# On Fast Converging Data-Selective Adaptive Filtering
## 2 Problem Description
Basically, to apply the threshold \(\tau\) in the coefficient updating (7), we need to calculate \(\sigma_{e}^{2}\). At this point, it should be mentioned that in the system identification application with sufficient order, the minimum MSE in steady-state is \(\sigma_{n}^{2}\), the variance of the measurement noise \(n(k)\). Hence, it is convenient to express \(\sigma_{e}^{2}\) as a function of the noise variance  
\[\sigma_{e}^{2}=(1+\rho)\sigma_{n}^{2} \tag{11}\]  
in which the excess MSE is rewritten as \(\rho\sigma_{n}^{2}\). The expression of \(\rho\) depends on the adaptive algorithm and is key to establish the prescribed probability of update \(P_{\text{up}}\). As a result, the coefficient updating is performed based on a scaled power noise, \(\tau(k)\sigma_{n}^{2}\)[14; 15; 1] so that an equivalent expression to Equation (8) can be rewritten as  
\[P_{\text{up}}(k)=P\left\{\frac{|e(k)|}{\sigma_{n}}>\sqrt{\tau(k)}\right\}-P \left\{\frac{|e(k)|}{\sigma_{n}}>\sqrt{\tau_{\text{max}}}\right\} \tag{12}\]  
resulting in the following modifications in Equations (9) and (10)  
\[P_{\text{up}}=2Q_{e}\left(\frac{\sigma_{n}\sqrt{\tau}}{\sigma_{e}}\right)-2Q _{e}\left(\frac{\sigma_{n}\sqrt{\tau_{\text{max}}}}{\sigma_{e}}\right)\text{ and }\sqrt{\tau}=\sqrt{(1+\rho)}Q_{e}^{-1}\left(\frac{P_{\text{up}}}{2}\right) \text{ for }\tau_{\text{max}}\rightarrow\infty. \tag{13}\]  
If the presence of outliers is known, a possible strategy to eliminate them consists of employing the first 20% of the data without taking into consideration the threshold \(\tau_{\text{max}}\), hence obtaining an estimate of the error behavior. For the remaining iterations, it is calculated by  
\[\sqrt{\tau_{\text{max}}}=\mathbb{E}[e(k)/\sigma_{e}]+3\text{Var}[e(k)/\sigma_ {e}]. \tag{14}\]  
Since the expression (5) represents a Gaussian distribution, we can use the empirical rule given by Equation (14) to identify the values that exceed the threshold as outliers. Under the considered assumptions regarding the error distribution, \(\mathbb{E}\{e(k)\}=0\) and thus,  
\[\sigma_{e}^{2}=\mathbb{E}[e^{2}(k)]=\xi(k) \tag{15}\]  
where \(\xi(k)\) for, \(k\rightarrow\infty\) is the steady-state MSE obtained by employed algorithm. The expression of \(\xi(k)\) depends on the filter application and the algorithm employed. In the following subsections, we compute the steady-state MSE for some adaptive filter applications. Alternatively, the error variance can be estimated by  
\[\sigma_{e}^{2}=(1-b)e^{2}(k)+(b)e^{2}(k-1), \tag{16}\]  
where \(b\) is a forgetting factor. Although not discussed here, for the cases the error distribution is not Gaussian we can determine the threshold based on measured data through the evaluation of tail probabilities, see [17]. It is  
and the MSE expression  
Figure 1:0  
give rise to a expression for the minimum MSE:  
\[\xi_{\min}(k)=r(0)-\mathbf{w}_{\mathrm{o}}^{T} \begin{bmatrix}r(L)\\ r(L+1)\\ \vdots\\ r(L+N)\end{bmatrix}  \tag{25}\]  
where \(\mathbf{w}_{\mathrm{o}}\) is the optimal coefficients of the predictor and \(r(l)=\mathbb{E}[x(k)x(k-l)]\). Since in the prediction case \(\xi(k)=\sigma_{\varepsilon}^{2}\approx\xi_{\min}\), Equation (25) can be used to obtain an estimate of \(\sigma_{\varepsilon}^{2}\) at iteration \(k\) by replacing \(\mathbf{w}_{\mathrm{o}}^{T}\) by \(\mathbf{w}(k)\) which are the coefficients of the adaptive filter at iteration \(k\). We can estimate \(r(l)\) through \(r(l)=\zeta r(l-1)+(1-\zeta)x(k)x(k-l)\) in which \(\zeta\) is a forgetting factor.