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Not Boring Introduction to Topological Data Analysis

What Is Topological Data Analysis?

Topography is the mathematical study of shapes, with the advent of big data analysis topography has started to be used to visualize what is happening with large data sets in multiple dimensions. Rather than just looking at how two variables form a simple line when compared on a graph topological data analysis unlike a quantitative data analysis allows you to view data in other dimensions to visualize the actual shapes that the data creates also what holes form in those shapes allowing you to discover more interesting knowledge from the data that you have.

statistical data analysis schema

We are able to collect huge amounts of data from areas such as supermarket loyalty schemes and medical records but with larger data sets the methods of analysis have to evolve also. Data mining and even computer vision are now using these methods to help better understand the data that they are trying to analyze.

So How Does Topological Data Analysis and Visualization Work?

How to do data analysis? Topological data analysis uses software to create visuals of the data sets that you are using. While the formula that the software uses can be quite complex the software ensures that you can quickly and easily visualize the data and start looking for patterns within it to enhance your understanding of what the data can tell you. While we may be used to seeing simple visualizations such as a simple x-y plot the software can use  These plots allow you to see distinct;

  • Clusters
  • Loops
  • Holes
  • Tunnels
  • Tendrils

By being able to see distinct shapes within the data it is then possible to infer the relationships between the data enabling you to increase your knowledge of a specific area or allowing you to go off and conduct more specific research in that area.  Having clusters of information allows you to partition those clusters to do additional research in those specific sets of data to gain a better understanding of what the data is telling you.

Getting Technical with Topological Data Analysis

Topology data analysis involves some very complicated theories and applications of statistics and other areas of mathematics. While this article is not going to get involved in the specifics of these theories you can get a pointer as to the areas that are covered through this article. The areas that most applications of topological data analysis cover for analysis of big data are;

  • Homology
  • Morse theory
  • 3-manifold topology and knot theory

Topological data analysis (TDA) is a relatively new field but is one that is growing rapidly in popularity with its ability to effectively provide visualizations of big data and aid understanding. Being able to use it effectively and understand what it can show you is becoming more and more important as researchers seek to find more and more meaning from the enormous amounts of data that they are now able to access. While other fields may be able to show numbers and indices to suggest a relationship between different areas of your data; topological data analysis allows you to see clear visualizations of that data in a manner that can be understood by almost everyone; even if the mathematics behind the analysis would baffle many.

While topological data analysis may have a long way to go it is already providing real insights into many areas of research!