Python Mapper http danifold netmapper See also mappersummary

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Python Mapper: http: //danifold. net/mapper/ See also mappersummary 2 a. pdf - A quick

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 :

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

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.

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

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.

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

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.

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)

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 :

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? ? ?

Increasing threshold What are the clusters? ? ?

Filter Function: k. NN distance

Filter Function: k. NN distance

mapper. filters. k. NN_distance(data, k, metricpar={}, callback=None) The distance to the k-th nearest neighbor

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)

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

knn distance with k = 5

knn distance with k = 5

Python Mapper: http: //danifold. net/mapper/ See also mappersummary 2 a. pdf - A quick

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 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, 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 3 intervals, 20% overlap [ ( ) ]

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

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

knn distance with k = 5 10 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 = 5 100 intervals, 50% overlap

knn distance with k = 50

knn distance with k = 50

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

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

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

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

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

View Screen shot to output png of figure

View Screen shot to output png of figure

Save figure to output pdf of figure

Save figure to output pdf of figure

Metric: Euclidian Filter Function: Eccentricity with exponent = 1 Cover: Uniform 1 -d cover

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