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Register a new userOn the Poisson Trick and its Extensions for Fitting Multinomial Regression Models
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Added by
Jarod Yan Liang Lee
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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This article is concerned with the fitting of multinomial regression models using the so-called Poisson Trick. The work is motivated by Chen & Kuo (2001) and Malchow-M{o}ller & Svarer (2003) which have been criticized for being computationally inefficient and sometimes producing nonsense results. We first discuss the case of independent data and offer a parsimonious fitting strategy when all covariates are categorical. We then propose a new approach for modelling correlated responses based on an extension of the Gamma-Poisson model, where the likelihood can be expressed in closed-form. The parameters are estimated via an Expectation/Conditional Maximization (ECM) algorithm, which can be implemented using functions for fitting generalized linear models readily available in standard statistical software packages. Compared to existing methods, our approach avoids the need to approximate the intractable integrals and thus the inference is exact with respect to the approximating Gamma-Poisson model. The proposed method is illustrated via a reanalysis of the yogurt data discussed by Chen & Kuo (2001).
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This paper considers the problem of variable selection in regression models in the case of functional variables that may be mixed with other type of variables (scalar, multivariate, directional, etc.). Our proposal begins with a simple null model and sequentially selects a new variable to be incorporated into the model based on the use of distance correlation proposed by cite{Szekely2007}. For the sake of simplicity, this paper only uses additive models. However, the proposed algorithm may assess the type of contribution (linear, non linear, ...) of each variable. The algorithm has shown quite promising results when applied to simulations and real data sets.
During the last decades, many methods for the analysis of functional data including classification methods have been developed. Nonetheless, there are issues that have not been adressed satisfactorily by currently available methods, as, for example, feature selection combined with variable selection when using multiple functional covariates. In this paper, a functional ensemble is combined with a penalized and constrained multinomial logit model. It is shown that this synthesis yields a powerful classification tool for functional data (possibly mixed with non-functional predictors), which also provides automatic variable selection. The choice of an appropriate, sparsity-inducing penalty allows to estimate most model coefficients to exactly zero, and permits class-specific coefficients in multiclass problems, such that feature selection is obtained. An additional constraint within the multinomial logit model ensures that the model coefficients can be considered as weights. Thus, the estimation results become interpretable with respect to the discriminative importance of the selected features, which is rated by a feature importance measure. In two application examples, data of a cell chip used for water quality monitoring experiments and phoneme data used for speech recognition, the interpretability as well as the selection results are examined. The classification performance is compared to various other classification approaches which are in common use.
Motivated by the current Coronavirus Disease (COVID-19) pandemic, which is due to the SARS-CoV-2 virus, and the important problem of forecasting daily deaths and cumulative deaths, this paper examines the construction of prediction regions or intervals under the Poisson regression model and for an over-dispersed Poisson regression model. For the Poisson regression model, several prediction regions are developed and their performance are compared through simulation studies. The methods are applied to the problem of forecasting daily and cumulative deaths in the United States (US) due to COVID-19. To examine their performance relative to what actually happened, daily deaths data until May 15th were used to forecast cumulative deaths by June 1st. It was observed that there is over-dispersion in the observed data relative to the Poisson regression model. An over-dispersed Poisson regression model is therefore proposed. This new model builds on frailty ideas in Survival Analysis and over-dispersion is quantified through an additional parameter. The Poisson regression model is a hidden model in this over-dispersed Poisson regression model and obtains as a limiting case when the over-dispersion parameter increases to infinity. A prediction region for the cumulative number of US deaths due to COVID-19 by July 16th, given the data until July 2nd, is presented. Finally, the paper discusses limitations of proposed procedures and mentions open research problems, as well as the dangers and pitfalls when forecasting on a long horizon, with focus on this pandemic where events, both foreseen and unforeseen, could have huge impacts on point predictions and prediction regions.
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.
Use copula to model dependency of variable extends multivariate gaussian assumption. In this paper we first empirically studied copula regression model with continous response. Both simulation study and real data study are given. Secondly we give a novel copula regression model with binary outcome, and we propose a score gradient estimation algorithms to fit the model. Both simulation study and real data study are given for our model and fitting algorithm.
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