Principal component analysis, classifier complexity, and robustness of sonographic breast lesion classification

“Principal Component Analysis” (PCA) is an established linear technique for dimensionality reduction. It performs an orthonormal transformation to replace possibly correlated variables with a smaller set of linearly independent variables, the so-called principal components, which capture a large portion of the data variance. The problem of finding the optimal number of principal components has been widely studied for offline PCA. However, when working with streaming data, the optimal number changes continuously. This requires to update both the principal components and the dimensionality in every timestep. While the continuous update of the principal components is widely studied, the available algorithms for dimensionality adjustment are limited to an increment of one in neural network-based and incremental PCA. Therefore, existing approaches cannot account for abrupt changes in the presented data. The contribution of this work is to enable in neural network-based PCA the continuous dimensionality adjustment by an arbitrary number without the necessity to learn all principal components. A novel algorithm is presented that utilizes several PCA characteristics to adaptivly update the optimal number of principal components for neural network-based PCA. A precise estimation of the required dimensionality reduces the computational effort while ensuring that the desired amount of variance is kept. The computational complexity of the proposed algorithm is investigated and it is benchmarked in an experimental study against other neural network-based and incremental PCA approaches where it produces highly competitive results.

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

Improved PCA + LDA Applies to Gastric Cancer Image Classification Process

In data analyzing and machine learning, PCA and LDA are widely used tools, these are linear transformation methods to reduce the dimension observation, and these are used in many practical applications like compression and data visualization in machine learning. Here, I have applied these two methods that are principal component analysis (PCA) and linear discriminant analysis (LDA) in handwritten digit recognition. The main motto of this paper is to present a simple and clear understanding of these methods. Here, this paper depicts that LDA can outperform PCA when training data set is huge, and PCA can outperform LDA when training data set is small.

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 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%.

Thyroid nodules are common clinical entities, with a significant proportion being malignant. Early, accurate, and non-invasive tools to differentiate benign and malignant nodules can optimize patient management and reduce unnecessary surgery. This study aimed to evaluate the efficacy and accuracy of near-infrared spectroscopy (NIRS) in distinguishing benign from malignant thyroid nodules. A diffuse reflectance spectrum for a total of 20 thyroid nodule samples (10 samples as colloid goiter and 10 samples as thyroid cancer), were acquired in the wavelength range from 1000 to 2500 nm. Spectral data from NIRS were analyzed by means of principal component analysis (PCA), quadratic discriminant analysis (QDA), and linear discriminant analysis (LDA) to classify and differentiate thyroid nodule samples. The present study found that NIRS effectively distinguished colloid goiter and thyroid cancer using the first two principal components (PCs), explaining 90% and 10% of the variance, respectively. QDA discrimination plot displayed a clear separation between colloid goiter and thyroid cancer with minimal overlap, aligning with reported 95% accuracy. Additionally, applying LDA to seven PCs from PCA achieved a 100% accuracy rate in classifying colloid goiter and thyroid cancer from near-infrared spectral data. In conclusion, NIRS offers a promising, non-invasive complementing diagnostic tool for differentiating benign from malignant thyroid nodules with high accuracy. Future work should integrate these results into predictive model development, emphasizing external validation, alternative performance metrics, and protecting against potential overfitting translation of a machine learning model to a clinical setting.

Remote sensing Hyperspectral Image (HSI) comprises significant information about the earth’s surface which is actually acquired by hundred of narrow and adjacent spectral bands. The intended performance of classification accuracy does not attain due to the volume of the original HSI dataset and the enormous quantity of spectral bands. As such, dimensionality reduction approaches using feature extraction and selection are typically adopted to enhance classification performance. The unsupervised Principal Component Analysis (PCA), as well as the supervised Linear Discriminant Analysis (LDA), are commonly used as linear feature extraction methods for feature subspace detection. However, due to considering the effects of global variation, both PCA and LDA fail to extract local characteristics of HSI. In this paper, we propose a segmented LDA-based (SLDA) feature extraction where we apply the LDA in a segmented way to extract better local characteristics as well as global characteristics from the HSI. Per-pixel classification using a Support Vector Machine (SVM) is applied to our proposed SLDA method, PCA, Segmented-PCA (SPCA), and LDA on the Indian Pines agricultural HSI dataset. The experimental results show that the overall classification performance of SLDA (90.60%) remarkably outperforms all the other investigated methods: PCA (85.55%), SPCA (86.96%), LDA (86.45%), and the complete original dataset without employing any feature reduction method (83.10%). The proposed SLDA also requires the least amount of space complexity in different implementation scenarios.

To evaluate the usefulness of differentially expressed proteins from colorectal cancer (CRC) tissues for differentiating cancer and normal tissues. A Proteomic approach was used to identify the differentially expressed proteins between CRC and normal tissues. The proteins were extracted using Tris buffer and thiourea lysis buffer (TLB) for extraction of aqueous soluble and membrane-associated proteins, respectively. Chemometrics, namely principal component analysis (PCA) and linear discriminant analysis (LDA), were used to assess the usefulness of these proteins for identifying the cancerous state of tissues. Differentially expressed proteins identified were 37 aqueous soluble proteins in Tris extracts and 24 membrane-associated proteins in TLB extracts. Based on the protein spots intensity on 2D-gel images, PCA by applying an eigenvalue > 1 was successfully used to reduce the number of principal components (PCs) into 12 and seven PCs for Tris and TLB extracts, respectively, and subsequently six PCs, respectively from both the extracts were used for LDA. The LDA classification for Tris extract showed 82.7% of original samples were correctly classified, whereas 82.7% were correctly classified for the cross-validated samples. The LDA for TLB extract showed that 78.8% of original samples and 71.2% of the cross-validated samples were correctly classified. The classification of CRC tissues by PCA and LDA provided a promising distinction between normal and cancer types. These methods can possibly be used for identification of potential biomarkers among the differentially expressed proteins identified.

An electronic nose (E-nose) was developed to distinguish different stored eggs by using principal component analysis (PCA),linear discriminant analysis (LDA),BP neural network (BPNN) and the combination of a genetic algorithm and BP neural network (GANN). Results showed that the E-nose can distinguish eggs of different storage time under cool and room-temperature storage by LDA and PCA. LDA preceded PCA. Better prediction values were obtained by GANN than by BPNN. Relationships were established between the E-nose signal and egg quality indices by quadratic polynomial step regression (QPSR). The prediction models indicated good prediction performance with correlation coefficient between predicted and measured values greater than 0.90.

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.

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.

Evaluation of an artificial olfactory system for grain quality discrimination

A feature extraction and fusion algorithm was constructed by combining principal component analysis (PCA) and linear discriminant analysis (LDA) to detect a fault state of the induction motor. After yielding a feature vector with PCA and LDA from current signal that was measured by an experiment, the reference data were used to produce matching values. In a diagnostic step, two matching values that were obtained by PCA and LDA, respectively, were combined by probability model, and a faulted signal was finally diagnosed. As the proposed diagnosis algorithm brings only merits of PCA and LDA into relief, it shows excellent performance under the noisy environment. The simulation was executed under various noisy conditions in order to demonstrate the suitability of the proposed algorithm and showed more excellent performance than the case just using conventional PCA or LDA.

We propose a new method to compress the geometry component of 3D animation sequence. It is based on the linear discriminant analysis (LDA) of the animation geometry data. The redundancy across the animation frames has been exploited by using the LDA in the temporal direction. Owing to the redundancy between the frames of a class, the covariance matrix of that class for the LDA computation may become singular. To overcome this drawback, we first transform the data into a new basis using the principal component analysis (PCA) and then apply the LDA on a few principal components. The reconstruction is simple and involves two stages: firstly for the LDA and then for the PCA. The experimental results show that the proposed method has the advantage of better reconstruction error at high compression ratios.