Python Mapper http danifold netmapper See also mappersummary

Python Mapper: http: //danifold. net/mapper/ See also mappersummary 2 a. pdf - A quick introduction to Mapper for linux computers in B 5 MLH

A) Data Set Example: Point cloud data representing a hand. B) Function f : Data Set R Example: x-coordinate f : (x, y, z) x C) Put data into overlapping bins. Example: f-1(ai, bi) D) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters http: //www. nature. com/srep/2013/130207/srep 01236/full/srep 01236. html

http: //scikit-learn. org/stable/auto_examples/cluster/plot_cluster_comparison. html

Hierarchical clustering Data http: //en. wikipedia. org/wiki/ File: Clusters. svg Dendrogram http: //en. wikipedia. org/wiki/File: Hiera rchical_clustering_simple_diagram. svg

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

Hierarchical clustering Data http: //en. wikipedia. org/wiki/ File: Clusters. svg Dendrogram http: //en. wikipedia. org/wiki/File: Hiera rchical_clustering_simple_diagram. svg

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

Different type of hierarchical clustering What is the distance between 2 clusters? http: //en. wikipedia. org/wiki/File: Hiera rchical_clustering_simple_diagram. svg http: //www. multid. se/genex/hs 515. htm

http: //statweb. stanford. edu/~tibs/Elem. Stat. Learn/ The Elements of Statistical Learning (2 nd edition) Hastie, Tibshirani and Friedman

A) Data Set Example: Point cloud data representing a hand. B) Function f : Data Set R Example: x-coordinate f : (x, y, z) x C) Put data into overlapping bins. Example: f-1(ai, bi) D) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters http: //www. nature. com/srep/2013/130207/srep 01236/full/srep 01236. html

Increasing threshold What are the clusters? ? ?

Filter Function: k. NN distance

mapper. filters. k. NN_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. d 1(x) = d 2(x) = 5 z 4 d 3(x) = d 4(x) = x 3 y http: //danifold. net/mapper/filters. html#filter-functions-in-python-mapper

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

knn distance with k = 5

Python Mapper: http: //danifold. net/mapper/ See also mappersummary 2 a. pdf - A quick introduction to Mapper for linux computers in B 5 MLH

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

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

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

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

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

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

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

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

knn distance with k = 50

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

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

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

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

View Screen shot to output png of figure

Save figure to output pdf of figure

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