Journal PaperThis paper studies multiscale stochastic processes which are random processes organized on the nodes of a tree. The random variables at different levels on the tree represent time series of samples of a stochastic process at different temporal or spatial cales. We focus on classes of multiscale processes with additional statistical structure connecting scales and seek an optimal linear estimator of coarse scale nodes using an incomplete set of nodes at a finer time scale. We prove that the optimal solution for any tree with so-called independent innovations is readily given by a polynomial-time algorithm which we term the water-filling algorithm. The optimal solutions vary dramatically with the correlation structure of the multiscale process. For so-called scale-invariant trees and processes with positive correlation progression through scales, uniformly spaced leaves are optimal and clustered leaves are the worst possible. For processes with negative correlation progression, uniformly spaced leaves are the worst possible. Our results have implications for network traffic estimation, sensor network design, and environmental monitoring.Texas InstrumentsDefense Advanced Research Projects AgencyNational Science Foundatio
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