Considering Theorem 3.1, (m) steps of the weighted Arnoldi process have been run on (7).
Having in mind Theorem 3.3, suppose that (m) steps of the weighted Arnoldi process have been performed on (6) and (x_{m} = [s_{m},t_{m} ]^{text{T}}) is the exact solution of the correction Eq. (10).
Having in mind the Theorem 3.1, now suppose that (m) steps of the weighted Arnoldi process [7] have been performed on the following matrix: left( {begin{array}{*{20}c} {I - uu^{text{T}} } & 0 0 & {I - B^{text{T}} Bww^{text{T}} } end{array} } right)left( {begin{array}{*{20}c} { - theta I} & A {A^{text{T}} } & { - theta B^{text{T}} B} end{array} } right).
Probability indices (PI) were calculated for each area at each node by counting the number of weighted gain steps (GSW) and weighted loss steps (LSW) on the cladogram resulting from the BEAST analyses (also based on matrix B).
The output of this step is a set of weighted target samples, which will be referred to as "the weighted target dataset," hereafter (6): leftlbraceleft {breve{mathbf{X}}}^{(n)}_{k+1}, breve{pi}^{(n)}_{k+1}right) rightrbrace_{n=1,..,breve{N}_{k+1}} (6).
Normalization and weighted steps of these parameters are important to reduce bias and uncertainty in the final result.
For each read, an amplicon instance was selected randomly (assuming evenl representation of all amplicons in the pool), and a step number was randomly selected for that amplicon with the probabilities of various steps weighted as specified.
In the step of feature extraction, weighted local clustering coefficients of each ROI in relation to the remaining ROIs are extracted from all the constructed brain networks to quantify the prevalence of clustered connectivity around the ROIs.
The first two time steps of each series comprise proton density weighted images that may be used for intensity inhomogeneity correction (see, e.g., [ 11]); however, this intensity correction is not considered here.
The use of weighted residuals greatly reduces the search space by focusing the downstream steps on genes that are significantly associated with the clinical factors.
