The Theory Of Pictographically Wrapping Manifold (PWM)and Its Application In Pattern Classification

Transportation vehicles (such as cars, ships, etc.) will lost its intended function or lead to serious catastrophic accidents if a fault occurs. Therefore, effective condition monitoring and fault diagnosis is an important means to guarantee their normal operation and fault warning.Pattern recognition is an important aspect in the field of artificial intelligence. After years of development, the pattern recognition technology has been applied in many industries. In short, the core of pattern recognition is to classify. It involves supervised pattern recognition and unsupervised pattern recognition. To use pattern recognition techniques in condition monitoring and fault diagnosis of transportation vehicles is important in theory and practice to monitor the working state of them intelligently and efficiently in real-time.The status and characteristics of four important pattern recognition methods for discrete data are analyzed and researched on, which are artificial neural network (ANN), support vector machine (SVM), manifold learning and biomimetic pattern recognition. In both ANN and SVM, the classification boundary in the sample dataset of different classes is needed to find. There exist some problems in ANN:besides local minimum problem, some parameters can only be determined by experiences. The quadratic programming in SVM is complex and difficult if the size of a sample dataset is large. These problems hinder the developments of ANN and SVM. But, at the same time, the exploration of other methods is triggered and promoted. Manifold learning and are different from ANN and SVM. They are based on exploration and use of the geometrical structure (an intrinsic property) of data. Manifold learning aims to solve problems of high dimensional data and the core is to find the low dimensional manifold on which the data points distribute. Biomimetic pattern recognition expresses the geometrical structure by constructing covers of data points. Both methods deepen pattern recognition into intrinsic properties of data, so they are the pattern recognition methods revealing the intrinsic nature of data.However, these two methods also have their drawbacks, for instance, the hypothesis that high dimensional data points should distribute on a low dimensional manifold is not always reasonable. It is more difficult to determine whether the data points distribute on a low dimensional manifold and to determine the dimension of this manifold if there exists a low dimensional manifold.Inspired by these methods and to solve the problems in them, this dissertation proposes the theory of pictographically wrapping manifold (PWM). This theory is based on the analysis of these methods, topology and differential geometry. A smooth manifold is used to wrap a high dimensional discrete dataset to express the geometric structure of the dataset efficaciously in this theory. The relevant analyses and proofs are shown in this dissertation. Axis of Arbitrary Direction (AAD) is proposed to analyze and solve some problems of a high dimensional PWM in2-dimensional space. And PWM pattern classification is also put forward and verified by experiments.The main work for this dissertation can he summarized as follows:(1) The PWM function and the PWM equation are constructed. The solution set of the PWM equation is a smooth closed manifold which is called PWM. The PWM can express the geometric structure of a dataset in the original space and keep the nature shape of the dataset without finding the dimension of a low dimensional manifold. The extended PWM equation is also constructed. The two examples of in this dissertation show that the PWM can express the geometric structure of the dataset effectively.(2) A method, axis of arbitrary direction (AAD), is proposed to simplify the discussion about a higher dimensional dataset and its PWM in a two-dimensional plane. The PWM is proved to be homeomorphic with a spherical surface in n dimensional Euclidean space when the parameters of the manifold satisfy certain conditions. The fact is expounded that a PWM can split or merge when its parameters change. Moreover, the number of PWM components is equal to the number of the dataset and each component is homeomorphic with a n dimensional spherical surface with appropriate manifold parameters.(3) An important application of PWM in supervised pattern classification is proposed, which is named as PWM classification. This algorithm uses PWMs of different classes as a classifier directly. There is no need to calculate the manifold, thus the computational cost is largely reduced. This is the most fundamental difference between this algorithm and the most supervised pattern classification algorithms. Since only addition, subtraction, division and power are included, the complexity of the algorithm is much lower especially for a high dimensional, large and nonlinear dataset. The PWM classification is a supervised pattern classification method to display the intrinsic nature of data. The experiment to classify data points with helix shape shows the feasibility and effectiveness of PWM classification.Connectivity of two seed points is discussed. A method to choose the initial values of parameters and evaluating indicators are given.(4) Inner distance and outer distance are defined since manifolds may overlap sometimes. These definitions can be used to determine the relative position of a data point and a manifold to avoid complex calculation on distance in high dimensional space. Inner distance and outer distance simplify the calculation of PWM classification. Three methods are introduced to solve the manifold overlap problem.The classification experiments are conducted on a real fault dataset. The results show that PWM classification performs quite well when appropriate parameters are taken.