K-BestMatch

Mirco Nanni e-mail: mirco.nanni@isti.cnr.it Roberto Trasarti e-mail: roberto.trasarti@isti.cnr.it KDD Lab, ISTI-CNR, Pisa, Italy SSTDM-09 Miami, 2009

Diffusion of devices with GPS technology leads to a large amount of data for vehicles and individuals .
Applications on urban context require to map this data on a street network .

Geometric Map Matching : Consider a set of timestamps T = {0, 1, . . . , t} and a function Pr = (latitude, longitude) which describes the real position of a user at time r ∈ T. Then, given a set of georeferenced map objects O = {o1, . . . , on}, a Map Matching function is deﬁned as follows: F( Pr) -> oj where oj ∈ O.
There are three classes of approches:
Point – to – point
The original points are mapped on nodes of the network
Point – to – segments
The original points are mapped on segments of the network
Segments – to – Segments
Considering every two consicutive original point the obtained segments are mapped on segments of the network

Accuracy of the devices :
Errors in the positioning create ambiguity during the process of map matching.
Usually solved by heuristics .
Sampling rate :
The sample rate (or storing rate) of the
device can produce a disconnected path.
How to complete the path?

Best Match : Let M be a street map, composed of:
a set of nodes M.nodes ,
a set of oriented segments M.segments ⊆ M.nodes ×M.nodes,
and a cost function that associates each segment with a real value M.cost : M.segments -> R.
Then, given a sequence of segments S = <s 1 , . . . , s n >, the match set of S over M is deﬁned as follows:
and the best match of S over M is:
where

From a set of disconnected segments to a connected path on the street network:

Point-to-segment
When the two nearest segments of a point have distances from it that are equal up to a given tolerance, the segment whose starting vertex is closest to the end point of the previous segment of the trajectory is chosen.
K-BestMatch
We use a more flexible approach which considers the k-optimal alternatives paths between two disconnected segments of the initial set.

The previous representation of a path on the street network becomes a multipath

Item-frequency representation : Given a street map M, a sequence of segments S = <s 1 , . . . , s n > and a positive integer k, the Item-Frequency representation IF(S,M, k) of S over M w.r.t. k is defined as a pair (I, f) such that:
The frequencies can be computed locally for each gap in S.

Freqeuncy:
Red = 1
Orange >=.75
Yellow >=.5
Green >=.25
Blue <=.25
K = 1 (BestMatch) K =4 As obvious, segments already contained in the original dataset have frequency 1 in both the cases.

Point-to-segment : Compares a set of points P with all the segments of the street network M.segments, therefore the complexity is O(|P|m) , where m = |M.segments|
K-BestMatch : considering G as the set of gaps in the dataset, the complexity is O(|G|kn(m +nlogn)) where k is the k-bestmatch parameter and n = |M.nodes| (See [1]).
The overall complexity is O(|G|kn(m+nlogn) + |P|m). If we assume that the road network M is ﬁxed, and therefore n and m are constant factors, the complexity reduces to O(|G|k + |P|) .
[1] Ernesto Q. V. Martins and Marta M. B. Pascoal. A new implementation of yen ranking loopless paths algorithm. In 4OR: A Quarterly Journal of Operations Research.

Given the Item-Frequency representations of two trajectories, IF1 = (I1, f1) and IF2 = (I2, f2), we define the following distances between IF1 and IF2: where:

As a ﬁrst validation of the method, we provide a visual account of the effects of the k−BestMatch reconstruction with the Jaccard distance on k-Nearest Neighbor queries (kNN):
Q: Returning the top 10 objects that are closest to a chosen pivot object
Our solution contains a larger core of segments but there are not the outlier paths that occur in the BestMatch based.
Pivot object BestMatch Result K-BestMatch Result

In order to test the method on a clustering task, a generic agglomerative hierarchical clustering algorithm was adapted to work with the item-frequency representation of trajectories.
A Cluster using K-BestMatch Same set of trajectory using BestMatch. (It’s not a cluster)

In this experiment the k−BestMatch reconstruction was performed with several different values for k, then comparing the resulting clusters against those obtained by adopting the BestMatch approach (k = 1).
The comparison between clustering results is performed by the standard F-measure:

A first exploration of the effects that a more flexible map matching approach can have on the comparison, query and mining of trajectories.
Preliminary results are encouraging, and suggest that overcoming the limits of standard best-match reconstruction strategies can have beneficial effects on the successive analysis to be performed on such data.
The work also rose several open issues:
the need for refined methods to select the k-optimal alternative paths, for instance trying to limit path redundancy ;
the need for considering also the order of visit of the segments, thus moving from the item-frequency representation to a more complex one

Thank you.
Questions?
Mirco Nanni e-mail: mirco.nanni@isti.cnr.it Roberto Trasarti e-mail: roberto.trasarti@isti.cnr.it KDD Lab, ISTI-CNR, Pisa, Italy

K-BestMatch

by Roberto Trasarti on May 06, 2010

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K-BestMatch — Presentation Transcript