Optimal Sampling Strategies for Multiscale Stochastic Processes

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

2. Motivation probe packets 0 T Global (space/time) average Limited number of local samples • 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 How to optimally place N samples to estimate the global quantity?

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

4. Independent Innovations Trees split N • 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

5. 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 cj(only a function of distance), covariance between root and any leaf node is constant,  • Positively correlation progression: cj>cj+1 • Negatively correlation progression: cj<cj+1

6. Covariance Tree Result • Optimality proof:Simply construct an independent innovations tree with similar correlation structure • Worst case proof: Based on eigenanalysis

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

8. 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

9. Water-Filling • : arbitrary set of leaf nodes; : size of X • : leaves on left, : leaves on right • : linear min. mean sq. error of estimating root using X 0 1 2 4 3 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 fL(l) fR(l) 0 1 2 3 4 0 1 2 3 4

10. 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 SX Using eigen analysis show that this implies that uniform choice minimizes sum of elements of S-1X