| This paper mainly studies the Variational Bayes Expectation Maximization(VBEM)algorithm of Infinite Erlang Mixture Model(InErMM).In the sense of weak convergence,the mixed Erlang distribution is dense in the positive continuous variable space.Therefore,the multivariate Erlang mixture model constitutes a widely applicable and easy to analyze distribution,which can effectively handle non-negative random variables.Especially in the field of actuarial science,the Erlang mixture distri’bution class can not only be preserved in many risk theories but also can calculate various types of insurance indicators analytically.The main work of this paper is to propose the InErMM,introduce the Dirichlet process as a priori distribution of its mixed components and deduce the CMM-VBEM algorithm for estimating the parameters of InErMM in detail.The CMM-VBEM algorithm mainly includes the following steps:(1)The setting of iterative initial value:CMM algorithm.In this process,we combine moment estimation and K-means clustering algorithm to set iterative initial values.(2)The estimation of posterior distribution:VBEM algorithm.It combines Bayesian estimation and variational principle,calculates the estimation of the true posterior distribution of all variables iteratively,and then obtains the parameter estimation of InErMM.(3)The adjustment of shape parameters:OSF-B algorithm.Since the shape parameter of the InErMM is a matrix and each element is a positive integer,the prior distribution is not easy to give.In the VBEM process,we assume that it is given.Now we update the shape parameters by plus one or minus one to each element in turn,and the parameters of the corresponding posterior distribution are updated.(4)The selection of mixed numbers:BIC criteria.The Bayesian Information Criterion has greater penalties and better effect for the mixed model.The CMM-VBEM algorithm proposed in this paper has several advantages:First,the algorithm have a good estimation effect,especially when the dimension is relatively high.Second,compared with the EP algorithm,the running time efficiency of this algorithm is much higher.Third,compared with random approximation methods such as MCMC,the difficult problem of judging whether to converge to a stable distribution in high-dimensional space is avoided.Fourth,compared with the EM algorithm,this method can prevent the over-fitting problem.When the amount of data is small,the superiority is even more prominent.Finally,the effectiveness of the CMM-VBEM algorithm of InErMM is verified by several simulation experiments with different data quantities in different dimensions and a real example.And we shows the fitting effect vividly through the density function graph,distribution function graph,Q-Q graph and P-P graph. |
