# Directed Clustering of Multivariate Data Based on Linear or Quadratic Latent Variable Models
## 8 Discussion  
We presented an approach for clustering multivariate data that enables the ordering of clusters along a curve parameterized by a latent variable, estimated alongside the clustering process. We illustrated this concept in the context of linear and quadratic latent variable models. We showed how to choose between these models, how to implement the estimators, and how to interpret the results. Particular attention was devoted to an application in traffic engineering, where the latent variable had a meaningful interpretation, corresponding to the traffic density. The application of the proposed methodology is not restricted to this field; it has potential applicability across various scientific fields. While the idea of a latent variable representing some 'underlying governing process' seems more natural in the physical sciences, the concept of latent variables is home to many other sciences, including fields like education or economics, as alluded to in the introduction. However, it is not actually necessary for the researcher to identify or justify the existence of a governing process in order to use the presented approach. At a minimum, there should be a belief that, grounded in subject matter expertise, the relative ordering of the clusters is informative. In doing so, the existence of a latent variable spanning the centers is postulated--and then the methodology estimates it. Our work contributes to the development of clustering algorithms that account for the governing process underlying the data at hand. As pointed out by a referee, mathematical clustering models are ineffective if they do not consider important driving factors that contribute to the generation of patterns. The proposed methodology allows the identification of a potentially existing latent variable driving the data-generating process and also allows for the further processing of this latent variable through its posterior random effects. This could include applications such as regression or correlation with actually measured physical variables. Conceptually, it would be even more desirable to directly include the knowledge of the governing process of the data in the clustering algorithm. To a certain extent, this could be achieved by adjusting the clustering process for the presence of influencing variables. For instance, in the context of the traffic examples, one may want to adjust for spatial or temporal factors. This would require the inclusion of covariates into the model (3). In the case of the linear model relationship (1), this was developed in [16], and could be extended to the more general framework presented here. It is apparent that the restriction to linear and quadratic latent variable models is a possible limitation of the presented methodology. To overcome this restriction, more complex shapes of latent variable models could be considered, such as:  
\[x_{i}=\sum_{j=0}^{p}\alpha_{j}B_{j}(z_{i})+\varepsilon_{i}, \tag{13}\]  
where \(x_{i}\in\mathbb{R}^{m}\) and \(z_{i}\in\mathbb{R}\) play the same roles as before, \(B_{j}(z_{i})\) can be regarded as any real-valued basis function, such as a B-spline or polynomial, and \(\alpha_{j}\) is a \(m\)-variate parameter vector. These still fit into the notational framework defined by (3), but now with \(\theta=(\alpha_{0}^{T},\ldots,\alpha_{p}^{T})^{T}\). The potential power of such an approach becomes evident when considering the LPC fit in Figure 12, which shows that there is still considerable scope for improvement in fitting capability for sufficiently complex data patterns. However, given the difficulties of even estimating the quadratic model (2), requiring a combination of iterative statistical algorithms (EM) and numerical root finders, it will be a considerable challenge to devise an algorithm to estimate this model in full generality. To prevent doubt, we would like to highlight that we do not advocate this methodology as a superior or favorable clustering technique for generic clustering applications.