The orthogonal functions wm(n) are derived from the candidate functions pm(n) using the Gram Schmidt (GS) orthogonalization algorithm. The GS algorithm starts by setting the first orthogonal function w0(n) equal to the first candidate function p0(n):w0(n)=p0(n)(3)The www.selleckchem.com/products/INCB18424.html next orthogonal function w1(n) is found by subtracting a weighted value of w0(n) from the second candidate function p1(n) as given by:w1(n)=p1(n)?��10w0(n)(4)where ��10 is the GS weight.
Now w0(n) and w1(n) are orthogonal to each other, so the correlation of these function should equal zero as given by:w1(n)w0(n)��=p1(n)w0(n)��?��10w02(n)��=0(5)where Inhibitors,Modulators,Libraries Inhibitors,Modulators,Libraries the overbar indicates the average over the N samples of the function, x��=1N��n=0N?1x(n)Solving Equation (5) results in:��10=p1(n)w0(n)��w02(n)��=p1(n)p0(n)��w02(n)��(6)Subsequent orthogonal functions are found by subtracting weighted values of all the previously fitted orthogonal functions from the kth candidate function as given by:wm(n)=pm(n)?��r=0m?1��mrwr(n)(7)where the GS weights can be shown to be given by:��mr=pm(n)wr(n)��wr2(n)��(8)The Inhibitors,Modulators,Libraries Inhibitors,Modulators,Libraries orthogonal functions wm(n) are implicitly defined by the Gram-Schmidt coefficients ��mr and do not need to be computed point-by-point. Using the same procedure as in Equation (5), the Gram-Schmidt coefficients ��mr can be found recursively using the equations [5,6]:D(m,0)=pm(n)p0(n)��(9)D(m,r)=pm(n)pr(n)��?��i=0r?1��riD(m,i)(10)and��mr=pm(n)wr(n)��wr2(n)��=D(m,r)D(r,r)(11)Note, it can be shown that [5,6]:wm2(n)��=D(m,m)(12)and this was used in simplifying Equation (11).
The next step in FOS is to compute the weights of the orthogonal functional expansion gm in Equation (2) that minimize the mean squared error between the functional expansion and the input y(n). The mean squared error is given by:��2(n)��=(y(n)?��m=0Mgmwm(n))2��(13)By Drug_discovery taking the derivative with respect to gm and solving, it can be shown that the values of the gm that minimize the MSE are given by:gm=y(n)wm(n)��wm2(n)��(14)The correlation between the input y(n) and the orthogonal functions wm(n) can be found recursively using the equations:C(0)=y(n)p0(n)��(15)and:C(m)=y(n)pm(n)��?��r=0m?1��mrC(r)(16)Using Equation (12) and Equation (14), the weights gm that minimize the MSE of the orthogonal functional expansion can be found using:gm=C(m)D(m,m)(17)In its last stage, FOS calculates the weights of the original functional expansion am (Equation (1)), from the weights of the orthogonal series expansion, gm, and the weights ��ir.
The value of am can be found recursively using:am=��i
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