After having read Gunnar Carlsson's 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 thing about the current state and purpose of TDA (I do understand TDA's main advantages are not for the social sciences though):

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?

Any bibliographical reference or explanation about why this is not possible is appreciated.