A qth-order tree-based method for multiscale modeling of stochastic dynamic processes

In this paper, based on using an extremely efficient and highly parallelizable algorithm for optimal estimate of multiscale stochastic model, we develop how the multiscale model of q^th-order tree can be used to represent 1-D reciprocal process defined on the unit interval. At first, the initial state of multiscale recursive model is defined as a (q+1)-dimensional vector at the coarsest scale of the q^th-order tree. Second, new interpolation points are produced at 1/q point of the child-interval which is generated from the unit interval by equally dividing into q parts. The corresponding conditional probability density functions are gained. Combining with Markovian we can produce the parameters of multiscale stochastic model by recursion this process and the property of conditional independence. Thus, the q^th-order tree multiscale representation of a class of stochastic process is constructed in detail. A more general q^th-order tree multiscale stochastic model is constructed for the signal and process that have Markovian, which in turn provides the theory basis for solution of the practical problems.