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It can be visualized as a 3D matrix of values called weights (W).
(A_{pop}) refers to the matrix of values that are available for all population subjects (in a survey, these are usually the attributes used in the design of the sample), (B_{obs}) refers to the part of the matrix B with available (observed) values and (B_{mis}) refers to the part with missing (unobserved) values.
And then the resulting collection of functions of the single variable y, i.e., ∀ai: Φ ai, y), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" bi substituted in place of variable y: :∀bj∀ai: Φ ai, bj).
This matrix of values can naturally be viewed as a weighted graph, in which the nodes are residues and the links between the residues are assigned weights according to the results of the pair analysis.
The package takes as input a matrix of values for genes versus samples, and sets up the pipeline for the learning process: initialisation, training and many auxiliary functions.
An unrooted neighbour-joining cladogram was constructed from the genetic distance matrix of values for all pairs of breeds using the R package APE [ 34].
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Let S ∗ be the similarity matrix of value pairs in V i.
(V_{t}) is a diagonal matrix of value-added coefficients, and (F_{t}) is the diagonal matrix of final demand by country of origin/destination.
What's interesting about deep learning is that while there's nothing magical or genie-like about it — as Geordie Wood points out in Wired, "it's really just simple math executed on an enormous scale" — its "programs" are ultimately matrixes of values that have been trained, rather than lines of code which are written (although many lines of traditional code go into the training, of course).
Second, we used DISTLM forward (McArdle and Anderson 2001) to perform multivariate multiple regression with permutation tests on locus-specific pairwise FST distance matrices versus pairwise matrices of values for physical variables (Carmichael et al. 2007; Olsen et al. 2010; Hess et al. 2011; Matala et al. 2011).
This resulted in a 1 1 matrix of expression values and aCGH cbs values, which were used for downstream statistical analysis.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com