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 在网上用中文搜索“ARMA模型的自协方差”怎么也找不到想要的东西,出来的结果大都是要付钱的期刊论文。我只想找一个入门级的教程而已。

后来突然想到用英文搜索,在http://www.google.com的英文界面里,我才输入“ARMA auto”,联想框就出现了我想要的"ARMA autocovariance function",于是我点击进去,第一个链接就正是我想要的,而且后面的链接都很不错。

原文是PDF文档,有一些下标错误。我将它修正了一下,分享出来:

http://docs.google.com/View?id=d4rdnxm_253d5c8hmf4

 

预览:


 

Finding the Autocovariance Function (AVCF) for a general ARMA model

(from the lecture 1.3.2006)

 

In a general ARMA model we have:

 

, where  for convenience of notation.                                          (1)
 

Let  and . Note that  for , since future noise values are independent of the observed  values.

 

Determining the cross covariances 

 

Multiplying both sides of (1) with  and taking expectations, we obtain:

,                                                                                                                               (2)

Since  unless .

Since  for , (2) is a triangular system of linear equations, and we get:

                                                       (3)

Thus, the cross-covariances can be obtained by a simple recursion.

 

Determining the covariances 

 

Multiplying both sides of (1) with  and taking expectations, we obtain:

                                                                                                  (4)

The right hand sum in (4) goes from  (not ), since  for .

(4) is a system of linear equations for the unknown covariances . Note that the same covariance can occur twice in each equation: the first few equations look as follows:

                                                                                                                                                                                                                                                                         (5)

Note that the right hand sides contain only known values, after the  have been found using (3).

Collecting the terms that contain the same covariances, and rearranging into an equation system for the , (5) can be rewritten:

                                                                                                                (6)

Where the  denotes the right hand sand of the equations given by (5). Solving (6) gives the required covariances.

 

Eivind Damsleth

2006-3-1