Geometric Approaches to Reconstructing Times Series Project Outline 15 February 2007 CSC/Math 870 Computational Discrete Geometry Connie Phong

Problem Statement How to reconstruct a time ordering from data without explicit time information (i.e. time indices)? When does such a scenario arise?

Accurate Time Series for Biological Processes are Difficult to Obtain Members of a population are not synchronized Members within a population often have different process rates Consider carcinogenesis – delayed identification of cancer cells –to investigate early stages must sample from a cell population to possibly reconstruct temporal sequence of events

Problem Formulation f(t) = [x 1 (t), x 2 (t), …, x d (t)] is a continuous vector function V = {f 1, f 2, …, f n } s i unknown time index for f i permutation  of the index set {1, 2, …, n} is a temporal ordering of the points V = {f 1, f 2, …, f n } if  (i) ≤  (j)  s i ≤ s j for all i, j in index set Find  given V

Magwene et al: Ordering Observations Using MSTs f(t) is a 1-D curve embedded in the space of the measurements Assumptions –Distance measured using standard Euclidean inner product (norms) –The embedding distance is monotonically related to the geodesic distance (shortest distance)

Magwene et al: Ordering Observations Using MSTs Let G = {v,e} be a weighted, complete graph –V represents sampled observations – edge weights are distances in the embedding geometry Rules for estimating arc length distances from distribution of observed data:

Magwene et al: Ordering Observations Using MSTs Find G mst –If G mst is a path, then the best estimate of the ordering is G mst –Else If noise is low and sampling intensity is high, the estimated ordering is the diameter path of G mst –Diameter path is the longest shortest path between any two vertices If noise high and/or sampling intensity is low, things become a little more complicated.

Objective To improve Magwene et al.’s algorithm To get there: –In-depth analysis of the algorithm What’s the intuition for exploiting the MST? MST algorithms –Implement algorithm, replicate results on given data –Test on other empirically derived data What realistic data scenarios cause hang-ups? –Look to the related problem of curve reconstruction for tips?