PyMC - 方差-协方差矩阵估计

我读了下面的论文（http://www3.stat.sinica.edu.tw/statistica/oldpdf/A10n416.pdf http://www3.stat.sinica.edu.tw/statistica/oldpdf/A10n416.pdf）其中，他们将方差-协方差矩阵 Σ 建模为：

Σ = diag(S)*R*diag(S)（论文中的公式1）

S 是标准差的 k×1 向量，diag(S) 是对角元素为 S 的对角矩阵，R 是 k×k 相关矩阵。

我如何使用 PyMC 来实现这一点？

这是我编写的一些初始代码：

import numpy as np
import pandas as pd
import pymc as pm

k=3
prior_mu=np.ones(k)
prior_var=np.eye(k)
prior_corr=np.eye(k)
prior_cov=prior_var*prior_corr*prior_var

post_mu = pm.Normal("returns",prior_mu,1,size=k)
post_var=pm.Lognormal("variance",np.diag(prior_var),1,size=k)
post_corr_inv=pm.Wishart("inv_corr",n_obs,np.linalg.inv(prior_corr))


post_cov_matrix_inv = ???

muVector=[10,5,-2]
varMatrix=np.diag([10,20,10])
corrMatrix=np.matrix([[1,.2,0],[.2,1,0],[0,0,1]])
cov_matrix=varMatrix*corrMatrix*varMatrix

n_obs=10000
x=np.random.multivariate_normal(muVector,cov_matrix,n_obs)
obs = pm.MvNormal( "observed returns", post_mu, post_cov_matrix_inv, observed = True, value = x )

model = pm.Model( [obs, post_mu, post_cov_matrix_inv] )
mcmc = pm.MCMC()

mcmc.sample( 5000, 2000, 3 )

Thanks

[edit]

我认为可以使用以下方法来完成：

@pm.deterministic
def post_cov_matrix_inv(post_sdev=post_sdev,post_corr_inv=post_corr_inv):
    return np.diag(post_sdev)*post_corr_inv*np.diag(post_sdev)

这是为了偶然发现这篇文章的人的利益而提供的解决方案：

p=3
prior_mu=np.ones(p)
prior_sdev=np.ones(p)
prior_corr_inv=np.eye(p)


muVector=[10,5,1]
sdevVector=[3,5,10]
corrMatrix=np.matrix([[1,0,-.1],[0,1,.5],[-.1,.5,1]])
cov_matrix=np.diag(sdevVector)*corrMatrix*np.diag(sdevVector)

n_obs=2000
x=np.random.multivariate_normal(muVector,cov_matrix,n_obs)

prior_cov=np.diag(prior_sdev)*np.linalg.inv(prior_corr_inv)*np.diag(prior_sdev)

post_mu = pm.Normal("returns",prior_mu,1,size=p)
post_sdev=pm.Lognormal("sdev",prior_sdev,1,size=p)
post_corr_inv=pm.Wishart("inv_corr",n_obs,prior_corr_inv)

#post_cov_matrix_inv = pm.Wishart("inv_cov_matrix",n_obs,np.linalg.inv(prior_cov))
@pm.deterministic
def post_cov_matrix_inv(post_sdev=post_sdev,post_corr_inv=post_corr_inv,nobs=n_obs):
    post_sdev_inv=(post_sdev)**-1
    return np.diag(post_sdev_inv)*cov2corr(post_corr_inv/nobs)*np.diag(post_sdev_inv)

obs = pm.MvNormal( "observed returns", post_mu, post_cov_matrix_inv, observed = True, value = x )

model = pm.Model( [obs, post_mu, post_sdev ,post_corr_inv])
mcmc = pm.MCMC(model)

mcmc.sample( 25000, 15000, 1,progress_bar=False )