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Clustering of location-based data

Mohammad Rezaei

May 2013

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Data mining and Clustering

- Huge amount of location-based Data

- Need for mechanisms to extract knowledge

- Clustering as an important field in spatio-temporal data mining

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Clustering

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Some applications

RoutingInteresting placesRecommendation of servicesMarketing managementUsers with same interestsVisualization

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Clustering Problems in Mopsi

Clutter of markers on the mapSimilar services or photos in a

listCategorization of servicesDistribution of users’ locationsTimeline view of photosClustering of events

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Clutter of markers

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Search results

Clustering

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Photos

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Users

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Solutions

Grid based clustering

Distance based clustering

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Google Maps version 3.0- Using location in pixels for grid-base

clustering- 22 zoom levels- 256*256 in zoom level 0 to 536870912*

536870912 in zoom level 21- ≈ 60*1012 cells in the zoom level 21

with cell size(60,80)

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Some issues

- Photos are added or deleted dynamically

- Querying for a certain time, certain user or according to photo description

- Different zoom levels, moving map

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Hierarchical Clustering on server

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Hierarchical Clustering on server

Individual clustering for different zoom levels

Clustering of whole data

How to extract clusters for a specific query?

Are clusters for a lower zoom level can be derived from higher level?

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Client side clustering

- Query from server (Resulting N objects)

- Take the zoom view

Not too many cells

- Taking objects in the zoom view and do

clustering only for them (M objects)

- It takes O(N) to find out the objects in

the zoom view!

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Grid based clustering

Input location (lat, lon) of markers Width and height of markers (Hm,Wm) Width and height of cells in the grid (H, W)

Output Location of clusters

Location of the marker

W

H

Wm

Hm

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Representation - Middle of cell

-No overlap

-Locations can be misleading

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Representation- First object

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Representation – Average Location

Proposed approach

- Grids start from beginning of the whole map

- Extend the grid in current zoom view

By moving map clusters do not

change

- Average location for representative

By moving map clusters

do not change

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W

H

(xmin, ymin)

(xmax, ymax)

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Algorithm

1. nRow = ceil((xmax-xmin)/W)

2. nColumn = ceil((ymax-ymin)/H)

3. nCell = nRow * nColumn 4. Clusters = all cells // empty clusters5. For all the markers6. row = floor((y-ymin)/gridHeight)

7. column = floor((x-xmin)/gridWidth)

8. cellNum = row*nColumn + column9. Add the marker to Clusters[cellNum]10. Update the cluster:

Clusters[cellNum]

W

H

(xmax, ymax)

(xmin, ymin)

(x,y)

1 2 3 4 5

1

2

3

4

5

1 2 3 4 5

6 7 8 9 10

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25

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Cell number

18 20

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Merging algorithm- Average location as representative

1. MergeClusters(clusters)2. change the order of clusters descending according to the size of

clusters 3. set parent of each cluster, the same cluster4. k=1 (K is number of clusters)5. while (k < K )6. if ( k is not “processed” )7. checkNeighbors(k);8. mark the cluster k “processed”9. k=k+1

10. CheckNeighbors(k)11. cluster1=clusters[k]12. For all 8 neighbors13. cluster2 = one of the neighbors // 14. if cluster2 is not an empty cell15. checkNeighbor(cluster1, cluster2)

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Merging algorithm1. checkNeighbor(cluster1, cluster2)2. find the distance d between the two clusters3. if d<T // distance threshold T4. while ( cluster2 is “processed” ) // means it has been merged5. cluster2 = clusters[cluster2.parent]6. MergeClusters(cluster1, cluster2);

7. MergeClusters(cluster1, cluster2)8. n1 and n2: size of the clusters9. (x1,y1) and (x2,y2): location of clusters10. x=(n1*x1+n2*x2)/(n1+n2) 11. y=(n1*y1+n2*y2)/(n1+n2) 12. x1 x and y1 y 13. mark the second cluster “processed”14. cluster2.parent = k

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Grid based clustering

Width and height of a cell H>Hm and W>Wm

Minimum distance of the markers to avoid overlap 22

mm HWd

d

Wm

Hm

Marker

Location of marker

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Distance based clustering

Input location (lat, lon) of markers Width and height of markers (Hm, Wm)

Output location of clusters

Time complexity: O(N2)

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Algorithm1. i= 0;2. While (i<N) // N=number of markers3. if ( marker i is not clustered )4. Label marker i as clustered5. Calculate distance (dj) to other non-clustered

markers6. for all markers j7. If dj<T // T: distance threshold

8. merge the markers i and j9. Label marker j as clustered10. i = i+1;

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Timeline view of photosDisplaying n photos in a limited space

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Timeline view of photos

Input Timestamps Number of clustersOutput PartitionsAlgorithm K-means

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Location clusters

Homes of usersShop

Walking street

Marketplace

Swimhall

Sciencepark

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Clustering of trajectories

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Similarity or distance

Start and end of the routes

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Similarity or distance

Speed, length, accelaration, time, etc

70 km/h 72 km/h

50 km/h

30 km/h

60 km/h

These two routes are more similar in speed than others

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Similarity or distance

Closeness of points and shape(Comparing whole route or segments of the routes)

t1T1

t2t3 t4

t5 t6t7

t8

T2t1

t2

t3

t4

t1T1

t2 t3 t4t5 t6

t7t8

T2t1

t2

t3

t4

Closest pair distance

Sum of pair distance

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Cluttering problem for routes