Project Lachesis: Parsing and Modeling Location Histories Daniel Keeney CS 4440

Introduction Location History is a record of an entity’s location in geographical space over time Archaeologists and historians look at migrations and census data to reconstruct location histories New technologies such as GPS allow us to enhance the accuracy and resolution greatly

Resolution Old temporal resolutions ranged from a decade to a century Old spatial resolutions ranged from tens to hundreds of kilometers GPS accuracy opens up a completely different type of analysis

Goal By tracking locations in real time, new types of analysis can be performed Goal: condense, understand, and predict the movements of an object over a period of time

Stays and Destinations Stay is a single instance of an object spending some time in one place Destination is any place where one or more objects have experienced a stay Trip occurs between two adjacent stays made by the same object Path is a representation of the description of a set of trips between destinations

Calculating Stays The roaming distance, is how far an object can stray while being counted as a stay The stay duration, is how long an object must remain within the roaming distance to count as a stay Medoid is the data point nearest to the “center” of the set

Calculating Stays

Worst case: O(n 2 ) for n data points, due to medoid and diameter working on all pairs In practice, clusters which require computation are far smaller than n, effectively yielding O(n)

Calculating Destinations Geographic scale, determines how close two stays can be and still be considered the same destination Destinations are represented by a location as well as the scale used:

Calculating Destinations

Example

Creating Probabilistic Models Assumptions: At the beginning of a given time interval, an object is at exactly one destination During any given time interval, an object makes exactly one transition between destinations Self-transitions are allowed

Creating Probabilistic Models Models are similar to Hidden Markov Models Critical difference from HMM is the incorporation of time-dependence, where transition probabilities are conditioned on recurring time intervals

Creating Probabilistic Models Model consists of three probability matrices Probability of the object starting time interval at destination is Probability of transition from to during interval is Observation probability: observing object at when actually at

Calculating π

Calculating A

Calculating B

Calculating Probabilistic Models Together as these tables represent a probabilistic model This model can be used to solve problems such as finding the most likely destination occupied at a particular time, determining the relative likelihood of a location history sequence, or generating a location history sequence

Calculating Probabilistic Models Using λ we estimate the relative likelihood of a new location history This is done using a Non-Markovian Solution and a Markovian Solution

Non-Markovian Solution

Markovian Solution

Experiment Results

“I always felt more productive on Tuesdays.” - Subject A

Experiment Results

A typical (left) and an atypical (right) week from Subject A.

Experimental Results Plots of synthesized weeks, using Non-Markov (left) and Markov (right) models

Markov vs. Non-Markov Markovian model showed an atypical week to have an unexpectedly high probability This could be mitigated by “training” on larger data sets, but generally the Non- Markovian model is sufficient

Conclusions Proposed rigorous definitions for location histories, stays, and destinations, as well as accompanying algorithms Non-Markovian is better suited for evaluating likelihoods of a location history Markovian is better for stochastically generating a history Future papers will examine trips and paths