Aggregating a series of correlated lognormal variables

Generating observations and log-normally distributed random errors

We generate 10000 Observations of a sum of 100 random variables with mean 10 and multiplicative standard deviation of 1.7.

nObs <- 100; nRep <- 10000
#nObs <- 1000; nRep <- 100
xTrue <- rep(10, nObs)
sigmaStar <- rep(1.7, nObs) # multiplicative stddev of 1.2
theta <- getParmsLognormForExpval(xTrue, sigmaStar)
# generate observations with correlated errors
acf1 <- c(0.4,0.1)
corrM <- setMatrixOffDiagonals(
  diag(nrow = nObs), value = acf1, isSymmetric = TRUE)
xObsN <- exp(mvtnorm::rmvnorm(
  nRep, mean = theta[,1]
  , sigma = diag(theta[,2]) %*% corrM %*% diag(theta[,2])))

A single draw of the autocorrelated 100 variables looks like the following.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   723.2   946.1   996.6  1000.6  1051.0  1350.7

Estimating the correlation matrix and effective number of parameters

The autocorrelation used to generate the sample was:

c(1, acf1)

## [1] 1.0 0.4 0.1

The effective one estimated from the sample is:

(effAcf <- computeEffectiveAutoCorr(ds$xErr))

## [1] 1.000000000 0.365928081 0.003976514

(nEff <- computeEffectiveNumObs(ds$xErr))

## [1] 57.72567

Due to autocorrelation, the effective number of parameters is less than nObs = R nObs.

Computing the mean and its standard deviation

First we compute the distribution parameter of the sum of the 100 variables. The multiplicative uncertainty has decreased.

#coefSum <- estimateSumLognormal( theta[,1], theta[,2], effAcf = effAcf )
coefSum <- estimateSumLognormal( theta[,1], theta[,2], effAcf = c(1,acf1) )
exp(coefSum["sigma"])

##    sigma 
## 1.077687

Its expected value corresponds to the expected value (100*10).

(sumExp <- getLognormMoments( coefSum[1], coefSum[2])[1,"mean"])

## mean 
## 1000

The lognormal approximation of the distribution of the sum, is close to the distribution of the 10000 repetitions.