Optimal Sampling Strategies for Multiscale Stochastic Processes Vinay Ribeiro Rolf Riedi, Rich Baraniuk (Rice University)

Motivation Probing for RTT (ping, TCP), available bandwidth (pathload, pathChirp) Packet trace collection –Traffic matrix estimation, overall traffic composition Routing/Connectivity analysis –Sample few routing tables Sensor networks –deploy limited number of sensors Global (space/time) average Limited number of local samples How to optimally place N samples to estimate the global quantity? 0 T probe packets

Multiscale Stochastic Processes Nodes at higher scales – averages over larger regions Powerful structure – model LRD traffic, image data, natural phenomena root – global average, leaves – local samples Choose N leaf nodes to give best linear estimate (in terms of mean squared error) of root node Bunched, uniform, exponential? root leaves Scale j Quad-tree

Independent Innovations Trees Each node is linear combination of parent and independent random innovation Recursive top-to-bottom algorithm Concave optimization for split at each node Polynomial time algorithm O(N x depth + (# tree nodes)) Uniformly spaced leaves are optimal if innovations i.i.d. within scale n N-n split N

Covariance Trees Distance : Two leaf nodes have distance j if their lowest common ancestor is at scale j Covariance tree : Covariance between leaf nodes with distance j is c j (only a function of distance), covariance between root and any leaf node is constant,  Positively correlation progression : c j >c j+1 Negatively correlation progression : c j <c j+1

Covariance Tree Result Optimality proof: Simply construct an independent innovations tree with similar correlation structure Worst case proof: Based on eigenanalysis optimalworst-case Positive correlation progression uniformbunch Negative correlation progression bunch (conjecture) uniform

Numerical Results Covariance trees with fractional Gaussian noise correlation structure Plots of normalized MSE vs. number of leaves for different leaf patterns Positive correlation progression Negative correlation progression

Future Directions Sampling –more general tree structures –non-linear estimates –non-tree stochastic processes –leverage related work in Statistics (Bellhouse et al) Internet Inference –how to determine correlation between traffic traces, routing tables etc. Sensor networks – jointly optimize with other constraints like power transmission

: arbitrary set of leaf nodes; : size of X : leaves on left, : leaves on right : linear min. mean sq. error of estimating root using X Water-Filling f L (l)f R (l) N= Repeat at next lower scale with N replaced by l * N (left) and (N-l * N ) (right) Result: If innovations identically distributed within each scale then uniformly distribute leaves, l * N = b N/2 c

Covariance Tree Result Result: For a positive correlation progresssion choosing leaf nodes uniformly in the tree is optimal. However, for negatively correlation progression this same uniform choice is the worst case! Optimality proof: Simply construct an independent innovations tree with similar correlation structure Worst case proof: The uniform choice maximizes sum of elements of S X Using eigen analysis show that this implies that uniform choice minimizes sum of elements of S -1 X