Nontraditional Approaches to Statistical Classification: Some Perspectives on Lp-Norm Methods

The body of literature on classification method which estimate boundaries between the groups (classes) by optimizing a function of the L_{p}-norm distances of observations in each group from these boundaries, is maturing fast. The number of published research articles on this topic, especially on mathematical programming (MP) formulations and techniques for L_{p}-norm classification, is now sizable. This paper highlights historical developments that have defined the field, and looks ahead at challenges that may shape new research directions in the next decade. In the first part, the paper summarizes basic concepts and ideas, and briefly reviews past research. Throughout, an attempt is made to integrate a number of the most important L_{p}-norm methods proposed to date within a unified framework, emphasizing their conceptual differences and similarities, rather than focusing on mathematical detail. In the second part, the paper discusses several potential directions for future research in this area. The long-term prospects of L_{p}-norm classification (and discriminant) research may well hinge upon whether or not the channels of communication between on the one hand researchers active in L_{p}-norm classification, who tend to have their roots primarily in decision sciences, the management sciences, computer sciences and engineering, and on the other hand practitioners and researchers in the statistical classification community, will be improved. This paper offers potential reasons for the lack of communication between these groups, and suggests ways in which L_{p}-norm research may be strengthened from a statistical viewpoint. The results obtained in L_{p}-norm classification studies are clearly relevant and of importance to all researchers and practitioners active in classification and discrimination analysis. The paper also briefly discusses artificial neural networks, a promising nontraditional method for classification which has recently emerged, and suggests that it may be useful to explore hybrid classification methods that take advantage of the complementary strengths of different methods, e.g., neural network and L_{p}-norm methods

Using Biophysical Geospatial and Remotely Sensed Data to Classify Ecological Sites and States

Monitoring and identifying the state of rangelands on a landscape scale can be a time consuming process. In this thesis, remote sensing imagery has been used to show how the process of classifying different ecological sites and states can be done on a per pixel basis for a large landscape. Twenty-seven years\u27 worth of remotely sensed imagery was collected, atmospherically corrected, and radiometrically normalized. Several vegetation indices were extracted from the imagery along with derivatives from a digital elevation model. Dominant vegetation components from five major ecological sites in Rich County, Utah, were chosen for study. The vegetation components were Aspen, Douglas-fir, Utah juniper, mountain big sagebrush, and Wyoming big sagebrush. Training sites were extracted from within map units with a majority of one of the five ecological sites. A Random Forests decision tree model was developed using an attribute table populated with spectral biophysical variables derived from the training sites. The overall out-of-bag accuracy for the Random Forests model was 97.2%. The model was then applied to the predictor spectral and biophysical variables to spatially map the five major vegetation components for all of Rich County. Each vegetation class had greater than 90% accuracies except for Utah juniper at 81%. This process is further explained in chapter 2. As a follow-on effort, we attempted to classify vegetation ecological states within a single ecological site (Wyoming big sagebrush). This was done using field data collected by previous studies as training data for all five ecological states documented for our chosen ecological site. A Maximum Likelihood classifier was applied to four years of Landsat 5 Thematic Mapper imagery to map each ecological state to pixels coincident to the map units correlated to the Wyoming big sagebrush ecological site. We used the Mahalanobis distance metric as an indicator of pixel membership to the Wyoming big sagebrush ecological site. Overall classification accuracy for the different ecological states was 64.7% for pixels with low Mahalanobis distance and less than 25% for higher distances

Selecting a Flexible Manufacturing System Using Multiple Criteria Analysis

This paper describes a visually interactive decision support framework designed to aid the decision maker, typically top management, in selecting the most appropriate technology and design, when planning a flexible manufacturing system (FMS). The framework can be used in the pre-investment stage of the planning process, after the decision in principle has been made to build an FMS. First, both qualitative and quantitative criteria are used to narrow the set of alternative system configurations under consideration down to a small number of most attractive candidates. After this pre-screening phase, a multiobjective programming model is formulated for each remaining configuration, allowing the manager to explore and evaluate the costs and benefits of various different scenarios for each configuration separately by experimenting with different levels of batch sizes and production volumes. The system uses visual interaction with the decision maker, graphically displaying the relevant tradeoffs between such relevant performance criteria as investment and production costs, manufacturing flexibility, production volume and investment risk, for each scenario. Additional criteria, when relevant, can be included as well. The ease of use and interpretation and the flexibility make the proposed system a powerful analytical tool in the initial FMS design process. The insights gained from experimenting with the different scenarios form the basis of understanding the anticipated impact of techno-economic factors on the performance of the FMS configuration, and provide valuable information for the implementation stage of building the FMS. An example using real data from a case study in the Finnish metal product industry is provided to illustrate the methodology

A Unified Mathematical Programming Formulation for the Discriminant Problem

In recent years, much research has been done on the application of mathematical programming (MP) techniques to the discriminant problem. While very promising results have been obtained, many of these techniques are plagued by a number of problems associated with the model formulation including unbounded, improper and unacceptable solutions as well as solution instability under linear transformation of the data. Some have attempted to prevent these problems by suggesting overly complex formulations which can be difficult to solve. Others have suggested formulations which solve certain problems but which create new ones. In this paper we develop a simple MP model which unifies many features of previous formulations and appears to avoid any solution problems. This approach also considers a classification gap often encountered in the related statistical techniques

Mathematical Programming Formulations for Two-group Classification with Binary Variables

In this paper, we introduce a nonparametric mathematical programming (MP) approach for solving the binary variable classification problem. In practice, there exists a substantial interest in the binary variable classification problem. For instance, medical diagnoses are often based on the presence or absence of relevant symptoms, and binary variable classification has long been used as a means to predict (diagnose) the nature of the medical condition of patients. Our research is motivated by the fact that none of the existing statistical methods for binary variable classification -- parametric and nonparametric alike -- are fully satisfactory. The general class of MP classification methods facilitates a geometric interpretation, and MP-based classification rules have intuitive appeal because of their potentially robust properties. These intuitive arguments appear to have merit, and a number of research studies have confirmed that MP methods can indeed yield effective classification rules under certain non-normal data conditions, for instance if the data set is outlier-contaminated or highly skewed. However, the MP-based approach in general lacks a probabilistic foundation, an ad hoc assessment of its classification performance. Our proposed nonparametric mixed integer programming (MIP) formulation for the binary variable classification problem not only has a geometric interpretation, but also is consistent with the Bayes decision theoretic approach. Therefore, our proposed formulation possesses a strong probabilistic foundation. We also introduce a linear programming (LP) formulation which parallels the concepts underlying the MIP formulation, but does not possess the decision theoretic justification. An additional advantage of both our LP and MIP formulations is that, due to the fact that the attribute variables are binary, the training sample observations can be partitioned into multinomial cells, allowing for a substantial reduction in the number of binary and deviational variables, so that our formulation can be used to analyze training samples of almost any size. We illustrate our formulations using an example problem, and use three real data sets to compare its classification performance with a variety of parametric and nonparametric statistical methods. For each of these data sets, our proposed formulation yields the minimum possible number of misclassifications, both using the resubstitution and the leave-one-out method

A Nonlinear Multicriteria Model for Strategic FMS Selection Decisions

The strategic decision of selecting an optimal flexible manufacturing system (FMS) configuration is a complicated question which involves evaluating tradeoffs between a number of different, potentially conflicting criteria such as annual production volume, flexibility, production and investment costs, and average throughput of the system. Recently, several structured approaches have been proposed to aid management in the FMS selection process. While acknowledging the nonlinear nature of a number of the relationships in the model, notably between batch size and the number of batches produced of each part, these studies used linear simplifications to illustrate the decision dynamics of the problem. These linear models were shown to offer useful analytical tools in the FMS pre-design process. Due to the nonlinearities of the true relationships, however, the tradeoffs between the criteria could not fully be explored within the linear framework. This paper builds on the two-phase decision support framework proposed by Stam and Kuula (1989), and uses a modified nonlinear multicriteria formulation to solve the problem. The software used in the illustration can easily be implemented, is user-interactive and menu-driven. The methodology is applied to real data from a Finnish metal product company, and the results are compared with those obtained in previous studies

Motivations and experiences of UK students studying abroad

This report summarises the findings of research aimed at improving understanding of the motivations behind the international diploma mobility of UK student