math 4910 & 5910 topological data analysis · 1 prediction methods (supervised learning):...

“What’s topology got to do with data analysis?”

MATH 4910 & 5910Topological Data Analysis

Instructor: Mehmet Aktas

January 9, 2018

1 / 26 Instructor: Mehmet Aktas MATH 4910 & 5910 Topological Data Analysis

Topology is “Pure” math!

http://xkcd.com/435/


Credit

This presentation is inspired by the following:

Robert Ghrist, Barcodes: The Persistent Topology of Data, 2008

Lecture Notes of Sara Kalisnik Verovsek


Outline

1 What is Topology?

2 What is data?

3 Topological Data Analysis


What is Topology?

Outline

1 What is Topology?

2 What is data?



What is Topology?

Topology

Pure branch of mathematics that dates back to 1700’.

Euler in Konigsberg

Konigsberg was a city in Prussia situated on the Pregel river (modern dayKaliningrad, a major industrial center of western Russia). Seven bridgesspanned the various branches of the river as depicted in the picture.

Is possible to cross all seven bridges exactly once and return to a startingpoint in a single stroll?


What is Topology?

What is Topology?


What is Topology?

Why Topology?

Three key ideas:

1 Invariance under deformation

2 Coordinate freeness

3 Compressed representations


What is data?

Outline

1 What is Topology?

2 What is data?



What is data?

Data is Big!

Big Data

Data is everywhere.


https://www.youtube.com/watch?v=1i6uESo98Yo

What is data?

Data Growth

An article by Forbesstates that

Data is growing fasterthan ever beforeBy the year 2020,about 1.7 megabytesof new informationwill be created everysecond for everyhuman being on theplanet.


What is data?

What do data scientists do?

Make discoveries while swimming indata


What is data?

What do data scientists do?

Make discoveries while swimming indata

The statistics represent this significantand growing demand for data scientists.

Data mining tops LinkedIns list of the“hottest skills of 2014”Best Job in USA for 20163,433: Number of Job Openings in2016#16 Highest Paying Job in Demandin 2016

Average Base Salary : $105,395:


What is data?

What is Data science/mining?

Non-trivial extraction of implicit,previously unknown, and potentiallyuseful information from data


What is data?



Storing, organizing and integrating hugeamount of unstructured data


What is data?




a.k.a. KDD (knowledge discovery indatabases)


What is data?





Types of Data


What is data?





Types of Data

Time-series data,


What is data?





Types of Data

Time-series data,

Sequence data,


What is data?





Types of Data

Time-series data,

Sequence data,

Graphs, social networks,


What is data?





Types of Data

Time-series data,

Sequence data,


Multimedia, WWWdata,


What is data?





Types of Data

Time-series data,

Sequence data,


Multimedia, WWWdata,

Text data.13 / 26 Instructor: Mehmet Aktas MATH 4910 & 5910 Topological Data Analysis

What is data?

Application of Data Science

Internet search


What is data?


Internet search

Recommender systems


What is data?


Internet search

Recommender systems

Biological Classification


What is data?


Internet search

Recommender systems

Biological Classification

...


What is data?

Data Mining Tasks

1 Prediction Methods (Supervised Learning): Predict unknown orfuture values of the data using other known data


What is data?

Data Mining Tasks


Classification: Is this A or B?


What is data?

Data Mining Tasks



Anomaly detection: Is this weird?


What is data?

Data Mining Tasks



Anomaly detection: Is this weird?

Regression: How much? Howmany?


What is data?

Data Mining Tasks - Continued

2 Description Methods (Unsupervised learning): Findhuman-interpretable (previously unknown) patterns that describe thedata (unlabeled)


What is data?



Clustering: How is dataorganized?


What is data?



Clustering: How is dataorganized?

Association rule mining:Are these related?


What is data?

What is data?

Collection of data objects andtheir attributes

Simple Case : n × d matrix

n objects with d dimensioneach,d columns are called variables,features or attributes ofobjects


Topological Data Analysis

Outline

1 What is Topology?

2 What is data?




Data

In recent years, data is complex



Data


It is “Big Data”.



Data


It is “Big Data”.It has also very rich features.



Data



Usually both!



Data



Usually both!

The problem in both cases is that thereis not a single story happening in yourdata




Topological Data Analysis (TDA) is the tool that summarizes out theirrelevant stories to get at something interesting.

TDA has applications in

BiologyMedical SciencesScience of VotingMusicsSports...



Basic Idea in TDA

Data has “shape”,



Basic Idea in TDA


shape has “meaning”,



Basic Idea in TDA


shape has “meaning”,

meaning drives “values”.



Data has “shape”

Convert the data into a graph (moregenerally a simplical complex)





Use the data points as the vertices ofthe graph






To locate edges;

Choose a radius r and draw circlescentered at data points






To locate edges;

Choose a radius r and draw circlescentered at data pointsLocate edges between pairs of pointswhen their circles intersect



Which radius?

How many groups are there in this data?



Shape has “meaning” - Persistence Homology

Persistence homology tracks the evolution of the topological featuresof data across scales

Betti numbers present these topological features of the dataquantitatively: β0 is the number of components, β1 is the number ofholes.



Persistence Barcodes



Persistence Barcodes

Barcodes offer an optimum balance between encoding rich shapeinformation and not being computationally intensive.



Meaning drives “values” - Bottleneck Distance

We have persistence barcodes for each data.

How can we compare two persistence barcodes?

A robust metric between two barcodes: bottleneck distance


math 4910 & 5910 topological data analysis · 1 prediction methods (supervised learning):...

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