topological time series analysis - tum...methods to signal analysis, foundations of computational...

Topological Time Series Analysis

Jose Perea

Lecture 2: Persistent Homology of Sliding Window Point Clouds

Sliding window embedding

Step/delay

Window size

Dimension

window

Sliding window point-cloud

Sliding Windows and Persistence: An application of topology to signal analysis, J. Perea and J. Harer, 2015

Periodicity Circularity

Period ( ) Roundness (window size )

# of prominent harmonics ( ) Ambient Dimension ( )

# of non-commensurate frequencies Intrinsic Dimension ( )

Today: Persistent Homology of Sliding Window Point Clouds

Persistence Barcode

SW


Rips filtration

Activity 1

• Open the jupyter notebook “2-PersistentHomology”

• Is there a relation between window size and maximum persistence ?

???

Conjecture

Maximum persistence is maximized when

Theorem (Adams, Adamaszek, 2015)

Let denote the circle of unit circumference with geodesic distance. Then

for

The Vietoris-Rips Complex of a Circle, H. Adams and M. Adamasezk, 2015

SW1PerS: Sliding Windows and 1-Persistence Scoring

SW1PerS: Sliding Windows and 1-Persistence Scoring

mp

Sample size

number of periods

SW1PerS: Sliding Windows and 1-Persistence Scoring, J. Perea et. al., 2016

Yeast Metabolic Cycle Data


Rankings of genes in the top 10% (out of 9,330 ) according to SW, and not in the top 10% for any other algorithm

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Activity 2

•Can you differentiate sums of harmonics, from sums of non-commensurate frequencies, using persistence?

Goal:

Given , understand

Persistent homology Rips

filtrationSliding window point cloud

Field of coefficients

Strategy

• Replace by its -truncated Fourier Series

• Understand the geometry of

• Take the limit of the resulting 1D-diagrams as


Theorem 0 (P. and Harer)

Let , and let be finite.

If and are the persistence diagrams of

and , respectively , then

Bottleneck distance

k-th derivative


1. is non-degenerate for

2. is roundest when

Let be s.t. for all ( )

and so that .

Theorem 1 (P. and Harer)


On Convergence…

Let , , and

Pointwise mean-center and normalize

Let , , and

Then and

are Cauchy with respect to , and

Theorem 2 (P. and Harer).


Let be s.t. for all ( )

and so that .

As , with and -dense,

then , with rational coefficients, satisfies

Theorem 3 (P. and Harer).


The Field of Coefficients…

Exercise: Let

Compute the maximum 1D-persistence of

with and coefficients.


Activity 3

• Why are the persistence diagrams different?

If where

, and , then

up to isometry

Why does this happen…


The bounding 2-chain for

Mobius strip Disk

Disk Disk


J. Perea and J. Harer, Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis, Foundations of Computational Mathematics, 2015.

Thanks!

J. Perea, A. Deckard, S. Haase and J. Harer, SW1PerS: Sliding Windows and 1-Persistence Socring; Discovering Periodicity in Time Series Data, Preprint, 2016.

topological time series analysis - tum...methods to signal analysis, foundations of computational...

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