TDA and Persistent Homology: a new method for analysing temporal - - PowerPoint PPT Presentation

▶

tda and persistent homology a new method for analysing

TDA and Persistent Homology: a new method for analysing temporal - - PowerPoint PPT Presentation

Apr 20, 2023 578 likes •1.12k views

TDA and Persistent Homology: a new method for analysing temporal graphs Marco Piangerelli - Emanuela Merelli marco.piangerelli@unicam.it Algorithmic Aspects on Temporal Graphs II ICALP2019 - Patras 08/07/2019 1 Outline Complex Systems

slide-1

SLIDE 1

TDA and Persistent Homology: a new method for analysing temporal graphs

Marco Piangerelli - Emanuela Merelli marco.piangerelli@unicam.it

1

Algorithmic Aspects on Temporal Graphs II ICALP2019 - Patras 08/07/2019

slide-2

SLIDE 2

2

Outline

Complex Systems
From Complex System to temporal graphs
Why Topological Data Analysis?
Topology, Filtration & Homology
Persistent Entropy
Results

slide-3

SLIDE 3

3

Complex Systems

The Human Brain The Stock Market

slide-4

SLIDE 4

4

Complex Systems

The Human Brain The Stock Market

Extracting Emerging GLOBAL behaviors

slide-5

SLIDE 5

5

Complex Systems

The Human Brain (Epileptic Seizures (1h)) The Stock Market (Dow Jones (1980-2017))

slide-6

SLIDE 6

6

Temporal Graphs

t = 0

slide-7

SLIDE 7

…

7

Temporal Graphs

t = 0 t = 1

slide-8

SLIDE 8

…

8

Temporal Graphs

t = 0 t = 1 t = n-1

slide-9

SLIDE 9

…

9

Temporal Graphs

t = n t = 0 t = 1 t = n-1

slide-10

SLIDE 10

10

Temporal Graphs

… 1 2 TIME n-1 n

slide-11

SLIDE 11

11

Temporal Graphs

… 1 2 TIME n-1 n

slide-12

SLIDE 12

Data (Global) Information Knowledge

12

→ →

Why topological data analysis?

slide-13

SLIDE 13

Data (Global) Information Knowledge

13

→ →

Why topological data analysis?

Simplicial Complex Graph

slide-14

SLIDE 14

What is topology?

14

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations:

Allowed: Stretching, Twisting, Bending
Forbidden: Cutting, Gluing

slide-15

SLIDE 15

What is topology?

15

slide-16

SLIDE 16

16

Topological Data Analysis (TDA)

A simplicial complex is a discrete topological space, obtained from the union of simplicies

0-simplex

slide-17

SLIDE 17

17

Topological Data Analysis (TDA)

A simplicial complex is a discrete topological space, obtained from the union of simplicies

0-simplex 1-simplex

slide-18

SLIDE 18

18

Topological Data Analysis (TDA)

A simplicial complex is a discrete topological space, obtained from the union of simplicies

0-simplex 1-simplex 2-simplex

slide-19

SLIDE 19

19

Topological Data Analysis (TDA)

A simplicial complex is a discrete topological space, obtained from the union of simplicies

0-simplex 1-simplex 2-simplex 3-simplex

slide-20

SLIDE 20

20

Topological Data Analysis (TDA)

A simplicial complex is a discrete topological space, obtained from the union of simplicies

0-simplex 1-simplex 2-simplex 3-simplex Simplicial Complex

slide-21

SLIDE 21

21

Topological Data Analysis (TDA)

Homology allows to compute the number of n-dimesional holes

slide-22

SLIDE 22

22

Topological Data Analysis (TDA)

Homology allows to compute the number of n-dimesional holes

A connected component is a 0-dimensional hole

slide-23

SLIDE 23

23

Topological Data Analysis (TDA)

Homology allows to compute the number of n-dimesional holes

A connected component is a 0-dimensional hole A loop of more than 3 vertices is a 1-dimensional hole

slide-24

SLIDE 24

24

Topological Data Analysis (TDA)

Homology allows to compute the number of n-dimesional holes

A connected component is a 0-dimensional hole An empty solid is a cavity, or a tunnel, and it is a 2-dimensional hole A loop of more than 3 vertices is a 1-dimensional hole

slide-25

SLIDE 25

25

Topological Data Analysis (TDA)

Homology allows to compute the number of n-dimesional holes

A connected component is a 0-dimensional hole An empty solid is a cavity,

r a tunnel, and it is a

2-dimensional hole A loop of more than 3 vertices is a 1-dimensional hole 3-dimensional hole

?

slide-26

SLIDE 26

Topological Data Analysis (TDA)

We want to recover the space of origin of our data
We want to obtain some quantity for characterizing the space
Those quantities are the topological invariants
Many topological invariants exist:
A. Euler Characteristics
B. Betti Numbers (ß0, ß1 , …)
C. Torsion Coefficients
D. …

26

slide-27

SLIDE 27

Persistent Homology

27

Hk = ker∂k(Ck) Img∂k+1(Ck) = Zn Bn

rank(Hk): = βk

slide-28

SLIDE 28

Persistent Homology

28

Hk = ker∂k(Ck) Img∂k+1(Ck) = Zn Bn

rank(Hk): = βk

Linear Algebra

slide-29

SLIDE 29

Persistent Homology

29

Hk = ∂ker(Bk) ∂Img(Bk+1)

Hk = βk

Linear Algebra

slide-30

SLIDE 30

Filtration

30

Cech & Vietoris Rips Filtration
Clique Weighted Rank Filtration

slide-31

SLIDE 31

Vietoris Rips Filtration

31

Point Cloud

slide-32

SLIDE 32

Vietoris Rips Filtration

32

Point Cloud

slide-33

SLIDE 33

Vietoris Rips Filtration

33

Point Cloud

slide-34

SLIDE 34

Vietoris Rips Filtration

34

Point Cloud

slide-35

SLIDE 35

Vietoris Rips Filtration

35

Point Cloud

slide-36

SLIDE 36

Vietoris Rips Filtration

36

Point Cloud

slide-37

SLIDE 37

Clique Weighted Rank Filtration

37

A k-Clique is equivalent to a (k-1)-simplex

2-simplex 3 - clique Graphs

slide-38

SLIDE 38

Clique Weighted Rank Filtration

38

A k-Clique is equivalent to a (k-1)-simplex

2-simplex 3 - clique

Bron-Kerbosch (O(3n/3))

slide-39

SLIDE 39

Clique Weighted Rank Filtration

39

slide-40

SLIDE 40

Clique Weighted Rank Filtration

40

slide-41

SLIDE 41

Clique Weighted Rank Filtration

41

slide-42

SLIDE 42

Clique Weighted Rank Filtration

42

slide-43

SLIDE 43

Clique Weighted Rank Filtration

43

slide-44

SLIDE 44

Clique Weighted Rank Filtration

44

slide-45

SLIDE 45

Clique Weighted Rank Filtration

45

slide-46

SLIDE 46

Barcodes & Diagrams

46

Ghrist, 2008,BARCODES: THE PERSISTENT TOPOLOGY OF DATA

slide-47

SLIDE 47

Persistent Entropy

47

PEHk = −

n=Nk

∑

i

li Ltot log li Ltot

PETot = ∑

k

PEHk

li = [deathi − birthi]; LTot = ∑

i

li

slide-48

SLIDE 48

48

Results I

slide-49

SLIDE 49

49

Results I

Merelli, Rucco, Piangerelli, & Toller, D. (2015). A topological approach for multivariate time series characterization: the epilepsy case study.

slide-50

SLIDE 50

50

Results II

slide-51

SLIDE 51

51

Results II

Piangerelli, Tesei, Merelli. (2019). A Persistent Entropy Automaton for the Dow Jones Stock Market. FSEN 2019

slide-52

SLIDE 52

Take home message

TDA is a new paradigm for data analysis
TDA allows to go behind the graph representation
TDA is versatile but computationally expensive
TDA sliding window-based, naturally, tracks, the evolution

in time of the global behavior (Persistent Entropy)

52

slide-53

SLIDE 53

Thank you!

53