| Topological Data Analysis（TDA）is a kind of data analysis method which is a combination of topology and data statistics.Compared with traditional data analysis methods,topological data analysis makes effective use of the knowl-edge of module theory in algebra,and takes persistent homology as the basic computing tool for research,to study the topological characteristics of the given data sets.We figuratively call this topological feature a“shape”of the data set,which is also what we call a topological invariant.The“shape”structure analysis of data has a wide range of applications and practical significance in various fields.But many theories need to be further improved.At present persistence homology is an important and widely used tool in topology data analysis.It is mainly based on module theory.Through a nested filtered complex sequence{VR4)（）}4∈Z)which is constructed on a given point cloud data set.Namely,We can obtain the corresponding nested complex{(14)（）}4∈Z)from the given metric(9 and a family of distance parameters{4)}4∈Z),when the finite point cloud date setis fixed.A continuous change of the homology class on the complex is then induced by the inclusion relation of the complexes in the nested complex sequence with the increasing of the dis-tance parameterin the simplest case when all4)’s are equal.In other words,it is a continuous process of change from the generation of a homology class to its demise.We can use a family interval to represent this process of continuous change.And this family of interval is what we call the barcode in persistence homology visualization.Visually speaking,a 0-dimensional homology class rep-resents a path-connected component in a topological space（in our case,a sim-ple complex）,while a 1-dimensional homology class represents a 1-dimensional“hole”in a topological space.Similarly,higher-dimensional homology classes denote higher-dimensional holes.From the barcode we can count the num-ber of path-connected components and one-dimensional or higher-dimensional“holes”over time.The homology theory has the characteristics of being eas-ier to calculate and easier to implement in practice.In this thesis,the theory of persistence homology is summarized,and the visualization of the theory is realized for a concrete case.Finally,the author puts forward the prospect of specific problems as the goal for future study.The thesis is divided into four chapters,with the following structure:In chapter one,we introduces the background and main ideas and methods.In chapter two,we introduce some necessary knowledge for the application.In chapter three,we introduce the basic principle of persistent homology.In chapter four,we introduce the main conclusion and the application implementation of persistent homology... |
