Download Optimal

Optimal Sampling Strategies for
Multiscale Stochastic Processes
Vinay Ribeiro
Rolf Riedi, Rich Baraniuk
(Rice University)
Motivation
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
probe
packets
How to optimally
placeTN samples
0
to estimate the global quantity?
Multiscale Stochastic Processes
root
Scale
j
leaves
• 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?
Quad-tree
Independent Innovations Trees
split
N
n
N-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
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
Covariance Tree Result
Positive
correlation
progression
Negative
correlation
progression
optimal
worst-case
uniform
bunch
bunch
(conjecture)
uniform
• Optimality proof: Simply construct an independent
innovations tree with similar correlation structure
• Worst case proof: Based on eigenanalysis
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
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
•
•
•
3
4
2
1
N= 0
• 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)
0 1 2 34
fR(l)
0 1 2 34
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