Multinomial logit models with implicit variable selection

The multinomial logit model is the most widely used model for the unordered multi-category responses. However, applications are typically restricted to the use of few predictors because in the high-dimensional case maximum likelihood estimates frequently do not exist. In this paper we are developing a boosting technique called multinomBoost that performs variable selection and fits the multinomial logit model also when predictors are high-dimensional. Since in multi-category models the effect of one predictor variable is represented by several parameters one has to distinguish between variable selection and parameter selection. A special feature of the approach is that, in contrast to existing approaches, it selects variables not parameters. The method can also distinguish between mandatory predictors and optional predictors. Moreover, it adapts to metric, binary, nominal and ordinal predictors. Regularization within the algorithm allows to include nominal and ordinal variables which have many categories. In the case of ordinal predictors the order information is used. The performance of boosting technique with respect to mean squared error, prediction error and the identification of relevant variables is investigated in a simulation study. The method is applied to the national Indonesia contraceptive prevalence survey and the identification of glass. Results are also compared with the Lasso approach which selects parameters.     

A comparison of generalized multinomial logit and latent class approaches to studying consumer heterogeneity with some extensions of the generalized multinomial logit model

We calibrate and contrast the recent generalized multinomial logit model and the widely used latent class logit model approaches for studying heterogeneity in consumer purchases. We estimate the parameters of the models on panel data of household ketchup purchases, and find that the generalized multinomial logit model outperforms the best‐fitting latent class logit model in terms of the Bayesian information criterion. We compare the posterior estimates of coefficients for individual customers based on the two different models and discuss how the differences could affect marketing strategies (such as pricing), which could be affected by applying each of the models. We also describe extensions to the scale heterogeneity model that includes the effects of state dependence and purchase history. Copyright © 2011 John Wiley & Sons, Ltd.     

基于Logit离散选择模型的品类优化问题综述

品类优化问题(Assortment Optimization Problem)是收益管理的经典问题.它研究零售商在满足运营约束的前提下,应如何从给定产品集合中选择一个子集提供给消费者,以最大化预期收益.该问题的核心在于如何准确地刻画消费者在面对细分产品时的选择行为、建立相应的优化模型并设计高效率的求解算法.基于Logit离散选择模型的品类优化问题:首先,介绍了基于Multinomial Logit模型的品类优化问题.然后介绍了两个更复杂的变种:第一个是基于两层以及多层Nested Logit模型的品类优化问题,这类问题可合理刻画细分产品之间的"替代效应";第二个是基于Mixtures of Multinomial Logits模型的品类优化问题,这类问题可充分考虑消费者群体的异质性.随后,介绍了数据驱动的品类优化问题的相关进展.最后,指出该问题未来可能的若干研究方向.     

Assortment optimization with repeated exposures and product-dependent patience cost

In this paper, we study the assortment optimization problem faced by many online retailers such as Amazon. We develop a cascade multinomial logit model, based on the classic multinomial logit model, to capture the consumers' purchasing behavior across multiple stages. Unlike most of existing studies, our model allows for repeated exposures of a product. In addition, each consumer has a patience budget that is sampled from a known distribution and each product is associated with a patience cost, which is the required amount of the cognitive efforts on browsing that product. We propose an approximation solution to the assortment optimization problem under cascade multinomial logit model.     

Multinomial estimation procedures for two stochastically ordered distributions

Standard estimation procedures in a multinomial setting are the methods of maximum likelihood, minimum chi-square, Neyman modified minimum chi-square, minimum discrimination information, and the Freeman-Tukey criteria. In this paper it is shown that if the parameter region of interest is restricted to be that of stochastic ordering between two multinomial parameters, then all these procedures can also be related as in Dykstra and Lee (1991) and the corresponding estimates can be expressed in terms of a simple weighted least square projection.     

Competitive Price Equilibrium with Consumer Reservation Utility

This paper studies a competitive price equilibrium in the market of a product category where consumers are homogeneous with a reservation utility below which they will not purchase the product. The impact of the reservation utility on the price equilibrium is of particular interest, because the reservation utility may change according to the business cycle and economic environments. Using multinomial logit model to describe market response, we study the comparative statics of the prices, profits and market shares of firms, each of which produces one brand in the product category, with respect to the reservation utility in the Nash equilibrium. It is shown that, as the reservation utility increases, the prices as well as the profits at Nash equilibrium decrease. Also, in the case of duopoly market, the firm with lower cost structure will increase its market share as the reservation utility increases.     

Evaluating the demand for supergauge trucks-on-trains services in Western Europe

We investigate the potential diversion of freight traffic from road to a new rail service in Western Europe. The research aims to predict broad estimates of the traffic diverting to the rail service using a network model and algorithms which optimise the use of time and hence cost for each journey. Traffic diversion is evaluated using a multinomial logit model which allocates volume across the set of shortest paths relevant to a given journey. Results from a case study are presented.     

A Model for Evaluation of Transport Policies in Multimodal Networks with Road and Parking Capacity Constraints

This paper presents a model for evaluation of transport policies in multimodal networks with road and parking capacity constraints. The proposed model simultaneously considers choices of travelers on route, parking location and mode between auto and transit. In the proposed model, it is assumed that auto drivers make a simultaneous route and parking location choice in a user equilibrium manner, and the modal split between auto and transit follows a multinomial logit formulation. A mathematical programming model with capacity constraints on road link and parking facilities is proposed that generates optimality conditions equivalent to the requirements for multimodal network equilibrium. An augmented Lagrangian dual algorithm embedded by partial linearization approach is developed to solve the proposed model. Numerical results on two example networks are presented to illustrate the proposed methodology. The results show that the service level of transit, parking charges, road link and parking capacities, and addition of a new parking location may bring significant impacts on travelers’ behavior and network performance. In addition, transport policies may result in paradoxical phenomenon.     

Visualization of Categorical Response Models: From Data Glyphs to Parameter Glyphs

The multinomial logit model is the most widely used model for nominal multi-category responses. One problem with the model is that many parameters are involved, and another that interpretation of parameters is much harder than for linear models because the model is nonlinear. Both problems can profit from graphical representations. We propose to visualize the effect strengths by star plots, where one star collects all the parameters connected to one term in the linear predictor. In simple models, one star refers to one explanatory variable. In contrast to conventional star plots, which are used to represent data, the plots represent parameters and are considered as parameter glyphs. The set of stars for a fitted model makes the main features of the effects of explanatory variables on the response variable easily accessible. The method is extended to ordinal models and illustrated by several datasets. Supplementary materials are available online.     

Penalized multinomial mixture logit model

Normal distribution based discriminant methods have been used for the classification of new entities into different groups based on a discriminant rule constructed from the learning set. In practice if the groups are not homogeneous, then mixture discriminant analysis of Hastie and Tibshirani (J R Stat Soc Ser B 58(1):155–176, 1996) is a useful approach, assuming that the distribution of the feature vectors is a mixture of multivariate normals. In this paper a new logistic regression model for heterogenous group structure of the learning set is proposed based on penalized multinomial mixture logit models. This approach is shown through simulation studies to be more effective. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using this penalized multinomial mixture logit model as compared to the classical discriminant rules. It also showed better results when applied to practical life data problems producing smaller errors.     

Linear regression with special coefficient features attained via parameterization in exponential,logistic, and multinomial–logit forms

Multiple linear regression with special properties of its coefficients parameterized by exponent, logit, and multinomial functions is considered. To obtain always positive coefficients the exponential parameterization is applied. To get coefficients in an assigned range, the logistic parameterization is used. Such coefficients permit us to evaluate the impact of individual predictors in the model. The coefficients obtained by the multinomial–logit parameterization equal the shares of the predictors, which is useful for interpretation of their influence. The considered regression models are constructed by nonlinear optimization techniques, have stable solutions and good quality of fit, have simple structure of the linear aggregates, demonstrate high predictive ability, and suggest a convenient way to identify the main predictors.     

A bayesian approach to the probability density estimation

Summary A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution. The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's Bayesian Information Criterion) is minimized. The basic procedure is developed under the assumption that the density is defined on a bounded interval. The handling of the general case where the support of the density function is not necessarily bounded is also discussed. The practical usefulness of the procedure is demonstrated by numerical examples. The Institute of Statistical Mathematics     

A fractional programming approach for retail category price optimization

We present a new mixed-integer programming (MIP) approach to study certain retail category pricing problems that arise in practice. The motivation for this research arises from the need to design innovative analytic retail optimization techniques at Oracle Corporation to not only predict the empirical effect of price changes on the overall sales and revenue of a category, but also to prescribe optimal dynamic pricing recommendations across a category or demand group. A multinomial logit nonlinear optimization model is developed, which is recast as a discrete, nonlinear fractional program (DNFP). The DNFP model employs a bi-level, predictive modeling framework to manage the empirical effects of price elasticity and competition on sales and revenue, and to maximize the gross-margin of the demand group, while satisfying certain practical side-constraints. This model is then transformed by using the Reformulation–Linearization Technique in tandem with a sequential bound-tightening scheme to recover an MIP formulation having a relatively tight underlying linear programming relaxation, which can be effectively solved by any commercial optimization software package. We present sample computational results using randomly generated instances of DNFP having different constraint settings and price range restrictions that are representative of common business requirements, and analyze the empirical effects of certain key modeling parameters. Our results indicate that the proposed retail price optimization methodology can be effectively deployed within practical retail category management applications for solving DNFP instances that typically occur in practice.     

The Construction and Properties of Boundary Kernels for Smoothing Sparse Multinomials

In recent years several authors have investigated the use of smoothing methods for sparse multinomial data. In particular, Hall and Titterington (1987) studied kernel smoothing in detail. It is pointed out here that the bias of kernel estimates of probabilities for cells near the boundaries of the multinomial vector can dominate the mean sum of squared error of the estimator for most true probability vectors. Fortunately, boundary kernels devised to correct boundary effects for kernel regression estimators can achieve the same result for these estimators. Properties of estimates based on boundary kernels are investigated and compared to unmodified kernel estimates and maximum penalized likelihood estimates. Monte Carlo evidence indicates that the boundary-corrected kernel estimates usually outperform uncorrected kernel estimates and are quite competitive with penalized likelihood estimates.     

Drawing inferences from logit models for panel data

Logit models have been widely used in marketing to predict brand choice and to make inference about the impact of marketing mix variables on these choices. Most researchers have followed the pioneering example of Guadagni and Little, building choice models and drawing inference conditional on the assumption that the logit model is the correct specification for household purchase behaviour. To the extent that logit models fail to adequately describe household purchase behaviour, statistical inferences from them may be flawed. More importantly, marketing decisions based on these models may be incorrect. This research applies White's robust inference method to logit brand choice models. The method does not impose the restrictive assumption that the assumed logit model specification be true. A sandwich estimator of the covariance ‘corrected’ for possible mis‐specification is the basis for inference about logit model parameters. An important feature of this method is that it yields correct standard errors for the marketing mix parameter estimates even if the assumed logit model specification is not correct. Empirical examples include using household panel data sets from three different product categories to estimate logit models of brand choice. The standard errors obtained using traditional methods are compared with those obtained by White's robust method. The findings illustrate that incorrectly assuming the logit model to be true typically yields standard errors which are biased downward by 10–40 per cent. Conditions under which the bias is particularly severe are explored. Under these conditions, the robust approach is recommended. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient formulations for pricing under attraction demand models

We propose a modeling and optimization framework to cast a broad range of fundamental multi-product pricing problems as tractable convex optimization problems. We consider a retailer offering an assortment of differentiated substitutable products to a population of customers that are price-sensitive. The retailer selects prices to maximize profits, subject to constraints on sales arising from inventory and capacity availability, market share goals, bounds on allowable prices and other considerations. Consumers’ response to price changes is represented by attraction demand models, which subsume the well known multinomial logit (MNL) and multiplicative competitive interaction demand models. Our approach transforms seemingly non-convex pricing problems (both in the objective function and constraints) into convex optimization problems that can be solved efficiently with commercial software. We establish a condition which ensures that the resulting problem is convex, prove that it can be solved in polynomial time under MNL demand, and show computationally that our new formulations reduce the solution time from days to seconds. We also propose an approximation of demand models with multiple overlapping customer segments, and show that it falls within the class of demand models we are able to solve. Such mixed demand models are highly desirable in practice, but yield a pricing problem which appears computationally challenging to solve exactly.     

Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.     

Reference priors for exponential families with simple quadratic variance function

Reference analysis is one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are often difficult to obtain. Recently developed theory for conditionally reducible natural exponential families identifies an attractive reparameterization which allows one, among other things, to construct an enriched conjugate prior. In this paper, under the assumption that the variance function is simple quadratic, the order-invariant group reference prior for the above parameter is found. Furthermore, group reference priors for the mean- and natural parameter of the families are obtained. A brief discussion of the frequentist coverage properties is also presented. The theory is illustrated for the multinomial and negative-multinomial family. Posterior computations are especially straightforward due to the fact that the resulting reference distributions belong to the corresponding enriched conjugate family. A substantive application of the theory relates to the construction of reference priors for the Bayesian analysis of two-way contingency tables with respect to two alternative parameterizations.     

A comparison of linear reformulations for multinomial logit choice probabilities in facility location models

In the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model. Three linear reformulations of the original non-linear model have been proposed so far. In this paper, we discuss these models in terms of solvability. We present empirical findings based on synthetic data.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.