Wavelet frames and redundant wavelet transforms for fault detection

Wavelet Analysis has been a very strong signal analysis technique in the last two decades compared to classical methods like Fourier and Short-Time Fourier transforms and widely used by researches on many different kinds of applications. Wavelets are very powerful in detecting and revealing signal characteristics of which other transforms lack or are not capable. Since there have been thousands of wavelet applications in the literature, it becomes too difficult to classify their fields of application. Hence, the most popular wavelet methods are compressing and de-noising signal applications. The most significant feature which put Wavelet Analysis aside from the other classical methods is that it provides both time and frequency information. It provides a fix to resolution problem that Short-Time Fourier transform suffers – the problem which arises from the usage of fixed window functions. However, wavelet functions can be chosen more freely as opposed to window functions in Short-Time Fourier analysis. Since they have an adaptive structure, they provide better time resolution in high frequencies which also makes them more selective methods. Wavelet applications in the literature usually do not touch base with wavelet functions which are used in applications as most of the researches are tend to utilize ready-to-use toolboxes. Some of these studies do not even mention which type of wavelet transform was used and the researchers need to read through all the text in order to understand which type of wavelet transform is considered. A vast majority of these studies do not literally question the usage of wavelet functions in terms of orthogonality. Furthermore, they do not make any pre-analysis for a given wavelet function and scale/level selection in order to optimize the analysis. Hence, this is a very important gap that the researchers have to fill. Since wavelet transforms are considered as very redundant methods, those considerations over scale/level, function and wavelet type selections will affect the redundancy of analysis. Redundancy term has two different usages in the literature. The first concept of redundancy is mentioned in Frame Theory as the redundancy ratio. Redundancy ratio is used a measure of orthogonality for a given frame which is defined in a Hilbert Space. The second concept of redundancy term has a broad definition and literally comprise the first concept of redundancy definition. It is used to define how robust a transform is by all means. For example, if a vector in a Euclidian space is defined with three dimensional vectors, this actually points the second concept redundancy whereas the first concept is explicitly applicable to Frame Theory. Even though the redundancy of wavelet functions can be related to the first concept of redundancy term, the redundancy of wavelet transforms should be related to the second concept. Daubechies did a calculation for frame boundaries (a.k.a. redundancy ratios) of her wavelet functions and the others in her book, Ten Lectures on Wavelet, where she had defined the mathematical structures of wavelet functions. She also encouraged the researchers to investigate the frames further. However, in this study, the second concept of redundancy definition will be considered and the most redundant wavelet transform which is the Continuous Wavelet Transform will be examined. The product of CWT transform which are the redundant subscale signals will be reconstructed under pre-defined conditions in order to build a more redundant signal representation compared to the original signal. It is argued that this so-called redundant signal representation provides more sensitivity to fault detection. A new method which is called as Partial Feature Construction (PFC), has also been introduced when reconstructing these new redundant signals. The flow chart of this method and its mathematical representations have also been explained throughout this study by testing it with both artificial and real data. The main goal is not a perfect reconstruction of the original signal but it is to construct redundant signal representations which will reveal faulty characteristics. Furthermore, the redundancies of the new signals are calculated with a variance based estimation. The first Chapter starts with a bit of historical background of wavelet transforms. It also includes a large literature survey of wavelet transforms on fault detection applications. Thus, it mentions about redundant wavelet transforms and their explorations by the researchers. The second Chapter talks about the mathematical background of wavelet transforms which is a very critical concept to understand the behavior of wavelet functions. It also mentions inner product spaces and orthogonality in Hilbert space and of course Frame Theory. The main goal of this chapter is to explain the researcher the orthogonality which produce redundancy in transforms. In addition, the general concepts of vector spaces are explained in Appendix A. In Chapter 3, wavelet function types are discussed as well as wavelet transforms. Continuous Wavelet Transform which is defined as the most redundant wavelet transform and which will be the core focus of this thesis is discussed at first. And next, Discrete Wavelet and Stationary Wavelet Transforms which are also be considered as a redundant transform are represented. Chapter 4 explains the proposed Partial Feature Construction method which helps to create redundant signal representations. The PFC algorithms' flow chart and the related mathematical equations are also given. Also, this chapter discusses the redundancy calculations for the first and seconds concept of redundancy terms. Chapter 5 includes artificial data applications using Continuous and Stationary Wavelet Transforms introduced in Chapter 4. Two artificial signals, stationary and non-stationary respectively, have been simulated. Each of them has been analyzed by Continuous and Stationary Wavelet Transform with using orthogonal and non-orthogonal wavelet functions. The effect of orthogonality has also been investigated. With the proposed PFC method, the entropy based functions are used to integrate redundant signals so that the redundancy effect of these transforms and functions are able to be investigated. Thus, as a comparison with wavelet transforms, the Short-Time Fourier application over these two signals are introduced in Appendix B in order to give the researcher to see the differences between a classical method and wavelets. In Chapter 6 and 7, the real data applications have been introduced with only using Continuous Wavelet Transform. In Chapter 6, a bearing damage vibration data from an electrical discharge machining is considered. In Chapter 7, the ferroresonance data from a power system is tested. Both data have been applied by Partial Feature Construction method. It is seen that the redundant signals which are generated by PFC can be used in fault detection as a sensitive method. In the last Chapter 8, the results and findings of these applications and proposed Partial Feature Construction method have been discussed. Based on the findings, Partial Feature Construction method seems very promising as a feature extraction method. In this manner, it can be planned to use with neural network applications. This study can be extended to be a real-time application for critical systems. It can be modelled as a condition monitoring tool and faults can be detected more sensitively by using redundancy feature. This would reduce the maintenance cost of industrial plants and factories where the continuous operations must be held.