If the ith row in the separating matrix produces the ��interestin

If the ith row in the separating matrix produces the ��interesting�� source signal originating from the direction Sorafenib Tosylate order ��i, it should maximize the gain of |ui(��i)|, that is:|wiHVa^i|>��>|wkHVa^i|k=1,2,?,Nk��i(13)where �� is a threshold measuring accuracy degree of the spatial constraint. Thus, an inequality constraint can be defined for the desired output component with the directivity pattern at ��i more than or equal to a threshold ��, that is:J2(w)=��-|wHVa^i|��0(14)Similarly, the problem of spatial constrained ICA can be modeled in the cICA framework as a constrained optimization problem:maxwJ1(w)s.t.J2(w)=��-|wHVa^i|J4(w)=��w��2-1=0��0(15)Firstly, we replace the inequality constraint by the equality constraint max(J2(w), 0) = 0 for simplicity and then a neural algorithm using the augmented Lagrange multipliers method and the gradient ascent learning approach can be derived to obtain the desired optimal solution.
The corresponding Lagrangian function L(w, ��) is given by:L(w,��)=J1(w)-��max|(J2(w),0)(16)The unit-norm constraint in Equation (15) is enforced by the projection of the estimated w on the unit-sphere in each iteration, that is:w=w/��w��(17)Following the strategy proposed in [24], the above optimization problem Inhibitors,Modulators,Libraries is addressed using alternative optimization. That is, given the current estimate ��, a new estimate for w is searched, and then given the estimated for w, we update ��.
The cICA algorithm reported here search Inhibitors,Modulators,Libraries for the optimal solution of w and the Lagrangian parameter �� by using conventional gradient descent for complex variable [24]:w��w+��1?wL�ˡ���-��2max(J2(w),0)(18)where Inhibitors,Modulators,Libraries ��1, ��2 are the corresponding learning rate and wL denotes the gradient vector of L(w, ��) (See the Appendix I):?wL=E2)+��2(sign(J2)+1)[Va^i(wHVa^i)?](19)where g(?) denotes the first-order derivative of G(?). In fact the selection of the learning rate Inhibitors,Modulators,Libraries (step size) is a crucial point and it has been a research focus for several decades where various step size selection schemes have been developed (see for example [25,26]).If there is a subset of SOIs with the same spatial constraint, we need to run the same procedure by re-initializing the extracting vector wi in order to identify the whole subset of SOIs. To prevent different vectors from converging to the same independent component, we must decorrelate the outputs w1Hz, w2Hz, �� As introduced in [6], there are two varieties of the FastICA algorithm: Cilengitide the one-unit, or deflation algorithm selleck inhibitor and the symmetric algorithm. So does our algorithm for extracting a desired subset of SOIs with the same cyclic frequencies.