Magnification Factors for the GTM

One of the most interesting consequences of the probabilistic definition of GTM is that the distortion caused by the nonlinear mapping can be explicitly quantified. Not only that: despite the fact that GTM, as SOM, is a discrete projection technique [9] in the sense that only a finite number of latent space points are considered, this distortion, known as Magnification Factors (MF) [22], can be calculated for any point in the latent space continuum.

As remarked in [241], the concept of MF has its origin in the field of computational neuroscience, where it refers to the mapping distortion between the spatial density of biological sensors and the two- dimensional spatial density of the corresponding topographic maps in the visual and somatosensory areas of the cortex. More specifically, the cortical MF would indicate the linear distance along the primary visual cortex concerned with each degree of visual field [211], although controversy remains on whether the cortical magnification of the central visual field reflects its selective amplification, or merely reflects the ganglion cell density of the retina [281]. As expressed in the context of vector quantization models [106], local magnification is the result of a specific connection of the density of model prototypes and stimuli (data).

For GTM, it is shown in [22] that the relationship between a differential area dA (for a 2-D representa- tion) in latent space and the corresponding area element in the generated manifold, dA′, can be expressed as dA = JdA′, where J is the Jacobian of the mapping transformation. This Jacobian can be written in terms of the derivatives of the basis functionsϕmas:

dA/dA′= J = det12₍ΨT_WT_WΨ), _(4.20)

whereΨ is a M × 2 matrix with elements φmi=∂ϕm/∂uiand uiis the ith coordinate (i = 1, 2) of a latent

point. Note that the MF as expressed by the Jacobian in equation (4.20) can be calculated for any value of uover the continuum.

From a practical viewpoint, the MFs for GTM were introduced as the geometrical functional equivalent to the distance matrix or U-matrix of the SOM [252]. These factors can provide useful information, such as areas of stretch in the manifold that separate different regions in the data space.

The MFs “add a dimension” to the visual representation of data by providing hints about their global cluster structure. It is easy to see why this should be so if we consider the Gaussian mixture model on the

GTM manifold. The EM algorithm will attempt to place the mixture components in regions of high data density and will move the components away from the regions with low data density. It can do this because the non-linear mapping from latent space to data space enables the manifold to stretch across regions of low data density. This stretching (or magnification) can be measured using techniques of differential geometry and plotting the MFs in latent space may allow the user to discover separation between clusters, if this exists.

Chapter 5

Supervised customer loyalty analysis

In this chapter we describe, following a supervised ML approach, the drivers towards customer satisfaction on the basis of a survey conducted amongst the customers of several Spanish petrol station brands. Such description is carried out through reasonably simple and actionable rules that could be applied in a real business environment. With this, we aim to achieve the necessary level of interpretability of the solutions that is often required from the application of ML methods [271].

A survey of several thousand customers was used to classify them according to satisfaction levels, using an artificial neural network (ANN) defined within a Bayesian framework [173]. An Automatic Rele- vance Determination (ARD) technique embedded in this model was used for supervised feature relevance quantification, leading to feature selection. The subset of selected features was used, in turn, to obtain a rule description of the classification performed by the ANN through the recently developed OSRE method [73]. OSRE was able to describe the factors driving customer satisfaction in a reasonably simple and interpretable manner that could be swiftly integrated in service management processes.

This brief chapter is structured as follows: the case study including the available data is briefly described in Section 5.1. This is followed by a summary technical description of the Bayesian ANN with ARD and the OSRE techniques in Section 5.2. Finally, the developed experiments and the obtained results are reported in Section 5.3, while some conclusions are drawn in Section 5.4.

Results of this study were presented at the 7th Intelligent Data Engineering and Automated Learn- ing International Conference (IDEAL 2006) [267]. This work provided, as envisaged, a first preliminary approximation to the prevention side of the customer retention vs. churn problem.

5.1

Petrol station customer satisfaction, loyalty and switching

barriers

As detailed in Chapter 2, efficient churn management requires a model of both preventive and treatment ac- tions: preventing dissatisfaction before it occurs and treating it when it has already set in. In this chapter we focus on the prevention side of customer loyalty management and, in particular, on customer satisfaction (see Figure 5.1). Satisfaction with the received service is likely to act as an antecedent to loyalty, consoli- dating customer permanence and avoiding substitution by a competitor. It might be a necessary condition for loyalty, but perhaps not sufficient. Therefore, the development by the company of active barriers should also be explored as an antecedent to customer loyalty.

The data analysed in this chapter correspond to a survey carried out among customers of Spanish petrol station main brands. This is a different and smaller data set as compared to the ones investigated in the following chapters. A total of 350 service stations of the Spanish market, sampled by location (urban vs. road) and type of service (with attendant vs. self-service), were selected for the exercise. The survey questionnaire was answered by over 3,500 clients during the last quarter of 2005.

Figure 5.1: Conceptual model of customer loyalty management. Customer satisfaction in prevention side of Customer

Continuity Management model (CCM).

The classification and rule extraction analyses described in the next section considered one binary dependent variable: overall satisfaction with two possible answers: satisfied / dissatisfied.

Overall satisfaction would measure the customer satisfaction construct. The questionnaire included 20 variables, listed in Table 5.1, measured in a Likert scale (values range from 1: very good to 5: very bad; value 6 means not answered (NA)). They fit into two qualitative categories: attributes of satisfaction with service and switching barriers.