After having read Gunnar Carlsson's [*Topology and Data*](http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf) I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge databases and I think it can be an interesting exercise to do TDA. I wonder two things about the current state and purpose of TDA (I do understand TDA's main advantages are not for the social sciences though):



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> 1. Are there TDA methods than can be used to establish a relationship among a variable of interest and a set of (possible) explanatory variables? That is, to do some sort of statistical inference?
> 2. Can we make predictions based on TDA tools?
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Any bibliographical reference or explanation about why this is not possible is appreciated.



Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?
Let me answer the broad question first: depending on what you actually want to do, the barcode-type invariants extracted by topological data analysis could be quite useful in your work. And it doesn't take too much prior knowledge to *use* the TDA tools. For instance, if all you want to do is show that two datasets are qualitatively different, you can just compute their barcodes (I've written [software](http://www.math.rutgers.edu/~vidit/perseus) to do this, [as](http://www.mrzv.org/software/dionysus/) [have](http://comptop.stanford.edu/programs/jplex/) [others](http://phat.googlecode.com/%E2%80%8E)) and calculate the difference between them. It's easy, fast and free so why not try something orthogonal and complementary to your existing techniques?




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The usual pipeline for TDA is as follows: starting with your data, you impose the structure of a filtered cell complex, compute the persistent homology, and output the barcode. The reason you may find it difficult to get precise answers to your questions is quite simple: everything depends on how the filtration is concocted! It is a bit of an art form to know exactly what to compute the persistent homology of, given the features that you actually care about.




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Here's a typical TDA approach to your first question: let the dependent variable be $x$ and the independent variables $y\_1,\ldots,y\_n$. 


Assume, by thresholding and binning if necessary, that each $y\_j$ attains only finitely many states. Construct the flag complex on the $n$-partite graph whose vertex-bins correspond to values attained by the $y\_j$s. Each simplex is weighted by the minimum $x$-value corresponding to the $y$-values fixed by its vertices. These weights give you a filtered simplicial complex, and generators of the $0$-dimensional persistence intervals of the super-levelset filtration tell you which configuration of $y$s correspond to which $x$ values.




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Regarding your second question, a lot depends on what you mean by "predictions". For two examples of using persistent homology for predictions, consider Liz Munch's PhD thesis available [here](http://www.elizabethmunch.com/math/research/ElizabethMunch-TimeVaryingPersistence.pdf). It is possible to predict -- up to an extent -- coverage failure in sensor networks modeled as devices with a given failure probability. Perhaps a modification of this model could make predictions in situations that are of interest to you.


