Elizabeth Munch SOED 2016

What does it mean for data to have shape?

Elizabeth Munch

University at Albany – SUNY:: Dept. of Mathematics & Statistics

Apr 7, 2016

Liz Munch (UAlbany) TDA Apr 7, 2016 1 / 24


Elizabeth Munch Data Point

University at Albany – SUNY:: Dept. of Mathematics & Statistics

Apr 7, 2016


(-0.02,-1.62) (-1.38,-0.93) (1.22,1.55) (-0.71,-1.48) (-0.17,-0.99)(0.25,-1.19) (-0.48,-1.71) (1.21,1.06) (-0.4,-1.73) (0.21,-1.87) (-0.09,1.23)(-0.95,0.33) (1.07,0.22) (1.87,-0.17) (-1.69,0.06) (-0.76,-0.9) (0.38,1.49)(-0.22,-1.31) (0.67,-1.58) (1.39,1.13) (-1.07,1.2) (1.26,1.02) (0.63,-1.01)(-1.13,0.37) (0.82,1.26) (0.92,0.46) (0.27,-1.22) (1.24,-1.56) (-1.38,1.0)(1.43,0.98) (-0.96,0.98) (1.77,-0.08) (-0.27,1.64) (1.48,1.2) (1.08,1.3)

(-1.16,-0.3) (-1.29,1.5) (-0.14,-1.93) (0.32,1.78) (-1.5,0.72) (-1.28,-0.63)(0.03,1.1) (1.57,-1.05) (-1.5,-0.34) (-0.22,-1.53) (0.39,-1.59) (-1.81,0.59)

(-0.38,-1.63) (-0.69,1.62) (-0.5,1.25) (-1.71,-1.03) (1.1,-0.11) (-0.02,-1.48)(-1.3,-0.25) (-1.37,0.84) (-0.88,-1.39) (-0.38,-1.77) (0.0,1.72) (-0.61,1.75)(0.15,1.74) (-0.11,-1.55) (-1.53,0.2) (-0.96,0.43) (-0.87,0.79) (-0.36,1.03)(1.59,0.15) (-0.13,1.18) (1.21,-0.35) (1.18,-0.85) (-1.2,1.27) (-1.43,-0.91)

(-1.44,-0.06) (-1.86,-0.55) (0.5,-1.24) (-1.78,-0.07) (0.48,-1.22)(-0.43,1.02) (1.37,-0.91) (-1.59,0.98) (1.15,-0.1) (-1.59,-0.6) (0.09,1.25)(0.32,1.53) (0.89,-1.43) (1.15,-1.22) (0.29,1.84) (-0.4,1.61) (-1.57,-1.07)

(-0.29,-1.55) (1.42,-0.99) (0.86,-1.81) (1.43,-1.15) (-0.53,1.65)(-1.18,-0.72) (-0.59,1.22) (-1.22,-0.61) (0.19,-1.26) (1.82,-0.84)

(-0.06,1.36) (-1.27,0.59)



Large Data Sets

Main goal of Topological Data Analysis (TDA)

Find and quantify structure in big data.

Goals of this talk

What tools are available?

How do we fit educational data into this pipeline?Spoiler alert: I don’t know how to do this....


Large Data Sets



Goals of this talk


How do we fit educational data into this pipeline?

Spoiler alert: I don’t know how to do this....


Large Data Sets



Goals of this talk


How do we fit educational data into this pipeline?Spoiler alert: I don’t know how to do this....


1 Persistent Homology

2 Reeb graphs and Mapper






Topology 6= Topography

Mathematical study of spacespreserved under continuousdeformations

stretching and bending

not tearing or gluing

Study of the shape andfeatures of the surface of theEarth


History

LeonhardEuler

(1707-1783)Euler Characteristic

Images: Wikipedia


History Pt 2

Esoteric field of study 1700-2000I Algebraic topologyI Applications/intersections with dynamical systemsI Would never be considered “applied” in traditional sense.

Topology, the pinnacle of human thought.In four centuries it may be useful.- Alexander Solzhenitzin, “The First Circle” 1968

Things change ca.2000I Introduction of Persistent Homology


History Pt 2





History Pt 2





Main questions

How do we quantify the structure we see?

Can we calculate something to represent the structure?


Very small radius is justdots.

Very large radius is just ablob.

Some range of radii lets ussee the big circle.

Some small circles appearand disappear quickly....maybe we get to just callthese noise!

How do we quantify this?
































Homology & Persistent Homology

What is Homology?

A topological invariant which assignsa sequence of vector spaces, Hk(X ),to a given topological space X .

What is Persistent Homology?

A way to watch how the homology ofa filtration (sequence) of topologicalspaces changes so that we canunderstand something about thespace.



What is Homology?






What is Homology?





Understanding a persistence diagram


Circles are useful when you least expect it.

Caveat:

Persistence does more than circles....


Circles are useful when you least expect it.

Caveat:

Persistence does more than circles....


Machining Dynamics

Workpiece

Stable

feed

Unstable

Images: Firas Khasawneh, SUNY Polytechnic Institute; and Boeing.


Chatter


Delay embedding

Definition

Given a time series X (t), the delay embedding is

ψmη : t 7−→ (X (t),X (t + η), · · · ,X (t + (m − 1)η)).


Differentiation by Max Persistence

100 120 140 160 180 200 220 240−1.0

−0.5

0.0

0.5

1.0

1.5

2.0 Signal, [0.9, 0.07]

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0Y(t)

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Y(t

+2.

13)

Takens Embedding, [0.9, 0.07]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Birth Radius

0.00.20.40.60.81.01.21.41.61.8

Dea

thR

adiu

s

Persistence Diagram, [0.9, 0.07]

70 80 90 100 110 120 130 140 150−1.0

−0.5

0.0

0.5

1.0

1.5

2.0 Signal, [1.42, 0.05]

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0Y(t)

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Y(t

+1.

62)


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Birth Radius

0.00.20.40.60.81.01.21.41.61.8

Dea

thR

adiu

s


60 70 80 90 100 110 120 130 140−1.0

−0.5

0.0

0.5

1.0

1.5

2.0 Signal, [1.48, 0.25]

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0Y(t)

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Y(t

+1.

56)


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Birth Radius

0.00.20.40.60.81.01.21.41.61.8

Dea

thR

adiu

s



Turning Model

0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25


Turning Model

Results

Warm colors⇒ High max persistence⇒ Chatter

Cool colors⇒ Low max persistence⇒ No Chatter

Combination withMachine LearningMethods⇒ 97% classificationaccuracy





Clustering


1-Dimensional Structure


1-Dimensional Structure


Original Reeb Graph construction


Original Reeb Graph construction


Mapper

Image: Nicolau Levine Carlsson, PNAS 2011


Mapper

Breast cancer gene expression data

Determine a good filter function

Run mapper

Found new type of breast cancer (c-MYB+) with high survival rateImage: Nicolau Levine Carlsson, PNAS 2011


Mapper

Image: Nicolau Levine Carlsson, PNAS 2011


Conclusions

Topology can help find structure in data that is not obvious by othermeans.

Lots of tools available, lots of open-source code for computation!I Mapper, Reeb graph, Contour Tree, Merge tree

F Python mapper - danifold.net/mapper/

I PersistenceF Perseus - sas.upenn.edu/~vnanda/perseus/F Dionysus - mrzv.org/software/dionysus/F R TDA - cran.r-project.org/web/packages/TDA/F PHAT - bitbucket.org/phat-code/phat

Input from domain scientists is imperative!I What is the right question?I What is the right tool?I How do we interpret the output?


danifold.net/mapper/

sas.upenn.edu/~vnanda/perseus/

mrzv.org/software/dionysus/

cran.r-project.org/web/packages/TDA/

bitbucket.org/phat-code/phat

Conclusions












Conclusions












Thank you!

CollaboratorsJose Perea (MSU)

Firas Khasawneh (SUNY Poly)

[email protected]


Elizabeth Munch SOED 2016

Education