three-cluster partition.

    Authors: We agree that a dedicated simulation study testing recovery from known parametric generating processes would provide stronger validation of the monotone-smoothing step. The existing sensitivity checks vary smoothing parameters but do not simulate from ground-truth trajectories. In the revised manuscript we will add such a simulation: generate discrete equivalence judgments from exponential, hyperbolic, and Weber-Fechner models, apply the full pipeline (monotone smoothing, normalization, FPCA, clustering), and report integrated squared error and cluster-recovery metrics between true and reconstructed trajectories. revision: yes

  2. Referee: [§4.2] §4.2 (functional clustering): The number of clusters is listed as a free parameter in the analysis pipeline. The manuscript reports bootstrap stability for the three-profile solution but does not describe an objective selection procedure (e.g., silhouette, gap statistic, or cross-validated prediction error) or show results for neighboring values of k. Because the low-dimensional claim is tied to the existence of these specific stable profiles, the justification for k=3 versus other values needs to be made explicit and load-bearing.

    Authors: The k=3 solution was chosen on the basis of bootstrap stability and profile interpretability. We acknowledge that formal model-selection criteria were not reported. In revision we will add gap-statistic and silhouette analyses for k=2 to 5, together with bootstrap stability results for k=2, 3, and 4, to demonstrate that the three-profile partition is both stable and preferred by these criteria. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical FDA pipeline is data-driven and externally validated

full rationale

The paper applies standard functional data analysis (monotone smoothing, FPCA, clustering) to questionnaire responses from 107 participants. Reported results (97.44% variance in first two FPCs, three stable clusters) are computed directly from the processed trajectories rather than being algebraically forced by any fitted parameter or self-citation. Bootstrap stability, sensitivity checks on smoothing parameters, and explicit comparison against exponential/Weber-Fechner/Stevens benchmarks are independent of the target functional summaries. No equations or uniqueness theorems reduce the central claims to the inputs by construction; the derivation chain remains self-contained against the observed data.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework rests on standard functional data analysis assumptions and domain-specific monotonicity of discount functions. Free parameters are limited to smoothing and clustering choices that are tuned during analysis. No new physical or theoretical entities are introduced.

free parameters (2)
  • monotone smoothing parameter
    Controls regularization of individual discount curves from discrete judgments; its value is chosen during fitting and affects recovered trajectory shape.
  • number of functional clusters
    Set to three based on component scores; choice determines the reported profile structure.
axioms (1)
  • domain assumption Discount functions derived from equivalence judgments are monotonically decreasing
    Invoked to justify monotone smoothing step that produces the input curves for FPCA.

pith-pipeline@v0.9.1-grok · 5791 in / 1438 out tokens · 31584 ms · 2026-06-28T20:23:11.709688+00:00 · methodology

discussion (0)

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