Forecasting model with asymmetric market response and its application to pricing of consumer package goods

This paper presents a dynamic forecasting model that accommodates asymmetric market responses to marketing mix variable—price promotion—by threshold models. As a threshold variable to generate a mechanism for different market responses, we use the counterpart to the concept of a price threshold applied to a representative consumer in a store. A Bayesian approach is taken for statistical modelling because of advantages that it offers over estimation and forecasting. The proposed model incorporates the lagged effects of a price variable. Thereby, myriad pricing strategies can be implemented in the time horizon. Their effectiveness can be evaluated using the predictive density. We intend to improve the forecasting performance over conventional linear time series models. Furthermore, we discuss efficient dynamic pricing in a store using strategic simulations under some scenarios suggested by an estimated structure of the models. Empirical studies illustrate the superior forecasting performance of our model against conventional linear models in terms of the root mean square error of the forecasts. Useful information for dynamic pricing is derived from its structural parameter estimates. This paper develops a dynamic forecasting model that accommodates asymmetric market responses to marketing mix variable—price promotion—by the threshold models. Copyright © 2005 John Wiley & Sons, Ltd.     

On the categorization of demand patterns

The categorization of alternative demand patterns facilitates the selection of a forecasting method and it is an essential element of many inventory control software packages. The common practice in the inventory control software industry is to arbitrarily categorize those demand patterns and then proceed to select an estimation procedure and optimize the forecast parameters. Alternatively, forecasting methods can be directly compared, based on some theoretically quantified error measure, for the purpose of establishing regions of superior performance and then define the demand patterns based on the results. It is this approach that is discussed in this paper and its application is demonstrated by considering EWMA, Croston's method and an alternative to Croston's estimator developed by the first two authors of this paper. Comparison results are based on a theoretical analysis of the mean square error due to its mathematically tractable nature. The categorization rules proposed are expressed in terms of the average inter-demand interval and the squared coefficient of variation of demand sizes. The validity of the results is tested on 3000 real-intermittent demand data series coming from the automotive industry.     

On the identification of sales forecasting models in the presence of promotions

Shorter product life cycles and aggressive marketing, among other factors, have increased the complexity of sales forecasting. Forecasts are often produced using a Forecasting Support System that integrates univariate statistical forecasting with managerial judgment. Forecasting sales under promotional activity is one of the main reasons to use expert judgment. Alternatively, one can replace expert adjustments by regression models whose exogenous inputs are promotion features (price, display, etc). However, these regression models may have large dimensionality as well as multicollinearity issues. We propose a novel promotional model that overcomes these limitations. It combines Principal Component Analysis to reduce the dimensionality of the problem and automatically identifies the demand dynamics. For items with limited history, the proposed model is capable of providing promotional forecasts by selectively pooling information across established products. The performance of the model is compared against forecasts provided by experts and statistical benchmarks, on weekly data; outperforming both substantially.     

Comparison of Algorithms for the Degree Constrained Minimum Spanning Tree

The Degree Constrained Minimum Spanning Tree (DCMST) on a graph is the problem of generating a minimum spanning tree with constraints on the number of arcs that can be incident to vertices of the graph. In this paper we develop three heuristics for the DCMST, including simulated annealing, a genetic algorithm and a method based on problem space search. We propose alternative tree representations to facilitate the neighbourhood searches for the genetic algorithm. The tree representation that we use for the genetic algorithm can be generalised to other tree optimisation problems as well. We compare the computational performance of all of these approaches against the performance of an exact solution approach in the literature. In addition, we also develop a new exact solution approach based on the combinatorial structure of the problem. We test all of these approaches using standard problems taken from the literature and some new test problems that we generate.     

The Logic of Analogy

How should a scientist argue when the data are insufficient to allow him to reason by classical or statistical models? After all, in most real world situations - in business or in war - that is the unhappy norm. In such cases the ordinary man instinctively argues by analogy, as Leibniz long ago showed; indeed if time presses, there is no alternative. The trouble, however, is that if we then include such arguments in our scientific reasoning, then, as we all know, this can lead to false conclusions. To escape from this dilemma, is there any alternative logical basis from which we can start our reasoning? What is proposed here is that instead of the well tried three valued logic of true, false or probable, we should adopt the three valued logic of true, false or possible. A rational system for analogue arguments can then be developed by these means, and with it the advantages brought by the use of symbols and so on. Such a method, however, includes many necessary changes as to how to structure our problems and how to apply new criteria; and it is some of these changes that are outlined in this note. For instance, it outlines the meaning of ‘causal relationships’ in analogue arguments, as well as how to define ‘rational choice’ in terms of analogue propositions. The advantage throughout is that this allows us to argue with less rather than more data.     

Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation

We study inexact subspace iteration for solving generalized non-Hermitian eigenvalue problems with spectral transformation, with focus on a few strategies that help accelerate preconditioned iterative solution of the linear systems of equations arising in this context. We provide new insights into a special type of preconditioner with “tuning” that has been studied for this algorithm applied to standard eigenvalue problems. Specifically, we propose an alternative way to use the tuned preconditioner to achieve similar performance for generalized problems, and we show that these performance improvements can also be obtained by solving an inexpensive least squares problem. In addition, we show that the cost of iterative solution of the linear systems can be further reduced by using deflation of converged Schur vectors, special starting vectors constructed from previously solved linear systems, and iterative linear solvers with subspace recycling. The effectiveness of these techniques is demonstrated by numerical experiments.     

Neuro-dynamic trading methods

Investment strategies are usually based on forecasting models, and these are optimized with respect to past predictive performance. However, the main goal of most investors is the optimization of a risk-adjusted performance measure, such as the well-known Sharpe index. This issue has been approached by a few different studies within the area of Neurocomputing. The present paper briefly describes and empirically compares some of the models and methods proposed in those studies. Such adaptive methods can be computationally demanding, and convergence to high-quality solutions can be difficult to achieve, yet they can be very useful in automated trading systems, namely for portfolio management. In particular, the Q-learning algorithm, when combined with neural networks for value function approximation, seems to be a reasonably competitive approach, although not overall superior to alternative ones.     

Concave minimum cost network flow problems solved with a colony of ants

In this work we address the Single-Source Uncapacitated Minimum Cost Network Flow Problem with concave cost functions. This problem is NP-Hard, therefore we propose a hybrid heuristic to solve it. Our goal is not only to apply an ant colony optimization (ACO) algorithm to such a problem, but also to provide an insight on the behaviour of the parameters in the performance of the algorithm. The performance of the ACO algorithm is improved with the hybridization of a local search (LS) procedure. The core ACO procedure is used to mainly deal with the exploration of the search space, while the LS is incorporated to further cope with the exploitation of the best solutions found. The method we have developed has proven to be very efficient while solving both small and large size problem instances. The problems we have used to test the algorithm were previously solved by other authors using other population based heuristics. Our algorithm was able to improve upon some of their results in terms of solution quality, proving that the HACO algorithm is a very good alternative approach to solve these problems. In addition, our algorithm is substantially faster at achieving these improved solutions. Furthermore, the magnitude of the reduction of the computational requirements grows with problem size.     

Judgement and supply chain dynamics

Forecasting demand at the individual stock-keeping-unit (SKU) level often necessitates the use of statistical methods, such as exponential smoothing. In some organizations, however, statistical forecasts will be subject to judgemental adjustments by managers. Although a number of empirical and ‘laboratory’ studies have been performed in this area, no formal OR modelling has been conducted to offer insights into the impact such adjustments may have on supply chain performance and the potential development of mitigation mechanisms. This is because of the associated dynamic complexity and the situation-specific nature of the problem at hand. In conjunction with appropriate stock control rules, demand forecasts help decide how much to order. It is a common practice that replenishment orders may also be subject to judgemental intervention, adding further to the dynamic system complexity and interdependence. The system dynamics (SD) modelling method can help advance knowledge in this area, where mathematical modelling cannot accommodate the associated complexity. This study, which constitutes part of a UK government funded (EPSRC) project, uses SD models to evaluate the effects of forecasting and ordering adjustments for a wide set of scenarios involving: three different inventory policies; seven different (combinations of) points of intervention; and four different (combinations of) types of judgmental intervention (optimistic and pessimistic). The results enable insights to be gained into the performance of the entire supply chain. An agenda for further research concludes the paper.     

Studying the effect of using low-discrepancy sequences to initialize population-based optimization algorithms

In this paper, we investigate the use of low-discrepancy sequences to generate an initial population for population-based optimization algorithms. Previous studies have found that low-discrepancy sequences generally improve the performance of a population-based optimization algorithm. However, these studies generally have some major drawbacks like using a small set of biased problems and ignoring the use of non-parametric statistical tests. To address these shortcomings, we have used 19 functions (5 of them quasi-real-world problems), two popular low-discrepancy sequences and two well-known population-based optimization methods. According to our results, there is no evidence that using low-discrepancy sequences improves the performance of population-based search methods.     

An asymptotical study of combinatorial optimization problems by means of statistical mechanics

The analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result: Under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem, the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely, as the size of the problem tends to infinity, so that the problem of optimization becomes trivial in some sense. Whereas this result can also be proven by purely probabilistic techniques, the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior.     

复杂数据统计过程的若干研究

统计过程控制(statistical process contor, SPC) 是应用统计方法对过程中的各个阶段进行监控,从而达到改进和保证质量的目的. 本文在一些重要的前沿问题上展开研究, 其中包括profile 数据过程的监控和诊断、监测drift 飘移的控制图、多元过程控制和多阶段过程的检测和诊断. 本文引入并开发各种新的统计技术, 紧密结合计算算法, 解决这些当前质量控制领域研究的难点问题.     

Hybrid genetic algorithms for bin-packing and related problems

The genetic algorithm (GA) paradigm has attracted considerable attention as a promising heuristic approach for solving optimization problems. Much of the development has related to problems of optimizing functions of continuous variables, but recently there have been several applications to problems of a combinatorial nature. What is often found is that GAs have fairly poor performance for combinatorial problems if implemented in a naive way, and most reported work has involved somewhat ad hoc adjustments to the basic method. In this paper, we will describe a general approach which promises good performance for a fairly extensive class of problems by hybridizing the GA with existing simple heuristics. The procedure will be illustrated mainly in relation to the problem ofbin-packing, but it could be extended to other problems such asgraph partitioning, parallel-machine scheduling andgeneralized assignment. The method is further extended by usingproblem size reduction hybrids. Some results of numerical experiments will be presented which attempt to identify those circumstances in which these heuristics will perform well relative to exact methods. Finally, we discuss some general issues involving hybridization: in particular, we raise the possibility of blending GAs with orthodox mathematical programming procedures.     

A Framework for Unbiased Model Selection Based on Boosting

Variable and model selection are of major concern in many statistical applications, especially in high-dimensional regression models. Boosting is a convenient statistical method that combines model fitting with intrinsic model selection. We investigate the impact of base-learner specification on the performance of boosting as a model selection procedure. We show that variable selection may be biased if the covariates are of different nature. Important examples are models combining continuous and categorical covariates, especially if the number of categories is large. In this case, least squares base-learners offer increased flexibility for the categorical covariate and lead to a preference even if the categorical covariate is noninformative. Similar difficulties arise when comparing linear and nonlinear base-learners for a continuous covariate. The additional flexibility in the nonlinear base-learner again yields a preference of the more complex modeling alternative. We investigate these problems from a theoretical perspective and suggest a framework for bias correction based on a general class of penalized least squares base-learners. Making all base-learners comparable in terms of their degrees of freedom strongly reduces the selection bias observed in naive boosting specifications. The importance of unbiased model selection is demonstrated in simulations. Supplemental materials including an application to forest health models, additional simulation results, additional theorems, and proofs for the theorems are available online.     

A generalized alternating direction method of multipliers with semi-proximal terms for convex composite conic programming

In this paper, we propose a generalized alternating direction method of multipliers (ADMM) with semi-proximal terms for solving a class of convex composite conic optimization problems, of which some are high-dimensional, to moderate accuracy. Our primary motivation is that this method, together with properly chosen semi-proximal terms, such as those generated by the recent advance of block symmetric Gauss–Seidel technique, is capable of tackling these problems. Moreover, the proposed method, which relaxes both the primal and the dual variables in a natural way with a common relaxation factor in the interval of (0, 2), has the potential of enhancing the performance of the classic ADMM. Extensive numerical experiments on various doubly non-negative semidefinite programming problems, with or without inequality constraints, are conducted. The corresponding results showed that all these multi-block problems can be successively solved, and the advantage of using the relaxation step is apparent.     

Dynamic mortality factor model with conditional heteroskedasticity

In most methods for modeling mortality rates, the idiosyncratic shocks are assumed to be homoskedastic. This study investigates the conditional heteroskedasticity of mortality in terms of statistical time series. We start from testing the conditional heteroskedasticity of the period effect in the naïve Lee-Carter model for some mortality data. Then we introduce the Generalized Dynamic Factor method and the multivariate BEKK GARCH model to describe mortality dynamics and the conditional heteroskedasticity of mortality. After specifying the number of static factors and dynamic factors by several variants of information criterion, we compare our model with other two models, namely, the Lee-Carter model and the state space model. Based on several error-based measures of performance, our results indicate that if the number of static factors and dynamic factors is properly determined, the method proposed dominates other methods. Finally, we use our method combined with Kalman filter to forecast the mortality rates of Iceland and period life expectancies of Denmark, Finland, Italy and Netherlands.     

Synthesis or selection of forecasting models

This report is relevant to the practical forecasting situation in which a decision-maker is faced with several feasible predictors for his variable of interest. If he has a substantial amount of data available on the performance of each of his predictors, then it is well known that a composite forecast can be suitably derived as an optimal forecasting procedure. Alternatively, if only a small amount of evidence is available on the predictors' performance, then there appear to be controversial recommendations upon whether it is still optimal to pursue a policy of synthesis leading to a composite predictor or whether it is better to attempt a selection of the singularly best forecasting model. This report discusses some of the associated issues and provides some experimental evidence on the performance of these two policies.     

Warping aerial photographs to orthomaps using thin plate splines

Using an image rectification application arising in the field of forest management, we demonstrate in this paper the practical feasibility of applying thin plate spline (TPS) techniques to real image warping problems. The use of TPS‐based warping in large problems can be limited by two factors: numerical instability in the calculation of TPS coefficients, and the intensive computation involved in evaluating TPS functions. Both drawbacks can be overcome by taking advantage of some recent advances in the numerical analysis of radial basis functions. Here we relate our experience in applying some of this work to realize successful TPS warping of large forestry images, and some graphical examples are given. Methods for automated control point selection and editing are also presented, and a cross‐correlation technique for evaluating the effectiveness of the warps is described. This experience has guided our development of an effective and efficient software package for control point selection and TPS warping of digital images. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some statistical challenges in automated driving systems

Automated driving systems are rapidly developing. However, numerous open problems remain to be resolved to ensure this technology progresses before its widespread adoption. A large subset of these problems are, or can be framed as, statistical decision problems. Therefore, we present herein several important statistical challenges that emerge when designing and operating automated driving systems. In particular, we focus on those that relate to request-to-intervene decisions, ethical decision support, operations in heterogeneous traffic, and algorithmic robustification. For each of these problems, earlier solution approaches are reviewed and alternative solutions are provided with accompanying empirical testing. We also highlight open avenues of inquiry for which applied statistical investigation can help ensure the maturation of automated driving systems. In so doing, we showcase the relevance of statistical research and practice within the context of this revolutionary technology.