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Utilizing Topological Data Analysis to Detect Periodicity Elizabeth - - PowerPoint PPT Presentation

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Utilizing Topological Data Analysis to Detect Periodicity Elizabeth Munch University at Albany - SUNY :: Department of Mathematics & Statistics Oct 2, 2016 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 1 / 30 Time series in biology

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SLIDE 1

Utilizing Topological Data Analysis to Detect Periodicity

Elizabeth Munch

University at Albany - SUNY :: Department of Mathematics & Statistics

Oct 2, 2016

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 1 / 30

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SLIDE 2

Time series in biology

Mitosis

Kredel et al. PLoS One 2009

Neuron Spike Trains

Curto et al. PLoS One 2008

Yeast gene expression

Deckard et al., Bioinformatics 2013

ECG

Goldberg et al. 2000 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 2 / 30

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SLIDE 3

Our definition of time series

Definition

A time series is a function f : R≥0 − → D for some topological space D.

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 3 / 30

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SLIDE 4

Our definition of time series

Definition

A time series is a function f : R≥0 − → D for some topological space D.

Choice for D

R - Classical time series analysis Rm×n - R-valued m × n matrices (movies) Pers - Persistence diagram valued time series (vineyards)

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 3 / 30

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SLIDE 5

Commonly used tools

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 4 / 30

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SLIDE 6

Common questions

Classification/Clustering

◮ Is this signal Type A or Type B? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

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SLIDE 7

Common questions

Classification/Clustering

◮ Is this signal Type A or Type B?

Periodicity

◮ Is this signal exhibiting periodic behavior? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

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SLIDE 8

Common questions

Classification/Clustering

◮ Is this signal Type A or Type B?

Periodicity

◮ Is this signal exhibiting periodic behavior?

Forecasting

◮ Given this previous signal, what do we expect to have happen next? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

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SLIDE 9

Common questions

Classification/Clustering

◮ Is this signal Type A or Type B?

Periodicity

◮ Is this signal exhibiting periodic behavior?

Forecasting

◮ Given this previous signal, what do we expect to have happen next?

Segmentation

◮ Which pieces of this signal come from similar systems? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

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SLIDE 10

Common questions

Classification/Clustering

◮ Is this signal Type A or Type B?

Periodicity

◮ Is this signal exhibiting periodic behavior?

Forecasting

◮ Given this previous signal, what do we expect to have happen next?

Segmentation

◮ Which pieces of this signal come from similar systems? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

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SLIDE 11

Idea:

Persistent homology and other TDA tools can be used to improve time series anaysis.

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 6 / 30

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SLIDE 12

Idea:

Persistent homology and other TDA tools can be used to improve time series anaysis.

This talk:

Mechanical engineering

◮ Firas Khasawneh ◮ Jose Perea

Atmospheric science

◮ Bill Dong ◮ Kristen Corbosiero ◮ Jason Dunion ◮ Ryan Torn Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 6 / 30

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SLIDE 13

1

Classification and Machining Dynamics

2

Periodicity and Hurricanes

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 7 / 30

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SLIDE 14

1

Classification and Machining Dynamics

2

Periodicity and Hurricanes

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 7 / 30

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SLIDE 15

Machining Dynamics

Workpiece Stable

feed

Unstable

Images courtesy Firas Khasawneh, SUNYIT; and Boeing.

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 8 / 30

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SLIDE 16

Deterministic model: ¨ y + 2ζ ˙ y + y = Kρα−1(1 + y(t − τ) − y(t))α Left side: standard linear

  • scillator

Right side: input based

  • n cutting forces

0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25

Khasawneh, F.A. & Mann, B. P. A spectral element approach for the stability of delay systems, International Journal for Numerical Methods in Engineering, 2011, 87, 566-592

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 9 / 30

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SLIDE 17

Chatter

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]

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]

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]

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 10 / 30

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SLIDE 18

Takens embedding

Definition

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

η : t −

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

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 11 / 30

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SLIDE 19

Persistent Homology of Point Cloud

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 12 / 30

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SLIDE 20

Noise resilience

5 10 15 t 1.5 1.0 0.5 0.0 0.5 1.0 1.5 X

Original Signals

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 13 / 30

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SLIDE 21

Noise resilience

5 10 15 t 1.5 1.0 0.5 0.0 0.5 1.0 1.5 X

Original Signals

1.5 1.0 0.5 0.0 0.5 1.0 1.5 X(t) 1.5 1.0 0.5 0.0 0.5 1.0 1.5 X(t +1. 32)

Delay Embedding

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

Persistence Diagrams

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 13 / 30

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SLIDE 22

Comparing signals using 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 2.0 Y(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 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.62)

Takens Embedding, [1.42, 0.05]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius Persistence Diagram, [1.42, 0.05]

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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.56)

Takens Embedding, [1.48, 0.25]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius Persistence Diagram, [1.48, 0.25] Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 14 / 30

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SLIDE 23

Comparing signals using 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 2.0 Y(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 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.62)

Takens Embedding, [1.42, 0.05]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius Persistence Diagram, [1.42, 0.05] 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.56)

Takens Embedding, [1.48, 0.25]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius Persistence Diagram, [1.48, 0.25]

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 14 / 30

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SLIDE 24

Overview

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 15 / 30

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SLIDE 25

Overview

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 15 / 30

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SLIDE 26

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 2.0 Y(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 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.62)

Takens Embedding, [1.42, 0.05]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius Persistence Diagram, [1.42, 0.05]

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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.56)

Takens Embedding, [1.48, 0.25]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Birth Radius

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Death Radius Persistence Diagram, [1.48, 0.25] Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 16 / 30

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SLIDE 27

Turning Model

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 2.0 Y(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 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.62) Takens Embedding, [1.42, 0.05] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius Persistence Diagram, [1.42, 0.05] 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 2.0 Y(t) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Y(t + 1.56) Takens Embedding, [1.48, 0.25] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Death Radius Persistence Diagram, [1.48, 0.25]

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 17 / 30

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SLIDE 28

Machine Learning

Adcock et al. Coordinates

Diagrams 0 and 1-dimensional of the form {(xi, yi)} xi(yi − xi) (ymax − yi)(yi − xi) x2

i (yi − xi)4

(ymax − yi)2(yi − xi)4 max{(yi − xi)}

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 18 / 30

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SLIDE 29

Machine Learning

Adcock et al. Coordinates

Diagrams 0 and 1-dimensional of the form {(xi, yi)} xi(yi − xi) (ymax − yi)(yi − xi) x2

i (yi − xi)4

(ymax − yi)2(yi − xi)4 max{(yi − xi)}

Results (Khasawneh, M, Perea)

Theoretical stability boundary for training Standard logistic classifier 97% accuracy

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 18 / 30

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SLIDE 30

Overview

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 19 / 30

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SLIDE 31

1

Classification and Machining Dynamics

2

Periodicity and Hurricanes

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 20 / 30

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SLIDE 32

1

Classification and Machining Dynamics

2

Periodicity and Hurricanes

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 20 / 30

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SLIDE 33

Hurricane Felix

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 21 / 30

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SLIDE 34

Diurnal cycle

3 hour difference

N(t) is IR matrix at time t N(t) − N(t − 3 hrs)

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 22 / 30

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SLIDE 35

Diurnal cycle

3 hour difference

N(t) is IR matrix at time t N(t) − N(t − 3 hrs)

Diurnal cycle

Sunset: cold ring, “diurnal pulse” Starts with radius ≤ 150km, spreads outward Warm ring forms behind this pulse and spreads outward Dunion et al. The Tropical Cyclone Diurnal Cycle of Mature Hurricanes. Monthly Weather Review, 2014.

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 22 / 30

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SLIDE 36

Sublevel Set Persistence

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 23 / 30

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SLIDE 37

Sublevel Set Persistence

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 23 / 30

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SLIDE 38

Sublevel Set Persistence

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 23 / 30

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SLIDE 39

Sublevel Set Persistence

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 23 / 30

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SLIDE 40

Why the obvious thing doesn’t work

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 24 / 30

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SLIDE 41

Plan B

Definition

Let Km×n = K be the m × n grid cubical complex.

Definition

Given M ∈ Rm×n, let M : K → R Mµ ⊂ K with function value ≥ µ. S : K → R defined by S(σ) = d(σ, Mµ) for dim(σ) = 2

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 25 / 30

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SLIDE 42

Plan B

Definition

Let Km×n = K be the m × n grid cubical complex.

Definition

Given M ∈ Rm×n, let M : K → R Mµ ⊂ K with function value ≥ µ. S : K → R defined by S(σ) = d(σ, Mµ) for dim(σ) = 2

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 25 / 30

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SLIDE 43

Plan B

Definition

Let Km×n = K be the m × n grid cubical complex.

Definition

Given M ∈ Rm×n, let M : K → R Mµ ⊂ K with function value ≥ µ. S : K → R defined by S(σ) = d(σ, Mµ) for dim(σ) = 2

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 25 / 30

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SLIDE 44

Resulting persistence diagrams

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 26 / 30

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SLIDE 45

Overview

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 27 / 30

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SLIDE 46

Overview

Death Radius Birth Radius Time

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 27 / 30

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SLIDE 47

Fourier spectrum of threshold

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 28 / 30

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SLIDE 48

Fourier spectrum of threshold

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 28 / 30

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SLIDE 49

Fourier spectrum of threshold

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 28 / 30

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SLIDE 50

Fourier spectrum of threshold

Results

23 hour day?

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 28 / 30

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SLIDE 51

General tools for TSA with TDA

Takens embedding → persistence

◮ Real-valued time series ◮ Can do classification, segmentation using persistence diagrams Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 29 / 30

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SLIDE 52

General tools for TSA with TDA

Takens embedding → persistence

◮ Real-valued time series ◮ Can do classification, segmentation using persistence diagrams

Image → sublevelset persistence

◮ Get a time series of persistence diagrams ◮ Pick out information from each diagram (max pers) to use standard

TSA methods

◮ Analyze speed ◮ Persistence of persistence

(Kramar, Levanger, et al. 2015 arXiv:1505.06168)

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 29 / 30

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SLIDE 53

General tools for TSA with TDA

Takens embedding → persistence

◮ Real-valued time series ◮ Can do classification, segmentation using persistence diagrams

Image → sublevelset persistence

◮ Get a time series of persistence diagrams ◮ Pick out information from each diagram (max pers) to use standard

TSA methods

◮ Analyze speed ◮ Persistence of persistence

(Kramar, Levanger, et al. 2015 arXiv:1505.06168)

Structures and behaviors that are easy to tease out

◮ Circles/holes ◮ Periodicty Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 29 / 30

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SLIDE 54

Thank you!

Hurricanes Kristen Corbosiero (Albany) Jason Dunion (Albany) Bill Dong (Guilderland High School) Ryan Torn (Albany) Machining Dynamics Firas Khasawneh (SUNY Poly) Jose Perea (MSU) FK and EM. Chatter detection in turning using persistent homology. Mechanical Systems and Signals Processing, 2016. elizabethmunch.com emunch@albany.edu

Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 30 / 30