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From the confusion matrix, we calculated the performance metrics defined in Table 2 for the two defined setups.
Using the phylotype abundance data matrix, we calculated the mean and standard deviation of the bacterial families and oxygen tolerance groups from pre-circumcision (n = 12) and post-circumcision (n = 12) samples in Microsoft Excel (Microsoft Corp., Seattle, USA).
To better reveal the relationships between the GO tips in the annotation matrix, we calculated the Jiang and Conrath's pairwise similarity distance [20] between any two GO tips using the GOSim package [30].
From this matrix we calculated a distance matrix by substracting each entry from the maximal entry.
The input for mothur is the distance matrix we calculated using the pairwise alignment instead of the default multiple alignment.
First, based on a positional frequency matrix, we calculated a corresponding score matrix and a cutoff threshold.
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To measure the sparseness of a matrix, we calculate its 'overall sparsity', which is defined as the percentage of nonzero elements in that matrix.
To measure the column-wise group sparseness of the eigenphone matrix, we calculate its 'column sparsity', which is defined as the percentage of nonzero columns in that matrix.
After building this null adjacency matrix, we calculate the RCA for each cell, applying the pre-process step of our methodology.
(8) r ij = C ij | C ii C jj | Finally, from the correlation matrix we calculate a distance matrix D whose elements are defined in Eq. (9).
From this matrix, we calculate the Gaussian kernel: (1) k (x, y ) = exp − γ | | x − y | | 2 The estimation of the value of γ is done using the method described below in Section 2.3.
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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