This paper analyzes customer product-choice behavior based on the recency and frequency of each customer's page views on e-commerce sites. Recently, we devised an optimization model for estimating product-choice probabilities that satisfy monotonicity, convexity, and concavity constraints with respect to recency and frequency. This shape-restricted model delivered high predictive performance even when there were few training samples. However, typical e-commerce sites deal in many different varieties of products, so the predictive performance of the model can be further improved by integration of such product heterogeneity. For this purpose, we develop a novel latent-class shape-restricted model for estimating product-choice probabilities for each latent class of products. We also give a tailored expectation-maximization algorithm for parameter estimation. Computational results demonstrate that higher predictive performance is achieved with our latent-class model than with the previous shape-restricted model and common latent-class logistic regression.
Latent-class (or mixture) regression @cite_14 @cite_1 is a traditional form of latent-class modeling @cite_2 @cite_36 . It is aimed at classifying a sample into latent classes and simultaneously forming a regression model within these classes. There are two main algorithms for maximum likelihood estimation of latent-class regression models: the Newton--Raphson algorithm @cite_28 and the EM algorithm @cite_35 @cite_30 . While these models are frequently used in marketing and business research to represent consumer heterogeneity, we apply latent-class modeling to the classification of a wide variety of products on e-commerce sites. For this purpose, we develop a latent-class model for estimating the product-choice probabilities that satisfy monotonicity, convexity, and concavity constraints. To the best of our knowledge, none of the existing studies have considered a latent-class model under such a variety of shape restrictions. We also demonstrate that our latent-class model is superior to latent-class logistic regression @cite_31 @cite_26 @cite_20 in terms of predictive performance based on clickstream data.
@cite_30 @cite_35 @cite_14 @cite_26 @cite_36 @cite_28 @cite_1 @cite_2 @cite_31 @cite_20
1604544535 2184753682 2070120478 2122189848
Preface Jacques A. Hagenaars and Allan L. McCutcheon Part I. Introduction: 1. Latent class analysis Leo A. Goodman 2. Basic concepts and procedures in singe- and multiple-group latent class analysis Allan L. McCutcheon Part II. Classification and Measurement: 3. Latent class cluster analysis Jeroen K. Vermunt and Jay Magidson 4. Some examples of latent budget analysis and its extensions Peter G. M. van der Heijden, L. Andries van der Ark and Ab Mooijaart 5. Ordering the classes Marcel Croon 6. Comparison and choice Ulf Bockenholt 7. Three-parameter linear logistic latent class analysis Anton K. Formann and Thomas Kohlmann Part III. 8. Use of categorical and continuous covariates in latent class analysis C. Mitchell Dayton and George B. Macready 9. Directed loglinear modelling with latent variables Jacques A. Hagenaars 10. Latent class models for longitudinal data Linda M. Collins and Brian P. Flaherty 11. Latent markov chains Rolf Langeheine and Frank van de Pol Part IV. Unobserved heterogeneity and non-response: 12. A latent class approach to measuring the fit of a statistical model Tamas Rudas 13. Mixture regression models Michael Wedel and Wayne S. DeSarbo 14. A general latent class approach to unobserved heterogeneity in the analysis of event history data Jeroen K. Vermunt 15. Latent class models for contingency tables with missing data Christopher Winship, Robert D. Mare and John Robert Warren Appendices Index. Latent variable models are a fundamental modeling tool in machine learning applications, but they present significant computational and analytical challenges. The popular EM algorithm and its variants, is a much used algorithmic tool; yet our rigorous understanding of its performance is highly incomplete. Recently, work in (2014) has demonstrated that for an important class of problems, EM exhibits linear local convergence. In the high-dimensional setting, however, the M-step may not be well defined. We address precisely this setting through a unified treatment using regularization. While regularization for high-dimensional problems is by now well understood, the iterative EM algorithm requires a careful balancing of making progress towards the solution while identifying the right structure (e.g., sparsity or low-rank). In particular, regularizing the M-step using the state-of-the-art high-dimensional prescriptions (e.g., Wainwright (2014)) is not guaranteed to provide this balance. Our algorithm and analysis are linked in a way that reveals the balance between optimization and statistical errors. We specialize our general framework to sparse gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, obtaining statistical guarantees for each of these examples. In the present paper the linear logistic extension of latent class analysis is described. Thereby it is assumed that the item latent probabilities as well as the class sizes can be attributed to some explanatory variables. The basic equations of the model state the decomposition of the log-odds of the item latent probabilities and of the class sizes into weighted sums of basic parameters representing the effects of the predictor variables. Further, the maximum likelihood equations for these effect parameters and statistical tests for goodness-of-fit are given. Finally, an example illustrates the practical application of the model and the interpretation of the model parameters. We describe a latent variable model for supervised dimensionality reduction and distance metric learning. The model discovers linear projections of high dimensional data that shrink the distance between similarly labeled inputs and expand the distance between differently labeled ones. The model's continuous latent variables locate pairs of examples in a latent space of lower dimensionality. The model differs significantly from classical factor analysis in that the posterior distribution over these latent variables is not always multivariate Gaussian. Nevertheless we show that inference is completely tractable and derive an Expectation-Maximization (EM) algorithm for parameter estimation. We also compare the model to other approaches in distance metric learning. The model's main advantage is its simplicity: at each iteration of the EM algorithm, the distance metric is re-estimated by solving an unconstrained least-squares problem. Experiments show that these simple updates are highly effective.