By Principe J., Liu W., Haykin S.

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**Example text**

20) Adjustment This equation shows that we can compute the minimum mean-square estimate w(i) of the state of a linear dynamic system by adding to the previous estimate w(i − 1), which is premultiplied by the transition matrix A, a correction term equal to k(i)e(i). The correction term equals the prediction error e(i) premultiplied by the gain vector k(i). 14) except for the step of premultiplying the estimate w(i − 1) by the transition matrix A. 21) The problem remains of ﬁnding a recursive way of computing the state-error correlation matrix P(i − 1).

1 ∞ ∞ + ∫−∞ ∫−∞ k2 (τ 1, τ 2 ) x (t − τ 1 ) x (t − τ 2 ) dτ 1dτ 2 2! 1 ∞ ∞ ∞ + ∫−∞ ∫−∞ ∫−∞ k3 (τ 1, τ 2, τ 3 ) x (t − τ 1 ) x (t − τ 2 ) x (t − τ 3 ) dτ 1dτ 2 dτ 3 3! 36) where the kn(τ1, τ2, …, τn) are called the Volterra kernels of the system. ” The determination of the Volterra kernels are generally complicated. Common methods include the harmonic input method, direct expansion method, and powers of transfer function method. ” If the system to be modeled is “strongly nonlinear,” then the Volterra series either takes a long time to converge or often diverges.

Akaike’s information criterion was developed by Hirotsugu Akaike under the name of “Akaike information criterion” in 1971 and proposed in Akaike [1974]. 32) where k is the number of the free parameters in the model and Lmax is the maximized value of the likelihood function for the model. Given a data set, several competing models may be ranked according to their AIC; the one with the lowest AIC is the best. 33) 22 BACKGROUND AND PREVIEW where N is the number of data points and MSE is the mean square error of the data by using the model.