Low rank approach in system identification using higher-order statistics

We consider the problem of designing low-rank estimators of higher-order statistics (HOS). In general, low rank estimators have smaller variance than the corresponding full rank estimators at the expense of increased bias. We propose a method for choosing the rank that minimizes the mean squared error associated with the low-rank HOS estimates, and derive analytical expressions for the mean squared error. We present simulation results of system reconstruction based on "best rank" low-rank HOS estimates of the system output, that indicate significant reduction in variance, when compared to the corresponding full-rank result. We also demonstrate that the full-rank mean-square error corresponding to some data length N can be attained by a low-rank estimator corresponding to a length significantly smaller than N.