1. Python Mapper: http://danifold.net/mapper/
See also mappersummary2a.pdf - A quick introduction to Mapper for linux computers in B5 MLH

2. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.
B) Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x
Put data into overlapping bins.
Example: f-1(ai, bi)
Cluster each bin & create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters

4. Hierarchical clustering
Data
Dendrogram

5. single linkage hierarchical clustering
Increasing threshold
Connect vertices whose distance is less than a given threshold
single linkage hierarchical clustering

6. Hierarchical clustering
Data
Dendrogram

7. Increasing threshold
Connect vertices (or clusters) whose distance is less than a given threshold

8. Different type of hierarchical clustering
What is the distance between 2 clusters?

9.
The Elements of Statistical Learning (2nd edition) Hastie, Tibshirani and Friedman

10. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.
B) Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x
Put data into overlapping bins.
Example: f-1(ai, bi)
Cluster each bin & create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters

11. Increasing threshold
What are the clusters???

12. Filter Function: kNN distance

13. mapper.filters.kNN_distance(data, k, metricpar={}, callback=None)
The distance to the k-th nearest neighbor as an (inverse) measure of density.
Note how the number of nearest neighbors is understood:
k=1, the first neighbor, makes no sense for a filter function since the first nearest neighbor of a data point is always the point itself, and hence this filter function is constantly zero.
The parameter k=2 measures the distance from xi to the nearest data point other than xi itself.
d1(x) =
d2(x) =
d3(x) =
d4(x) = z
x y 3

14. x y
If x is in a denser region than y, then dk(x) dk(y)

16. knn distance with k = 5

17. Python Mapper: http://danifold.net/mapper/
See also mappersummary2a.pdf - A quick introduction to Mapper for linux computers in B5 MLH

18. knn distance with k = 5 and 3 bins???
[ ( ) ) ] (

19. knn distance with k = 5 and 3 bins???
[ ( ) ) ] (

20. knn distance with k = 5
3 intervals, 50% overlap

21. knn distance with k = 5
3 intervals, 50% overlap
[ ( ) ) ] (

22. knn distance with k = 5
3 intervals, 20% overlap
[ ( ) ) ] (

23. knn distance with k = 5
5 intervals, 50% overlap

24. knn distance with k = 5
10 intervals, 50% overlap

25. knn distance with k = 5
100 intervals, 50% overlap

26. knn distance with k = 50

27. knn distance with k = 50
3 intervals, 50% overlap

28. knn distance with k = 50
5 intervals, 50% overlap

29. knn distance with k = 50
10 intervals, 50% overlap

30. knn distance with k = 50
100 intervals, 50% overlap

31. View Screen shot to output png of figure

32. Save figure to output pdf of figure

34. Filter Function: Eccentricity with exponent = 1
Metric: Euclidian
Filter Function: Eccentricity with exponent = 1
Cover: Uniform 1-d cover
Clustering: Single
modified slide from Maria Gommel