quantifying intrinsic value of information of trajectories
TRANSCRIPT
Quantifying Intrinsic Value of Information of TrajectoriesKien Nguyen
[email protected] of Southern CaliforniaLos Angeles, California, USA
John [email protected]
Microsoft ResearchRedmond, Washington, USA
Cyrus [email protected]
University of Southern CaliforniaLos Angeles, California, USA
ABSTRACTA trajectory, defined as a sequence of location measurements, con-tains valuable information about movements of an individual. Itsvalue of information (VOI) may change depending on the specificapplication. However, in a variety of applications, knowing theintrinsic VOI of a trajectory is important to guide other subsequenttasks or decisions. This work aims to find a principled frameworkto quantify the intrinsic VOI of trajectories from the ownerβs per-spective. This is a challenging problem because an appropriateframework needs to take into account various characteristics of thetrajectory, prior knowledge, and different types of trajectory degra-dation.We propose a framework based on information gain (IG) as aprincipled approach to solve this problem. Our IG framework trans-forms a trajectory with discrete-time measurements to a canonicalrepresentation, i.e., continuous in time with continuous mean andvariance estimates, and then quantifies the reduction of uncertaintyabout the locations of the owner over a period of time as the VOIof the trajectory. Qualitative and extensive quantitative evaluationshow that the IG framework is capable of effectively capturingimportant characteristics contributing to the VOI of trajectories.
CCS CONCEPTSβ’ Information systems β Geographic information systems;β’ Human-centered computingβ Ubiquitous computing.
KEYWORDSlocation measurements, trajectory, value of information, informa-tion gain, measurement uncertainty, location inference
ACM Reference Format:Kien Nguyen, John Krumm, and Cyrus Shahabi. 2021. Quantifying IntrinsicValue of Information of Trajectories. In 29th International Conference onAdvances in Geographic Information Systems (SIGSPATIAL β21), November2β5, 2021, Beijing, China. ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/3474717.3483912
1 INTRODUCTIONThe availability of mobile devices with location-tracking capabilityhas enabled individuals to generate a great amount of location datafrom various types of signals, e.g., GPS, Wi-Fi, or cell service. A
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trajectory, which is a sequence of location measurements, is animportant type of location data that contains valuable informationabout the ownerβs locations and movements. For example, a trajec-tory may indicate the ownerβs moving behavior [20], e.g., a trip tohome, office, or favorite shops and time spent there; it may helpidentify irregularities such as vacation days or house moving; it isimportant for traffic, speed, and route inference [21].
Each trajectory gives information of some kind. The value of theinformation (VOI) varies depending on the use and user of the data.For example, the same trajectory may have one value for the ownerand another for an enterprise, and that same trajectory may alsohave different values for different enterprises, such as a public orprivate, especially for an ad-targeting company.
However, a trajectory also contains an intrinsic, formulaic VOI,since it contains quantified locations of an individual over spaceand time. For example, it is reasonable to assume that a home-to-office trajectory with 1000 measurements from beginning to endof the trip tends to have more information than another home-to-office trajectory with only 2 measurements at home and office.Our goal is to find a principled approach to quantify the intrinsicVOI of trajectories from the ownerβs perspective. To the best of ourknowledge, this is the first attempt at such a quantification.
Given a trajectory, there are natural questions about sellingit [10], sharing it, using it to train machine learning, storing it, orexamining it more closely. Quantifying its VOI helps measure howvaluable, revealing, informative [19], distinct, and surprising [21]it is. A trajectoryβs VOI can be a critical piece of metadata thatindicates its intrinsic value for a variety of tasks, both stand-aloneand as part of a collection.
This problem presents several interesting challenges:β’ First, a trajectory has many characteristics contributing to itsVOI, and effectively capturing their complex relationships isnon-trivial. Examples of these characteristics are the numberofmeasurements, temporal duration, and howmeasurementsdistribute spatially and temporally.
β’ Second, a trajectory can be degraded for different purposes,e.g., measurements can be perturbed by adding random noiseor completely removed to enhance the ownerβs privacy be-fore releasing or selling. It is challenging to capture the effectof different types of degradation on the VOI of the trajectory.
β’ Third, the owner may assume different prior knowledgeabout a trajectory, which may change its VOI from theownerβs perspective. For example, if the owner has neverreleased any trajectory data before, meaning their trajectorywould give more information about their locations than ifthey already released a perturbed version of the trajectory.
Some straightforward methods, e.g., using a trajectoryβs charac-teristics such as size or duration, fail to capture other characteris-tics or prior knowledge. Previous work based on Spatial Privacy
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Pricing [12] can be adapted to sum the values of degraded mea-surements of the trajectory. However, it also fails to capture somecharacteristics (e.g, how measurements distribute over space) andprior knowledge, because it ignores the mutual information of theconstituent points. The notion of correctness [17] also appearspromising. For this method, based on a degraded version π of atrajectory π and prior knowledge, a probabilistic prediction is madefor each measurement. Then the correctness of the prediction, indi-cating how close it is to the actual measurement, is aggregated toderive the VOI of π . However, this method is not applicable whenπ = π , because correctness is not available when there is no actualmeasurement with which to compare the prediction.
The core idea behind our approach is that instead of computingthe VOI of a trajectory from its discrete location measurements overtime, we view VOI as how much the trajectory data helps recon-struct locations of the owner. More specifically, the VOI correspondsto how much the trajectory data helps reduce the uncertainty ofestimating its ownerβs locations in continuous time. We use infor-mation gain (IG) to quantify that reduction. Even though IG is aknown measure [14], it is not obvious how it should be used tocompare discrete-time trajectories with widely different charac-teristics: long and short, dense and sparse, clean and noisy. Ourinnovation here is that we transform each trajectory to a canonicalrepresentation, i.e., continuous in time with continuous estimatesof mean and variance, which is necessary for comparing widelydifferent trajectories and comparing new data against prior data. IGcan only then be applied after all those steps are taken. We realizethis transformation by employing a reconstruction method thatcan produce continuous-time probabilistic predictions along thetrajectory.
Consequently, we propose an IG framework as a principled wayto quantify the intrinsic VOI of a trajectory. The main idea is toquantify the reduction of uncertainty about the ownerβs continuous-time locations, comparing a new trajectory to a previously releaseddegraded version or to prior information. Thus the IG frameworkutilizes a reconstruction method to produce continuous probabilis-tic predictions over time, and then it calculates the reduction ofuncertainty from those predictions compared to a reconstructionfrom prior knowledge. The uncertainty is measured by (differential)entropy. Our IG framework accepts any reasonable probabilisticreconstruction method. Gaussian process is used as the reconstruc-tion method in this paper due to its popularity and flexibility, butis not necessarily the only choice.
Specifically, our contributions are:β’ Propose the problem of quantifying intrinsic VOI of a trajec-tory from the ownerβs perspective
β’ Define characteristics that should be captured by an appro-priate quantification method
β’ Show how alternate methods fail, even when they appearreasonable initially
β’ Introduce a method to transform each trajectory to a canon-ical representation, allowing us to examine the VOI with acontinuous location reconstruction and to compare trajecto-ries with widely different characteristics
β’ Develop an IG framework over transformed trajectories as aprincipled approach capable of effectively capturing varioustrajectory characteristics, prior knowledge, and degradation
β’ Evaluate the proposed framework both qualitatively andquantitatively with extensive experiments on a large, real-world trajectory dataset
By providing a standard, comprehensive method for assessing theVOI of a trajectory, we enable a deeper understanding of trajectoriesand their utility for a wide variety of purposes.
2 PROBLEM SETTINGThis section introduces the problem of quantifying the intrinsicVOI of a trajectory from the ownerβs perspective. We focus on themost basic form of a trajectory, which is a sequence of locationmeasurements. Therefore, incorporating other information, such ascensus data or points of interest, is beyond the scope of this work.
The owner can be anyone having access and right to use the data,e.g., one whose phone recorded this trajectory, or a data collectorwho aggregates location data from individuals. Without loss ofgenerality and to ease the discussion, from here on, we assume theowner is the individual whose device recorded the trajectory.
A trajectory π is a sequence of (potentially noisy) location mea-surements π = π1, π2, . . . , π |π |. Each measurement or data pointπ (bold symbol) is a tuple π =< lon, lat, t, π > where π .lon and π .latare the longitude and latitude, π .t is the timestamp, and π .π is theaccuracy or uncertainty of the measurement. It is reasonable toassume Gaussian noise for location measurements [5] such as GPSpoints, so π .π can be considered as the standard deviation of inde-pendent Gaussian noise of longitude and latitude. Measurementsin π are ordered by their timestamps, i.e, βππ , π π β π, 1 β€ π β€ π β€|π |, ππ .t β€ π π .t.
The owner can assume the potential recipient has some priorknowledge Ξ© about the location ππ , e.g., by gathering public dataor from some data already released from the owner before. Thisprior knowledge is represented as a prior distribution π (ππ |Ξ©) andshould be respected by the proposed methods. Intuitively, priorknowledge about a point would tend to reduce the VOI of the point.
In addition to releasing raw trajectories, owners can also offertheir data at different quality levels, potentially enhancing theirprivacy while reducing the trajectoryβs VOI, e.g, a trajectory canbe degraded by adding noise or subsampling. More details aboutdegradation are discussed in Section 2.3. The methods to quantifythe VOI also need to reflect this quality degradation.
Our goal is to find a principled approach to quantify the VOI of atrajectory from the ownerβs perspective. It should capture differentcharacteristics of the raw trajectory, the prior knowledge, and theeffect of various degradation processes. Next, we described the char-acteristics that an appropriate method should aim to capture. Thisis an extensive but not exhaustive list of desirable characteristics,thus we expect future work will study more characteristics. We willthen illustrate how several baseline methods fail to capture thesecharacteristics and how our new proposed framework succeeds.
2.1 Trajectory CharacteristicsFrom the raw trajectory π , certain characteristics can be derived thatshould contribute to its VOI. These includes size, duration, spatialdistribution, temporal distribution, and measurement uncertainty.
Size. The size |π | is the number of measurements of π . A largersize often means there is more data to learn about the ownersβ
Quantifying Intrinsic Value of Information of Trajectories SIGSPATIAL β21, November 2β5, 2021, Beijing, China
locations and movements. Therefore, it is reasonable to assume thata trajectory with a larger size tends to have more information.
Duration. The duration of a trajectory π is the time differencebetween the first and last measurements of π , i.e, π· (π) = π |π | .π‘ βπ1 .π‘ . Similar to the size, a trajectory with a longer duration oftenmeans it may have more information about the ownerβs locationsthan one with a shorter duration.
Spatial distribution. The spatial distribution of π indicateshow its measurements are distributed over space. For example, onetrajectory may have all measurements at one place, while anothertrajectory may have measurements at multiple places (e.g, solelyat home vs. at home, then at a coffee shop, then at the office). Atrajectory visiting more places tends to give more informationabout locations of the owner. The spatial distribution of π canbe measured using spatial entropy π»π (π) over a grid of cells, say10π Γ 10π, covering the area of interest. π»π (π) is calculated bycreating a histogram of number of measurements of π belongingto each cell, converting this histogram to probabilities, and thencomputing the Shannon entropy for the probabilities.
Temporal distribution. Similarly, the temporal distribution ofa trajectory π indicates how its measurements are distributed overtime. For example, one trajectory may have many measurementsin the first few minutes and then a long gap before the next one.Another trajectory may be distributed more uniformly over time,e.g., one measurement every 30 seconds. Intuitively, when measure-ments distribute more evenly over a longer period of time, theytend to give more information. The temporal distribution of π canbe measured using the temporal entropy π»π (π) calculated during[π1 .t, π |π | .t] by computing a histogram of the number of measure-ments of π belonging to each temporal bin, say every 1 minute,from π1 .t to π |π | .t, converting the histogram to probabilities, andthen computing the Shannon entropy for the probabilities.
Measurement uncertainty.Measurements may have their ownuncertainty depending on how they were taken, e.g, an uncertaintyof several hundred meters with cell towers [2] or 1 to 5 meters forGPS-enabled smartphones [13]. Since Gaussian noise is a reasonableassumption for GPS [5], measurement uncertainty is defined as thestandard deviation π of independent Gaussian noise of longitudeand latitude. In general, less accurate measurements (i.e, larger π)tend to give less information about locations of the owner.
2.2 Prior KnowledgeAs mentioned, the owner can assume there exists some availableprior knowledge about the location ππ , e.g., from public data or fromprevious releases from this owner. This knowledge is expressedas a prior distribution π (ππ |Ξ©) for each timestamp of interest. Ingeneral, better prior knowledge often means the trajectory givesless information compared to the case with poorer prior knowledge.
2.3 Trajectory DegradationThe owner can offer their trajectories at lower quality. The processof lowering the quality of a trajectory is called degradation. The rea-son for the owner to produce degraded trajectories is that a recipientcan still benefit from data at a certain quality depending on theirspecific applications [12]. For example, estimating a neighborhood-level origin-destination matrix may not need extremely accurate
measurements, thus the recipient with this application can poten-tially spend less to purchase lower quality trajectories. In general, alower quality trajectory would give less information than a higherquality one. A degraded version of a trajectory π is denoted as π .
In this work, we consider three types of degradation: perturbation,truncation, and subsampling. Other types of degradations and othervariations of these degradations are beyond the scope of this paperand considered as part of future work.
Perturbation. The perturbation process degrades a trajectoryπ by adding independent random Gaussian noise with standarddeviation ππ§ to each ππ β π . Formally, a perturbed trajectory π of πconsists of measurements π§π s.t.
π§π .lon = ππ .lon + [1 (1)π§π .lat = ππ .lat + [2 (2)π§π .t = ππ .t (3)
π§π .π =
β(ππ .π)2 + π2π§ (4)
[1, [2 βΌ N(0, π2π§ ) (5)
Noise magnitude π§π .π is called the total noise of π .Truncation. The truncation process degrades a trajectory π by
truncating π and only keeping a fraction πΌπ‘ β [0, 1] of the firstmeasurements of π . For example, if |π | = 100, an πΌπ‘ = 0.2 meansthat the temporally first 20% of measurements are kept, which areπ1, . . . , π20. The fraction πΌπ‘ is called the truncation ratio. Theretained measurements are kept in their raw form. Also, there is atleast one measurement retained.
Subsampling. Similarly, the subsampling process degrades atrajectory π by uniformly subsampling π with probablity πΌπ β [0, 1],called the subsampling ratio. More specifically, a measurement ππ βπ is retained with probability πΌπ ; otherwise, ππ is discarded. Hence,in expectation, a fraction πΌπ of measurements of π are retained. Theretained measurements are also kept in their raw form and at leastone measurement is retained.
3 BASELINESThis section discusses baseline techniques to quantify the VOI of atrajectory. Each baseline is described along with which desirablecharacteristics from Section 2 they can capture. Table 1 summarizeswhich characteristics the baselines and our proposed method canfaithfully represent. Our proposed method, based on informationgain and shown in the last row of Table 1, is described in detail inSection 4. A cross ( )/exclamation mark ( )/checkmark ( ) indi-cates that the method in that row cannot/can partially/can fullycapture the characteristic in that column, respectively.
The following analysis shows how the baseline methods, whileappearing initially reasonable, fail to represent some importantcharacteristics of the VOI. For each characteristic, a method is firstevaluated qualitatively. In some cases, it is obvious that a methodcan or cannot capture a characteristic, e.g., a size-based methodcan capture the size but not prior knowledge. In other cases, it maynot be as clear. In those cases, the method is said to be capable ofcapturing the characteristic if there is a strong correlation betweenthe output of the method and the characteristic. The correlation isexamined using Spearmanβs rank correlation coefficient π , whichquantifies strictly monotonic relationships between two variables
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Size Duration SpatialDistribution
TemporalDistribution
MeasurementUncertainty
PriorKnowledge Degradation
Fixed ValueSize-basedDuration-basedTravel Distance
Entropy-based * *SPP-based [12]
Correctness-based [17] * *Information Gain
Table 1: Potential methods to quantify the intrinsic VOI of a trajectory and the characteristics they can or cannot capture. Ourproposed method based on information gain can capture almost all of the desirable characteristics.
and is relatively robust against outliers [16]. An absolute magnitude|π | in [0.4, 0.7) and [0.7, 1] indicates moderate and strong correla-tion, respectively. The cutoff points are based on previous work [16].The values of π are calculated from a large real-world trajectorydataset described in Section 5.1.
3.1 Fixed Value MethodOne potential method is to set the same value for all trajectories.While this method is straightforward and, in fact, was used in somesurveys about values of location data from a sellerβs perspective [4,18], this method ignores the fact that each trajectory may contain adifferent VOI. For example, a 40-mile long commute from home towork can be very different from a short trip to a nearby store. Hence,this method does not capture any of the desirable characteristics.
3.2 Size-based MethodThe size-based method uses the size |π | of a trajectory π to quantifythe VOI of π . It is reasonable to say that a trajectory with moremeasurements tends to have more information. While this methodmay distinguish trajectories with many or few measurements, thesize alone would fail to fully capture the VOI of a trajectory, becausethe size |π | depends heavily on the sampling rate and the duration.For example, the same trip from home to office if sampled every1 second would have a size 5 times larger than if sampled every5 seconds, while having roughly similar information about thelocations of the person along the trip.
Consequently, the size-based method successfully captures thesize characteristic of a trajectory, but not other aspects. For theduration characteristic, because it only depends on the first andlast measurements of the trajectory, the size cannot fully capture it,e.g., a trajectory with two measurements can be arbitrarily shortor long. However, the size can partially capture the duration if thesampling rate is relatively similar among trajectories. The correla-tion coefficient π between size and duration of trajectories in ourdataset is 0.86, which indicates a strong correlation between them.
The size |π | also does not capture the spatial nor temporal dis-tribution of π , because it does not indicate how the measurementsare distributed. The same number of measurements can happenat nearly the same place/time if the sampling rate is high, or at
different places/times if the sampling rate is low. It is also clear that|π | does not indicate measurement uncertainty nor prior knowledge.In fact, any method using solely trajectory characteristics wouldfail to capture prior knowledge because prior knowledge is notconsidered in that method. The size can capture some degradationsuch as truncation, but not other degradation such as perturbation.
3.3 Duration-based MethodAnother method is to use the duration π· (π) of π , i.e, π· (π) =
π |π | .π‘ β π1 .π‘ , to quantify its VOI, as it is reasonable to assumethat a temporally longer trajectory tends to have more information.However, using only duration would fail to fully capture the VOIsince it ignores all information between the start and end points.
Similar to the size-based method explained before, this methodcan fully capture the duration but can only partially capture the size.The duration also does not represent how the measurements distrib-ute over space and time, and is unable to take into account measure-ment uncertainty and prior knowledge. Duration can capture theeffect of truncation but not that of perturbation nor subsampling,thus, only partially capturing the effects of degradation.
3.4 Travel Distance MethodThis method computes the travel distance πΏ(π) of π by summingthe distances between each pair of consecutive measurements. It isreasonable to assume that a trajectory with longer distance tendsto have more information than a shorter one.
With the same mode of transportation (e.g., with bikes or cars),travel distance likely reflects the duration. Thus, travel distancecan partially capture duration and the size of π . With a longerdistance, the trajectory tends to go through more places and timeperiod. In our dataset, a strong correlation 0.89 and 0.78 between thetravel distance and spatial and temporal entropy indicate that traveldistance can partially capture the spatial and temporal distribution.
However, using travel distance would fail take into account mea-surement uncertainty or any prior knowledge. It can capture somedegradation, e.g., truncation, but not others, thus, only partiallycapturing the effect of degradation.
Quantifying Intrinsic Value of Information of Trajectories SIGSPATIAL β21, November 2β5, 2021, Beijing, China
3.5 Entropy-based MethodThe spatial and temporal entropy π»π (π) or π»π (π) of π , computedfrom the Shannon entropy of the probabilities converted from thehistogram of number of measurements of π belonging to each gridcell or temporal bin, can be good candidates to quantify the VOIgiven its extensive applications in information theory. A higherspatial/temporal entropy likely indicates that the trajectory givesmore information about a personβs locations over space/time.
Sinceπ»π (π) orπ»π (π) are used to measure the spatial and tempo-ral distribution of π , respectively, a method combining both π»π (π)or π»π (π) can capture these two characteristics. However, it is un-clear how they should be combined, which is why there are asterisksfor the entropy-based row in Table 1.
Whileπ»π (π) andπ»π (π) do not fully capture the size and duration(e.g., a trajectory with only two points but far apart from each othercan have a long duration but low entropy values), when π has arelatively high sampling rate, π»π (π) would be highly correlatedwith the size and duration. In our dataset, the correlation coefficientsbetweenπ»π (π) and size and duration are 0.89 and 0.97, respectively,which indicate a very strong correlation.
However, since entropy reflects the uncertainty, when the mea-surements have higher uncertainty (i.e., larger π .π), both π»π (π)and π»π (π) tend to increase. This is opposite of what one mightexpect, because a more uncertain measurement means less informa-tion. Thus, entropy does not capture measurement uncertainty nordegradation such as perturbation. Computing π»π (π) and π»π (π)also ignores all prior knowledge. Another issue is that the entropycan change significantly when arbitrary parameters for computingentropy (i.e., size of grid cells or length of time bins) change.
3.6 Spatial Privacy Pricing-based MethodThis method is based on the previous work on Spatial Privacy Pric-ing [12] (SPP) where each measurement has the same value definedby the owner, and the value can be reduced when the measurementis perturbed by noise. The VOI of the trajectory is then calculatedby summing up the values of each individual measurement.
With the ability to change the value based on the noise in themeasurements, this method can capture the measurement uncer-tainty and represent degradation. Summing the individual valuesmeans this method is similar to the size-based method, but is alsosensitive to the perturbation parameters. Therefore, this methodhas similar characteristics as the size-based method, which meansfully capturing size, partially capturing duration, and unable tocapture spatial and temporal distributions and prior knowledge.
3.7 Correctness-based MethodThis method is based on the correctness of reconstructing the rawmeasurements from available data [17]. Roughly speaking, whencalculating the VOI of a degraded version π of π , the owner canassume the recipient is attempting to reconstruct eachmeasurementππ β π based on π . The correctness is the expected error betweenthe actual points ππ and the probabilistically reconstructed pointsπ . If the expected error is lower, then it would be reasonable toassume a higher VOI of π .
While this concept was proposed for a different problem setting,some techniques can be used to adapt it to our problem. In fact,
our proposed framework, discussed later, also has a reconstructionstep that can capture prior knowledge and degradation. Thus acorrectness-based method can capture the prior knowledge anddegradation. The asterisks in the correctness-based row in Table 1indicate that a correctness-based method needs some modificationsfor the problem setting and techniques to be fit for our problem.
The main drawback of using correctness is that it requires someground-truth measurements of π being available to evaluate thecorrectness of the reconstructed trajectory made from the degradedversion π and prior knowledge. So, it cannot measure the VOI ofthe full, raw trajectory π , because there is no ground-truth measure-ment exists to evaluate correctness in that case. Consequently, sincethe full, raw trajectory cannot be fully captured, which means thereare often some characteristics not fully captured (e.g., subsamplingchanges the size and truncation changes the duration), this methodcan only partially capture characteristics of a trajectory.
Another issue is that this method relies on the correctness ofdiscrete-time predictions, and it is unclear how the correctness ofeach prediction should be aggregated to obtain the correctness ofthe whole trajectory. For example, the correctness derived froman 80% subsampled version is evaluated on each measurement ofthe remaining 20% data, while the correctness derived from a 60%subsampled version is evaluated on the remaining 40%. It is unclearhow the correctness should bemodified to reasonably quantify both,and how the correctness of each prediction should be aggregated.
3.8 Other Potential QuantitiesThere are other potential quantities contributing to the VOI of atrajectory. However, these are not considered as baselines becausethey are either orthogonal (i.e., they can be used in conjunctionwith other methods) or complicated (i.e., finding these quantitiesrequires techniques beyond the scope of this work). Examples aretime period, subjective sensitivity, and visits.
Time period. The time period that a trajectory π was taken,e.g.,weekdays or weekend, can be a factor contributing to its VOI. Forexample, a person may often have commute-related trajectoriesduring weekdays but more leisure-related trajectories on weekends.While time period does not represent actual locations, it can be usedin conjunctionwith other methods, e.g., an owner can have differentvalues for trajectories during weekdays compared to weekendsbecause of privacy concerns.
Subjective sensitivity. Each person may have their own sensi-tivity for different types of locations, which may lead to differentsensitivity for different trajectories. For example, one may feel theirworkplace is more sensitive than their favorite coffee shop, thus hav-ing a higher sensitivity for the trajectory from home to work thanthe one to the coffee shop. While this information can contributeto the VOI of a trajectory, incorporating it requires additional in-formation about the location measurements, which is not the focusof this work. When this information is available, it can be usedin conjunction with the proposed methods in this work to betterquantify the VOI tailored to the ownerβs subjective reasoning.
Visits. This method bases on the number of visits of a trajectoryto quantify the VOI. Its main drawback is how to define a visit. Itis often not feasible for the owner to manually define all visits forall of their trajectories. On the other hand, complicated techniques
SIGSPATIAL β21, November 2β5, 2021, Beijing, China Nguyen, et al.
to automatically find visits [20] are often used to segment a longsequence of measurements into trajectories, which is beyond thescope of this work. These techniques also often require additionalinformation and/or a complex set of parameters where a smallchange of some parameters may result in a significantly differentnumber of visits, which is not desirable.
4 THE INFORMATION GAIN FRAMEWORKThis section describes the proposed framework to quantify the VOIof a trajectory. The framework is based on the notion of informa-tion gain (IG) which quantifies the reduction of uncertainty whennew information is available. Adopting IG to trajectories is notstraightforward, and we enable it by transforming each trajectoryto a canonical representation. This transformation is done by em-ploying a reconstruction method that can produce continuous-timeprobabilistic predictions along the trajectory. The Gaussian pro-cess (GP) is used in this work as the reconstruction method anddiscussed in the next section, but we emphasize that our IG frame-work accepts any reasonable method of producing probabilisticlocation inferences.
4.1 Trajectory Information GainWe define information gain IGπ (π,Ξ©) of a degraded version π of atrajectory π as the total reduction of uncertainty about locations ofthe owner over time period π compared to the uncertainty fromthe prior knowledge Ξ©. For a typical human trajectory, there arepotentially several reasonable choices for π . In this work, π isdefined as the entire day covering the trajectory, because a typicaltrajectory would not extend beyond a day. We propose IGπ (π,Ξ©) toquantify the intrinsic VOI of π and/or its degraded version π . Thissection describes how IGπ (π,Ξ©) is derived. In Section 5, we willexplain how IGπ (π,Ξ©) satisfies almost all of the criteria in Table 1,making it a better choice than the previous methods we described.
Recall that the owner can assume that the recipient obtainedsome prior knowledge Ξ©, e.g., from public data or from previousnoisier release of the same trajectory π . From Ξ©, the owner canassume that the recipient can derive a prior probability distributionπ (π |Ξ©) for the location of the owner at a specific time π .t.
The owner can then assume that after receiving π , the recipientcan use a model to reconstruct (or predict/interpolate) locations ofthe owner in continuous time. This assumption is made, becausewithout knowledge of the recipientβs intent, the owner should actconservatively and assume the recipient will exploit the new datafully, such as with a maximally accurate reconstruction. Such aninference can be represented as a posterior distribution π (π |π,Ξ©).
Subsequently, the information gain IGt (π,Ξ©) at timestamp t =π .t indicates the reduction of uncertainty from π (π |Ξ©) to π (π |π,Ξ©).The uncertainty is measured by differential entropy [3] (or contin-uous entropy), because both the prior and posterior distributionsare likely continuous distributions in space.
The differential entropy β(π ) of a random variable π with prob-ability density function π whose support is a set X is defined as
β(π ) = ββ«Xπ (π₯) log π (π₯)ππ₯ (6)
Several popular probability distributions have a closed-form expres-sion for their differential entropy, e.g, if a π is a Gaussian random
variable with distribution N(`, π2), its differential entropy is
β(π ) = 12log 2πππ2 (7)
The entropy values of latitude and longitude of π are calculated sep-arately using Equation 6, and summed to get β(π |Ξ©) and β(π |π,Ξ©).
Thus the IG at time π‘ , IGt (π,Ξ©), can be calculated as
IGt (π,Ξ©) = β(π |Ξ©) β β(π |π,Ξ©) (8)
Figure 1a illustrates IGt (π,Ξ©) where the black dot shows theprediction mean and the blue area shows the uncertainty as thecircle with a radius which is twice the standard deviation of thepredicted distribution π (π |Ξ©). Ifπ is available, the uncertainty is re-duced to the smaller blue circle on the right representing π (π |π,Ξ©).IGt (π,Ξ©) quantifies the reduction and is illustrated as the red ring.
(a) IGt (π,Ξ©) illustration
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(b) IGπ (π,Ξ©) illustration
Figure 1: Illustrations of information gain (a) at a singletimestamp and (b) over a time period. The reduced uncer-tainty is shown in the red area minus the blue area.
Note that π does not need to be in any trajectory, because theprobabilistic reconstruction is continuous in time. For example, πcan be in between two measurements ππ , ππ+1 β π , thus, knowingππ and ππ+1 would help reduce the uncertainty of where π can be.
Subsequently, the information gain IGπ (π,Ξ©) over a time periodπ can be calculated by integrating IGt (π,Ξ©) for all time t β π , i.e.,
IGπ (π,Ξ©) =β«tβπ
IGt (π,Ξ©)πt (9)
Figure 1b illustrates IGπ (π,Ξ©) computed over a time periodπ = [0, 10] with prior π (π |Ξ©) = N(0, 100) for each timestamp.Over this period, the red line shows the prediction mean of theprior distributions, and the red area shows the prior uncertainty.When π , shown as two black crosses, is available, the uncertainty isreduced to the blue area. The amount of such reduction is shown asthe red area minus the blue area and is quantified using IGπ (π,Ξ©).
4.2 Reconstruction MethodTo compute IGπ (π,Ξ©), a probabilistic reconstruction method isneeded to reconstruct locations of the owner given Ξ© and/or π . AGaussian process (GP) is used in this work as the reconstructionmethod because of its flexibility to incorporate different types ofinformation. However, we emphasize that reconstruction is not the
Quantifying Intrinsic Value of Information of Trajectories SIGSPATIAL β21, November 2β5, 2021, Beijing, China
focus of this paper. The IG framework accepts any reasonable prob-abilistic reconstruction method. Thus, we briefly discuss importantaspects of the GP. More details about GPs can be seen in [15].
For a scalar function π (π‘), a GP implies that any subset of pointssampled from π (π‘) is distributed according to a multidimensionalGaussian. Two independent GPs are created for longitude and lati-tude prediction. The input for a GP are pairs < π‘, π (π‘) > where π (π‘)is longitude or latitude of the measurement at time π‘ . The outputfor a set of timestamps T β are predictions π β (π‘) for each π‘ β T β.
A GP depends on a scalar covariance kernel/function π (π‘, π‘ β²)defining how much a measurement at time π‘ correlates with ameasurement at time π‘ β². In general, the correlation decreases to zeroas |π‘ β π‘ β² | gets larger. A good kernel can help incorporate differenttypes of information into the model, especially when kernels canbe combined together. In our implementation, two common kernelsare summed together: the main kernel is a MatΓ©rn kernel thatcaptures the relationship betweenmeasurements as well as the priorknowledge, and a white kernel to capture measurement uncertainty.
The formula of the MatΓ©rn kernel used in this work is
πMatΓ©rn (π‘, π‘ β²) = π2π
(1 +
β3π|π‘ β π‘ β² |
)exp
(ββ3π|π‘ β π‘ β² |
)(10)
Setting ππ to the standard deviation of the prior distribution canhelp capture the prior knowledge, so that when the model becomesmore uncertain, the standard deviation of the prediction will grad-ually reach this value. The length scale π is trained from data.
The formula of a white kernel is
ππ€βππ‘π (π‘, π‘ β²) =π2π if t = tβ0 otherwise
(11)
Setting ππ to π .π or π§.π helps capture the measurement uncertainty.The final kernel is
π (π‘, π‘ β²) = πMatΓ©rn (π‘, π‘ β²) + ππ€βππ‘π (π‘, π‘ β²) (12)
Finally, a GP also depends on amean functionπ(π‘) which definesthe expected mean values of the measurements. In our adaption, themean functionπ(π‘) of a GP is set to be the regression line obtainedby running a linear regression on the data in Ξ©.
5 EVALUATIONS OF THE IG FRAMEWORKThis section provides an evaluation of how the IG framework cancapture each characteristic from Table 1, both qualitatively andquantitatively. The quantitative evaluation is performed on theGeolife dataset, which is a large, real-world trajectory dataset.
5.1 DatasetThe Geolife dataset [22] is used for experiments to quantitativelyjustify claims in the paper. This is a trajectory dataset collected by182 people carrying GPS loggers and GPS-phones in the Beijingarea from April 2007 to August 2012. A trajectory is representedby a sequence of measurements containing the latitude, longitude,and timestamp, with a variety of sampling rates. With this large,real-world dataset, covering a large span in both space and time,different types of devices, and a variety of sampling rates, we expectit is representative of the observations in other real-world datasets.
Measurements in the Geolife dataset are filtered to retain onlyones within the Bejing area with longitude from 116.20 to 116.55
degrees and latitude from 39.80 to 40.06 degrees. This area coversalmost all measurements and, in total, more than 16 million mea-surements were retained from all 182 people. The latitude/longitudecoordinates in each measurement are converted to local Euclideancoordinates (π₯,π¦) in meters with the reference origin (πππ0, πππ‘0)arbitrarily chosen as the center of the aforementioned area. In lo-cal Euclidean coordinates, the area has a lower left coordinate of(β15000,β15000) and an upper right coordinate of (15000, 15000).
Measurements of each individual are separated into trajectoriesby iterating through the ordered measurements and creating a newtrajectorywhenever the time gap between the currentmeasurementand the next measurement is more than π seconds. The maximumtime gap π is arbitrarily chosen at 300 seconds or 5 minutes. Theactual value of π does not significantly change the experimentalresults. We also emphasize that finding trajectories is not the focusof this work. Thus, more sophisticated methods to find trajecto-ries from measurements can also be used as an alternative to thissegmentation approach. The final dataset has 45,831 trajectories.
5.2 Experiment SetupThe experiments were conducted on the aforementioned datasetwith various sets of parameters. The results, such as the correlationcoefficient or regression line, are computed from all trajectories.Several outliers are removed for visual presentation purpose but stillincluded in all computations. The default size of grid cells/temporalbins to calculate spatial/temporal entropy is 10 meters/one minute.
The raw measurements do not include uncertainty. However,because they were recorded by GPS-equipped devices and the noisefor GPS-equipped smartphones is about 3 meters [13], each mea-surement is assumed to have ππ .π = 3. For perturbation degra-dation, the noise is added such that the total noise (i.e., π§π .π) isin 3, 10, 100, 200, 300, 400 meters. The ratios for truncation andsubsampling degradation (i.e., πΌπ‘ and πΌπ ) are 0.8, 0.6, 0.4, 0.2, 0.05,which mean 80%, 60%, 40%, 20%, 5% of the raw trajectory. The sub-sampling is implemented so that the subsampled data with a higherratio is a superset of the subsampled data with a smaller ratio.
A measurement is assumed to be inside the Beijing area. Thischoice is reasonable for this dataset. However, it is not crucial to theframework where any reasonable prior knowledge can be used (e.g.,measurements can be anywhere on Earth) and would only changethe scale of IG. This prior knowledge is represented by a Gaussiandistribution with extremely high variance where the mean is at thecenter of the area (i.e., coordinates (0, 0)) and the standard deviationis π0 = 7500 meters (i.e., the distance from the center to an edge ofthe area is twice this standard deviation). This is an uninformativeprior and called the Gaussian prior, i.e., for each timestamp, thelocation is assumed to be N(0, π20 ) for each dimension.
It is also natural to consider any previously released data as priorknowledge when computing the VOI of the data the owner stillretains. So we also consider previous releases as priors. This alsodemonstrates an advantage of the IG framework, where any previ-ous release can be naturally considered when computing the VOI ofthe ownerβs remaining data. To illustrate these cases, depending onthe nature of a degradation, different informative priors based onprevious releases are considered. For perturbation, two additionalpriors are 400m noise and 300m noise priors, illustrating the cases
SIGSPATIAL β21, November 2β5, 2021, Beijing, China Nguyen, et al.
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Figure 2: An illustration of how IG can capture trajectory characteristics. The figures from left to right showing the effect ofmeasurement uncertainty, duration, size, temporal, and spatial distribution of a trajectory.
where the owner released their trajectories at those noise levels be-fore. For example, with the 400m noise prior, when computing theIG for a degraded versionπ100 with 100m noise of a trajectory π , theowner may consider the prior knowledge is the degraded versionπ400 with 400m noise of π . Thus, there are three priors for perturba-tion in total: the uninformative Gaussian prior and two informativepriors 400m noise and 300m noise. Similarly, there are three priorsfor truncation/subsampling: the uninformative Gaussian prior andtwo informative 5% trajectory and 20% trajectory priors, illustratingthe cases when the owner released trajectories with 0.05 or 0.2truncation/subsampling ratio before. Other prior knowledge, e.g.,previously purchased trajectories, can also be considered as priors,provided that they can be expressed as prior location distributions.
The GP is trained on the prior Ξ© (e.g., 300m prior), except whenthe prior is the uninformative Gaussian prior, when it is then trainedon the new degraded version π . The degraded version π and priorΞ© are combined and then provided to the GP. For perturbation,measurements π and Ξ© of each timestamp are combined usinginverse-variance weighting [7]. For truncation and subsampling,because the higher-ratio degraded version is always a supersetof a lower-ratio degraded version (e.g, π5% β π20% β π40%), thecombined data is π . This is a reasonable approach for GP. There canbe other methods to better combine a prior and new data, especiallyif another prediction model is used instead of GP; however, this isnot the focus of the paper.While we expect similar trends, especiallyfrom models similar to GP such as a Kalman filter [9] or a particlefiler [6], absolute values from other models can be slightly different.
For GP kernels, π0 = 7500 is provided to the MatΓ©rn kernel. Totalnoise π§.π is provided to the white kernel. Length scale π can take val-ues in [0.01, 10] and is trained separately for each degraded version.IG over the time period is computed using numerical integrationwith the trapezoid rule. Finally, logarithms are base 2.
5.3 IG Capturing Trajectory CharacteristicsFor qualitative evaluation, Figure 2 illustrates the IG for differentcases in one spatial dimension. Starting from left to right with atrajectory having two noisy measurements close to each other inFigure 2a, the IG increases as expected when the measurements areless noisy as in Figure 2b. When the duration increases, illustratedby two measurements being farther apart in Figure 2c, the IG alsoincreases because these two measurements can also help reduceuncertainty for the longer time in between these two measurements.
Whenmore measurements are available as in Figure 2d, these pointshelp reduce more uncertainty around them, thus increasing theIG further. The IG keeps increasing when the measurements aretemporally distributed more evenly as shown in Figure 2e because,for most of the duration along the trajectory, there are temporallynearby measurements, thus helping reduce uncertainty. Finally, inFigure 2f, the IG also slightly increases when measurements appearin between the start and end locations (shown on the vertical axis),thus making the movement less abrupt. However, if the locationsjump unpredictably, the uncertainty tends to increase comparedto, e.g., when the person stays at the same place, thus decreasingthe IG for that case. Therefore, the IG can only partially capturethe spatial distribution and can capture the size, duration, temporaldistribution, and measurement uncertainty, as reflected in Table 1.
For quantitative evaluation using the Geolife data, the correla-tion coefficients of the IG with size, duration, and temporal entropyare 0.85, 0.97 and 0.96, respectively, which show very strong cor-relation. The correlation coefficients of the IG and spatial entropyis 0.68 which is also close to a strong correlation. Figure 3 showsthe histograms of the IG and these characteristics on a log scaleof the number of trajectories, along with a linear regression lineindicating a clear trend between the IG and these characteristics.The regression uses Huber loss [8] to mitigate the effect of outliers.Quantitative evaluation shown in Figure 4 (explained in detail later)also shows that the IG can capture measurement uncertainty, be-cause when a trajectory has large measurement uncertainty (i.e.,less informative), the IG tends to decrease. Computing IG tookaround 2-4 seconds per trajectory on a single machine.
5.4 IG Capturing Prior Knowledge andDegradation
The impact of prior knowledge and degradation can be effectivelycaptured by the IG framework. In general, when a trajectory isdegraded more (e.g., larger total noise), the IG tends to decrease,matching the intuition that a lower quality trajectory gives lower in-formation. Similar trends show in both the absolute value IG(π,Ξ©)and the ratio IG(π,Ξ©)
IG(π,Ξ©) of the IG compared to the raw trajectory π .We first discuss the results for perturbation. Figure 4 shows the IG
value IG(π,Ξ©) and the IG ratio IG(π,Ξ©)IG(π,Ξ©) in percentage with different
perturbations and prior knowledge. As discussed in Section 5.2, thetotal noise ranges from 3 to 400 meters, and the prior knowledge
Quantifying Intrinsic Value of Information of Trajectories SIGSPATIAL β21, November 2β5, 2021, Beijing, China
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Figure 3: Histogram of IG and (a) size, (b) duration, (d) spatial, (c) temporal entropy, and their correlation coefficients.
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Figure 4: (a) Absolute IG and (b) percentage IG change com-pared to original trajectories for different values of totalnoise and different types of prior knowledge.
scenarios are Gaussion, 400m noise, and 300m noise priors. Eachbox plot in Figure 4a (or 4b) shows the IG value (or the IG ratio) ofπ with the total noise shown on the π₯ axis given particular priorknowledge. For example, the red, dotted box plot at total noise 100in Figure 4a shows the IG values of a degraded version π100 ofπ with total noise 100 meters, given that the prior knowledge isanother previously-released degraded version π400 of π with totalnoise 400meters. It is clear that when total noise becomes larger, i.e.,more degradation, the IG value and the IG ratio tend to decrease.
With a more informative prior, the absolute IG value tends to besmaller. For example, the 400m noise prior is a more informativeprior than the Gaussian prior, thus the absolute IG value of the sameπ with the 400-noise prior is smaller than with the Gaussian prior.This also supports intuition: if one already has good informationabout the location of a person, getting more data does not increasesuch information as much as when one only has little information.This is an important property that does not exist in other baselines.
Truncation and subsampling degradation also show similar ob-servations as those from perturbation. Figures 5 and 6 show theIG value and the percentage of IG change for truncation and sub-sampling with different prior knowledge, respectively. The trun-cation/subsampling ratios range from 0.8 to 0.05, which means80% down to 5% of the original trajectory is retained. Because ofthe 0.05 (or 1
20 ) ratio, these figures show the results from almost
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Figure 5: (a) Absolute IG and (b) percentage IG change com-pared to original trajectories for different truncation ratiosand types of prior knowledge.
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Figure 6: (a) Absolute IG and (b) percentage IG change com-pared to original trajectories for different subsampling ra-tios and types of prior knowledge.
40,000 trajectories that have at least 20 measurements. As discussedin Section 5.2, the prior knowledge scenarios are Gaussion prior,5% trajectory prior and 20% trajectory prior. The IG also tends todecrease when the trajectory is degraded more (i.e., lower ratio)and when the prior gets better (e.g, 5% vs. 20% priors).
The decrease in IG with truncation is much stronger than withsubsampling. The reason is that knowing a measurement not only
SIGSPATIAL β21, November 2β5, 2021, Beijing, China Nguyen, et al.
reduces the uncertainty at the time of the measurement but also forthe time around that measurement. Thus when measurements aremore evenly distributed, they help reduce more uncertainty, whichmeans getting higher IG, as illustrated in Section 5.2. With the sameratio, which means the same number of measurements are retained,a truncation is similar to the case when measurements are close toeach other (Figure 2d) while (uniform) subsampling is similar to thecase when measurements are more evenly distributed (Figure 2e).This property also highlights the difference and advantage of IG forquantifying the VOI of a trajectory compared to other baselines.
Another observation is that by effectively capturing the impact ofdegradation, one can find the equivalence classes of different typesof degradation for the same trajectory. For instance, given π , whichtruncation ratio produces the same VOI compared to perturbingπ at total 300 meter noise? Figure 7 shows an example of suchVOI equivalence for the first trajectory of the first owner in ourdataset. The intersection between the (interpolated) truncation andperturbation lines indicates that truncating this trajectory at 65%gives roughly similar VOI to perturbing it with total 130m noise.
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6 RELATEDWORKQuantifying the value of location data has been an active line ofresearch. There have been several surveys of how individuals valuetheir location data [4, 18]. Aly et. al [1] showed how the value of alocation data point can be quantified from the buyerβs perspectivein a geo-marketplace. Also in a geo-marketplace context, Nguyen et.al [12] proposed a framework allowing sellers to offer a location datapoint at different qualities for different prices. However, previouswork focused on the monetary value of location data. To the bestof our knowledge, this is the first work that attempts to quantifythe intrinsic VOI of a trajectory in their most basic form, which isa sequence of location measurements.
There is also extensive work on quantifying location privacy,which also attempted to reconstruct the locations of individuals.Krumm [11] surveyed a variety of computational location privacyschemes and emphasized the importance of finding a single quan-tifier for location privacy. Shokri et. al [17] proposed correctnessas the metric for quantifying location privacy. Location privacyand the VOI of a trajectory can be related, e.g., a trajectory withhigh VOI may reveal more about a person than one with lower VOI,
thus leaking privacy. However, location privacy does not exactlycorrespond to the intrinsic VOI of a trajectory: the former concernsmore about semantic locations and is often more subjective, whilethe latter concerns more about the location coordinates over time.Correctness and subjective sensitivity, which are related to locationprivacy, were also discussed in detail in Section 3.7.
7 CONCLUSION AND FUTUREWORKThe intrinsic VOI of trajectories plays an important role in manyapplications. We proposed the IG framework and qualitatively andquantitatively demonstrated its capability to effectively captureimportant characteristics of raw trajectories, prior knowledge, andvarious types of degradation. This shows that the IG is an appropri-ate framework to quantify the intrinsic VOI of trajectories. Thereare several potential directions for future work such as incorporat-ing other features, or supporting other degradation types.
Acknowledgements: This research has been funded in part byNSF grants IIS-1910950 and CNS-2027794, the USC IntegratedMediaSystems Center, and unrestricted cash gifts from Microsoft. Anyopinions, findings, and conclusions or recommendations expressedin this material are those of the author(s) and do not necessarilyreflect the views of any of the sponsors such as the NSF.
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