logn_to_normal: Convert Log-Normal parameters to Normal parameters

In AlessandroDeCarlo27/mvlognCorrEst: Sampling from a multivariate Log-Normal distribution using Log-Normal parameters and Estimation of Indirect Correlations

Convert Log-Normal parameters to Normal parameters

Description

This function is used to convert mean vector and covariance matrix of a multivariate Log-Normal distribution to mean vector and covariance matrix of the associated multivariate Normal distribution. Two conditions are evaluated before computing the parameters of Normal distribution:

1. For each couple of variables, X1, X2, it is tested if \rho(X1,X2) satisfies the condition necessary to obtain a positive definite correlation/covariance matrix for the Normal distribution underlying the Log-Normal one. This condition is also sufficient for 2x2 matrix [1]. If this condition is not satisfied, a warning message is displayed but covariance matrix of the Normal distribution is still computed even if it is not positive definite.

2. For each couple of variables, X1, X2, it is tested if \rho(X1,X2)*cv1*cv2 > -1. If this condition is not satisfied for all the couples the covariance matrix of the underlying Normal distribution cannot be computed. In fact, the covariance between X1, X2 of the Normal distribution, \Sigma^2(X1,X2), is defined as ln((\rho(X1,X2)*cv1*cv2 )+1) [1]. In this case, a warning message will be displayed and a summary of the performed tests is returned.

Usage

logn_to_normal(mu, covMatrix)

Arguments

mu	Array object containing means of multivariate Log-Normal distribution.
covMatrix	Matrix object containing covariance matrix of multivariate Log-Normal distribution.

Value

A list containing:

muN	Array object. It contains mean of the multivariate Normal distribution associated to Log-Normal one.
sigmaN	Matrix object. It contains covariance matrix of the multivariate Normal distribution associated to Log-Normal one.
is_valid_logN_corrMat	Boolean flag which indicates if the input correlation matrix satisfies condition 1.
can_log_transf_covMat	Boolean flag which indicates if the input correlation matrix satisfies condition 2.
validation_res	Matrix object which contains a brief summary of the tested conditions on the input correlation matrix. It is composed by N(=number of Log-Normal variable couples) rows and 8 columns:
	var1: numerical index of the first Log-Normal variable of the couple, X1 var2: numerical index of the second Log-Normal variable of the couple, X2 lower: lower bound for the range of Log-Normal correlation between var1 and var2 upper: upper bound for the range of Log-Normal correlation between var1 and var2 tested_corr: input correlation value between var1 and var2 is_valid_bound: numerical flag. If 1 tested_corr is inside the range, 0 otherwise tested_for_logTransf: value computed for testing condition 2 (\rho(X1,X2)*cv1*cv2) can_logTransf: numerical flag. If 1 covariance of normal distribution between var1 and var2 can be computed, 0 otherwise.

Note

This function tries to compute the covariance matrix of the multivariate Normal distribution even if the Log-Normal covariance matrix is not positive definite. User should check that input matrix is positive definite, however in this situation logn_to_normal returns a warning message. Even if input matrix is positive definite and conditions 1 and 2 are fulfilled, covariance matrix of multivariate Normal distribution could not be positive definite because condition 1 is only necessary [1]. In order to fix this problem, nearPD of Matrix package could be used.

Author(s)

Alessandro De Carlo [e-mail]

References

[1] Henrique S. Xavier, Filipe B. Abdalla, Benjamin Joachimi, Improving lognormal models for cosmological fields, Monthly Notices of the Royal Astronomical Society, Volume 459, Issue 4, 11 July 2016, Pages 3693–3710, https://doi.org/10.1093/mnras/stw874

See Also

nearPD

get_logNcorr_bounds

validate_logN_corrMatrix

Examples

#define correlations
corr<- diag(rep(1,4))
corr[1,4] <- 0.9
corr[4,1]<-corr[1,4]
corr[2,4] <- -0.3
corr[4,2] <- corr[2,4]
corr[3,2] <- -0.2
corr[2,3] <- corr[3,2]

#define sd of variables
sd2 <- array(c(rep(1,4)))
#obtain covariance matrix
covMatrix2 <- sd2%*%t(sd2)*corr
#define mean vector
mu2 <- array(rep(2.5,4))
logn_to_normal(mu2,covMatrix2)

#output:

# $is_valid_logN_corrMat
# [1] TRUE

# $can_log_transf_covMat
# [1] TRUE
#
# $validation_res
#       var1 var2     lower upper tested_corr is_valid_bound tested_for_logTransf can_logTransf
# [1,]    1    2 -0.862069     1         0.0              1                0.000             1
# [2,]    1    3 -0.862069     1         0.0              1                0.000             1
# [3,]    1    4 -0.862069     1         0.9              1                0.144             1
# [4,]    2    3 -0.862069     1        -0.2              1               -0.032             1
# [5,]    2    4 -0.862069     1        -0.3              1               -0.048             1
# [6,]    3    4 -0.862069     1         0.0              1                0.000             1

# $muN
# [1] 0.8420807 0.8420807 0.8420807 0.8420807

# $sigmaN
# [,1]        [,2]        [,3]        [,4]
# [1,] 0.1484200  0.00000000  0.00000000  0.13453089
# [2,] 0.0000000  0.14842001 -0.03252319 -0.04919024
# [3,] 0.0000000 -0.03252319  0.14842001  0.00000000
# [4,] 0.1345309 -0.04919024  0.00000000  0.14842001

AlessandroDeCarlo27/mvlognCorrEst documentation built on March 23, 2023, 10:11 a.m.