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About 2,560 papers for “Multivariate random variable”

Variable Importance Measures for Multivariate Random Forests

Sharmistha Sikdar, Giles Hooker, Vrinda Kadiyali · Journal of Data Science · 2024

Multivariate random forests (or MVRFs) are an extension of tree-based ensembles to examine multivariate responses. MVRF can be particularly helpful where some of the responses exhibit sparse (e.g., zero-inflated) distributions, making borrowing strength from correlated features attractive. Tree-based algorithms select features using variable importance measures (VIMs) that score each covariate based on the strength of dependence of the model on that variable. In this paper, we develop and propose new VIMs for MVRFs. Specifically, we focus on the variable’s ability to achieve split improvement, i.e., the difference in the responses between the left and right nodes obtained after splitting the parent node, for a multivariate response. Our proposed VIMs are an improvement over the default naïve VIM in existing software and allow us to investigate the strength of dependence both globally and on a per-response basis. Our simulation studies show that our proposed VIM recovers the true predictors better than naïve measures. We demonstrate usage of the VIMs for variable selection in two empirical applications; the first is on Amazon Marketplace data to predict Buy Box prices of multiple brands in a category, and the second is on ecology data to predict co-occurrence of multiple, rare bird species. A feature of both data sets is that some outcomes are sparse — exhibiting a substantial proportion of zeros or fixed values. In both cases, the proposed VIMs when used for variable screening give superior predictive accuracy over naïve measures.

Cited by 3Open access

Multivariate random forest model standardized variable importance.

James M. Alfieri (8066966), Tavis K. Anderson (7380986) · Figshare · 2019

<p>Variable importance in the model predicting: direct life-cycle parasite community structure at the infracommunity scale (A); indirect life-cycle parasite community structure at the infracommunity scale (B); direct life-cycle parasite community structure at the component community scale (C); and indirect life-cycle parasite community structure at the component community scale (D).</p>

Cited by 0Open access

MulvariateRandomForestVarImp: Variable Importance Measures for Multivariate Random Forests

Sikdar Sharmistha, Hooker Giles · 2021

Calculates two sets of post-hoc variable importance measures for multivariate random forests. The first set of variable importance measures are given by the sum of mean split improvements for splits defined by feature j measured on user-defined examples (i.e., training or testing samples). The second set of importance measures are calculated on a per-outcome variable basis as the sum of mean absolute difference of node values for each split defined by feature j measured on user-defined examples (i.e., training or testing samples). The user can optionally threshold both sets of importance measures to include only splits that are statistically significant as measured using an F-test.

Cited by 0Open access

Variable selection in multivariate linear regression with random predictors

Alban Mbina Mbina, Guy Martial Nkiet, Assi N’Guessan · South African Statistical Journal · 2023

We propose a method for variable selection in multivariate regression with random predictors. This method is based on a criterion that permits to reduce the variable selection problem to a problem of estimating a suitable set. Then, an estimator for this set is proposed and the resulting method for selecting variables is shown to be consistent. A simulation study that permits to study several properties of the proposed approach and to compare it with existing methods is given.

Cited by 2Open access

Variable Importance Measures for Variable Selection and Statistical Inference in Multivariate Random Forests

Sharmistha Sikdar, Giles Hooker, Vrinda Kadiyali · Research Square · 2023

Abstract Multivariate random forests (or MVRFs) are an extension of tree-based ensembles to examine multivariate responses. MVRF can be particularly helpful where some of the responses exhibit sparse (i.e., zero-inflated) distributions, making borrowing strength from correlated features attractive. Tree-based algorithms select features using variable importance measures (VIMs) that score each covariate based on the strength of dependence of the model on that variable. In this paper, we develop and propose new VIMs for MVRFs. Specifically, we focus on the variable’s ability to achieve split improvement, i.e., the difference in the responses between the left and right nodes obtained after splitting the parent node, for a multivariate response. Our proposed VIMs are an improvement over the default VIM in existing software that are based on naïve measures like frequency of variable usage in an MVRF and allow us to investigate the strength of dependence both globally and on a per-response basis. Our simulation studies show that our proposed VIM recovers the true predictors better than naïve measures. We demonstrate usage of the proposed VIMs for variable selection in two empirical applications; the first is on Amazon Marketplace data to predict Buy Box prices of multiple brands in a category, and the second is on ecology data to predict co-occurrence of multiple, rare bird species. A feature of both data sets is that some outcomes are sparse – exhibiting a substantial proportion of zero values. In both cases the proposed VIMs when used for variable screening give superior predictive accuracy over naïve measures.

Cited by 1Open access

Finding PageRank (PR) Using Stochastic Matrix and Multivariate Random Variable

Sreekanth Kavuri Vedavathi Katneni · International Journal of Science and Research (IJSR) · 2023

In the modern world, information is an incredibly important aspect that must not be neglected.In the field of information, several different procedures or algorithms are developed and put to use to extract the required data. Google uses an algorithm called PageRank (PR) to determine where websites should be placed in search engine results. One of the creators of Google, Larry Pag e, was respected for the naming of the company's ranking algorithm. The relevance of every website on the web is quantified using an algorithm called Page Rank. Many issues in web usage mining have yet to be overcome. The greatest way to quantify anything is via a rating. Because there is an ever-increasing wealth of high-quality information available online. The search engine's effectiveness in yielding useful results has a direct bearing on users' levels of happiness. It becomes more challenging without any help s orting through it all.The issue is that it is not simple to locate the desired information.This study proposes an algorithm for determining the rank of mined web pages that uses a transition matrix and random vector to address these problems.The analysis of the experiment reveals that there were 34 separate iterations of 9 pages, each with a unique PR.

Cited by 0Open access

An importance sampling for a function of a multivariate random variable

J.A. Park, Hee-Geon Kang, Sunggon Kim · Communications for Statistical Applications and Methods · 2024

The tail probability of a function of a multivariate random variable is not easy to estimate by the crude Monte Carlo simulation.When the occurrence of the function value over a threshold is rare, the accurate estimation of the corresponding probability requires a huge number of samples.When the explicit form of the cumulative distribution function of each component of the variable is known, the inverse transform likelihood ratio method is directly applicable scheme to estimate the tail probability efficiently.The method is a type of the importance sampling and its efficiency depends on the selection of the importance sampling distribution.When the cumulative distribution of the multivariate random variable is represented by a copula and its marginal distributions, we develop an iterative algorithm to find the optimal importance sampling distribution, and show the convergence of the algorithm.The performance of the proposed scheme is compared with the crude Monte Carlo simulation numerically.

Cited by 0Open access

Advantages of free random variable calculus for main spectral asymptotic formulae in multivariate statistics

Musiał, Krzysztof · Jagiellonian University Repository (Jagiellonian University)

Wiele naturalnych zjawisk jest opisywane probabilistycznie. Obecnie oprócz klasycznej teorii prawdopodobieństwa, teoria nieprzemiennych zmiennych losowych zdobywa popularność. W tej pracy postaramy się pokazać zalety użycia teorii Wolnych Zmiennych Losowych w statystyce wielowymiarowej. Aby to uczynić wyprowadzimy gęstość spektralną tak zwanego wolnego rozkładu F oraz MANOVA używając jedynie narzędzi z teorii wolnego prawdopodobieństwa. Zaprezentujemy także relatywnie krótkie wprowadzenie do koniecznych pojęć.

Cited by 0

The causal effect of class size on academic achievement: Multivariate instrumental variable estimators with data missing at random

Yongyun Shin, Stephen W. Raudenbush · 2016

This article addresses three questions: Does reduced class size cause higher academic achievement in reading, mathematics, listening, and word recognition skills? If it does, how large are these effects? Does the magnitude of such effects vary significantly across schools? The authors analyze data from Tennessee’s Student/Teacher Achievement Ratio study (STAR) of 1985, where students and teachers are randomly assigned to a small or regular class. The authors propose a three-level multivariate simultaneous equation model with an instrumental variable (IV) and estima-tion via maximum likelihood (ML) to analyze the data under an assumption of data missing at random (MAR). The IV, random assignment of students to a small or regular class, reduces class size which, by hypothesis, improves academic achievement in these domains. The authors extend Rubin’s Causal Model (RCM) by involving a modified Stable Unit Treatment Value Assump-tion (SUTVA), requiring no interference between classrooms and intact schools. The method accommodates data with a general missing pattern and extracts full information for analysis from the STAR data. The authors investigate both homogenous and heterogenous causal effects of class size on academic achievement scores across schools. The results show that reducing class size improves reading, mathematics, listening, and word rec-ognition test scores from kindergarten to third grade, although the effects appear relatively small in second grade. The authors find no evidence that the causal effects vary across schools.

Cited by 0

High dimensional variable selection with clustered data: an application of random multivariate survival forests for detection of outlier medical device components

Guy Cafri, Peter Calhoun, Juanjuan Fan · Journal of Statistical Computation and Simulation · 2019

In many medical studies patients are nested or clustered within doctor. With many explanatory variables, variable selection with clustered data can be challenging. We propose a method for variable selection based on random forest that addresses clustered data through stratified binary splits. Our motivating example involves the detection orthopedic device components from a large pool of candidates, where each patient belongs to a surgeon. Simulations compare the performance of survival forests grown using the stratified logrank statistic to conventional and robust logrank statistics, as well as a method to select variables using a threshold value based on a variable's empirical null distribution. The stratified logrank test performs superior to conventional and robust methods when data are generated to have cluster-specific effects, and when cluster sizes are sufficiently large, perform comparably to the splitting alternatives in the absence of cluster-specific effects. Thresholding was effective at distinguishing between important and unimportant variables.

Cited by 3

On the regular conditional distribution of a multivariate Normal given a linear transformation

Rajeshwari Majumdar, Suman Majumdar · arXiv (Cornell University) · 2016

We show that the orthogonal projection operator onto the range of the adjoint of a linear operator T can be represented as UT, where U is an invertible linear operator. Using this representation we obtain a decomposition of a multivariate Normal random variable Y as the sum of a linear transformation of Y that is independent of TY and an affine transformation of TY. We then use this decomposition to prove that the regular conditional distribution of a multivariate Normal random variable Y given a linear transformation TY is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional multivariate Normal random variable, where k < n, the regular conditional distribution of the remaining (n - k)-dimensional component is a (n - k)-dimensional multivariate Normal distribution.

Cited by 0Open access

Inference on the Properties of Probability of Multivariate Fuzzy Random Variables- Part 2

Der-Chen Lin · 2015

For the inference of properties of probability of multivariate fuzzy random variables, the general phenomenon can be divided into determinacy phenomenon and indeterminacy phenomenon. The indeterminacy phenomenon can be divided into random condition and fuzzy condition. Sometimes the random variable concept shall be combined with fuzzy concept to discuss problems. In this article, the theory of random variable of theory of probability is combined with fuzzy set and crisp set to infer the properties of probability of multivariate fuzzy random variables and illustrate them.

Cited by 0

Generation of a Multivariate Distribution for Specified Univariate Marginals and Covariance Structure.

I. R. Goodman · Defense Technical Information Center (DTIC) · 1981

This paper addresses the problem of determining and generating outcomes for a joint multivariate distribution having specified arbitrary marginal distributions and correlation structure. in addition, a new technique is developed which generates outcome for essentially any given multivariate random variable from a more conveniently realizable random variable. This generalizes in a natural way the well known univariate transformation of probability sampling technique. (Author)

Cited by 0

Fast simulation for multifactor portfolio credit risk in the t-copula model

Wanmo Kang, Perwez Shahabuddin · Proceedings of the Winter Simulation Conference, 2005. · 2006

We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the t -copula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and a scaled chi-square random variable. The procedure consists of two steps. First, using the large deviations result for the Gaussian model in Glasserman, Kang, and Shahabuddin (2005a), we devise and apply a change of measure to the chi-square random variable. Then, conditional on the chi-square random variable, we apply the importance sampling procedure developed for the Gaussian copula model in Glasserman, Kang, Shahabuddin (2005b). We support our importance sampling procedure by numerical examples.

Cited by 11

The Gaussian Process

2009

This chapter contains sections titled: The Gaussian Random Variable The Bivariate Distribution The Multivariate Distribution The Gaussian Random Process The Narrow-band Gaussian Random Process Sine Wave Plus Narrow-band Gaussian Random Process Problems

Cited by 8