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Statistical System Identification and Classification Using Laguerre Statistical System Identification and Classification Using Laguerre Functions by Pramoda Sachinthana Jayasinghe A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of Master of Science Department of Statistics University of Manitoba Winnipeg Copyright c 2019 by Pramoda Sachinthana Jayasinghe Abstract This research focuses on developing classification methods to automatically identify partial discharge (PD) signals from multiple sources and developing new system identification techniques to help increase the accuracy of these classifications. We use a Laguerre functional basis to approximate PD signal waveforms, obtained through a lab experiment. To perform the approximation, we considered multiple approaches, such as using objective functional methods and a deterministic method. In the process of signal approximation, we developed methods of selecting a proper scaling factor for Laguerre functions. We evaluated the use of the Laguerre basis expansion coefficients to classify PD signals into their respective sources. Linear discriminant analysis (LDA), quadratic discriminant analysis (QDA) and support vector machines (SVM) were used as classifiers in this analysis. It was observed that these methods can classify partial discharge signals with high accuracies, even when the signals are visually indistinguishable. We also developed two methods for system identification, based on a deterministic approach in the form of a recursive formula and a stochastic method based on a group Lasso model. The aim of system identification was to improve the classification accuracies by removing the effect of the system from the observed PD waveforms. Through numerical evaluations we showed that there are situations where classification after system identification can improve classification accuracies. Acknowledgement I would like to express many thanks to my advisor Dr. Mohammad Jafari Jozani for the guidance and support given to me throughout the M.Sc. program. His suggestions and support was invaluable not only for this research but for achieving my future goals as well. I would also like to express my sincerer gratitude to my co-advisor, Dr. Behzad Kordi for his suggestions and comments throughout my research. Without their help, I would not have been able to realize my dream of obtaining a masters degree and continuing my higher studies. Their doors were always open whenever I hit encountered a problem or when I had a question about my research. Many thanks to my thesis committee members, Dr. Alexandre Leblanc and Dr. Saman Muthukumarana for their comments and suggestions to improve my research. I am also grateful to Dr. Saeed Shahabi and Mr. Ali Nasr Esfahani for their contribution in collecting laboratory data required for this project. Last but not least, I would like to thank my family and friends. Many sincere thanks to my lovely wife and my parents and my sister for their kindness and patience and their entire support while I was pursuing my studies. i Examining Committee This thesis was examined and approved by the following examining committee on August 16, 2019: • Dr. Mohammad Jafari Jozani (advisor) Department of Statistics University of Manitoba • Dr. Behzad Kordi (co-advisor) Department of Electrical & Computer Engineering University of Manitoba • Dr. Alexandre Leblanc (examiner) Department of Statistics University of Manitoba • Dr. Saman Muthukumarana (examiner) Department of Statistics University of Manitoba ii Dedication Page To my lovely and supportive parents. iii Contents Contents iv List of Tables viii List of Figures x 1 Introduction 1 1.1 Introduction to Systems and Classification . .1 1.2 Problem Definition and Motivation . .3 1.2.1 Output Classification . .3 1.2.2 Input Classification and System Identification . .4 1.3 Partial Discharge Classification and System Identification . .4 1.4 Research Contributions . .9 1.5 Publications . 11 1.6 Organization of the Thesis . 11 iv 2 Signal Approximation 13 2.1 Introduction . 13 2.2 Laguerre Functions . 15 2.3 Estimating Coefficients . 18 2.3.1 Least-Squares Objective Function . 19 2.3.2 Least Absolute Objective Function . 20 2.3.3 Lasso Objective Function . 20 2.4 Deterministic Method of Coefficient Estimation . 21 2.5 Examples of Signal Approximation . 23 2.5.1 Example 1 (Gaussian Function) . 24 2.5.2 Example 2 . 27 2.6 Selecting a Scaling Parameter . 28 3 Partial Discharge Source Classification 32 3.1 Experimental Setup . 33 3.2 Data Description . 34 3.3 Signal Reconstruction . 35 3.4 Removing Signal Delay . 37 3.5 Overview of Some Classification Methods . 41 3.5.1 Bayes Classifier . 42 3.5.2 Linear Discriminant Analysis . 43 3.5.3 Quadratic Discriminant Analysis . 45 v 3.5.4 Support Vector Machines . 46 3.6 Classification of Experimental Data . 49 3.7 Principal Component Analysis . 51 3.8 Normalizing the Signals . 52 4 System Identification with Laguerre Functions 55 4.1 System Identification . 55 4.2 Simulation Studies . 58 4.2.1 Example 1 . 59 4.2.2 Example 2 . 63 4.2.3 Example 3 . 65 4.3 System Identification Based on Noisy Signals . 66 5 Lasso Methodology for System Identification 70 5.1 Introduction . 70 5.2 Statistical Approach Towards System Identification . 72 5.3 Group Lasso Objective Function . 73 5.3.1 Group Lasso . 74 5.4 System Identification Using Mean Input and Output Signals . 76 5.5 Experimental Setup . 77 5.6 Estimated Systems using Group Lasso Methodology . 78 5.7 Reconstructed Output . 80 vi 5.8 Reconstructed Input . 81 5.9 Classification on Reconstructed Input . 82 5.10 Simulating for Classification using the Input Signal . 84 6 Conclusion and Future Work 90 6.1 Numerical Limitations . 93 6.2 Future Work . 93 A Smoothing the Signal Using Cubic Smoothing Splines 94 Bibliography 96 vii List of Tables 2.1 Interval for p for different orders. 31 3.1 The confusion matrix between the actual and predicted labels for the test data using the LDA model. 50 3.2 The confusion matrix between the actual and predicted labels for the test data using the QDA model. 50 3.3 The confusion matrix between the actual and predicted labels for the test data using the SVM model with a Gaussian kernel. 51 4.1 Resistors, capacitors and inductors in the time and Laplace domains. 59 5.1 The confusion matrix between the actual and predicted labels for the test data using the LDA model. 83 5.2 The confusion matrix between the actual and predicted labels for the test data using the QDA model. 83 5.3 The confusion matrix between the actual and predicted labels for the test data using the SVM model. 84 5.4 The confusion matrix between the actual and predicted labels for the test data using the QDA and SVM models of the input and the QDA classification in case 1. 88 viii 5.5 The confusion matrix between the actual and predicted labels for the test data using the SVM classifier of case 1. 88 5.6 The confusion matrix between the actual and predicted labels for the test data using the SVM classifier of case 2. 88 5.7 The confusion matrix between the actual and predicted labels for the test data using the SVM classifier of case 2. 88 ix List of Figures 1.1 A blackbox representation of a system. .1 2.1 A simple Resistor{Capacitor (RC) circuit which is time invariant if the values of R and C do not change over time. [Source: Oppenheim et al. (1996)] . 14 2.2 First 5 orders of (a) Laguerre polynomials in the y range of −10 to 20 and (b) Laguerre functions with p = 1. 16 2.3 Laguerre functions on order n = 4 for multiple scaling parameters (p). 17 2.4 Gaussian function f1(t) to be approximated using Laguerre functions. 24 2.5 Approximated Gaussian functions using (a) least square and (b) Lasso objective functions together with the actual function. 25 2.6 Approximated Gaussian function using least absolute objective for an initial value of (a) 1 and (b) 0. 26 2.7 Approximated Gaussian function using the exact method. 26 2.8 (a) Function f2(t) to be approximated. Laguerre approximation of f2(t) using (b) least-squares, (c) least absolute and (d) Lasso objective functions. 28 x 2.9 Approximated function f2(t) using the exact method. 29 1% 2.10 (a) Relationship of n and p with log fT (n; p) . (b) Fitted relationship 1% of n and p with log fT (n; p) , using a quadratic model. 30 3.1 Discharge sources, (a) a twisted pair of wires and (b) a needle-plane setup, used to obtain partial discharge signals in this research. 34 3.2 Sample of PD pulses for the sources (a) twisted pair of wires, (b) needle-plane setup and (c) combined, obtained in the lab. 35 3.3 (a) a regular PD pulse in the source with the twisted pair of wires. (b) { (f) anomaly pulses generated due to the positive half of the high voltage signal. 36 3.4 Sample PD pulses and the approximated signals obtained using the Laguerre basis approximation. 37 3.5 Sample of PD pulses where the delays are removed, and the approxi- mated signal overlaid. 41 3.6 Three separating hyperplanes, out of infinite possibilities. [Source: James et al. (2013).] . 47 3.7 Support vectors. 48 3.8 Errors in support vector classification. 49 3.9 First 3 Laguerre coefficients for the signals from all sources. 50 3.10 First 3 principal components of the Laguerre coefficients for the signals form all sources. 52 3.11 Sample of normalized PD pulses and the approximated signal overlaid. 53 xi 3.12 First 3 Laguerre coefficients for the scaled signals form all sources. 54 3.13 Misclassification error for the three sources with the number of selected coefficients.
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