Nonparametric Approaches for Generating Mobility Trajectories
Anonymous Author(s)
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Abstract
Mobility datasets are fundamental for evaluating algorithms pertaining to geographic information systems and facilitating experimental reproducibility.
Butprivacy implications restrict sharing such datasets, as even aggregated location-datais vulnerable to membership inference attacks.
Current synthetic mobility datasetgenerators attempt to superﬁcially match a priori modeled mobility characteristicswhich do not accurately reﬂect the real-world characteristics.
Modeling humanmobility to generate synthetic yet semantically and statistically realistic trajectoriesis therefore crucial for publishing trajectory datasets having satisfactory utility levelwhile preserving user privacy .
Speciﬁcally, long-range dependencies inherent tohuman mobility are challenging to capture with both discriminative and generativemodels.
In this paper, we benchmark the performance of recurrent neural architectures (RNNs), generative adversarial networks (GANs) and nonparametric copulasto generate synthetic mobility traces.
We evaluate the generated trajectories withrespect to their geographic and semantic similarity, circadian rhythms, long-rangedependencies, training and generation time.
We also include two sample teststo assess statistical similarity between the observed and simulated distributions,and we analyze the privacy tradeoffs with respect to membership inference andlocation-sequence attacks.
Introduction
The pervasiveness of mobile devices equipped with internet connectivity and global-positioningfunctionality has resulted in an increasingly large amount of location-data on individuals.
This data isbeneﬁcial to address and validate spatiotemporal data-based problems; predictive and kNN queries,object tracking, mobility modeling and location privacy among others.
Due to the sensitive nature ofdatasets containing mobility traces, sharing them with untrusted entities present privacy implications.
Trivial heuristics can be applied on such datasets to derive personally identiﬁable information ofindividuals, even at aggregate levels [35].
Publicly accessible mobility datasets [39, 24, 14] are usually not adequate for large scale experimentalevaluations, compromising scalability tests.
This issue incentivizes synthetic mobility trajectorygenerators that simulate the behavior of moving objects required to attain comprehensive performance valuations.
In this context, one typically considers rigid and unnatural mobility models, notguaranteeing the existence or even cardinality of patterns within the synthetic population.
Alternativeapproaches rely on parametric sequential models [16] and Markov processes [3] to learn and generatetrajectories.
Such techniques also ignore the presence of long-range dependencies[20] inherent tohuman mobility which features non-Markovian character [38, 17].
It is therefore imperative to generate context-dependent synthetic traces resembling the humanmobility behavior at satisfactory utility levels while preserving user privacy.
However, one of the
[Table]
major challenges is the absence of quantitative methods for evaluating the realistic nature of synthetictraces and the associated utility-privacy tradeoff.
To this end, we present several nonparametric approaches to generate large-scale synthetic trajectoriesby training the models on a real-world dataset followed by hallucinating trajectories using the trainedmodel.
We perform an extensive evaluation of the generated trajectories by assessing their geographicand semantic similarity compared to the actual dataset.
We use two sample metrics to obtain thestatistical similarity between datasets.
We then quantify the presence of long-range dependenciesby computing the mutual-information decay and conduct privacy-leakage tests on the generatedtrajectories.
We conclude with a discussion on appropriate strategies and applicable evaluationmetrics based on our experimental results and tackle open questions and challenges.
Related Work
Table 1 provides a summary of existing trajectory generators, where they formulate the synthetictrajectory simulation as an optimization problem, solved by genetic algorithms under the constraintof a priori determined parameters.
A fundamental issue is the selection and deﬁnition of theparameter space that controls the evolution of the moving objects.
The stringent and classiﬁednetwork connections thus inﬂuence the realistic nature of the generated trajectories.
In several cases,there is no correlation between the future direction of movement and the past locations.
Repeatedvisits to a given location within a short span of time are also observed due to the bounding parameters.
Therefore, the symbolic nature of these frameworks result in an implicit location-dependent context,which compromises the realistic nature of the generated activity patterns.
To address these drawbacksassociated with parametric modeling, Ouyang et al.
[28] propose a GAN-based approach to generatetrajectories, where the discriminator is based on a convolutional neural network (CNN) [19].
Similarly,we explore other deep learning architectures based on RNNs known to model sequential data betterthan CNNs [31].
We also investigate generative models based on the nonparametric copulas of [7].
Synthesizing Trajectories using Generative Modeling
First we explore the beneﬁts of applying deep learning architectures to synthesize mobility trajectories.
RNNs use their hidden memory representation to process input sequences and we select fourarchitectures: (1) Char-RNN (SRNN)
[11], (2) RNN-LSTM [12], (3) recurrent highway networks(RHN)
[40], and (4) pointer sentinel mixture model (PSMM)
For GANs, where two neuralnetworks compete in a zero-sum game framework, we select two architectures: (1) SGAN [37],and (2) RGAN [6].
These architectures differ in their capacity to manipulate their internal memoryrepresentation and propagate gradients along the network.
In addition to neural-network basedsolutions, we also evaluate copulas ; a seldom explored generative model in the machine learningcommunity.
Given a bivariate random vector ( X 1 , X 2 ) , Sklar’s theorem [33] states that the jointdensity 1 is f ( x 1 , x 2 ) =
f 1 ( x 1 ) f 2 ( x 2 ) c ( F 1 ( x 1 ) , F 2 ( x 2 )) , where f and f i are the marginal densities,
F i the marginal distributions, and c the copula density.
In other words, the bivariate density can beuniquely described by the product between its marginal densities and a copula density representing its
[Figure]
dependence structure.
A useful consequence of this representation is that, by taking the logarithm onboth sides, estimation of the joint density can be performed in two steps: the marginal distributionsﬁrst, and the copula afterwards.
In a nutshell, copulas allow to ﬂexibly specify the marginal and jointbehavior of random variables.
An important aspect in the generative context is that, because U = F ( X ) ∼ U (0 , 1) for anycontinuous random variable with distribution F , the copula is a distribution with uniform margins.
Hence, from a copula sample ( U 1 , U 2 ) , one obtains a sample on the original scale using the inversecumulative distributions via ( X 1 , X 2 ) =
( F − 1 1 ( U 1 ) , F − 1 2 ( U 2 )) .
For further details on copulas, werefer the reader to [13].
In this paper, we combine the kernel-based nonparametric copulas of [7] withkernel-based estimators for the margins to obtain highly ﬂexible models.
Data representation Given a dataset of n mobility trajectories, where a trajectory T u of an individualu is a temporally ordered sequence of tuples, such that, T u = (cid:104) ( l 1 , t 1 ) , ( l 2 , t 2 ) ...
( l n , t n ) (cid:105) , wherel i =
( lat i , lon i ) , 0 ≤ i ≤ n , the latitude-longitude coordinate pair and t , the timestamp such thatt
i +1 >
t i .
We ﬁrst transform the location data onto a uniform grid for dimensionality reductionusing a technique that preserves spatial locality 2 , thus translating T u into a 2-D trace S ( t ) =(cid:104)
( c 1 , t 1 ) , ( c 2 , t 2 ) ...
( c n , t n ) (cid:105) , where c i is the geo-hash of the projected cell ID and the timestamp t i .
Experiments, Results and Discussion
Experimental setup A complete trajectory sequence can be generated by iteratively feeding thecurrent output trajectory sequence as input for the next step to the trained model.
RNNs are trainedon the geo-hashes and timestamps of all the individuals present in the dataset in a deterministicframework.
GANs are ﬁrst trained to model and then successively reproduce the traces in thesame representation, which is mapped back to the ( lat i , lon i ) coordinates.
We use the standardimplementations of the predictive algorithms and hyper-parameters as described in their respectivepapers.
To use copulas as generative models, we rely on the rvinecopulib package [25], whose vineroutine implements the automatic kernel-based ﬁtting of the marginal distributions and dependencestructure.
Dataset Experiments are performed using the Nokia mobile dataset [18] that consists of mobilitytrajectories of individuals collected in Switzerland.
We use a total of 71,000 data points to train theconsidered models.
Evaluation We perform the evaluation of the generated trajectories using this dataset from fourdistinct dimensions: (1) geographic and semantic similarity, (2) statistical similarity (3) long-rangedependencies and (4) privacy tests.
In order to assess the geographic and semantic similarity, wecompare the probability distribution of visiting topN locations (visit-time and dwell-time) in thegenerated trajectories for each technique compared to the true dataset (see Figure 1).
Char-RNN,
RGAN and copulas have the closest ﬁt to the true distribution indicating that the topN locations arevery well preserved in the respective synthetic datasets.
To evaluate the statistical similarity, we use Mean Maximum Discrepancy (MMD) [10] to test whetherone can reject the null hypothesis that a synthetic sample has the same distribution as the data.
MMDworks by replacing the probability densities with embeddings that facilitate the computation ofdistances between distributions.
Note that deﬁning distance metrics in the context of time seriesdata such as mobility trajectories is challenging due to the alignment concerns [6].
We thus considerthe time axis for alignment as done by Esteban et al.
The results along with the training andgeneration time for each approach is shown in Table 2, where we observe that all approaches achievesimilar results in terms of MMD, with copulas standing out with a lower value. We can thus infer
[Figure]
that copulas can synthesize distributions with statistical characteristics closer to the observed ones.
Regarding the computational efﬁciency, copulas require a fraction of the time needed by NN-basedapproaches.
[Table]
Figure 2(a) shows the result of long-range dependency test, in terms of mutual information decay [21,20].
We observe a power-law decay in case of GANs, copulas and RNN-LSTM indicating that theyaccount for the long-range dependencies in mobility trajectories.
Figure 2(b) shows the results of twoprivacy tests: (1) location-sequence attack, and (2) membership interference attack.
Given a syntheticdataset, (1) answers to what level of accuracy can trajectories in the dataset be reconstructed [32], and(2) an adversary’s accuracy of inferring if a target individual contributed to the speciﬁc trajectory [30].
For these tests, we use the the location-privacy and mobility meter [32], where obfuscation isperformed using the location hiding mechanism.
Given a completely random distribution the accuracyof a recovered user-information is 0, we therefore suspect that the privacy-based score is biasedtowards representations which do not accurately capture the statistical properties of the true dataset.
Conclusion and Future Work
In this work, we propose and evaluate a variety of generative models to synthesize mobility trajectories.
To the best of our knowledge, this is the ﬁrst study to do so using seven different approaches whileevaluating their realism across four dimensions.
From the results and discussion, we observe thatregarding statistical and semantic properties, copulas have an advantage over all other methods.
Additionally, all NN-based methods are time consuming, which makes copulas favorable whencomputational efﬁciency is important to the end-users.
As future work, we will consider datasetscollected in bigger cities and generate larger synthetic datasets to evaluate the performance of thesemodels under high movement stochasticity.
From curbing the privacy leakage of the true datasetwhile maintaining utility, trajectory generation can be designed as an optimization problem with anobjective to jointly maximize statistical similarity and privacy.
But it is still not clear how to assesssuch property and adaptive/conﬁgurable metric, and is part of ongoing work [27].
While this paperrepresents an initial comparative study of various generative models, a deeper understanding of theirperformances will be needed to compute utility-privacy scores as applied to online services by Krauseand Horvitz [15] before publicly releasing the synthetic datasets.
Another interesting avenue forresearch is to apply transfer learning in order to map a mobility behavioral model captured in one cityon to another region.