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I am NOT an academic nor a graduate tudent somewhere. However, I read books on Topology out of personal interest. Does anyone here have any interesting links or white papers on using mathematical topological (metric space, etc.) approaches to data mining. After wtaching some PBS series on mathematics, I am also interested in looking at Euler's formulas for data mining as well.
Topology can be related to clustering problems through Voronoi diagrams and tessellations, in particular in the context of keyword taxonomy building in small dimensions. Jean-Yves Dexmier has developed business applications based on topology concepts.
Thanks, Vince. The only BASIC understanding I got recently was in the Tan, Steinbach and Kumar book, "Introduction to Data Mining" (Pearson Education, 2006), Section 7.5.4 (Candidate Generation). The clustering problems must somehow be linked with the association analysis for subgraphs. I have invited him on my LinkedIn account.
The application of topological spaces covers a very wide area of interests, including economics - my favorite. Topology, with respect to economics, is used in consumer preferences, utility and individual/society choice. The topology comes into play within the metric space in determining continuity of a function on an open set with properties that X and Y possess. A topological property with X that is invariant. X and Y would be the preference of say for example a X=big mac or a Y=whopper -- each of the burgers have properties that people must choose between(a set of attributes that follow the properties of a topological space)...hence a preference/choice or the duality pairing. The understanding of a consumer preference requires large amounts of data and performing data mining activities.
Topology is also used in game theory like Nash's work. You would also have to understand fixed point theorems. Go to google and type in "economic consumer preference topology" without the quotes. That search gave me a bunch of acrobat pdfs in the results - try to use the *.edu sites as much as possible.
A good reference for the above explanation would be "Real Analysis with Economic Applications" by Efe A Ok. That book is a very mathematically intensive book but I like reading it - you have been warned. To gain a better understanding of topology, I would recommend an introduction to "real analysis" book for your reading pleasure - it would only help in data mining. Everyone learns differently or in different ways.
Okay, maybe the burger thing was a lame example.
Have fun...I do.
Contrare, Lance. The example was fine. You just wet my appetite more so. The game theory thing is something worth exploring. I would love to borrow the Munkre book from soem college book store, as on Amazon it's over $100. Thanks for your input. :-)
Interesting article on some Air Force officer compiling (DE-CLASSIFIED hopefully) every record of bombs dropped since WW1.
This can sure be used for data mining analyses somehow.
Thanks for sharing. I wonder if the database is publicly available for review. Simplistically, I would like to see a world map with circles on it - the size of the circles would represent the number of bombs dropped. I got the idea from Vincent's credit card fraud map as I was logging into this site. Then correlate that information in different ways to answer a boat load of questions.
BTW: Analyticbridge.com (Vincent) must be doing a good job with SEO, the above phrase to use in a search for consumer preference and some references about game theory, this posting has changed the search results. In order to get what I saw earlier you will need to use the following on Google:
site:*.edu filetype:pdf economic game theory "topological space"
Then you will see the publications about game theory and topological spaces. This post and search results also illustrates another important concept in game theory that, in essence, our actions influence others. Like this posting, as an example, changed the search results enough that the search criteria needed to be changed to get closer results previously.
Algebraic topology associates algebraic invariants (integers, groups, rings, etc) to topological spaces. The field of applied algebraic topology is relatively new and is relevant to this question, I think. The "Topology and Data" by Gunnar Carlson is nice survey. The Wikipedia entry Topological Data Analysis has some more info. I have a couple of talks on my website. "Topological Data Analysis" is intended for undergraduates and "Intro to Applied Alg Top" was given in a seminar.
I find this a nice introduction to the topic of data analysis using topological features of data: http://normaldeviate.wordpress.com/2012/07/01/topological-data-anal...
Vanessa Robins (http://people.physics.anu.edu.au/~vbr110/) did lots of work on this topic. Her webpage has links to many papers.