On transformation discriminant analysis

Sufficient dimension-reduction methods provide effective ways to visualize discriminant analysis problems. For example, Cook and Yin showed that the dimension-reduction method of sliced average variance estimation (SAVE) identifies variates that are equivalent to a quadratic discriminant analysis (QDA) solution. This article makes this connection explicit to motivate the use of SAVE variates in exploratory graphics for discriminant analysis. Classification can then be based on the SAVE variates using a suitable distance measure. If the chosen measure is Mahalanobis distance, then classification is identical to QDA using the original variables. Just as canonical variates provide a useful way to visualize linear discriminant analysis (LDA), so do SAVE variates help visualize QDA. This would appear to be particularly useful given the lack of graphical tools for QDA in current software. Furthermore, whereas LDA and QDA can be sensitive to nonnormality, SAVE is more robust.

Linear and Quadratic Discriminant Analysis have been used widely in many areas of data mining, machine learning, and bioinformatics. Friedman proposed a compromise between Linear and Quadratic Discriminant Analysis, called Regularized Discriminant Analysis (RDA), which has been shown to be more flexible in dealing with various class distributions. RDA applies the regularization techniques by employing two regularization parameters, which are chosen to jointly maximize the classification performance. The optimal pair of parameters is commonly estimated via cross-validation from a set of candidate pairs. It is computationally prohibitive for high dimensional data, especially when the candidate set is large, which limits the applications of RDA to low dimensional data.In this paper, a novel algorithm for RDA is presented for high dimensional data. It can estimate the optimal regularization parameters from a large set of parameter candidates efficiently. Experiments on a variety of datasets confirm the claimed theoretical estimate of the efficiency, and also show that, for a properly chosen pair of regularization parameters, RDA performs favorably in classification, in comparison with other existing classification methods.

Acknowledgements. Preface. 1 Introduction. 1.1 Past, Present and Future. 1.2 About this Book. Bibliography. 2 Case Studies. 2.1 Introduction. 2.2 Datasets, Matrices and Vectors. 2.3 Case Study 1: Forensic Analysis of Banknotes. 2.4 Case Study 2: Near Infrared Spectroscopic Analysis of Food. 2.5 Case Study 3: Thermal Analysis of Polymers. 2.6 Case Study 4: Environmental Pollution using Headspace Mass Spectrometry. 2.7 Case Study 5: Human Sweat Analysed by Gas Chromatography Mass Spectrometry. 2.8 Case Study 6: Liquid Chromatography Mass Spectrometry of Pharmaceutical Tablets. 2.9 Case Study 7: Atomic Spectroscopy for the Study of Hypertension. 2.10 Case Study 8: Metabolic Profiling of Mouse Urine by Gas Chromatography of Urine Extracts. 2.11 Case Study 9: Nuclear Magnetic Resonance Spectroscopy for Salival Analysis of the Effect of Mouthwash. 2.12 Case Study 10: Simulations. 2.13 Case Study 11: Null Dataset. 2.14 Case Study 12: GCMS and Microbiology of Mouse Scent Marks. Bibliography. 3 Exploratory Data Analysis. 3.1 Introduction. 3.2 Principal Components Analysis. 3.2.1 Background. 3.2.2 Scores and Loadings. 3.2.3 Eigenvalues. 3.2.4 PCA Algorithm. 3.2.5 Graphical Representation. 3.3 Dissimilarity Indices, Principal Co-ordinates Analysis and Ranking. 3.3.1 Dissimilarity. 3.3.2 Principal Co-ordinates Analysis. 3.3.3 Ranking. 3.4 Self Organizing Maps. 3.4.1 Background. 3.4.2 SOM Algorithm. 3.4.3 Initialization. 3.4.4 Training. 3.4.5 Map Quality. 3.4.6 Visualization. Bibliography. 4 Preprocessing. 4.1 Introduction. 4.2 Data Scaling. 4.2.1 Transforming Individual Elements. 4.2.2 Row Scaling. 4.2.3 Column Scaling. 4.3 Multivariate Methods of Data Reduction. 4.3.1 Largest Principal Components. 4.3.2 Discriminatory Principal Components. 4.3.3 Partial Least Squares Discriminatory Analysis Scores. 4.4 Strategies for Data Preprocessing. 4.4.1 Flow Charts. 4.4.2 Level 1. 4.4.3 Level 2. 4.4.4 Level 3. 4.4.5 Level 4. Bibliography. 5 Two Class Classifiers. 5.1 Introduction. 5.1.1 Two Class Classifiers. 5.1.2 Preprocessing. 5.1.3 Notation. 5.1.4 Autoprediction and Class Boundaries. 5.2 Euclidean Distance to Centroids. 5.3 Linear Discriminant Analysis. 5.4 Quadratic Discriminant Analysis. 5.5 Partial Least Squares Discriminant Analysis. 5.5.1 PLS Method. 5.5.2 PLS Algorithm. 5.5.3 PLS-DA. 5.6 Learning Vector Quantization. 5.6.1 Voronoi Tesselation and Codebooks. 5.6.2 LVQ1. 5.6.3 LVQ3. 5.6.4 LVQ Illustration and Summary of Parameters. 5.7 Support Vector Machines. 5.7.1 Linear Learning Machines. 5.7.2 Kernels. 5.7.3 Controlling Complexity and Soft Margin SVMs. 5.7.4 SVM Parameters. Bibliography. 6 One Class Classifiers. 6.1 Introduction. 6.2 Distance Based Classifiers. 6.3 PC Based Models and SIMCA. 6.4 Indicators of Significance. 6.4.1 Gaussian Density Estimators and Chi-Squared. 6.4.2 Hotelling's T 2 . 6.4.3 D-Statistic. 6.4.4 Q-Statistic or Squared Prediction Error. 6.4.5 Visualization of D- and Q-Statistics for Disjoint PC Models. 6.4.6 Multivariate Normality and What to do if it Fails. 6.5 Support Vector Data Description. 6.6 Summarizing One Class Classifiers. 6.6.1 Class Membership Plots. 6.6.2 ROC Curves. Bibliography. 7 Multiclass Classifiers. 7.1 Introduction. 7.2 EDC, LDA and QDA. 7.3 LVQ. 7.4 PLS. 7.4.1 PLS2. 7.4.2 PLS1. 7.5 SVM. 7.6 One against One Decisions. Bibliography. 8 Validation and Optimization. 8.1 Introduction. 8.1.1 Validation. 8.1.2 Optimization. 8.2 Classification Abilities, Contingency Tables and Related Concepts. 8.2.1 Two Class Classifiers. 8.2.2 Multiclass Classifiers. 8.2.3 One Class Classifiers. 8.3 Validation. 8.3.1 Testing Models. 8.3.2 Test and Training Sets. 8.3.3 Predictions. 8.3.4 Increasing the Number of Variables for the Classifier. 8.4 Iterative Approaches for Validation. 8.4.1 Predictive Ability, Model Stability, Classification by Majority Vote and Cross Classification Rate. 8.4.2 Number of Iterations. 8.4.3 Test and Training Set Boundaries. 8.5 Optimizing PLS Models. 8.5.1 Number of Components: Cross-Validation and Bootstrap. 8.5.2 Thresholds and ROC Curves. 8.6 Optimizing Learning Vector Quantization Models. 8.7 Optimizing Support Vector Machine Models. Bibliography. 9 Determining Potential Discriminatory Variables. 9.1 Introduction. 9.1.1 Two Class Distributions. 9.1.2 Multiclass Distributions. 9.1.3 Multilevel and Multiway Distributions. 9.1.4 Sample Sizes. 9.1.5 Modelling after Variable Reduction. 9.1.6 Preliminary Variable Reduction. 9.2 Which Variables are most Significant?. 9.2.1 Basic Concepts: Statistical Indicators and Rank. 9.2.2 T-Statistic and Fisher Weights. 9.2.3 Multiple Linear Regression, ANOVA and the F-Ratio. 9.2.4 Partial Least Squares. 9.2.5 Relationship between the Indicator Functions. 9.3 How Many Variables are Significant? 9.3.1 Probabilistic Approaches. 9.3.2 Empirical Methods: Monte Carlo. 9.3.3 Cost/Benefit of Increasing the Number of Variables. Bibliography. 10 Bayesian Methods and Unequal Class Sizes. 10.1 Introduction. 10.2 Contingency Tables and Bayes' Theorem. 10.3 Bayesian Extensions to Classifiers. Bibliography. 11 Class Separation Indices. 11.1 Introduction. 11.2 Davies Bouldin Index. 11.3 Silhouette Width and Modified Silhouette Width. 11.3.1 Silhouette Width. 11.3.2 Modified Silhouette Width. 11.4 Overlap Coefficient. Bibliography. 12 Comparing Different Patterns. 12.1 Introduction. 12.2 Correlation Based Methods. 12.2.1 Mantel Test. 12.2.2 R V Coefficient. 12.3 Consensus PCA. 12.4 Procrustes Analysis. Bibliography. Index.

The objective of this study is to concentrate on the classification method of the logistic regression and the discriminant analysis by using the simulation dataset and the liver patients as the actual data. These datasets are used the binary dependent variable depending on the correlated independent variables or called multicollinearity data. The standard classification method is logistic regression, which uses the logit function’s probability to conduct the dichotomous dependent variable. The iteration process can be solved to estimate logit function parameters and explain the relationship between a dependent binary variable and independent variables. Discriminant analysis is a powerful classification based on linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), and regularized discriminant analysis (RDA). These methods consider the decision boundaries by building a classifier model on the multivariate normal distribution. LDA defines the standard covariance matrix, but QDA has an individual covariance matrix. RDA extends from QDA by setting the regularized parameter to estimate the covariance matrix. In the case of the simulation study, the independent variables are generated by defining the constant correlation on the multivariate normal distribution that made the multicollinearity problem. Then the binary response variable can be approximated from the logit function. For application to actual data, we expressed the classification of type liver and non-liver patients as the dependent variables and obtained patient personal information on the nine independent variables. The highest average percentage of accuracy determines the performance of these methods. The results have shown that the logistic regression was successful when using small independent variables, but the RDA performed when using large independent variables.

The performance of seven classification methods, binary logistic (BLR), probit (PR) and cumulative probit (CPR) regression, linear (LDA) and quadratic (QDA) discriminant analysis, artificial neural networks (ANN), and naïve Bayes classification (NBC), is examined in skeletal sex estimation. These methods were tested using cranial and pelvic sexually dimorphic traits recorded on a modern documented collection, the Athens Collection. For their implementation, an R package has been written to perform cross-validated (CV) sex classification and give the discriminant function of each of the methods studied. A simple algorithm that combines two discriminant functions is also proposed. It was found that the differences in the classification performance between BLR, PR, CPR, LDA, QDA, ANN, and NBC are overall small. However, LDA is simpler and more flexible than CPR, QDA, and ANN and has a small but clear advantage over BLR, NBC, and PR. Consequently, LDA may be preferred in skeletal sex estimation. Finally, it is striking that the combination of pelvic and cranial traits via their discriminant functions, determined either by BLR or LDA, removes practically any population-specificity and yields much better predictions than the individual functions; in fact, the prediction accuracy increases above 97%.

Many contemporary studies involve the classification of a subject into two classes based on n observations of the p variables associated with the subject. Under the assumption that the variables are normally distributed, the well-known linear discriminant analysis (LDA) assumes a common covariance matrix over the two classes while the quadratic discriminant analysis (QDA) allows different covariance matrices. When p is much smaller than n, even if they both diverge, the LDA and QDA have the smallest asymptotic misclassification rates for the cases of equal and unequal covariance matrices, respectively. However, modern statistical studies often face classification problems with the number of variables much larger than the sample size n, and the classical LDA and QDA can perform poorly. In fact, we give an example in which the QDA performs as poorly as random guessing even if we know the true covariances. Under some sparsity conditions on the unknown means and covariance matrices of the two classes, we propose a sparse QDA based on thresholding that has the smallest asymptotic misclassification rate conditional on the training data. We discuss an example of classifying normal and tumor colon tissues based on a set of p = 2; 000 genes and a sample of size n = 62, and another example of a cardiovascular study for n = 222 subjects with p = 2; 434 genes. A simulation is also conducted to check the performance of the proposed method.

Lung cancer is a type of cancer that starts from the lungs with abnormal cell growth. Lung cancer is a type of cancer with the largest number of sufferers. In this research, Regularized Discriminant Analysis (RDA) analysis is used to classify the types of lung cancer that exist. Regularized Discriminant Analysis (RDA) is a generalization of Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). If the alpha parameter is set to 1, this operator performs an LDA. Likewise if the alpha parameter is set to 0, this operator performs QDA. Discriminant analysis is used to determine which variables distinguish between two or more groups that occur naturally. With an accuracy value of 60%, it can be ascertained the type of lung cancer suffered by patients for ease of care and treatment.

We investigated three classifiers for the task of distinguishing between benign and malignant breast lesions. Classification performance was measured in terms of area under the ROC curve (AUC value). We compared linear discriminant analysis (LDA), quadratic discriminant analysis (QDA) and a Bayesian neural net (BNN) with 5 hidden units. For each lesion, 46 image features were extracted and principal component analysis (PCA) of these features was used as classifier input. For each classifier, the optimal number of principal components was determined by performing PCA within each step of a leave-one-case-out protocol for the training dataset (1125 lesions, 14% cancer prevalence) and determining which number of components maximized the AUC value. Subsequently, each classifier was trained on the training dataset and applied ‘cold turkey’ to an independent test set from a different population (341 lesions, 30% cancer prevalence). The optimal number of principal components for LDA was 24, accounting for 97% of the variance in the image features. For QDA and BNN, these numbers were 5 (70%) and 15 (93%), respectively. The LDA, QDA and BNN obtained AUC values of 0.88, 0.85, and 0.91, respectively, in the leave-one-case-out analysis. In the independent test – with AUCs of 0.88, 0.76, and 0.82 – only LDA achieved performance identical to that for the training set (lower bound of 95% non-inferiority interval -.0067), while the others performed significantly worse (p-values << 0.05). While the more complex BNN classifier outperformed the others in leave-one-case-out of a large dataset, LDA was the robust best-performer in an independent test.

Motor oil classification is important for quality control and the identification of oil adulteration. In this work, we propose a simple, rapid, inexpensive and nondestructive approach based on image analysis and pattern recognition techniques for the classification of nine different types of motor oils according to their corresponding color histograms. For this, we applied color histogram in different color spaces such as red green blue (RGB), grayscale, and hue saturation intensity (HSI) in order to extract features that can help with the classification procedure. These color histograms and their combinations were used as input for model development and then were statistically evaluated by using linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), and support vector machine (SVM) techniques. Here, two common solutions for solving a multiclass classification problem were applied: (1) transformation to binary classification problem using a one-against-all (OAA) approach and (2) extension from binary classifiers to a single globally optimized multilabel classification model. In the OAA strategy, LDA, QDA, and SVM reached up to 97% in terms of accuracy, sensitivity, and specificity for both the training and test sets. In extension from binary case, despite good performances by the SVM classification model, QDA and LDA provided better results up to 92% for RGB-grayscale-HSI color histograms and up to 93% for the HSI color map, respectively. In order to reduce the numbers of independent variables for modeling, a principle component analysis algorithm was used. Our results suggest that the proposed method is promising for the identification and classification of different types of motor oils.

For purposes of dimensionality reduction in classification Linear Discriminant Analysis (LDA) is probably the most common approach. In fact, LDA is a linear dimension reduction technique that also returns a classification rule. In the case of heteroscedasticity of the classes, Quadratic Discriminant Analysis (QDA) can be used to determine an appropriate classification rule, but QDA does not serve for dimensionality reduction. Sliced Average Variance Estimation (SAVE) has been shown to be adequate in such situations as implemented in R in the package dr. This paper presents an alternative approach for linear dimensionality reduction for situations of heteroscedastic intraclass covariances, namely Heteroscedastic Discriminant Analysis (HDA) as well as its R implementation. Furthermore, tests are suggested in order to determine the dimension for the discriminative data subspace and a generalization of HDA by regularization of the covariance matrix estimates is proposed. Examples for application of HDA in R are demonstrated as well as a small simulation study turning out that HDA is preferable to SAVE in a situation where the classes differ in both means and covariances.

ABSTRACT- Firmness is one of the most important quality indicators for apple fruits, which is highly correlated with the storage time. The acoustic impulse response technique is one of the most commonly used nondestructive detection methods for evaluating apple firmness. This paper presents a non-destructive method for classification of Iranian apple (Malus domestica Borkh. cv. Golab) according to the duration of storage. Several data preprocessing methods were tested: normalization, detrending, Savitzky-Golay smoothing, first and second derivatives, multiplicative scatter correction, standard normal variate and moving average. It was observed that the maximum average Fβ value of classification on the test dataset (0.84) belongs to non-preprocessing. In this study, principal component analysis (PCA) technique was performed to determine the key variables that explain most differences in the spectra. Seven principal components were used to calibrate linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA) classifiers. The classification accuracy for LDA and QDA models were about 80.56% and 83.33%, respectively. The results indicated that the acoustic impulse response method is potentially applicable for the detection of apple firmnes

We evaluated the directional asymmetry between right and left wings and quantified the intraspecific and interspecific variation of the wing shape of 601 specimens of the genus Diachlorus to determine to what extent the geometrical variation discriminates six species distributed in six protected areas of Colombia. Geometric analyses were performed, integrating Procrustes methods, principal component analyses, cluster analyses, linear and quadratic discriminant analyses, and evaluations of shape changes. In Diachlorus, left and right wings did not present significant asymmetry but a geometrical analysis was allowed for species identification and, in some cases, the origin of the specimens using the variation of wing shape; the best-assigned species was Diachlorus leticia Wilkerson & Fairchild, while the worst was Diachlorus jobbinsi Fairchild, which also had the highest intraspecific variation, while Diachlorus fuscistigma Lutz had the lowest variation. Diachlorus fuscistigma and Diachlorus leucotibialis Wilkerson & Fairchild were the most similar species, while D. leucotibialis and Diachlorus nuneztovari Fairchild & Ortiz were the most disimilar. The specimens with the most different wing shape belonged to Chocó (especially those of D. jobbinsi), the geographically farthest area from the others in the study; however, no correlation was observed between geometric and geographical distances. Linear discriminants were better than nonlinear (quadratic) discriminant analyses in predicting species membership, but the opposite was true for predicting area membership. Based on our data, we hypothesized that other species of Diachlorus could also be discriminated using geometric morphometry of the wing shape.

The abundant movement information carried by the surface electromyography (sEMG) signal can comprehensively reflect the muscle movement patterns on the surface of the human body, and the characteristics of physiological signals involving movement information have promoted the vigorous development in the field of human-computer interaction in machine learning. This paper takes subtle gestures as the research objects, and proposes a subtle gestures recognition system that uses two-channel sEMG signal and utilizes Quadratic Discriminant Analysis (QDA) and Linear Discriminant Analysis (LDA) which are parameter estimation classifiers based on probability density. First, combining the raw sEMG signal and the envelope sEMG signal, the number of channels for sensors and corresponding position of the muscle groups are preferably selected. Then, we propose to use the Pearson correlation coefficient to optimize the four types of features extracted in the time domain, frequency domain, time-frequency domain and AR model parameters, which effectively ensures the uniqueness of each gesture and eliminates individual differences. Finally, this paper compares four classifiers of QDA, LDA, Support Vector Machine (SVM), and K-Nearest Neighbor (KNN), and explores the pattern recognition classifier suitable for subtle actions. The experiment collects 9 kinds of subtle gestures from 8 subjects. The results show that the system can use only two channels of sEMG signal to recognize 9 kinds of subtle gestures which include multi-finger clicking or pinching or holding objects, and the most suitable classifiers QDA and LDA have the average recognition rates of 95.79% and 95.01%, respectively.

BackgroundPredictive models for delayed graft function (DGF) after kidney transplantation are usually developed using logistic regression. We want to evaluate the value of machine learning methods in the prediction of DGF.Methods497 kidney transplantations from deceased donors at the Ghent University Hospital between 2005 and 2011 are included. A feature elimination procedure is applied to determine the optimal number of features, resulting in 20 selected parameters (24 parameters after conversion to indicator parameters) out of 55 retrospectively collected parameters. Subsequently, 9 distinct types of predictive models are fitted using the reduced data set: logistic regression (LR), linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), support vector machines (SVMs; using linear, radial basis function and polynomial kernels), decision tree (DT), random forest (RF), and stochastic gradient boosting (SGB). Performance of the models is assessed by computing sensitivity, positive predictive values and area under the receiver operating characteristic curve (AUROC) after 10-fold stratified cross-validation. AUROCs of the models are pairwise compared using Wilcoxon signed-rank test.ResultsThe observed incidence of DGF is 12.5 %. DT is not able to discriminate between recipients with and without DGF (AUROC of 52.5 %) and is inferior to the other methods. SGB, RF and polynomial SVM are mainly able to identify recipients without DGF (AUROC of 77.2, 73.9 and 79.8 %, respectively) and only outperform DT. LDA, QDA, radial SVM and LR also have the ability to identify recipients with DGF, resulting in higher discriminative capacity (AUROC of 82.2, 79.6, 83.3 and 81.7 %, respectively), which outperforms DT and RF. Linear SVM has the highest discriminative capacity (AUROC of 84.3 %), outperforming each method, except for radial SVM, polynomial SVM and LDA. However, it is the only method superior to LR.ConclusionsThe discriminative capacities of LDA, linear SVM, radial SVM and LR are the only ones above 80 %. None of the pairwise AUROC comparisons between these models is statistically significant, except linear SVM outperforming LR. Additionally, the sensitivity of linear SVM to identify recipients with DGF is amongst the three highest of all models. Due to both reasons, the authors believe that linear SVM is most appropriate to predict DGF.

Regularized discriminant analysis for the small sample size problem in face recognition