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چكيده انگليسي :

A study on sparse estimation of inverse variance covariance matrices Nasrin Nadafi n nadafi@math iut ac ir Sep 20 2020 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Reyhaneh Rikhtehgaran r rikhtehgaran@cc iut ac ir2000 MSC 62H10 62J07 62J10 Keywords Variance covariance matrix graphical models Abstract This M S c thesis is based on the following papers Friedman J Hastie T and Tibshirani R Sparse inverse covariance estimation with the graphical lasso Bio statistics 9 3 432 441 2007 Tibshirani R Friedman J and Hastie T Applications of the lasso and grouped lasso to the estimation of sparse graphical models 2010 Today due to the advancement of technology it is possible to store and process high dimensional data The main featureof high dimensional data is the large number of variables for each individual The variance covariance matrix is the sim plest tool for measuring the dependence between several variables This matrix contains information about the pairwiserelationship between the components of a random vector The characteristic of this matrix is that it is negative definite which is a constraint on all elements of the matrix Increasing the number of variables rapidly increases the dimension ofthis matrix making it difficult to estimate Thus the sparse estimation of this matrix has been considered On the otherhand estimating the inverse of the variance covariance matrix also known as the accuracy matrix because it representsthe conditional dependencies and conditional independences between each pair of variables conditional on other variableshas been considered in many fields Estimation of this matrix is useful in obtaining joint distribution of variables in thehigh dimension in interpreting graphical models and also in the causal inference In recent years the issue of estimatingthe inverse of the covariance matrix in the high dimension has been an important and challenging task in many areas Usually a small number of variables play an important role in data analysis Sparse estimation by eliminating redundantvariables leads to better interpretability of the model and increasing the accuracy of predictions In this thesis shrinkagemethods have been introduced to obtain the sparse estimation In these methods attempts are made to avoid coefficientsclose to zero as much as possible so that only variables remain in the model that have a completely significant effecton the dependent variable In these methods by regularization the likelihood is penalized by adding a penalty functionincluding a coefficient called tuning parameter which controls the sparsity of the model The graphical lasso is a shrink age method which is usually used to obtain the sparse estimation of the variance covariance matrix and its inverse Thegraphical lasso is a probabilistic graphical model which uses a graph based representation as a basis for the factorizationof a multidimensional joint distribution based on a set of conditional independence In general two common types ofgraphical representations of distributions are called Bayesian networks directional graphical model and Markov net works unidirectional graphical model Both models include factorization properties and display of dependencies Ingraphical models each vertex symbolizes a random variable and the graph represents an image to understand the jointdistribution of a set of random variables In unidirectional graphs like those are used in the graphical lasso edges donot have a directional arrow In these graphs the absence of an edge between two vertices indicates the existence ofconditional independence of the variables related to those two vertices on the condition of the other variables Sparsegraphs have a relatively smaller number of edges Such graphs are used in a variety of topics such as social network

ندافي، نسرين

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ماتريس واريانس-كوواريانس , مدل گرافيكي