Discriminant analysis using fuzzy linear programming models

Fisher proposes Fisher’s linear discriminant function (LDF) based on maximization of variance ratio. If data satisfies Fisher’s assumption, the same LDF is easily derived by variance-covariance matrix. If variance-covariance matrices of two classes are not same, quadratic discriminant function (QDF) is proposed. These discriminant functions have three problems. First, all discriminant functions except for Revised IP-OLDF can’t discriminate the cases xi on the discriminant hyper plane correctly (the unresolved problem of discriminant analysis). Only Revised IP-OLDF can solve this problem in theoretically. Second, LDF and QDF can’t recognize linear separable data in many cases, and the numbers of misclassifications (NMs) of these functions are usually higher than logistic regression that is used in many users. Third, these are not obtained if the values of some independent variable are constant, because inverse matrix can’t be calculated. JMP prepare the regularized discriminant function if QDF can’t analyze the difficult data. These facts mean that LDF and QDF shouldn’t be used for an important discrimination. On the contrary, Revised IP-OLDF based on Minimum Number of Misclassifications (MNM) criterion resolves these problems. In addition to this, means of error rates of Revised IP-OLDF are less than those of LDF and logistic regression and S-SVM by 100-fold cross validation in many cases.

The stability of decision—Theoretic electrocardiographic classifiers based on the use of discretized features

In this study, seismic events in Northern and Central Egypt are inspected to discriminate quarry blasts from earthquakes. We examine a collection of 639 events in both time and frequency domains with local magnitudes of 1.5 ≤ ML ≤ 3.3 from the Egyptian Seismological Network’s seismic event catalogue between 2009 and 2015. The maximum S-wave to the maximum P-wave amplitude ratio, complexity (C), spectral ratio (Sr), and power of events (Pe) classifiers as well as two statistical approaches, linear discriminant function (LDF) and quadratic discriminant function (QDF), are used to distinguish between earthquakes and quarry blasts. The usage of the LDF and QDF forms did not result in any major differences in the discrimination. The results obtained by the LDF and QDF from (Pe-C) are the best of all approaches. The findings of all approaches were compared to get a final categorization for each event, and a decision was achieved when at least three of the four methods provided the same event category. In Northern Egypt, 243 earthquakes and 308 quarry blasts could be identified as final decisions, with two misclassified events, resulting in an overall success rate of 99.6%. In Central Egypt, 48 earthquakes and 36 quarry blasts were classified as a final decision, with two misclassified events, for an overall success percentage of 97.6%.

Epilepsy is a common neurological disorder involving spontaneous seizures. While electroencephalography (EEG) is a useful diagnostic approach of epilepsy that records the electrical activity of the brain, detection of epileptic seizures is still clinically difficult. This study proposes a segmental classification approach for the detection of epileptic seizure in EEG signals. Regularized least squares and smoothness priors methods are applied to minimize the nonstationary components in the signals. The optimal frequency band energy features are selected by using the sequential floating forward selectrion (SFFS) algorithm, with linear, quadratic and cubic discriminant function as classifiers. The results show that when the quadratic discriminant function is applied, the sensitivity and specificity of seizure detection reach a maximum of 98.1% and 95.6% respectively for discriminating health subjects against epileptic subjects in seizure period, and the overall classification rate is 97.2%.

This paper studies the distribution network design problem considering the uncertain information in the demand, capacities, costs and prices in a multi-product environment and multiple periods. We consider a fractional objective function that consist in maximize the ratio between total profit and total cost. We use a model that integrates a facility location problem with a distribution network problem with fuzzy constraints, technological coefficients, and costs. To solve the problem, we use a method that transform the fuzzy linear fractional programming model in an equivalent multi-objective linear fractional programming problem to calculate the upper, middle and lower bounds of the original problem.

This paper proposes a new mathematical programming approach to represent the dimensions of the discriminant space for the multiple-group classification problem. Few papers have investigated generalizations of two-group mathematical programming approaches for the classification of multiple groups. While several papers have proposed mathematical programming models for separating groups of observations, the issue of considering the classification problem by finding discriminant linear functions to describe the groups in fewer dimensions has not been addressed. The new mathematical programming approach proposed in this paper first solves the multiple-group problem using a single discriminant function, which essentially represents the separation of the groups in one dimension. Then the multiple-group problem is successively solved using single discriminant functions with the requirement that successive linear discriminant functions have a sample covariance equal to zero. An algorithm is proposed to classify observations from multiple groups using the linear discriminant functions from the mathematical programming approach in a reduced number of dimensions.

Chapter 4 - Comparing samples from two populations: The discriminant function

In this paper, steelmaking-continuous casting (SCC) scheduling problem is investigated and a mixed integer linear programming model is developed. Since the technical constraints in the mathematical model are not defined exactly, fuzzy sets are used to present them. With using fuzzy constraints, a compromise between different criteria is made based on expert’s ideas in steel industry. Since the mathematical model is suitable only for small size problems, a hybrid algorithm is developed to solve the SCC scheduling problem. The proposed algorithm is based on a combination of particle swarm optimization and fuzzy linear programming (FLP) methods. In this algorithm, the process of generating a solution consists of two phases. In the first phase, assigning and sequencing the charges on machines is done, while in the second phase, a FLP model is applied to determine start time of charges on assigned machines. To evaluate the proposed procedure, it is compared with an algorithm reported in the literature through solving different problems. Numerical results show that the proposed algorithm can obtain very good solutions for SCC scheduling problem. The results also reveal the higher efficiency of the proposed approach in decreasing waiting time of charges between processing.

Recent simulation‐based studies of linear programming models for discriminant analysis have used the Fisher linear discriminant function as the benchmark for parametric methods. This article reports experimental evidence which suggests that, while some linear programming models may match or even exceed the Fisher approach in classification accuracy, none of the fifteen models tested is as accurate on normally distributed data as the Smith quadratic discriminant function. At the minimum, further testing is warranted with an emphasis on data sets that arise from significantly non‐Gaussian populations.

Revised IP-OLDF (optimal linear discriminant function by integer programming) is a linear discriminant function to minimize the number of misclassifications (NM) of training samples by integer programming (IP). However, IP requires large computation (CPU) time. In this paper, it is proposed how to reduce CPU time by using linear programming (LP). In the first phase, Revised LP-OLDF is applied to all cases, and all cases are categorized into two groups: those that are classified correctly or those that are not classified by support vectors (SVs). In the second phase, Revised IP-OLDF is applied to the misclassified cases by SVs. This method is called Revised IPLP-OLDF.In this research, it is evaluated whether NM of Revised IPLP-OLDF is good estimate of the minimum number of misclassifications (MNM) by Revised IP-OLDF. Four kinds of the real data—Iris data, Swiss bank note data, student data, and CPD data—are used as training samples. Four kinds of 20,000 re-sampling cases generated from these data are used as the evaluation samples. There are a total of 149 models of all combinations of independent variables by these data. NMs and CPU times of the 149 models are compared with Revised IPLP-OLDF and Revised IP-OLDF. The following results are obtained: 1) Revised IPLP-OLDF significantly improves CPU time. 2) In the case of training samples, all 149 NMs of Revised IPLP-OLDF are equal to the MNM of Revised IP-OLDF. 3) In the case of evaluation samples, most NMs of Revised IPLP-OLDF are equal to NM of Revised IP-OLDF. 4) Generalization abilities of both discriminant functions are concluded to be high, because the difference between the error rates of training and evaluation samples are almost within 2%. Therefore, Revised IPLP-OLDF is recommended for the analysis of big data instead of Revised IP-OLDF. Next, Revised IPLP-OLDF is compared with LDF and logistic regression by 100-fold cross validation using 100 re-sampling samples. Means of error rates of Revised IPLP-OLDF are remarkable fewer than those of LDF and logistic regression.

In daily life events, there are many complexities arising from lack of information and uncertainty. Fuzzy linear programming model has been developed to reduce or eliminate this complexity. Fuzzy linear programming is the process of choosing the optimum solution from among the decision alternatives to achieve a specific purpose in cases where the information is not certain. One of the fields where the lack of information or uncertainty makes it difficult to decide is financial markets. Investors who have a certain amount of accumulations are aiming to increase in various ways as well as protecting the value of their income. While doing this, encounter the problem of deciding to which investment vehicle they need to invest in what extent. Therefore, investors apply to fuzzy linear programming model to eliminate this uncertainty and to create the optimal portfolio. In the portfolio selection process suggestions in the literature, the determination of criteria weights is based on triangular fuzzy numbers. In this study, as an alternative to the Enea and Piazza's portfolio selection model, which uses the triangular fuzzy numbers for criteria weighting, a new model that uses the trapezoidal fuzzy numbers for the same aim was proposed. With the solution of the linear programming model which is based on the determined weights, an alternative solution has been produced to the problem of which investment instrument will be invested at what proportion. The results obtained from the existing methods and the results obtained from the proposed model were compared.

We propose a new algorithm for the allocation of an individual to one of several possible groups or populations. The algorithm enables us to define a finite partition over the sample space, based on distance function. This partition is used, jointly with the application of a standard Bayesian decision rule, to allocate individuals to the populations. The algorithm also provides a measure of the allocation confidence for each individual, in a similar manner to that of logistic regression. The error rates for classification are also computed using the leave-one-out method. Results are compared with those obtained with other discriminant analysis techniques previously reported: Fisher's linear discriminant function, the quadratic discriminant function, logistic discrimination, and others.

The estimation of the error rates is of vital importance in classification problems, as this is used as a basis to choose the best discriminant function, i.e. the one with a minimum misclassification error. The quadratic discriminant function (QDF) has been in use for a long time for the purpose of classification. However, there is no exact analytical expression for the misclassification error rate based on QDF. Consequently, researchers depend on use of simulation trials to gauge the misclassification error. This paper provides exact formulae for the misclassification error rates associated with QDF involving two multivariate normal populations. The results are obtained for the general case where the means and the covariance matrices for the two populations are unequal.

The integration of traditional and heuristic techniques is considered for the reactive power/voltage control program. The steady-state reactive power problem is addressed. Fuzzy sets are used to modify the linear programming (LP) approach to voltage control and to incorporate some heuristic concepts of the expert system approach. Multiple objectives and soft constraints are modeled using fuzzy sets. Piecewise linear convex membership functions for the fuzzy sets are defined. Under this definition, the fuzzy optimization problem is reformulated as a standard linear programming problem. The objective function represents the compromise among the original competing objectives and the soft constraints. In addition, discrete constraints are considered. Numerical examples demonstrate the approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets that exist in the constraints and objective function. A stepwise interactive algorithm (SIA) was advanced to solve GFLP model and generate solutions expressed as fuzzy sets. To demonstrate its application, the developed GFLP method was applied to a regional sulfur dioxide (SO2) control planning model to identify effective SO2 mitigation polices with a minimized system performance cost under uncertainty. The results were obtained to represent the amount of SO2 allocated to different control measures from different sources. Compared with the conventional interval-parameter linear programming (ILP) approach, the solutions obtained through GFLP were expressed as fuzzy sets, which can provide intervals for the decision variables and objective function, as well as related possibilities. Therefore, the decision makers can make a tradeoff between model stability and the plausibility based on solutions obtained through GFLP, and then identify desired policies for SO2-emission control under uncertainty. Implications: The GFLP method is effective in addressing air quality management planning associated with fuzzy parameter in the constraints and objective. The stepwise interactive algorithm proposed to solving the GFLP model would not lead to complicated intermediate submodels. Thus it would be much applicable for large-scale air quality management problems. The applications of the method can help decision makers in (i) generating allocation schemes for SO2 emissions; (ii) identifying the plausibility of each allocation alternative; and (iii) estimating the total costs of different SO2 emission control policies.