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Longer Records, Broader Invariance: The Hidden Scaling Problem in Longitudinal Contrastive Learning

Rameen Mahmood, Xuhai "Orson" Xu, Zachary Beattie, Jeffrey Kaye, Danny Yuxing Huang

arXiv:2609.36409v1cs.LGcs.AI

TL;DR

Person-level contrastive learning can make longer records broaden the temporal reach of positive pairs, confounding more history with broader invariance. The paper separates record span from supervision span and finds that broader supervision suppresses changing-state information, while local positives still permit long-record loss that within-person negatives reverse.

  • Problem

    As records grow, same-person positive pairs can span increasingly distant behavioral states, confounding whether information loss reflects more history or broader supervision.

    Source: §1 · p. 1

  • Method

    The paper separates record span from supervision span and tests whether contrasting other observations from the same person preserves changing-state information.

    Source: §1 · pp. 1–2

  • Results

    Less than 10% of the untrained reference’s changing-state information remains at the broadest supervision span; local supervision still permits long-record loss, which within-person negatives reverse.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Record length and invariance need not scale together; tracking how supervision changes with added data can help preserve distinctions within a person’s history.

    Source: §1 · p. 2, §5 · p. 9

    The findings concern person-level contrastive learning in behavioral sensing, and the study does not establish whether they extend to larger models or other forms of self-supervision.

    Source: §5 · p. 9

Abstract

from arXiv · show

Longitudinal data are valuable because people change. Yet the objectives used to learn from these data can inadvertently erase that change. In person-level contrastive learning, observations from the same person are treated as positives; as records grow, those positives can span increasingly distant---and increasingly different---behavioral states. More history can therefore produce not only more data, but broader invariance. We show that this distinction is fundamental. We separate \emph{record span}, how much history the learner sees, from \emph{supervision span}, how far across that history positive-pair supervision reaches. Across in-home sensing records spanning up to 2.7 years, broader supervision systematically suppresses recoverable changing-state information, even when the available history is held fixed. At the broadest span, less than 10\% of the information recoverable from an untrained encoder remains. Yet keeping positives local is not sufficient: as records grow, even distant states that are never paired become increasingly similar. Explicitly contrasting other observations from the same person reverses this loss without shortening the record, revealing a second route by which longitudinal scale can broaden invariance. Finally, we prospectively reproduce the supervision-span effect in 199 GLOBEM participants. Longitudinal scale therefore presents a choice: more history need not mean more invariance. By controlling what is held invariant as records grow, we can preserve the change that made the longitudinal data valuable in the first place.

1 INTRODUCTION

In person-level contrastive learning, longer records can broaden the temporal reach of same-person positive pairs, confounding the effects of record span and supervision span. Experiments show that broader supervision suppresses changing-state information, while long records can also blur never-paired states; within-person negatives reverse that loss, and the supervision-span effect generalizes to GLOBEM.

1 INTRODUCTION

  • Because same-person observations can become months or seasons apart in long records, extending history can also extend the invariance imposed by the positive-pair rule.The same-person sampling rule can stay fixed even as its temporal relation broadens.Source: §1 · pp. 1–2
  • Record span is the amount of history available, whereas supervision span is the temporal reach over which the objective relates observations as positives.Ordinary same-person sampling couples them as records grow, making it unclear whether information loss comes from added history or changed supervision.Source: §1 · pp. 1–2
  • At the broadest supervision span, changing-state information falls below 10% of the untrained reference, even with record span held fixed.The DETECT records span up to 2.7 years and include 57 single-resident participants and 44 multi-resident households.Source: §1 · p. 2
  • With local supervision, longer records still reduce changing-state information and make even never-paired distant observations increasingly similar.Local supervision limits direct alignment across time but does not guarantee local invariance.Source: §1 · p. 2
  • Treating other observations from the same person as negatives restores changing-state information to the short-record level without shortening the record, while identity remains strongly decodable.The recovery does not depend on temporal segmentation: random groupings work, whereas partitioning without within-person discrimination does not.Source: §1 · p. 2
  • The supervision-span effect was prospectively reproduced across all five seeds and both prespecified readouts in 199 GLOBEM participants.GLOBEM is a later cohort from a different population and mobile phone-sensing study.Source: §1 · p. 2

2 SETUP: MEASURING INFORMATION BEYOND IDENTITY

The setup measures how much a learned history representation predicts future behavior beyond a fixed person-level behavioral reference. It uses longitudinal sensing cohorts and controls prediction anchors and targets while varying the history property under study.

2 SETUP: MEASURING INFORMATION BEYOND IDENTITY

  • DETECT comprises 57 single-resident participants and 44 multi-resident households with records up to 2.7 years, represented by 48 daily behavioral features from eight sensing channels.Canonical analyses aggregate consecutive days into non-overlapping 5-day windows; GLOBEM is reserved for prospective replication.Source: §2 · p. 3
  • For each anchor, the history precedes time t and the target is the subsequent 5-day behavioral vector; experiments hold anchors, targets, and observation budgets fixed.The prediction horizon equals the target window width, or one day in daily-resolution analyses.Source: §2 · p. 3, Figure 1 · p. 3
  • High future-prediction accuracy alone cannot show that a representation captures changing state, because it may encode a person’s stable behavioral profile.The setup therefore measures predictive information added beyond a fixed person-level reference computed from training-era data.Source: §2 · p. 3
  • The operational measure of changing-state information is the representation’s incremental predictive information beyond the person-level reference, reported as ΔI in nats per output dimension.Positive ΔI means the representation improves prediction of future behavior beyond what the typical-behavior reference predicts.Source: §2 · pp. 3–4
  • Matched ridge decoders are primary, and the reported declines persist with nonlinear MLP decoders, recency-matched baselines, and a residualized estimator.Comparisons with the untrained reference use the ridge decoder throughout.Source: §2 · p. 4
  • Chronological train-test splits use a buffer to prevent histories or targets from crossing the split, and uncertainty is estimated by resampling seeds and independent units.Effects are called consistent across cohorts only when their signs agree and confidence intervals exclude zero in both.Source: §2 · p. 4

3 LONGITUDINAL HISTORY IS NOT TEMPORALLY EXCHANGEABLE

Equal-sized histories carry different amounts of information about future behavior depending on when they were observed; behavioral-state information fades much faster than identity. Recent-state prediction also depends partly on temporal order.

3.1 THE SAME HISTORY BECOMES STALE

  • Older histories contain 84–98% less changing-state information than recent histories across both cohorts, despite an equal 20-day history budget.The advantage persists when predicting directly from raw history, indicating that the temporal asymmetry is present in the underlying behavior.Source: §3.1 · p. 5

3.2 STATE FADES; IDENTITY PERSISTS

  • Over 60 days, changing-state information falls by 59% in the single-resident cohort and 72% in households, while identity accuracy remains nearly unchanged even at 180 days.Figure 2 tracks changing-state information and identity decoded from displaced histories.Source: §3.2 · p. 5, Figure 2 · p. 5
  • As behavior drifts, recent history becomes more informative than older history, including within the same individual over time in the single-resident cohort.Source: §3.2 · p. 5

3.3 RECENT STATE LIVES PARTLY IN TEMPORAL ORDER

  • Shuffling the same recent windows reduces changing-state information by 0.023–0.046 nats/dim, showing that useful state information resides partly in temporal order.The order-free set representation closely tracks the shuffled condition.Source: §3.3 · p. 6, Table 2 · p. 6

3 LONGITUDINAL HISTORY IS NOT TEMPORALLY EXCHANGEABLE

  • Identity remains a valid match across time even as behavioral states diverge, so longer records expand the range of states contained within a same-person relation.Source: §3.3 · p. 6

4 LONGITUDINAL SCALE BROADENS LEARNED INVARIANCE

The experiments separate record span from supervision span, showing that each can broaden learned invariance in a different way. Within-person discrimination recovers long-record state information, and the direct supervision-span effect also transfers prospectively to GLOBEM.

4 LONGITUDINAL SCALE BROADENS LEARNED INVARIANCE

  • The design separates coupled scaling by keeping supervision local as records grow, fixing records as supervision broadens, and testing within-person discrimination at full record span.Source: §4 · p. 6, Figure 3 · p. 7

4.1 LOCAL SUPERVISION DOES NOT GUARANTEE LOCAL INVARIANCE

  • With local supervision, extending records from 30 days to full length reduces changing-state information by 33% and 47%, while identity decodability rises in both cohorts.States more than 180 days apart become increasingly similar even when they were never paired during training.Source: §4.1 · pp. 6–7, Figure 3 · p. 7

4.2 BROADER SUPERVISION DIRECTLY SUPPRESSES CHANGING STATE

  • At the broadest supervision span, less than 10% of changing-state information remains; with one positive per anchor, increasing span still reduces it by 83% and 89% across the two cohorts.The available history is held fixed while positive-pair supervision reaches farther through time.Source: §4.2 · p. 7, Figure 4 · p. 7

4.3 WITHIN-PERSON DISCRIMINATION RECOVERS THE LOST STATE

  • Within-person discrimination raises changing-state information by 54–56% in single-resident participants and 96–121% in households, returning both cohorts to their 30-day levels without shortening records.Random groupings work equally well, but partitioning without contrasting across groups does not recover information; the intervention provides no benefit on short records.Source: §4.3 · p. 8

4.4 THE SUPERVISION-SPAN EFFECT TRANSFERS PROSPECTIVELY

  • In GLOBEM, broadening realized supervision span from 3.2 to 24.6 days reduces changing-state information by 0.0070 nats/dim, with all five seeds agreeing in direction.The effect is reproduced on a prespecified secondary readout and transfers across population, dataset, and sensing setting.Source: §4.4 · p. 8

5 DISCUSSION

The discussion distinguishes the amount of history available from the invariance imposed by the learning objective. Its conclusions are limited to person-level contrastive learning in behavioral sensing and do not establish the same behavior for other forms of self-supervision.

5 DISCUSSION

  • Longer records can broaden invariance even under local supervision, while requiring states within a person’s history to remain distinct can recover lost information without shortening the record.Source: §5 · p. 9
  • Record length is a property of the data, whereas invariance is a property of the learning problem; they need not scale together.Scaling self-supervised systems therefore calls for tracking how the meaning of supervision changes as data are added.Source: §5 · p. 9
  • The evidence establishes these effects for person-level contrastive learning in behavioral sensing, not self-supervised learning generally or substantially larger model scales.GLOBEM reproduces the direct supervision-span effect, but its shorter records do not test the long-record loss targeted by within-person discrimination.Source: §5 · p. 9

REPRODUCIBILITY STATEMENT

The reproducibility statement locates evaluation, implementation, and cohort details in appendices and summarizes the datasets’ access restrictions.

REPRODUCIBILITY STATEMENT

  • The evaluation protocol, chronological splits, leakage audits, and ∆I estimator are specified in Appendix B.Source: REPRODUCIBILITY STATEMENT · p. 9
  • Appendix J gives architectures, optimization settings, positive and negative sampling, and compute; Appendix A describes cohort construction, household deduplication, and features.Source: REPRODUCIBILITY STATEMENT · p. 9
  • Reported effects are averaged over five seeds with unit-clustered bootstrap intervals, and DETECT and GLOBEM cannot be redistributed.Source: REPRODUCIBILITY STATEMENT · p. 9

AI USE STATEMENT

The appendix documents the sensing cohorts and preprocessing behind the analyses, along with checks on household dependence and input-feature observation. It also reports the authors’ AI-use disclosure.

AI USE STATEMENT

  • The authors disclose using large language models to draft sections and aid or polish writing, with all assisted content reviewed, verified, and edited by the authors.The authors take responsibility for the final manuscript.Source: AI USE STATEMENT · p. 9

APPENDIX OVERVIEW

  • DETECT provides the in-home sensing data used throughout, while GLOBEM is reserved for the replication described in Appendix G.The appendix overview assigns data, evaluation, estimator, supporting analyses, robustness checks, scope-delimiting results, and implementation details to Appendices A–J.Source: A DETECT: DATA, COHORTS, AND… · p. 11, APPENDIX OVERVIEW · p. 11

A.1 DAILY FEATURE CONSTRUCTION

  • Each retained day is represented by 48 features from eight sensing channels, including three daily statistics per channel and an additional 24-bin hourly motion profile.Thirty dimensions are motion-derived.Source: §A.1 · p. 11

A.2 RETENTION AND TEMPORAL AGGREGATION

  • Days require at least 18 hours of motion coverage, and retained five-day windows require four qualifying days; feature standardization uses training-era windows only.Daily vectors are averaged within non-overlapping five-day windows.Source: §A.2 · p. 12

A.3 COHORT CONSTRUCTION AND HOUSEHOLD DEDUPLICATION

  • The multi-resident cohort contains 87 subject records from 44 homes, but the analysis retains one stream per home to avoid duplicating shared sensing data.The retained record prioritizes sensing modalities, then recorded hours.Source: §A.3 · p. 12
  • Co-resident streams share major home-level signals, so the household—not each subject record—is the independent unit; individual-level state-change claims are not identifiable in this cohort.Motion and weight are duplicated across paired homes, and bed-derived channels are near-duplicated; steps and medication events behave as individual channels.Source: Table 3 · p. 12, §A.3 · p. 12
  • 0.245 to 0.215 nats per dimension: household deduplication narrows, but does not reverse, the recent-history advantage.Deduplication reduces effective sample size and widens uncertainty.Source: §A.3 · p. 13, Table 4 · p. 13

A.4 FEATURE OBSERVATION AND MOTION-ONLY ROBUSTNESS

These appendices test robustness of the longitudinal findings and specify the evaluation and inference procedures. Motion-only analyses, temporal controls, and identity probes delimit what the reported effects mean.

A.4 FEATURE OBSERVATION AND MOTION-ONLY ROBUSTNESS

  • Recent-history advantage persists in both cohorts using 30 motion-derived dimensions, but the reduced magnitude cannot isolate the share attributable to missingness.Non-motion features can encode both behavior and device availability.Source: §A.4 · p. 13, Table 6 · p. 14

B.1 CHRONOLOGICAL SPLITS AND LEAKAGE PREVENTION

  • Permuting future targets reduces estimates to at most 0.0002 nats per dimension, under 0.1% of the recent-history value of 0.282.The history advantage collapses when history–future correspondence is destroyed.Source: §B.1 · p. 15, Table 8 · p. 15
  • An eight-window train–test buffer preserves anchor eligibility as distance grows; a stricter 24-window buffer shifts curve levels and induces an apparent 20-day zero crossing, but the monotone decline remains.The zero crossing is absent under the eight-window construction on the same data.Source: §B.1 · p. 15, Table 9 · p. 15

B.2 CONDITIONAL USABLE PREDICTIVE INFORMATION

  • ∆I measures future-prediction information beyond a stable person- or household-specific baseline; learned estimates agree across canonical and residualized scoring within 0.0022 nats/dim.Both predictors use the same ridge predictive family and identical held-out targets.Source: §B.2 · p. 16

B.3 BOOTSTRAP INFERENCE AND SEED AGGREGATION

  • Hierarchical bootstrap resampling over independent units and seeds turns the separation-matched households arm into a null, while the uniform arms still decline.The household arm is +0.0037 under both resampling schemes, but its hierarchical interval includes zero.Source: §B.3 · p. 17

B.4 IDENTITY PROBING

  • Distant history remains a strong identity cue despite contributing little incremental prediction, whereas recent history is nearly chance-identifying under a linear probe yet carries the largest ∆I.The distant block is fixed across anchors for each participant, so its near-perfect identity identifiability is largely by construction, not evidence of temporal persistence.Source: §B.4 · p. 18
  • Shifted history appears unidentifying to a linear probe but reaches 0.510 and 0.541 accuracy under an MLP probe, showing that a linear probe alone can miss identity information.The MLP values are higher than recent history under the same probe.Source: §B.4 · p. 18

B.5 NUMERICAL CONDITIONING AND RIDGE SELECTION

The appendix audits decoder conditioning and penalty selection, then compares learned representations with raw and untrained history baselines. It finds robust declines under uniform history sampling while qualifying how representation comparisons should be read.

B.5 NUMERICAL CONDITIONING AND RIDGE SELECTION

  • Held-out validation MSE avoids optimistic leave-one-out penalty selection on autocorrelated windows; varying only the selection rule reproduces the pipeline difference.The selected held-out penalties also forecast better on untouched test rows.Source: §B.5 · pp. 18–19
  • Held-out penalty selection predicts better in 8 of 8 arms across both cohorts, by 3.7% to 25.1% of leave-one-out test MSE, with every interval excluding zero.The comparison uses the same design, split, and seeds, with test rows used only for evaluation.Source: §B.5 · p. 19, Table 13 · p. 19
  • An SVD/pseudoinverse ridge reproduces Cholesky estimates to 10^-15 without changing signs or intervals, making the solver immaterial for these selected penalties.The normal-equation form is less stable in principle because it squares the condition number.Source: §B.5 · p. 19
  • Across decoder grids and validation splits, the uniform record-span decline is negative in 24/24 cells; matched arms show no consistent effect.The single-resident uniform decline has an interval excluding zero in 12/12 cells; household significance varies by split.Source: §B.5 · p. 19

C.1 HISTORY SOURCES AND REPRESENTATION-FREE BASELINES

  • Raw-block recency gaps (+0.3097, +0.2543) exceed learned gaps (+0.2371, +0.2167); random projection preserves the ordering at roughly a quarter of the magnitude, and training recovers much of that loss.No learned representation exceeds raw history in the reported comparison, but raw history has 96 dimensions versus 32 for learned representations.Source: §C.1 · p. 20, Table 14 · p. 20
  • The raw distant-history value of +0.0597 is a fixed-block artifact: the earliest block is constant within each unit and can be exploited by an unbottlenecked decoder.The authors therefore prefer the anchor-relative far arm.Source: §C.1 · p. 20

C.2 BEHAVIORAL-DRIFT OPERATIONALIZATIONS

The analyses distinguish how far behavior has moved from how quickly it is changing, then examine how conditioning and longer locally supervised records affect recoverable changing-state information. Longer records reduce that information even when positive-pair reach stays local, and several tested alternatives do not explain the decline.

C.2 BEHAVIORAL-DRIFT OPERATIONALIZATIONS

  • Within-unit recency advantage tracks contemporaneous displacement from a unit’s past in both cohorts, not rate of change; lagged displacement replicates only in households.The within-person claim is restricted to displacement, rather than all three drift definitions.Source: §C.2 · pp. 21–22

C.3 CONDITIONED AND UNCONDITIONED PERSISTENCE

  • Conditioned predictive information falls to 0.411 and 0.276 by 60 days, while unconditioned information retains 0.666 and 0.575; much remaining distant prediction reflects stable person-level information.The analysis interprets the conditioned quantity as deviation from a person’s baseline, not behavioral predictability in general.Source: §C.3 · p. 23, Table 18 · p. 23

D.1 NESTED-HISTORY CONSTRUCTION

  • Nested histories share an end cut and equal unit anchor counts, while positive separation remains capped at 30 days, with realized means of 20.1–20.7 days.This construction adds older observations without extending positive-pair reach.Source: §D.1 · p. 23

D RECORD SPAN WITH LOCAL SUPERVISION

  • 33% and 47%: changing-state information falls from a 30-day to a full record under local supervision, with declines in all five seeds in both cohorts.Most of the decline arrives by 120 days and the curve then flattens.Source: Table 19 · p. 24, §D.2 · p. 24

D.2 CHANGING STATE AND IDENTITY ACROSS RECORD SPAN

  • As record span grows, identity prediction rises from 0.546 to 0.613 and from 0.668 to 0.757, although positive pairs never span more than a month.Source: Table 19 · p. 24, §D.2 · p. 24

D.4 ALTERNATIVE EXPLANATIONS FOR THE RESIDUAL DECLINE

  • The decline persists with fixed anchor counts and blocked within-batch cross-period discrimination; input-shift and heterogeneity tests found no strong relationships, while age-shift estimates span zero.The input-shift and heterogeneity tests had power only against strong relationships.Source: §D.4 · pp. 24–25

D.3 GEOMETRY OF DISTANT SAME-PERSON STATES

  • Far same-person pairs more than 180 days apart, never used as positives, become relatively closer (0.637 to 0.576; 0.539 to 0.458); household far pairs contract more than near pairs.Between-person dispersion relative to within-person variation rises as changing-state information falls across the record-span grid.Source: §D.3 · p. 24, Table 20 · p. 25

D.5 ALTERNATIVE RECORD-SPAN CONSTRUCTIONS

The alternative constructions separate history available to the encoder from the temporal reach of positive supervision. Across fixed-record and growing-record tests, broader supervision suppresses changing-state information, while the record-span decline persists under local supervision and robustness checks.

D.5 ALTERNATIVE RECORD-SPAN CONSTRUCTIONS

  • At local positive separation near 20 days, increasing history still reduces changing-state information by 33% and 47% at the reported encoder capacity.The nested-history construction isolates the cost of more history; unlike the expanding and fixed-anchor sweeps, it varies encoder history while holding supervision local.Source: §D.5 · pp. 25–27

D.6 RECENCY-MATCHED BASELINES

  • With recency-matched baselines, the record-span decline appears in all 12 baseline, decoder, and cohort combinations, ranging from 55% to 84%.The separation-matched contrast persists, though the matched arm is close to flat only in the single-resident cohort; absolute ∆I levels depend on the baseline.Source: Table 25 · p. 28, §D.6 · p. 28, Table 26 · p. 29

E.1 FIVE-DAY WINDOWS

  • At fixed record span, changing-state information falls 79% and 83% from the tightest to unrestricted supervision, while identity decodability rises in both cohorts.The single-resident series is monotone; the household series has one small inversion within its interval.Source: Table 27 · p. 29, §E.1 · p. 29

E.2 DAILY WINDOWS

  • Only the two-day daily-window condition reaches its untrained reference; its apparent advantage does not survive matching embedding scale.The two-day gaps are +0.0457 and +0.0503 before scale matching; after L2 normalization, both intervals include zero.Source: Table 28 · p. 30, §E.2 · p. 30

E.3 SUPERVISION SPAN AS AN EXPLICIT PARAMETER

  • With full records and scale-matched representations, changing-state information declines from τ = 2 onward, with no interior optimum; only τ = 2 reaches the untrained floor.Identity remains flat across capped spans and rises only with unrestricted sampling.Source: Table 29 · p. 31, §E.3 · p. 31

E.4 FIXED-CARDINALITY CONTROL

  • With one positive per anchor, increasing separation from τ = 2 to τ = 128 reduces ∆I by 83% and 89%, and broader pools add a further cohort-consistent benefit when separation is matched.The fixed-cardinality arm exceeds cap-sampling at every matched separation in both cohorts.Source: §E.4 · pp. 31–32, Table 30 · p. 32

E.5 DISTANCE-WEIGHTED POSITIVE SAMPLING

  • Distance-weighted sampling also loses changing-state information as realized separation grows, but its persistent distant-positive tail prevents it from reproducing the tightest cap.At matched realized separations, kernel and cap results are similar; no kernel arm reaches the untrained floor.Source: §E.5 · p. 32, Table 31 · p. 33

E.6 TARGET-OVERLAP AUDIT

  • Excluding target-overlapping positives leaves the record-span decline essentially unchanged: 63% versus 63% in single-resident homes and 65% versus 63% in households.This control does not automatically extend to daily windows, where overlap at the tightest cap reaches 50%.Source: Table 32 · p. 33, §E.6 · p. 33

E.7 SEPARATION VERSUS POOL BREADTH

Within-person negatives recover changing-state information at full record span while preserving strong identity decodability. The separation effect is consistent across cohorts, whereas any additional effect of pool breadth is unresolved in these data.

E.7 SEPARATION VERSUS POOL BREADTH

  • At K=1, ∆I falls 69% and 64% from four days to roughly 100 days of realized separation across the two cohorts.The decline holds in all three seeds in both cohorts.Source: §E.7 · p. 34, Table 33 · p. 34
  • The pool-size effect is cohort-dependent and unresolved: matched-separation comparisons show no effect in single-resident homes but a substantial effect in households.The figure plots availability and retained state against realized separation; its wide-pool curves have normalization artifacts when band widening changes their starting separation.Source: §E.7 · pp. 34–35, Table 34 · p. 35, Figure 6 · p. 35

F.1 CONSTRUCTION

  • Within-person discrimination changes the positive identity to a participant–group pair, placing that participant’s observations from other groups in the negatives.The remaining construction keeps anchors, locally capped positives, and the training and evaluation setup fixed.Source: §F.1 · p. 36

F.2 STATE RECOVERY AND IDENTITY PRESERVATION

  • At full record span, within-person discrimination recovers the record-span loss: 30-day groups are statistically indistinguishable from the short-record level in single-resident homes and slightly exceed it in households.Identity accuracy falls modestly but remains far above chance.Source: F WITHIN-PERSON DISCRIMINATION · p. 35, §F.2 · p. 36

F.3 RANDOM-GROUP AND MASKING CONTROLS

Randomized grouping shows that within-person negatives—not temporal segmentation—drive the recovery, while the geometry and prospective replication analyses qualify where that result applies. The GLOBEM test supports transfer of the supervision-span effect, but not the other longitudinal mechanisms.

F.3 RANDOM-GROUP AND MASKING CONTROLS

  • Randomly grouped windows improve changing-state information by +0.0530 and +0.0580 against the unsegmented objective, matching contiguous groups because distant windows repel.At L=30, the relabeling does not help, indicating the intervention is specific to long records.Source: §F.3 · pp. 36–37
  • Masking cross-group pairs reproduces the unsegmented objective exactly, with a ∆I difference of +0.0000 and a zero-width interval.The unsegmented objective already masks same-person pairs; only the added repulsive term links distant windows.Source: §F.3 · p. 36

F.4 REPRESENTATION GEOMETRY

  • Within-person discrimination moves representation geometry toward short-record values, with a complete reversal in households and a partial reversal in the single-resident cohort.In households, the between-to-within ratio returns to 3.40–3.51 from 4.74, near the 30-day value of 3.55; distant-window geometry also recovers.Source: §F.4 · p. 37, Table 36 · p. 37

G PROSPECTIVE REPLICATION IN GLOBEM

  • The prospective GLOBEM test reproduces the supervision-span decline across five seeds and both prespecified readouts in a different population and sensing setting.The test is limited to the supervision-span mechanism; its 60–97-day records do not support an informative record-span manipulation, and within-person discrimination was not tested.Source: G PROSPECTIVE REPLICATION IN GLOBEM · p. 38

G.1 GLOBEM: SELECTING VARIABLES THAT CAN CARRY STATE

  • Coverage-ranked GLOBEM variables mostly lack persistent state, so the analysis instead selects features using persistence and predictability.The coverage-ranked set had median lag-1 autocorrelation 0.169 and median 2.49% of baseline error removable from two-day context; the selected set raises the latter to 26.3%.Source: §G.1 · p. 38, §G.2 · p. 38

G.2 GLOBEM: A PROSPECTIVE STATE-FEATURE RULE

  • The prospective feature rule uses Wave 1 variance, autocorrelation, half-life, and univariate r2 criteria, without using SSL results or learned representations.Features must meet all four criteria; survivors are ranked by r2, with thresholds left unchanged if fewer than 48 survive.Source: §G.2 · p. 38

G.3 THE RULE TRANSFERS

  • On unseen Wave 2 participants, the Wave 1 rule still selected features in the intended state-predictive regime.Median lag-1 autocorrelation was 0.409, 41 of 46 features met C2, and median error removed was 26.9%.Source: §G.3 · p. 38

G.4 RESULT

Across GLOBEM and robustness analyses, the results examine how supervision span and record span relate to recoverable changing-state information, with record-span effects depending on whether records provide enough separation. They also compare learned representations with capacity-matched PCA and TNC, finding that TNC partly recovers information lost under unrestricted person-contrastive training but remains below the raw-history baseline.

G.6 QUALIFICATIONS

  • GLOBEM record-span arms showed no effect, with all intervals containing zero; records were only 60–97 days, while positive-pool breadth also confounded separation in the supervision sweep.Mean positive separation rose only from 14.4 to 24.5 days with record span, versus a 7.7-fold change under the direct cap; supervision span transferred, but record span as its proxy did not. The person mean added little beyond an intercept on Wave 2.Source: §G.4 · p. 39, §G.6 · pp. 39–40

G.5 GLOBEM: SINGLE-POSITIVE CONTROL

  • With one positive per anchor, changing-state information declined 62% as τ widened from 4 to 32 days; the unrestricted single-positive cell was less damaging than four moderately distant positives.The unrestricted cell selected the farthest eligible candidate, at 51.2 days, and had ∆I = +0.0154; the result indicates pool breadth contributes alongside separation.Source: §G.5 · p. 39

G.4 RESULT

  • Contrastive learning removed only 0.14% of baseline test error and yielded 0.779 participant decodability, versus 0.159 for masked reconstruction and 0.085 for an untrained encoder.Across representations, baseline test error removed ranged from 12.8% to 21.6%, except for the person-contrastive encoder; its absolute state information remained small relative to PCA-32.Source: §G.4 · p. 39, Table 37 · p. 40
  • Widening same-person positives from 3.2 to 24.6 realized days reduced state information by −0.0070 [−0.0121, −0.0033] primary and −0.0064 [−0.0120, −0.0026] secondary, both across 5/5 seeds.The readouts agreed in sign; state information fell while identity rose across the sweep.Source: §G.4 · p. 39, Table 38 · p. 40

H ROBUSTNESS ACROSS MODELS AND READOUTS

  • The scale effect persisted with motion-only features and nonlinear decoding: fixed-record supervision widening reduced state information, while near-21-day separation left it nearly unchanged.In motion-only data, broadening supervision reduced state information by 50% in the single-resident cohort and 53% in the household cohort; holding separation near 21 days changed it by −2% and +3%. The MLP decoder also retained record-span declines, though their magnitudes differed from ridge.Source: §H.1 · pp. 40–41, §H.2 · p. 41, Table 39 · p. 41

H.3 SEQUENCE-AWARE ENCODER

  • With a GRU, predictive information decayed with temporal distance in both cohorts, while last-state and mean-pooled readouts diverged in sign as averaged history expanded to eight windows.The distance effect was slightly steeper than with the MLP; the recent–non-recent separation also persisted, but the ordering among non-recent arms did not replicate across cohorts.Source: §H.3 · pp. 42–43, Table 41 · p. 43

H.4 TEMPORAL ORDER AT DAILY RESOLUTION

  • At daily resolution, correct temporal order had a larger advantage than at five-day resolution under both matchings, while shuffled and set representations remained indistinguishable.The ordered–shuffled advantage was 1.6 to 3 times larger and held across 5/5 seeds in both cohorts; absolute ∆I was not comparable across resolutions.Source: §H.4 · p. 43, Table 42 · p. 44

H.5 REPRESENTATION DIMENSION AND CAPACITY; H.6 A TEMPORALLY AWARE BASELINE

  • The state–identity dissociation persisted across representation dimensions, and scale-matched TNC recovered part, but not all, of the changing-state information lost under unrestricted supervision.TNC recovered 63%, 47%, 50% and 64% of the unrestricted-to-PCA gap across four build-by-cohort cells; its exact recovery fraction was unstable, and PCA-32 still exceeded it in all four. The TNC neighborhood rule alone did not outperform an untrained encoder under scale matching.Source: §H.5 · p. 44, Table 44 · p. 45, §H.6 · p. 46

H.6 A TEMPORALLY AWARE BASELINE

  • TNC recovers roughly half to two thirds of what unrestricted person-contrastive training destroys across four cells and both readouts.It extracts no state information absent from an untrained encoder of the same shape, and neither TNC component does so alone once representation scale is matched.Source: §H.6 · p. 46
  • +0.0727 [+0.0525, +0.0947] and +0.0962 [+0.0719, +0.1254] on five-day, and +0.0928 [+0.0690, +0.1180] and +0.0700 [+0.0460, +0.0932] on daily, are PCA-32d’s advantages over TNC.All four intervals exclude zero across 5/5 seeds; TNC’s ADF criterion selects the shortest admissible neighborhood—the candidate-grid floor imposed by non-overlap—so this indicates stationarity fails beyond that floor, not an estimated optimum.Source: §H.6 · p. 46

H.7 CAPACITY-MATCHED REPRESENTATION-FREE BASELINES

  • +0.2095 [+0.1763, +0.2516] and +0.1751 [+0.1242, +0.2242] are the paired-bootstrap advantages of capacity-matched PCA-32d over the strong learned representation.PCA retained 81.7% and 84.1% of block variance and 89% and 97% of raw-block ∆I, so dimensionality explains almost none of the gap.Source: §H.7 · p. 47, Table 47 · p. 47
  • 0.0995 for an untrained three-layer MLP versus 0.2034 for a same-dimension linear random projection shows the untrained floor is architecture-specific.With an MLP decoder, the learned-representation contrast remains +0.1631 [+0.1356, +0.1948] and +0.1646 [+0.1370, +0.1956]; the MLP decoder lowers the raw-block score from 0.3013 to 0.2332, so the true gap likely lies between the two decoders.Source: §H.7 · p. 47

H.8 CONTRASTIVE IMPLEMENTATION AND PRETEXT DIFFICULTY

Harder negatives and larger negative pools do not remove the decline in changing-state information as records grow, while pretext-task metrics contradict saturation. Global representation similarity also fails to track future-relevant information, which resides substantially in less-shared directions.

H.8 CONTRASTIVE IMPLEMENTATION AND PRETEXT DIFFICULTY

  • Hard-negative mining uses fixed nearest other-unit blocks, while the in-batch condition dynamically resamples other-unit negatives; both preserve the record-span decline.The in-batch pool averages 63.3 and 51.8 negatives per anchor, rather than the nominal 127.Source: §H.8 · pp. 47–48
  • Extending records makes the pretext task harder rather than easier, opposing the saturation explanation.Under standard sampling, held-out InfoNCE rises from 2.21 to 2.74 and from 2.13 to 2.45, while top-1 accuracy falls from 0.595 to 0.353 and from 0.625 to 0.443.Source: §H.8 · pp. 47–48, Table 48 · p. 48
  • With mined negatives, ∆I still declines from 30 days to the full record by 40% and 29%; with in-batch negatives, declines are 56% and 48%, versus 63% and 65% for the standard pool.The larger pool attenuates the effect slightly but does not explain it.Source: §H.8 · p. 48, Table 49 · p. 48

H.9 GLOBAL REPRESENTATION SIMILARITY

  • Non-recent representations retain only 2%–10% of recent-history predictive information while reaching 54%–61% of the recent–recent CKA ceiling, so global similarity does not track predictive information.The distant-history encoder retains 5% and 2% while reaching 57% and 60% of the ceiling; an input-reconstruction autoencoder also reaches 0.845 and 0.916 of that ceiling.Source: §H.9 · p. 49, Table 50 · p. 49
  • Removing the four most shared canonical directions leaves 5-day predictive information essentially unchanged, while the less-shared component restores approximately 90% of the predictive gap for a 40-day encoder.The full representation yields 0.2837 nats/dim, compared with 0.2831 after removing those directions; the shared component alone yields 0.1285.Source: §H.9 · p. 49
  • CKA characterizes global overlap but cannot establish that longitudinal representations preserve the same future-relevant information.Directions contributing most to global similarity need not carry the temporal information used for prediction.Source: §H.9 · p. 49

I.1 HELD-OUT CHANNEL EVALUATION

Withholding time-in-bed statistics tests whether record-span effects extend beyond the behavioral target used in training. The results also examine cohort size, alternative temporal objectives, and what learned temporal weights reflect.

I.1 HELD-OUT CHANNEL EVALUATION

  • The evaluation withholds time-in-bed statistics from encoder inputs, objectives, and baseline, fixes training anchors, and is a same-home held-out-feature test—not external clinical validation.The target comprises the log daily total, overnight fraction, and hourly-distribution entropy.Source: §I.1 · pp. 49–50
  • ∆I drops from +0.2030 to −0.0115 and from +0.2177 to −0.0018 under full-record person-level contrastive learning, leaving time-in-bed prediction no better than the person baseline.Both full-record intervals include zero.Source: Table 51 · p. 50, §I.1 · p. 50

I.2 COHORT-SIZE ANALYSIS

  • State-information loss is flat across cohort size, but identity gain rises monotonically; separation-matched state and identity changes remain near zero even with 15 identities.The state result is more portable across cohort sizes than the identity increase.Source: §I.2 · pp. 50–51, Table 52 · p. 51

I.3 ALTERNATIVE TEMPORAL OBJECTIVES

  • Horizon-weighted prediction, stable/local factorization, horizon conditioning, and context-dependent retrieval did not robustly improve prediction across cohorts.Effects were inconsistent across cohorts or within the unmodified objective’s interval.Source: §I.3 · p. 51

I.4 DIAGNOSTICS AND IMPLEMENTATION DISCLOSURES

  • Initially, per-bin InfoNCE loss rises with temporal distance; after training it varies only 1.4%, while learned weights track alignment difficulty, not measured informativeness.The learned decay should not be read as recovering the temporal informativeness curve measured by ∆I.Source: §I.4 · pp. 51–52

J.1 MODEL ARCHITECTURES

  • The encoder uses a three-layer ReLU MLP with a 32-dimensional representation, matching parameter count as representation width changes.Representations are L2-normalized before similarity computation.Source: §J.1 · p. 52

J.2 POSITIVE AND NEGATIVE SAMPLING

  • Positives are other windows from the same unit, while negatives come from other units; separation-matched sampling restricts positives to the 30-day reference separation distribution.The span experiments use four positives and eight negatives per anchor.Source: §J.2 · p. 52

J.3 OPTIMIZATION AND HYPERPARAMETERS

  • Training holds the anchor pool fixed across spans, while held-out validation and disjoint test anchors support evaluation; experiments run on CPU.The encoder uses Adam for 60 epochs; held-out contrastive metrics use validation anchors, separate from training and test anchors.Source: §J.3 · pp. 52–53, §J.4 · p. 53

J.5 STRONG-CONTRASTIVE ENCODER

A stronger contrastive learner tests whether the supervision-span effect reflects an underpowered encoder. The effect persists, although the stronger model does not outperform its untrained floor in one cohort.

J.5 STRONG-CONTRASTIVE ENCODER

  • The stress test uses four times the parameters, in-batch negatives, five times the training, and behavioral augmentations, holding these settings fixed while changing only the positive pool.The strong encoder has 159,562 parameters; the weak encoder has 40,000.Source: §J.5 · p. 53
  • Short minus unrestricted remains +0.0193 and +0.0171 (33% and 36% losses), yet the strong encoder does not beat its untrained floor in households and is below it in one cohort.The untrained-minus-trained contrast interval includes zero in the single-resident cohort and is positive in households.Source: Table 54 · p. 53, §J.5 · p. 53
  • Broadening supervision raises held-out InfoNCE from 7.34 to 8.33 and from 6.38 to 7.99, while top-1 falls from 0.347 to 0.214 and from 0.461 to 0.280.The pretext task gets harder, not easier.Source: §J.5 · p. 53
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