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New steepest descent algorithms for adaptive filtering and have been devised which allow error minimization in the mean fourth and mean sixth, etc., sense. During adaptation, the weights undergo exponential relaxation toward their optimal solutions. Time constants have been derived, and surprisingly they turn out to be proportional to the time constants that would have been obtained if the steepest descent least mean square (LMS) algorithm of Widrow and Hoff had been used. The new gradient algorithms are insignificantly more complicated to program and to compute than the LMS algorithm. Their general form is W J+l = w, t 2plqK-lx,, where W, is the present weight vector, W, + 1 is the next weight vector, r, is the present error, X, is the present input vector, u is a constant controlling stability and rate of convergence, and 2 K is the exponent of the error being minimized. Conditions have been derived for weight-vector convergence of the mean and of the variance for the new gradient algorithms. The behavior of the least mean fourth (LMF) algorithm is of special interest. In comparing this algorithm to the LMS algorithm, when both are set to have exactly the same time constants for the weight relaxation process, the LMF algorithm, under some circumstances, will have a substantially lower weight noise than the LMS algorithm. It is possible, therefore, that a minimum mean fourth error algorithm can do a better job of least squares estimation than a mean square error algorithm. This intriguing concept has implications for all forms of adaptive algorithms, whether they are based on steepest descent or otherwise.

The data demonstrate an excellent sensitivity (100%) of the FFS values in patients with uncomplicated fatty infiltration. This was also the only group of patients in whom the FFS score was superior to the radiologists' best score. The FFS method can be used as a tool to follow up the response to a clinical or research treatment and to obtain standardization of pattern interpretation independently of the individual reader.

A fundamental relationship exists between the quality of an adaptive solution and the amount of data used in obtaining it. Quality is defined here in terms of "misadjustment," the ratio of the excess mean square error (mse) in an adaptive solution to the minimum possible mse. The higher the misadjustment, the lower the quality is. The quality of the exact least squares solution is compared with the quality of the solutions obtained by the orthogonalized and the conventional least mean square (LMS) algorithms with stationary and nonstationary input data. When adapting with noisy observations, a filter trained with a finite data sample using an exact least squares algorithms will have a misadjustment given by ME?, number of weights N number of training samples If the same adaptive filter were trained with a steady flow of data using an ideal "orthogonalized LMS" algorithm, the misadjustment would be M=!L= number of weights 4 Tmse number of training samples Thus, for a given time constant rmse of the learning process, the ideal orthogonalized LMS algorithm will have about as low a misadjustment as can be achieved, since this algorithm performs essentially as an exact least squares algorithm with exponential data weighting. It is well known that when rapid convergence with stationary data is required, exact least squares algorithms can in certain cases outperform the conventional Widrow-Hoff LMS algorithm. It is shown here, however, that for an important class of nonstationary problems, the misadjustment of conventional LMS is the same as that of orthogonalized LMS, which in the stationary case is shown to perform essentially as an exact least squares algorithm.