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i2,Y)=(I+XpVwXp��)e.i2 Raf inhibitor (A9) It is necessary to solve matrix products and inverse matrices with dimension (p �� p). A.2. Informative Prior Distribution Furthermore, in the following time points t, if last parameter estimation is ?[t ? 1], s2[t ? 1] and Vw[t ? 1] and we have new data X[t] and Y[t], new parameters estimation at this point of time can be computed by treating the prior as additional data points, and then weighting their contribution to the new estimation [20]. To perform the computations, for each prediction value Y[t].i (i = 1, ��, q)it is necessary to construct a new vector of observations Y.i* with new data and last parameter estimations, and predictor matrix X*, and weight matrix �� based on previous variance parameters estimation as follows: Y.i*=[Y[t].iW^[t?1].i] (A10) X*=[X[t]Ip] (A11) ��=[In00Vw[t?1]si2[t?1]] (A12) New parameters estimation at time t could (-)-p-Bromotetramisole Oxalate be written as: W^[t].i=(X*?��?1X*)?1X*?��?1Y.i*i=1,��,q (A13) Vw[t]=(X*?��?1X*)?1 (A14) si2[t]=n0s0i2+n1s1i2n0+n1 (A15) where s1i2=(Y[t].i?X[t]W^[t].i)?(Y[t].i?X[t]W^[t].i)n1?p (A16) moreover, s02 is the variance estimation at time t ? 1 and n0 is its degrees of freedom, and n1 is the degrees of freedom in the new data variance estimation. Computational cost is higher than first step, with more matricial products and more inverse matrices calculus. The Bayesian standard parameters estimation is a simple process but with high resources requirements. B. On-Line Back-Propagation Derivation for ANN Models This section mathematically formalizes the BP algorithm, which is widely known in ANN related literature. These equations are described for completeness and to help the understanding of memory requirements stated at Section 4. For any ANN model, with zero or more hidden layers, the procedure of computing its output ? given its inputs x is known as forward step. The following equations show the computation needed during forward step: h0=x (B1) hj=s(Wj?hj?1+bj),for1��jStattic concentration to compute the loss L(?, y) of the ANN output respect to the given desired output y by using the mean square error. The derivation of this loss function respect to every output and hidden layer is computed by the backprop step by means of the next equations: L(y^,y)=12��y^?y��22+?2��j=1N�Ʀء�Wj��2 (B4) ��N=?L(x,y)?y^=y^?y (B5) ��j=?L(x,y)?hj=hj��?(Wj+1??��j+1)for1��j
