Topological data analysis (TDA) is a recent methodology for extracting topological and geometrical features from data of complex geometric structures. Persistent homology, a new mathematical notion proposed by Edelsbrunner (2002), provides a multiscale descriptor for the topology of data, and has been recently applied to a variety of data analysis. In this work I will introduce a machine learning framework of TDA by combining persistence homology and kernel methods. A method of kernel embedding of persistence diagrams to obtain their vector representation is presented, which enables one to apply any kernel methods in topological data analysis. As a special kernel to persistence diagrams, the persistence weighted Gaussian kernel is proposed and some theoretical properties are discussed. The methods are applied to change point detection and time series analysis in the field of material sciences and biochemistry.