Skip to main content
Cross Validated

Return to Answer

replaced http://stats.stackexchange.com/ with https://stats.stackexchange.com/
Source Link
Community Bot
Community Bot
  • 1

General ideas are from here and herehere. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000

General ideas are from here and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000

General ideas are from here and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
Source Link
Community Bot
Community Bot
  • 1

General ideas are from herehere and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000

General ideas are from here and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000

General ideas are from here and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000
Source Link
zlon
zlon
  • 718
  • 6
  • 21

General ideas are from here and here. #R code

  1. You first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $

     NVariables=5
 VariableLen=1000
 Z=matrix(rnorm(NVariables*VariableLen), ncol=NVariables)

  2. Create covariance matrix $\Sigma$. Let all variables be correlated with neighbor as 0.5.

     Sigma=matrix(data=0, ncol=NVariables-1,nrow=NVariables-1)
 diag(Sigma)<-0.5
 Sigma=cbind(matrix(data=0,nrow=NVariables-1),Sigma)
 Sigma=rbind(Sigma,matrix(data=0,ncol=NVariables))
 diag(Sigma)<-1

  3. Find a square root of $\Sigma$. Cholesky decomposition is common choice

     C=chol(Sigma)

  4. To obtain random variables with given correlation matrix $\Sigma$ multiply $\bf C$ and $\bf Z$

     Y=Z%*%C

  5. Use inverse CDF method to obtain any distribution You wish. Here it is lognormal

     Ylog=qlnorm(pnorm(Y))

#Results enter image description here

Correlation Matrix

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000  0.52817152 -0.01887624 -0.07113405 -0.05551355
[2,]  0.52817152  1.00000000  0.49392903 -0.03233261 -0.01504632
[3,] -0.01887624  0.49392903  1.00000000  0.50604908  0.04029076
[4,] -0.07113405 -0.03233261  0.50604908  1.00000000  0.49000229
[5,] -0.05551355 -0.01504632  0.04029076  0.49000229  1.00000000