Functional Support Vector Machine In Reproducing Kernel Hilbert Space And Its Applications

With the continuous progress of today's science and technology,the quantity and form of data obtained by people has also been greatly developed,and functional data,such as onedimensional or multi-dimensional functions in the time domain or region,are increasingly used by people in finance,economy,medicine,e-commerce and other fields are widely used.The classification of functional data is closely related to traditional human life.For example,glaucoma,the second most common blind eye disease in the world,is aided by the fundus OCT image data obtained by computer,and the fat content of minced meat samples is studied by using the absorbance value of near-infrared light.In recent years,the incidence of cardiovascular diseases has been increasing year by year.In order to effectively prevent such diseases,it's necessary to make a preliminary diagnosis of the patient through the electrocardiogram.However,under normal circumstances,manual discrimination of ECG by experts in the medical field is not only time-consuming and labor-intensive,but also prone to errors.Therefore,the use of computer-aided diagnosis of electrocardiogram is very important.The use of computer-aided diagnosis of whether there is an abnormality in the patient's electrocardiogram can actually be transformed into a binary classification problem of functional data.But when we try to use classical classifiers to classify functional data,we often encounter some bottlenecks.The reason is that functional data is inherently infinite-dimensional,and most algorithms are used to deal with low-dimensional vectors.The existing classical classifiers cannot be directly applied or have poor performance due to the curse of dimensionality.In order to solve this problem,this thesis firstly constructs a linear function-type support vector machine model,and based on a reproducing kernel Hilbert space H(K)that defines the reproducing kernel K,by finding the minimum value of the regularized empirical risk function.Then,random numerical simulations are carried out for finite samples with three different settings,and the results show that they all have good finite-sample properties.Finally,the model is applied to ECG data and spectral data.The results show that the linear function support vector machine proposed in this thesis is effective and practical.Support vector machine has been widely used in the classification of finite-dimensional data,but its application in the classification of functional data is rare.Most of the research results in the existing literature are based on the functional data after dimensionality reduction,that is,construct a support vector machine for the extracted feature vectors without preserving the integrity of the original data.Or based on Hilbert space,it simply mentioned the use of support vector machines to classify functional data,but did not discuss the specific form of the model and parameter estimation in depth.Therefore,this thesis discusses the above problems and improves the limitations of the existing theoretical research.In this thesis,a linear functional SVM model is constructed by projecting functional data into a specific direction with the help of kernel tricks,and then using SVM to classify it.Among them,the projection direction is identified by minimizing the empirical risk function containing a specific loss function in the support vector machine classifier,and the risk function is built on the reproducing kernel Hilbert space.At present,there is no literature to construct a functional support vector machine model based on the reproducing kernel Hilbert space.In addition,with the help of some theorems,this thesis deduces the expression of the estimator to find the optimal solution of the unknown parameters,and transforms the infinite-dimensional minimization problem into a finite-dimensional problem.The existing functional data classification methods are extended and innovated.Nowadays,functional data has been widely used in various fields.Since the functional data can be classified into useful information,its classification problem has also become an important task in scientific research.Therefore,by using support vector machines with strong generalization ability,we can further study and expand the efficient classification of functional data.The method has certain practical significance.The linear functional support vector machine proposed in this thesis can avoid the problem of overfitting and has the excellent characteristics of the support vector machine classifier.In addition,the use of a reproducing kernel Hilbert space also enables one to control the roughness of the estimated projection direction.Therefore,compared with the traditional functional logistic regression and central classifier,the classifier proposed in this thesis can improve the prediction accuracy.