key: cord-0303068-9pld91ci authors: Nolasco, O.; QuezadaTellez, L. A.; Salazar Flores, Y.; Diaz Hernandez, A. title: COMORBIDITY ANALYSIS: OVERLAPPING SEMICIRCLES WITH WIGNER LAW AND RANDOM MATRIX THEORY date: 2021-08-26 journal: nan DOI: 10.1101/2021.08.23.21262184 sha: 1a6cfe5773a29ecc3d69719ca6a712cd03cfb0cc doc_id: 303068 cord_uid: 9pld91ci In December 2019 COVID-19 appeared as a new pandemic that has claimed the lives of millions of people around the world. This article presents a regional analysis of COVID-19 in Mexico. Due to the comorbidities of Mexican society, the new pandemic implies a higher risk for the population. The study period runs from April 12 to October 5, 2020 (761 665 Patients). In this proposal we apply a unique methodology of random matrix theory in the moments of a probability measure that appears as the limit of the empirical spectral distribution by the Wigner semicircle law. The graphical presentation of the results is done with Machine Learning methods in the SuperHeat maps. With this is possible to analyze the behavior of patients who tested positive for COVID-19 and their comorbidities. We conclude that the most sensitive comorbidities in hospitalized patients are the following three: COPD, Other Diseases and Renal Diseases. Throughout its history, humanity faced different pandemics were millions of people lost their lives in the world. The recently epidemic of those SARS-CoV and MERS-CoV stand out [1] . Currently in December 2019 in the city of Wuhan-China, a series of cases were reported that met criteria for pneumonia with severe characteristics. Due to them, the local health authorities noticed that in the patients an epidemiological relationship with a wholesale seafood market, where wild animals were also sold [2] . For December 31 it was notified to the Chinese Center for Disease Control and Prevention an epidemiological investigation, like at first security measure was the closing of the seafood market to the public on January 1, 2020. Later on January 9, the Chinese government reported the discovery of the new coronavirus; and on January 12 they released their genomic sequence of nCoV-2019. Initially, the epidemic growth rate was 0.10 per day (95% CI) and it was doubling time in 7.4 days. On January 11, the first death was reported in China [1] . On January 13 in Thailand, the first imported case was registered in a 61-year-old patient from Wuhan. The USA reported its first confirmed case on January 20 in a 35-year-old patient who traveled to Wuhan. It was until January 30 that the WHO declared the nCoV-2019 infection an international public health emergency. On February 11, the name of the disease officially changed to COVID-19 (coronavirus disease). The name of the virus, after genomic analysis of the sequences, is SARS-CoV-2 [3] . Wigner has been used with operators in large data dimensions with independent input the Random Matrix Theory (RMT) [14] . Wigner has also analyzed the distribution of the gaps in the energy levels, where they found that they were independent of the underlying matter; surprisingly, this gap distribution is successfully reproduced by RMT [15] . This study focuses on a particular case in Mexico and it is undoubtedly that this methodology is applicable to many countries in the world. One of the approaches proposed in this research is through the Random Matrix Theory (RMT) approach. RMT has its origin and application when John Wishart analyzed properties in multivariate normal populations [16] . Also in the predictions in quantum mechanics, the energy levels can be able calculated by the eigenvalues together with the RMT elements [17] . In general the RMTs can work to analyze the multivariate behavior of data as it is done in this work. One of the most important contributions of this document is Wigner and RMT in comorbidities of patients with COVID-19. The effect of the COVID-19 pandemic in Mexico has been investigated by some authors, like [18] , they used data mining for data analysis. In [19] they analyzed the risk factors for COVID-19 and managed to rank the most determining factors using a multivariate logistic regression. In [20] in 2021 they found a high incidence of comorbidities in deaths that occurred up to August 2020. In and another analysis, [21] in 2021 give predictions on the spread of the pandemic using Bayesian inference. However, as far as we know, there is no study that uses our methodology that has analyzed the Mexican case. This methodology describes a tool which helps to infer weak convergence: with the method of moments in a probability measure. We propose to apply this method for both deterministic and random probability measures. This document studies in depth the concepts of weak convergence of probability measures and random probability measures with the Wigner law. In this application it is important to know the moments of a probability measure or at least some properties of the moments in combination with the fact that a real symmetric matrix is positive definite in the real sense. Oftentimes it will not be of interest if a sequence of numbers really belongs to a probability measure, since we automatically obtain this result when employing the method of moments. This method a priori assuming that the target distribution to have specific moments, so, it can be used to check convergence to a random probability measure. In any case, the essential for the method of moments is the knowledge about the uniqueness of a distribution with given moments, that is, there is at most one distribution with a given sequence of moments. In random matrix theory, the probability measure that appears as the limit of the empirical spectral distribution is a naturally of the semicircle distribution. What we mean by naturally? is that it appears in Wigner's semicircle law, which is the easiest non-trivial random matrix ensemble, for it has standardized entries which are independent up to the symmetry constraint. It is safe to say that the role of the semicircle distribution in random matrix theory is as large as the role of the standard normal distribution in probability theory. To remind the reader, the semicircle distribution is the probability measure. The Wigner matrices are unit matrices, written in an irreducible unit group (SU) and their rotationally (SO) matrices [11] : where Jx, Jy, and Jz are generators of the Lie algebra of the previous groups [22] , that is, there is a non-associative vector of space g, with an alternate bilinear map: g x g ⇥g; (x, y) ⇥ [x, y], satisfying the Jacobi identity, which means that the sum of all even permutations is zero. So these three operators are the components of a vector operator, known as angular momentum. A square matrix called Hermitian matrix if it has the property of A * = A, where A * denotes the conjugate transpose (or Hermitian transpose) of A, that is, where the subscripts i , j are formally defined by . An important property of these matrices is that each Hermitian matrix is diagonalizable and its eigenvalues are real and its eigenvectors are two by two orthogonal [11] . The probability density function f x (t) of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. A distribution has a density function if and only if its cumulative distribution function F X (x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its derivative can be used as probability density [13] : Definition 3. The empirical measure If X 1 , X 2 , ... be a sequence of identically distributed independent random variables with values in R. Where is denoted by P their probability distribution. The empirical measure of P n is a measurable subset A  R. The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko-Cantelli theorem. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function [13] : where 1A is the indicator function. Please note that if it chooses A =[-, then P n (A) is the distribution of the empirical function. A Wigner matrix W n M n (C) is a Hermitian matrix where (X i, j ) with subscripts i