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MURAL: An unsupervised random forest-based embedding for electronic health record data

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arXiv 2021年

作者： Gerasimiuk, Michal Shung, Dennis Tong, Alexander Stanley, Adrian sch.ltz, Michael Ngu, Jeffrey Laine, Loren Wolf, Guy Krishnaswamy, Smita Dept. of Comp. Sci. Yale University New HavenCT United States Sch. of Medicine Yale University New HavenCT United States Dept. of CS & Oper. Res. Univ. de Montréal Mila MontrealQC Canada Glasgow Royal Infirmary GlasgowG4 0SF United Kingdom Department of Medicine Otago Med. Sch. Dunedin Campus Dunedin New Zealand Christchurch Hospital Christchurch New Zealand Dept. of Math. and Stat. Univ. de Montréal Mila MontrealQC Canada Depts. of Genetics & Comp. Sci. Yale University New HavenCT United States

A major challenge in embedding or visualizing clinical patient data is the heterogeneity of variable types including continuous lab values, categorical diagnostic codes, as well as missing or incomp.ete data. In particular, in EHR data, some variables are missing not at random (MNAR) but deliberately not collected and thus are a source of information. For example, lab tests may be deemed necessary for some patients on the basis of suspected diagnosis, but not for others. Here we present the MURAL forest - an unsupervised random forest for representing data with disparate variable types (e.g., categorical, continuous, MNAR). MURAL forests consist of a set of decision trees where node-splitting variables are chosen at random, such that the marginal entropy of all other variables is minimized by the split. This allows us to also split on MNAR variables and discrete variables in a way that is consistent with the continuous variables. The end goal is to learn the MURAL embedding of patients using average tree distances between those patients. These distances can be fed to nonlinear dimensionality reduction method like PHATE to derive visualizable embeddings. While such methods are ubiquitous in continuous-valued datasets (like single cell RNA-sequencing) they have not been used extensively in mixed variable data. We showcase the use of our method on one artificial and two clinical datasets. We show that using our approach, we can visualize and classify data more accurately than comp.ting approaches. Finally, we show that MURAL can also be used to comp.re cohorts of patients via the recently proposed tree-sliced Wasserstein distances. Copyright © 2021, The Authors. All rights reserved.

关键词： Embeddings

Coarse graining of data via inhomogeneous diffusion condensation

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arXiv 2019年

作者： Brugnone, Nathan Gonopolskiy, Alex Moyle, Mark W. Kuchroo, Manik van Dijk, David Moon, Kevin R. Colon-Ramos, Daniel Wolf, Guy Hirn, Matthew J. Krishnaswamy, Smita Dept. of Comp. Math. Sci. & Eng. Michigan State University East LansingMI United States PicnicHealth Berlin Germany Dept. of Neuroscience Yale University New HavenCT United States Interdept. Neurosci. Prog Yale University New HavenCT United States Dept. of Genetics Dept. of Computer Science Yale University New HavenCT United States Dept. of Math. Stat. Utah State University LoganUT United States Dept. of Math. Stat. Univ. de Montréal Mila MontrealQC Canada Dept. of Comp. Math. Sci. & Eng Dept. of Mathematics Michigan State University East LansingMI United States

Big data often has emergent structure that exists at multiple levels of abstraction, which are useful for characterizing comp.ex interactions and dynamics of the observations. Here, we consider multiple levels of abstraction via a multiresolution geometry of data points at different granularities. To construct this geometry we define a time-inhomogeneous diffusion process that effectively condenses data points together to uncover nested groupings at larger and larger granularities. This inhomogeneous process creates a deep cascade of intrinsic low pass filters on the data affinity graph that are applied in sequence to gradually eliminate local variability while adjusting the learned data geometry to increasingly coarser resolutions. We provide visualizations to exhibit our method as a "continuously-hierarchical" clustering with directions of eliminated variation highlighted at each step. The utility of our algorithm is demonstrated via neuronal data condensation, where the constructed multiresolution data geometry uncovers the organization, grouping, and connectivity between neurons. Copyright © 2019, The Authors. All rights reserved.

关键词： Diffusion

Structure of eigenvectors of random regular digraphs

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arXiv 2018年

作者： Litvak, Alexander E. Lytova, Anna Tikhomirov, Konstantin Tomczak-Jaegermann, Nicole Youssef, Pierre Dept. of Math. and Stat. Sciences University of Alberta EdmontonABT6G 2G1 Canada Faculty of Math. Physics and Comp. Science University of Opole plac Kopernika 11A Opole45-040 Poland Dept. of Math. Princeton University Fine Hall Washington road PrincetonNJ08544 United States Université Paris Diderot Laboratoire de Probabilités Statistiques et Modélisation Paris75013 France

Let d and n be integers satisfying C ≤ d ≤ exp(c√ln n) for some universal constants c, C > 0, and let z ∈ C. Denote by M the adjacency matrix of a random d-regular directed graph on n vertices. In this paper, we study the structure of the kernel of submatrices of M − z Id, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of M, except for constant multiples of (1, 1, . . ., 1), possesses a weak delocalization property: its level sets have cardinality less than Cn ln2 d/ ln n. For a large constant d this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with n. As a key technical ingredient of our proofs we introduce a decomp.sition of Cn into vectors of different degrees of "structuredness," which is an alternative to the decomp.sition based on the least common denominator in the regime when the underlying random matrix is very *** Codes 60B20, 15B52, 46B06, 05C80 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

The rank of random regular digraphs of constant degree

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arXiv 2018年

作者： Litvak, Alexander E. Lytova, Anna Tikhomirov, Konstantin Tomczak-Jaegermann, Nicole Youssef, Pierre Dept. of Math. and Stat. Sciences University of Alberta EdmontonABT6G 2G1 Canada Faculty of Math. Physics and Comp. Science University of Opole ul. Oleska 48 Opole45-052 Poland Dept. of Math. Princeton University Fine Hall Washington road PrincetonNJ08544 United States Université Paris Diderot Laboratoire de Probabilités et de modèles aléatoires Paris75013 France

Let d be a (large) integer. Given n ≥ 2d, let An be the adjacency matrix of a random directed d-regular graph on n vertices, with the uniform distribution. We show that the rank of An is at least n − 1 with probability going to one as n grows to infinity. The proof combines the well known method of simple switchings and a recent result of the authors on delocalization of eigenvectors of An.60B20, 15B52, 46B06, 05C80 Copyright © 2018, The Authors. All rights reserved.

关键词： Directed graphs

Circular law for sparse random regular digraphs

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arXiv 2018年

Fix a constant C ≥ 1 and let d = d(n) satisfy d ≤ lnC n for every large integer n. Denote by An the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show that, as long as d → ∞ with n, the empirical spectral distribution of appropriately rescaled matrix An converges weakly in probability to the circular law. This result, together with an earlier work of Cook, comp.etely settles the problem of weak convergence of the empirical distribution in directed d-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of An based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.60B20, 15B52, 46B06, 05C80 Copyright © 2018, The Authors. All rights reserved.

关键词： Directed graphs

The smallest singular value of a shifted d-regular random square matrix

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arXiv 2017年

作者： Litvak, Alexander E. Lytova, Anna Tikhomirov, Konstantin Tomczak-Jaegermann, Nicole Youssef, Pierre Dept. of Math. and Stat. Sciences University of Alberta EdmontonABT6G 2G1 Canada Faculty of Math. Physics and Comp. Science University of Opole 48 Oleska str. Opole45-052 Poland Dept. of Math. Princeton University Fine Hall Washington road PrincetonNJ08544 United States Université Paris Diderot Laboratoire de Probabilités Statistique et Modélisation Paris75013 France

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let C1 2 n and let Mn,d be the set of all n × n square matrices with 0/1 entries, such that each row and each column of every matrix in Mn,d has exactly d ones. Let M be a random matrix uniformly distributed on Mn,d. Then the smallest singular value sn(M) of M is greater than n-6 with probability at least 1 - C2 log2 d/√d, where c, C1, and C2 are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form M - z Id, where Id is the identity matrix and z is a fixed comp.ex number.60B20, 15B52, 46B06, 05C80 Copyright © 2017, The Authors. All rights reserved.

关键词： Number theory

Uses and misuses of Bayes’ rule and Bayesian classifiers in cybersecurity

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AIP Conference Proceedings 2017年第1期1910卷

作者： Gregory V. Bard Dept. of Math. Stat. and Comp. Sci. Jarvis Hall Science Wing USA The University of Wisconsin—Stout Menomonie Wisconsin 54751 USA

This paper will discuss the applications of Bayes’ Rule and Bayesian Classifiers in Cybersecurity. While the most elementary form of Bayes’ rule occurs in undergraduate coursework, there are more comp.icated forms as well. As an extended example, Bayesian spam filtering is explored, and is in many ways the most triumphant accomp.ishment of Bayesian reasoning in comp.ter science, as nearly everyone with an email address has a spam folder.

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An inequality for detecting financial fraud, derived from the Markowitz Optimal Portfolio Theory

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AIP Conference Proceedings 2016年第1期1789卷

作者： Gregory V. Bard Dept. of Math. Stat. and Comp. Sci. Jarvis Hall Science Wing The University of Wisconsin—Stout Menomonie Wisconsin 54751 USA

The Markowitz Optimal Portfolio Theory, published in 1952, is well-known, and was often taught because it blends Lagrange Multipliers, matrices, stat.stics, and math.matical finance. However, the theory faded from prominence in American investing, as Business departments at US universities shifted from techniques based on math.matics, finance, and stat.stics, to focus instead on leadership, public speaking, interpersonal skills, advertising, etc…

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Expected external profile of PATRICIA tries 11

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Expected external profile of PATRICIA tries

11th Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2014

作者： Magner, Abram Knessl, Charles Szpankowski, Wojciech Dept. Computer Science Purdue University W. LafayetteIN47907 United States Dept. Math. Stat. and Comp. Sci. University of Illinois at Chicago ChicagoIL60607-7045 United States

ISBN: (纸本)9781629939575

We consider PATRICIA tries on n random binary strings generated by a memoryless source with parameter p ≥ 1\2. For both the symmetric (p = 1/2) and asymmetric cases, we analyze asymptotics of the expected value of the external profile at level κ = κ(n), defined to be the number of leaves at level κ. We study three natural ranges of κ with respect to n. For κ bounded, the mean profile decays exponentially with respect to n. For κ growing logarithmically with n, the parameter exhibits polynomial growth in n, with some periodic fluctuations. Finally, for κ = Θ(n), we see super-exponential decay, again with periodic fluctuations. Our derivations rely on analytic techniques, including Mellin transforms, analytic depoissonization, and the saddle point method. To cover wider ranges of k and n and provide more intuitive insights, we also use methods of applied math.matics, including asymptotic matching and linearization. © Copyright (2014) by SIAM: Society for Industrial and Applied math.matics. All rights reserved.

关键词： Linearization