The analysis of biological data
Our experiments show that vicus is a more robust alternative to traditional laplacian matrix for network tions: laplacian vs this section we consider predetermined 2d and 3d structures, represent them as a graph and analyze the performance of local vicus as compared to traditional laplacian in the task of graph-based dimensionality , let us consider a particular type of protein fold that has a complex structure in which four pairs of antiparallel beta sheets, only one of which is adjacent in sequence, are wrapped in three dimensions to form a barrel shape. The multiplicity of the eigenvalue 0 of both and equals the number of connected components in the n is the number of nodes in the network.
Analysis of biological data
Main reason we chose these four single-cell datasets is that their ground-truth labels have been validated either experimentally or computationally in their original studies. The partition indicator matrix with hij = 1 if the node i is classified with cluster j and hij = 0 the stability measure on time t is defined in terms of the clustered auto-covariance the stability curve of the network is obtained by maximizing this measure over all possible partitions:A good clustering over time t will have large stability, with a large trace of rt over such a time variation is defined in terms of the asymptotic stability induced by going from the ‘finest’ to the ‘next finest’ partitions is:Where u2 is the normalized fiedler eigenvector with its corresponding eigenvalue λ2.
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We postulate, that in tasks where it is important to take into account local network information, spectral-based methods should be using vicus matrix in place of on: wang b, huang l, zhu y, kundaje a, batzoglou s, goldenberg a (2017) vicus: exploiting local structures to improve network-based analysis of biological data. An illustrative example of comparison between laplacian and vicus to illustrate their sensitivity to hyper-parameters used in the construction of similarity first column shows the groundtruth of the data distribution.
An illustrative example showing vicus is more robust to noise and outliers compared to a shows the underlying ground-truth network heatmap consisting of 3 connected components. Number of objects in a pair are placed in the same group in u and in different groups in v;.
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It is no surprise that networks became a representation of choice for many problems in biology and medicine including gene-gene and protein-protein interaction networks [1], diseases [2] and their interrelations [3], cancer subtyping [4], genetic diversity [5], image retrieval [6], dimensionality reduction [7, 8] and many other applications. Unfortunately, the laplacian does not take into account intricacies of the network’s local structure and is sensitive to noise in the network.
Moreover, recently algorithms designed to capture the local structure of the data have been shown to significantly outperform global methods [9, 10]. Note that βi is the last row of the transition kernel , hence we have , considering the sum of each row of li is all one.
J mach learn res, 11(18):2837–2854, wm objective criteria for the evaluation of clustering methods journal of the american statistical association, 66(336), 846–850, gx, terry jm, belgrader p, ryvkin p, bent zw, wilson r, et al. The final grouping of datapoints into clusters is achieved by performing k-means clustering on q as in [29].
It is consistently observed across six datasets that vicus can select better features than sionthe proposed vicus matrix for weighted networks exhibits greater power to represent the underlying cluster structures of the networks than the traditional global laplacian. Using vicus in place of the laplacian matrix helps spectral decomposition to transform the original data to the latent space with reduced complexity while preserving the contiguity and the cluster memberships of the original 1.
Eigenvectors associated with the laplacian matrix of the weighted network are used in many tasks (e. Inside the definition of stability (materials and methods), the laplacian is used in a markov process on the network which allows to compare and rank partitions at each analyze the stability of our method we partition a protein-protein interaction(ppi) network, which consists of 7,613 interactions between 2,283 escherichia coli proteins [24].
This task is more challenging than traditional clustering problems due to the intrinsic complexity of the cell captured by the ppi network. Networks entail important topological features and patterns critical to understanding interactions within complicated biological systems.
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Here, βi represents the convergence of the label propagation for the datapoint i (note that the original matrix was constructed as the concatenation of the neighborhood of i and datapoint i as the last row). We define our vicus score (vs) analogously to laplacian score:For each data set presented in table 2, we rank the features by laplacian score and vicus score.
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Using vicus in place of the laplacian allows spectral methods to exploit local structures and makes them a lot more relevant to a variety of biological this paper we introduce vicus and compare its performance to the laplacian across a wide range of tasks. Visualization of low-dimensional representations for single cells learned by vicus and global columns represent the embedding results for buettner data, kolodziejczyk data, pollen data, and usoskin data respectively.
We applied our method on a scrna-seq data consisting of 2700 peripheral blood mononuclear cells (pbmc). Proceedings of the 30th annual international acm sigir conference on research and development in information retrieval, 119–126, 2007.
In all our experiments, as suggested in [28]) and is the scaled cluster indicator vector of the subnetwork . The unique challenges associated with single-cell rna-seq data include large noise in quantification of transcriptomes and high dropout rates, therefore reducing the usability of traditional unsupervised clustering methods.
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These approaches aim to reconstruct each data point using its local neighbours and have been shown to be robust and powerful for unweighted networks. Pmid:er f, natarajan kn, casale fp, proserpio v, scialdone a, theis fj, et al.