Inferring missing locations in human mobility data using memory effects

Pablo Rosillo-Rodes; Antònia Tugores; David Sánchez; José J. Ramasco

cs.LGOct 31, 2026

Inferring missing locations in human mobility data using memory effects

Authors: Pablo Rosillo-Rodes, Antònia Tugores, David Sánchez, José J. Ramasco

Organizations: Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca, Spain

Abstract

Empirical datasets used to characterize human mobility are sparse in space and time, owing to technological factors such as coverage and battery, or to the erratic nature of online interactions. Missing data can bias the conclusions drawn from them, so filling the gaps in incomplete timelines is a key challenge in computational social science. Here, we propose a simple imputation method in which the informativeness of an observation decays with its bidirectional temporal distance from the missing entry, at a rate set by a single parameter per individual. On synthetic and real Twitter data, a power-law decay, selected by Bayesian model comparison, yields accurate imputations (macro-AUROC above 0.9 overall, and above 0.8 for the most mobile users) and matches a gap-aware continuous-time Markov chain, outperforming it among the most mobile users. The method is robust to moderate noise and generalizes across countries and European administrative regions.

Figures & tables

Figure 1: Sketch of the method . A human travels along different locations ( a ), which are mapped onto a timeline diagram ( b ). While certain locations are included in the mobility dataset (depicted as blue dots), others are missing (depicted as a gray rectangle or gray dots). Our goal is to infer those missing locations from a simple model. c We introduce a temporal weighting scheme that models the influence of past and future observations on a missing location by assigning weights based on their temporal distance $\Delta t$ , according to a specified function $f_{\lambda}\left(\Delta t\right)$ , where $\lambda$ is a decay parameter. d The negative log composite likelihood of the validation days, $-\log\mathcal{C}$ , as a function of $\lambda$ ; its minimum defines $\lambda_{*}$ . Three characteristic shapes appear. In Case 1 the minimum lies at $\lambda_{0}$ , where the model reduces to predicting the most frequently visited location. In Case 2 the minimum is interior, and the timeline is best described by a genuine decay of informativeness with temporal separation. In Case 3 the curve decreases monotonically towards $\lambda_{\infty}$ , where the model reduces to copying the location of the temporally nearest observed day (persistence).

Figure 2: The empirical dataset. Distributions over the

10^{6}

country-level timelines of a the number of observed days

n_{i}

, b the number of distinct countries

N_{c}

, c the fraction of observed days spent at the modal country, and d the change rate

p

. e Per-individual median temporal distance from a test day to its nearest training day, under random hold-out (median over individuals

3.0

days) and under

14

-day block hold-out (

14.0

days).

Figure 3: Synthetic generators. a–d One analysed trajectory from each generator at noise

p\approx 0.13

: a SG1, alternation between two locations with fixed residence intervals (

P=5

\Delta t_{1}=18

\Delta t_{2}=54

); b SG2, heavy-tailed inter-event times, with a switch of location at each observation with probability

p

; c SG3, four locations with a change of home location and jittered excursions; d SG4, four locations with log-normal, non-periodic residence durations. Colors and vertical positions indicate the location. e–h Median test-fold macro-AUROC as a function of the noise parameter

p

for the three kernels, the nearest observed day and the gap-aware Markov chain; bands span the interquartile range over the 500 trajectories at each value of

p

, and the dotted line marks chance level. The meaning of

p

differs between generators (Table V).

Generator	Stratum	$n$	PL	EXP	GAUSS	MostFreq.	LastObs.	NearestObs.	Markov	MarkovCT
SG1	$0.02\leq p<0.10$	1103	0.981	0.980	0.978	0.500	0.913	0.944	0.913	0.993
	$0.10\leq p<0.30$	2447	0.839	0.840	0.821	0.500	0.748	0.780	0.748	0.828
	$p\geq 0.30$	6450	0.550	0.541	0.500	0.500	0.545	0.556	0.535	0.557
SG2	$p<0.02$	166	1.000	1.000	1.000	0.500	1.000	1.000	1.000	1.000
	$0.02\leq p<0.10$	1479	0.962	0.963	0.948	0.500	0.889	0.929	0.889	0.989
	$0.10\leq p<0.30$	3743	0.800	0.754	0.636	0.500	0.717	0.758	0.715	0.833

Table 1: Performance on the synthetic generators. Median test-fold macro-AUROC of the three kernels and the five baselines, stratified by the realized change rate

p

;

n

is the number of trajectories in each stratum (500 trajectories at each of 20 values of the noise parameter per generator, random hold-out). Only SG2 populates

p<0.02

. MostFrequent assigns the same distribution to every day, so its macro-AUROC is 0.5 by construction. The best value in each row is in bold. The fraction of correct inferences, the log-loss and the remaining kernels are reported in Supplementary Note 5.

Generator	Stratum	$n$	EXP	GAUSS	STREXP
SG1	$0.02\leq p<0.10$	1103	1.62 (0.76)	4.06 (0.77)	$-$ 0.01 (0.47)
	$0.10\leq p<0.30$	2447	2.95 (0.98)	6.93 (1.00)	0.61 (0.73)
	$p\geq 0.30$	6450	3.36 (1.00)	6.83 (1.00)	0.95 (0.93)
SG2	$p<0.02$	166	$-$ 0.01 (0.26)	$-$ 0.01 (0.28)	0.00 (0.17)
	$0.02\leq p<0.10$	1479	1.41 (0.63)	3.81 (0.70)	0.00 (0.46)
	$0.10\leq p<0.30$	3743	3.41 (0.93)	7.34 (0.97)	0.87 (0.77)

Table 2: Kernel comparison on the synthetic generators. Median over the trajectories in each stratum of

\log\mathrm{PsBF}^{\mathrm{PL}}_{j}

, the log pseudo-Bayes factor of the power law against kernel

j

(Eq. ( 14 )), stratified by the realized change rate

p

. In parentheses, the fraction of trajectories favoring the power law. Positive values favor the power law. The truncated power law and the log-normal decay are reported in Supplementary Note 2.

	Random hold-out				Block hold-out
Method	$T$	micro-AUROC	macro-AUROC	log-loss	$T$	micro-AUROC	macro-AUROC	log-loss
PL	1.000	0.996	1.000	0.050	1.000	0.960	0.864	0.072
EXP	1.000	0.994	0.994	0.057	1.000	0.957	0.818	0.072
GAUSS	0.996	0.988	0.974	0.061	1.000	0.938	0.500	0.077
STREXP	1.000	0.996	1.000	0.054	1.000	0.960	0.850	0.072
MostFrequent	0.970	0.909	0.500	0.194	1.000	0.882	0.500	0.160

Table 3: Comparison with baselines on real trajectories. Median over individuals of test-fold metrics at country level, for random hold-out and for

14

-day block hold-out. The memory kernels use the point estimate

\lambda_{*}

; for STREXP the shape parameter is fixed at

\beta=0.5

. MostFrequent and NearestObserved are the

\lambda\to 0

and

\lambda\to\infty

limits of the memory model (Methods); MostFrequent assigns the same distribution to every day, so its macro-AUROC is

0.5

by construction.

Figure 4: Comparison with baselines. Test-fold macro-AUROC of the three memory kernels and the four baselines on the country-level timelines, a under random hold-out and b under

14

-day block hold-out. Boxes span the interquartile range across individuals and whiskers extend to

1.5

times that range. A dagger marks the baseline that corresponds to a limiting case of the memory model; the other, MostFrequent, is omitted because, by construction, its macro-AUROC is 0.5. c Median macro-AUROC and log-loss when the kernel sum of Eq. ( 3 ) is restricted to observations within

W

days of the imputed day, for the three main kernels, and “full” denotes the unrestricted timeline.

Figure 5: Performance stratified by mobility. Median test-fold fraction of correct inferences

T

(top) and median macro-AUROC (bottom) for selected kernels and baselines, stratified by the change rate

p

( a ), the number of distinct countries

N_{c}

( b ) and the number of observed days

n_{i}

( c ).

	Random hold-out				Block hold-out
$j$	$p<0.02$	$0.02$ – $0.10$	$0.10$ – $0.30$	$p\geq 0.30$	$0.10$ – $0.30$	$p\geq 0.30$
EXP	$0.00$ (0.30)	$0.00$ (0.46)	3.30 (0.74)	4.73 (0.93)	3.82 (0.74)	4.66 (0.93)
GAUSS	$0.00$ (0.32)	$0.00$ (0.47)	6.95 (0.76)	8.61 (0.94)	7.12 (0.75)	7.71 (0.94)
STREXP	$0.00$ (0.33)	$0.00$ (0.43)	0.85 (0.67)	1.54 (0.86)	1.16 (0.70)	1.54 (0.89)

Table 4: Kernel comparison on real trajectories. Median over individuals of

\log\mathrm{PsBF}^{\mathrm{PL}}_{j}

, the log pseudo-Bayes factor of the power law against kernel

j

(Eq. ( 14 )), stratified by change rate

p

, at country level. In parentheses, the fraction of individuals favoring the power law. Positive values favor the power law.

Figure 6: Kernel selection across mobility strata. Median log pseudo-Bayes factor of the power law against each competing kernel, stratified by a the change rate

p

, b the number of distinct countries

N_{c}

and c the number of observed days

n_{i}

. Positive values favor the power law. The kernels can be discriminated only among mobile individuals, where the light-tailed kernels are clearly disfavored and the stretched exponential, which requires an additional shape parameter, remains close to the power law. The truncated power law (TPL) and the log-normal decay (LOGNORM), discussed in Supplementary Note 2, are shown for reference.

$p$	Meaning
SG1	Chance that any individual day nominally assigned to a location is replaced by the other.
SG2	Probability that the user switches location relative to the previous observation.
SG3–4	Chance that any observed day is replaced by a location drawn uniformly among the $K$ locations.
Real data	Fraction of consecutive observations that imply a change of location.

Table 5: Meanings of the noise parameter

p

across generators and in the real data.

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