2IMA20 Algorithms for Geographic Data Spring 2016 Lecture 4: Segmentation

Motivation: Geese Migration Two behavioural types stopover migration flight Input: GPS tracks expert description of behaviour Goal: Delineate stopover sites of migratory geese

Abstract / general purpose questions Single trajectory simplification, cleaning segmentation into semantically meaningful parts finding recurring patterns (repeated subtrajectories) Two trajectories similarity computation subtrajectory similarity Multiple trajectories clustering, outliers flocking/grouping pattern detection finding a typical trajectory or computing a mean/median trajectory visualization

Problem For analysis, it is often necessary to break a trajectory into pieces according to the behaviour of the entity (e.g., walking, flying, …). Input: A trajectory T, where each point has a set of attribute values, and a set of criteria. Attributes: speed, heading, curvature… Criteria: bounded variance in speed, curvature, direction, distance… Aim: Partition T into a minimum number of subtrajectories (so-called segments) such that each segment fulfils the criteria. “Within each segment the points have similar attribute values”

Criteria-Based Segmentation Goal: Partition trajectory into a small number of segments such that a given criterion is fulfilled on each segment

Example Trajectory T sampled with equal time intervals Criterion: speed cannot differ more than a factor 2?

Example speed Trajectory T sampled with equal time intervals Criterion: speed cannot differ more than a factor 2? Criterion: direction of motion differs by at most 90⁰? speed

Example speed speed Trajectory T sampled with equal time intervals Criterion: speed cannot differ more than a factor 2? Criterion: direction of motion differs by at most 90⁰? speed speed heading

Example speed speed Trajectory T sampled with equal time intervals Criterion: speed cannot differ more than a factor 2? Criterion: direction of motion differs by at most 90⁰? speed speed speed & heading heading

Decreasing monotone criteria Definition: A criterion is decreasing monotone, if it holds on a segment, it holds on any subsegment. Examples: disk criterion (location), angular range (heading), speed… Theorem: A combination of conjunctions and disjunctions of decreasing monotone criteria is a decreasing monotone criterion.

Greedy Algorithm Observation: If criteria are decreasing monotone, a greedy strategy works.

Designing greedy algorithms try to discover structure of optimal solutions: what properties do optimal solutions have ? what are the choices that need to be made ? do we have optimal substructure ? optimal solution = first choice + optimal solution for subproblem do we have greedy-choice property for the first choice ? prove that optimal solutions indeed have these properties prove optimal substructure and greedy-choice property use these properties to design an algorithm and prove correctness proof by induction (possible because optimal substructure)

lemma obviously holds, so assume OPT does not include ai. Lemma: Let k be the smallest index such that 𝜏 𝑡 0 , 𝑡 𝑘 fails for a monotone decreasing criterion 𝜏. If there is a solution, then there is an optimal solution including 𝜏 𝑡 0 , 𝑡 𝑘−1 as segment. Proof. Let OPT be an optimal solution for A. If OPT includes ai then the lemma obviously holds, so assume OPT does not include ai. We will show how to modify OPT into a solution OPT* such that (i) OPT* is a valid solution (ii) OPT* includes ai (iii) size(OPT*) ≤ size(OPT) Thus OPT* is an optimal solution including 𝜏 𝑡 0 , 𝑡 𝑘−1 , and so the lemma holds. To modify OPT we proceed as follows. standard text you can basically use in proof for any greedy-choice property quality OPT* ≥ quality OPT here comes the modification, which is problem-specific

Greedy Algorithm Observation: If criteria are decreasing monotone, a greedy strategy works. For many decreasing monotone criteria Greedy requires O(n) time, e.g. for speed, heading…

Greedy Algorithm Observation: For some criteria, iterative double & search is faster. Double & search: An exponential search followed by a binary search.

Criteria-Based Segmentation Boolean or linear combination of decreasing monotone criteria Greedy Algorithm incremental in O(n) time or constant-update criteria e.g. bounds on speed or heading double & search in O(n log n) time for non-constant update criteria [M.Buchin et al.’11]

Motivation: Geese Migration Two behavioural types stopover migration flight Input: GPS tracks expert description of behaviour Goal: Delineate stopover sites of migratory geese

Case Study: Geese Migration Data Spring migration tracks White-fronted geese 4-5 positions per day March – June Up to 10 stopovers during spring migration Stopover: 48 h within radius 30 km Flight: change in heading <120° Kees Adri

Comparison manual computed

Evaluation Few local differences: Shorter stops, extra cuts in computed segmentation

Criteria A combination of decreasing and increasing monotone criteria stopover migration flight Within radius 30km At least 48h Change in heading <120° AND OR

Decreasing and Increasing Monotone Criteria Observation: For a combination of decreasing and increasing monotone criteria the greedy strategy does not always work. Example: Min duration 2 AND Max speed range 4 5 speed 1

Non-Monotone Segmentation Many Criteria are not (decreasing) monotone: Minimum time Standard deviation Fixed percentage of outliers For these Aronov et al. introduced the start-stop diagram Example: Geese Migration

Start-Stop Diagram Algorithmic approach input trajectory compute start-stop diagram compute segmentation

Start-Stop Diagram Given a trajectory T over time interval I = {t0,…,t} and criterion C The start-stop diagram D is (the upper diagonal half of) the n x n grid, where each point (i,j) is associated to segment [ti,tj] with (i,j) is in free space if C holds on [ti,tj] (i,j) is in forbidden space if C does not hold on [ti,tj] A (minimal) segmentation of T corresponds to a (min-link) staircase in D forbidden space free space staircase

Start-Stop Diagram A (minimal) segmentation of T corresponds to a (min-link) staircase in D. 12 11 10 9 8 7 6 5 4 3 2 1 6 7 8 5 9 11 10 4 3 12 2 123456789101112 1 2 3 4 5 6 7 8 9 10 11 12 1

Start-Stop Diagram A (minimal) segmentation of T corresponds to a (min-link) staircase in D. 123456789101112 1 2 3 4 5 6 7 8 9 10 11 12 12 11 10 9 8 7 6 5 4 3 2 1 6 7 8 5 9 11 10 4 3 12 2 1

Start-Stop Diagram Discrete case: A non-monotone segmentation can be computed in O(n2) time. 123456789101112 1 2 3 4 5 6 7 8 9 10 11 12 12 11 10 9 8 7 6 5 4 3 2 1 4 5 8 11 1 12 6 9 10 7

Start-Stop Diagram Discrete case: A non-monotone segmentation can be computed in O(n2) time. 123456789101112 1 2 3 4 5 6 7 8 9 10 11 12 12 11 10 9 8 7 6 5 4 3 2 1 4 5 8 11 1 12 6 9 10 7

Stable Criteria Definition: A criterion is stable if and only if 𝑖=𝑜 𝑛 𝑣 𝑖 =𝑂(𝑛) where 𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i], …, [i-1,i]

Stable Criteria Definition: A criterion is stable if and only if 𝑖=𝑜 𝑛 𝑣 𝑖 =𝑂(𝑛) where 𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i], …, [i-1,i] Observations: Decreasing and increasing monotone criteria are stable. A conjunction or disjunction of stable criterion are stable.

Compressed Start-Stop Diagram For stable criteria the start-stop diagram can be compressed by applying run-length encoding. Examples: decreasing monotone increasing monotone 9

Computing the Compressed Start-Stop Diagram For a decreasing criterion consider the algorithm: ComputeLongestValid(crit C, traj T) Algorithm: Move two pointers i,j from n to 0 over the trajectory. For every trajectory index j the smallest index i for which 𝜏 𝑡 𝑖 , 𝑡 𝑗 satisfies the criterion C is stored. 9 i j

Computing the Compressed Start-Stop Diagram For a decreasing criterion ComputeLongestValid(crit C, traj T): Move two pointers i,j from n to 0 over the trajectory 9 i j Requires a data structure for segment [ti,tj] allowing the operations isValid, extend, and shorten, e.g., a balanced binary search tree on attribute values for range or bound criteria. Runs in O(nc(n)) time where c(n) is the time to update & query. Analogously for increasing criteria.

Computing the Compressed Start-Stop Diagram The start-stop diagram of a conjunction (or disjunction) of two stable criteria is their intersection (or union). The start-stop diagram of a negated criteria is its inverse. The corresponding compressed start-stop diagrams can be computed in O(n) time. 9

Attributes and criteria Examples of stable criteria Lower bound/Upper bound on attribute Angular range criterion Disk criterion Allow an approximate fraction of outliers …

Computing the Optimal Segmentation Observation: The optimal segmentation for [0,i] is either one segment, or an optimal sequence of segments for [0,j<i] appended with a segment [j,i], where j is an index such [j,i] is valid. Dynamic programming algorithm for each row from 0 to n find white cell with min-link That is, iteratively compute a table S[0,n] where entry S[i] for row i stores last: index of last link count: number of links so far runs in O(n2) time n 10, 4 8, 3 8, 3 7, 2 0, 1 0, 0+1=1 nil, ∞ S nil, ∞ nil, ∞ nil, ∞ nil, ∞ nil, 0

Computing the Optimal Segmentation More efficient dynamic programming algorithm for compressed diagrams Process blocks of white cells using a range query in a binary search tree T (instead of table S) storing index: row index last: index of last link count: number of links so far augmented by minimal count in subtree T [Alewijnse et al.’14]

Methodology for augmenting a data structure Choose an underlying data structure. Determine additional information to maintain. Verify that we can maintain additional information for existing data structure operations. Develop new operations. R-B tree min-count[x] theorem next slide count(block) using range query

Theorem Augment a R-B tree with field f, where f[x] depends only on information in x, left[x], and right[x] (including f[left[x]] and f[right[x]]). Then can maintain values of f in all nodes during insert and delete without affecting O(log n) performance. When we alter information in x, changes propagate only upward on the search path for x …

Computing the Optimal Segmentation More efficient dynamic programming algorithm for compressed diagrams Process blocks of white cells using a range query in a binary search tree T (instead of table S) storing index: row index last: index of last link count: number of links so far augmented by minimal count in subtree runs in O(n log n) time T [Alewijnse et al.’14]

Beyond criteria-based segmentation When is criteria-based segmentation applicable? What can we do in other cases?

Excursion: Movement Models

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic?

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic? Space-time Prisms

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic? Space-time Prisms

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic? Space-time Prisms

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic? Space-time Prisms Random Motion Models (e.g. Brownian bridges)

Movement Models Movement data: sequence of observations, e.g. (xi,yi,ti) Linear movement realistic? Space-time Prisms Random Motion Models (e.g. Brownian bridges)

Brownian Bridges - Examples no bridges

Segment by diffusion coefficient

Summary Greedy algorithm for decreasing monotone criteria O(n) or O(n log n) time [M.Buchin, Driemel, van Kreveld, Sacristan, 2010] Case Study: Geese Migration [M.Buchin, Kruckenberg, Kölzsch, 2012] Start-stop diagram for arbitrary criteria O(n2) time [Aronov, Driemel, van Kreveld, Löffler, Staals, 2012] Compressed start-stop diagram for stable criteria O(n log n) time [Alewijnse, Buchin, Buchin, Sijben, Westenberg, 2014]

References References S. Alewijnse, T. Bagautdinov, M. de Berg, Q. Bouts, A. ten Brink, K. Buchin and M. Westenberg. Progressive Geometric Algorithms. SoCG, 2014. M. Buchin, A. Driemel, M. J. van Kreveld and V. Sacristan. Segmenting trajectories: A framework and algorithms using spatiotemporal criteria. Journal of Spatial Information Science, 2011. B. Aronov, A. Driemel, M. J. Kreveld, M. Loffler and F. Staals. Segmentation of Trajectories for Non-Monotone Criteria. SODA, 2013. M. Buchin, H. Kruckenberg and A. Kölzsch. Segmenting Trajectories based on Movement States. SDH, 2012.