Rank estimation of NTD-2

Koki Tsuyuzaki; Manabu Ishii; Itoshi Nikaido

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Rank estimation of NTD-2

KT Koki Tsuyuzaki

MI Manabu Ishii

IN Itoshi Nikaido

This method is extracted from research article: BMC Bioinformatics, Nov 2023

Sctensor detects many-to-many cell–cell interactions from single cell RNA-sequencing data

DOI: 10.1186/s12859-023-05490-y

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To extract the CaHs, scTensor estimates the NTD-2 ranks for each matricized CCI-tensor ( $X^{(n)}$ , $n = 1$ or 2). To be able to focus only on the dimensions that are informative and are not noisy, we used an ad hoc approach for NTD-2 rank estimation.

Because NMF is performed in each matricized CCI-tensor in scTensor, we estimated each rank of NMF based on the residual sum of squares (RSS) [122] as

where ${RSS}_{max}$ is the RSS by full rank NMF, ${RSS}_{min}$ is the RSS by rank-1 NMF, ${RSS}_{k}$ is the RSS by rank-k NMF, and ${thr}_{rank}$ is the threshold value, ranging 0 to 1 (the default value is 0.8). RSS by rank-k NMF is calculated between a data matrix X and the reconstructed matrix from W and H calculated by multiplicative updating rule [60] as follows:

RSS by full-rank and rank-1 NMF is calculated by setting k as J and 1, respectively. With the estimated ranks $(\hat{R 1}, \hat{R 2})$ , NTD-2 was performed, and only the pairs (r1,r2) with large core tensor values are selected as CaHs. In its default mode, scTensor selects CaHs that explain the top 20 pairs sorted by the core tensor values.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

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