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###Questions About my Project

We have D datasets, each with Ni subjects, for i = 1...D.
Each dataset contains graphs, Gni = (V, E, A, Y) for ni=1...Ni.
V is in the set of \mathcal{V}, with cardinality |V|. We know that \mathcal{V} is the same across all datasets.
E is in the set of \mathcal{E} = {V x V: Vm ~ Vp}, for m,p = 1...|V|.
A is an edge attribute which indicates weight and belongs to the set \mathbb{R}+.Implicitly, |A| = |E|.
Y are class labels and belong to the set {0,1}, and indicate subject gender.

Descriptive

  • What is Ni for all i?
  • What is |V|?
  • Are there graphs Gni that are expected but not present?
  • Do the graphs Gni contain any values of A that cannot be processed traditionally (i.e. inf, NaN)?
  • How sparse, |Eni| / |Vni|x|Vni|, are the graphs?

Exploratory

  • What is mean(|E|) for each dataset i?
  • What is the average graph, where average here means the graph consiting of edges and weights corresponding to the average weight of a given potential edge across the all the datasets?
  • What is the distribution of max(A)-min(A) (i.e. dynamic range) for each dataset i?

Inferential

  • If graphs Gni and Gnj for all i != j are processed the same way, is descriminability maximized?

If our brains are samples Xi in \mathcal{X}, then the observed graphs are Yiq = fq(Xi), for processing strategies fq where q=1,...,Q, and where fq: Xi \sim Yi. If we are asking whether or not using the same functional f improves descriminability, our alernate and hypothesis and null become:

p( ||Axy - Ax'y|| \leq || Axy - Ax'y' || ), where x is the graph observed and y is the label associated with the observed graph. If a superscript indicates processing technique, we have:

HA: p( ||Yijq - Yij'q|| < || Yijq - Yi'j'q' || )
H0: p( ||Yijq - Yij'q|| \geq || Yijq - Yi'j'q' || )

Predictive

  • What is the best classifier, h, of the form h in \mathcal{H} such that h: Gni=(V, E, W) \to Yni?*

Causal

  • How does gender (i.e. Y={0,1}) influence the structure of the brain?

Mechanistic

  • How does a difference in gene expression across genders thoughout development influence the structure of the brain?