The robust EM-type algorithms for log-concave mixtures of regression models

Hu, Hao; Yao, Weixin; Wu, Yichao

doi:10.1016/j.csda.2017.01.004

Cited by 17 publications

(14 citation statements)

References 44 publications

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“…By extending the idea of Hu et al (), Hu et al () proposed a robust EM‐type algorithm for mixture regression models by assuming the component error densities are log‐concave. A density g ( x ) is log‐concave if its log‐density,

ϕ false(x false) = normallog g false(x false)

, is concave.…”

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

“…The likelihood function for the log‐concave mixture of regression model can be presented as

ℓ false(θ, normalg false| normalX, normaly false) = true \sum_{i = 1}^{n} normallog true \sum_{j = 1}^{m} π_{j} g_{j} false(y_{i} - {boldx}_{i}^{T} β_{j} false),

where θ =( π 1 , β 1 ,…, π m , β m ) T , and

g_{j} false(x false) = \exp false{ϕ_{j} false(x false) false}

for some unknown concave function ϕ j ( x ). Hu et al () proposed an EM algorithm to maximise the objective function . Specifically, in the ( k +1)‐th step, the error density g j in the M‐step can be updated by

g_{j}^{false(k + 1 false)} \leftarrow \arg \max_{g_{j} \in double-struckG} true \sum_{i = 1}^{n} p_{i j}^{false(k + 1 false)} normallog g_{j} false(y_{i} - {boldx}_{i}^{T} {\overset{β}{false}}_{j}^{false(k + 1 false)} false), j = 1, …, m,

where

double-struckG

is the family of all log‐concave densities.…”

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

“…Unlike the methods mentioned in Section , the mixture regression model proposed by Hu et al () is semiparametric in the sense that it does not require a specific parametric assumption of error densities and can be adaptive to different error densities with weak log‐concave assumption.…”

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

“…Yu et al () proposed a robust mixture regression via mean‐shift penalisation approach to conduct simultaneous outlier detection and robust mixture model estimation. Hu et al () proposed a robust EM‐type algorithm by assuming the component error densities are log‐concave. In this article, we aim to give a selective overview of some recently developed robust methods for mixture regression models and use a simulation study to compare the performance of some of them.…”

Section: Introductionmentioning

confidence: 99%

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A Selective Overview and Comparison of Robust Mixture Regression Estimators

Yao

Yang

2019

Int Statistical Rev

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Summary Mixture regression models have been widely used in business, marketing and social sciences to model mixed regression relationships arising from a clustered and thus heterogeneous population. The unknown mixture regression parameters are usually estimated by maximum likelihood estimators using the expectation–maximisation algorithm based on the normality assumption of component error density. However, it is well known that the normality‐based maximum likelihood estimation is very sensitive to outliers or heavy‐tailed error distributions. This paper aims to give a selective overview of the recently proposed robust mixture regression methods and compare their performance using simulation studies.

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ϕ false(x false) = normallog g false(x false)

, is concave.…”

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

“…The likelihood function for the log‐concave mixture of regression model can be presented as

ℓ false(θ, normalg false| normalX, normaly false) = true \sum_{i = 1}^{n} normallog true \sum_{j = 1}^{m} π_{j} g_{j} false(y_{i} - {boldx}_{i}^{T} β_{j} false),

where θ =( π 1 , β 1 ,…, π m , β m ) T , and

g_{j} false(x false) = \exp false{ϕ_{j} false(x false) false}

g_{j}^{false(k + 1 false)} \leftarrow \arg \max_{g_{j} \in double-struckG} true \sum_{i = 1}^{n} p_{i j}^{false(k + 1 false)} normallog g_{j} false(y_{i} - {boldx}_{i}^{T} {\overset{β}{false}}_{j}^{false(k + 1 false)} false), j = 1, …, m,

where

double-struckG

is the family of all log‐concave densities.…”

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

Section: Robust Mixture Regression Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Selective Overview and Comparison of Robust Mixture Regression Estimators

Yao

Yang

2019

Int Statistical Rev

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show abstract

“…Ma et al (2018) extended the identifiability result for the model (3.7) by allowing different component error densities and further established the consistency and asymptotic normality of their proposed estimators as well as the estimator proposed by Hunter and Young (2012). Hu et al (2017) assumed the error densities to be log-concave. That is, the model has the same form as (3.7), whereas g c (x) = exp{φ c (x)} for some unknown concave function φ c (x).…”

Section: Nonparametric Errorsmentioning

confidence: 99%

An Overview of Semiparametric Extensions of Finite Mixture Models

Xiang¹,

Yao²,

Yang³

2019

Statist. Sci.

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Finite mixture models have been a very important tool for exploring complex data structures in many scientific areas, for example, economics, epidemiology, finance. In the past decade, semiparametric techniques have been popularly introduced into traditional finite mixture models, and so semiparametric mixture models have experienced exciting development in methodologies, theories and applications. In this article, we provide a selective overview of newly-developed semiparametric mixture models, discuss their estimation methodologies, theoretical properties if applied, and some open questions. Recent developments and some open questions are also discussed.

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Robust structured heterogeneity analysis approach for high‐dimensional data

Sun

Luo

Fan

2022

Statistics in Medicine

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Revealing relationships between genes and disease phenotypes is a critical problem in biomedical studies. This problem has been challenged by the heterogeneity of diseases. Patients of a perceived same disease may form multiple subgroups, and different subgroups have distinct sets of important genes. It is hence imperative to discover the latent subgroups and reveal the subgroup‐specific important genes. Some heterogeneity analysis methods have been proposed in the recent literature. Despite considerable successes, most of the existing studies are still limited as they cannot accommodate data contamination and ignore the interconnections among genes. Aiming at these shortages, we develop a robust structured heterogeneity analysis approach to identify subgroups, select important genes as well as estimate their effects on the phenotype of interest. Possible data contamination is accommodated by employing the Huber loss function. A sparse overlapping group lasso penalty is imposed to conduct regularization estimation and gene identification, while taking into account the possibly overlapping cluster structure of genes. This approach takes an iterative strategy in the similar spirit of K‐means clustering. Simulations demonstrate that the proposed approach outperforms alternatives in revealing the heterogeneity and selecting important genes for each subgroup. The analysis of Cancer Cell Line Encyclopedia data leads to biologically meaningful findings with improved prediction and grouping stability.

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The robust EM-type algorithms for log-concave mixtures of regression models

Cited by 17 publications

References 44 publications

A Selective Overview and Comparison of Robust Mixture Regression Estimators

A Selective Overview and Comparison of Robust Mixture Regression Estimators

An Overview of Semiparametric Extensions of Finite Mixture Models

Robust structured heterogeneity analysis approach for high‐dimensional data

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