Optimal multilinear estimation of a random vector under constraints of causality and limited memory

A new technique is provided for random vector estimation from noisy data under the constraints that the estimator is causal and dependent on at most a finite number p of observations. Nonlinear estimators defined by multilinear operators of degree r are employed, the choice of r allowing a trade-off between the accuracy of the optimal filter and the complexity of the calculations. The techniques utilise an exact correspondence of the nonlinear problem to a corresponding linear one. This is then solved by a new procedure, the least squares singular pivot algorithm, whereby the linear problem can be repeated reduced to smaller structurally similar problems. Invertibility of the relevant covariance matrices is not assumed. Numerical experiments with real data are used to illustrate the efficacy of the new algorithm.