| Title: | Functions for the Lognormal Distribution |
|---|---|
| Description: | The lognormal distribution (Limpert et al. (2001) <doi:10.1641/0006-3568(2001)051%5B0341:lndats%5D2.0.co;2>) can characterize uncertainty that is bounded by zero. This package provides estimation of distribution parameters, computation of moments and other basic statistics, and an approximation of the distribution of the sum of several correlated lognormally distributed variables (Lo 2013 <doi:10.12988/ams.2013.39511>) and the approximation of the difference of two correlated lognormally distributed variables (Lo 2012 <doi:10.1155/2012/838397>). |
| Authors: | Thomas Wutzler |
| Maintainer: | Thomas Wutzler <[email protected]> |
| License: | GPL-2 |
| Version: | 0.1.9 |
| Built: | 2024-11-04 05:05:06 UTC |
| Source: | https://github.com/bgctw/lognorm |
Estimate vector of effective components of the autocorrelation
computeEffectiveAutoCorr(res, type = "correlation")computeEffectiveAutoCorr(res, type = "correlation")
res |
numeric of autocorrelated numbers, usually observation - model residuals |
type |
type of residuals (see |
Returns all components before first negative autocorrelation
numeric vector: strongest components of the autocorrelation function
Zieba 2011 Standard Deviation of the Mean of Autocorrelated
Observations Estimated with the Use of the Autocorrelation Function Estimated
From the Data
# generate autocorrelated time series res <- stats::filter(rnorm(1000), filter = rep(1,5), circular = TRUE) res[100:120] <- NA (effAcf <- computeEffectiveAutoCorr(res))# generate autocorrelated time series res <- stats::filter(rnorm(1000), filter = rep(1,5), circular = TRUE) res[100:120] <- NA (effAcf <- computeEffectiveAutoCorr(res))
Compute the effective number of observations taking into account autocorrelation
computeEffectiveNumObs( res, effAcf = computeEffectiveAutoCorr(res), na.rm = FALSE, exact.na = TRUE )computeEffectiveNumObs( res, effAcf = computeEffectiveAutoCorr(res), na.rm = FALSE, exact.na = TRUE )
res |
numeric of autocorrelated numbers, usually observation - model residuals |
effAcf |
autocorrelation coefficients. The first entry is fixed at 1 for zero distance. |
na.rm |
if not set to TRUE will return NA in there are missings in the series |
exact.na |
if set to FALSE then do not count and correct for missing in the sum of autocorrelation terms. This is faster, but results are increasingly biased high with increasing number of missings. |
Assumes records of all times present. DO NOT REMOVE OR FILTER NA records before. The length of the time series is used.
Handling of NA values: The formula from Zieba 2011 is extended to subtract the number of missing pairs in the count of correlation terms. If 'exact.na=false' the original formula is used (after trimming edge-NAs).
integer scalar: effective number of observations
Zieba & Ramza (2011)
Standard Deviation of the Mean of Autocorrelated
Observations Estimated with the Use of the Autocorrelation Function
Estimated From the Data.
Metrology and Measurement Systems,
Walter de Gruyter GmbH, 18 10.2478/v10178-011-0052-x
Bayley & Hammersley (1946)
The "effective" number of independent observations in an autocorrelated
time series.
Supplement to the Journal of the Royal Statistical Society, JSTOR,8,184-197
# generate autocorrelated time series res <- stats::filter(rnorm(1000), filter = rep(1,5), circular = TRUE) res[100:120] <- NA # plot the series of autocorrelated random variables plot(res) # plot their empirical autocorrelation function acf(res, na.action = na.pass) #effAcf <- computeEffectiveAutoCorr(res) # the effective number of parameters is less than number of 1000 samples (nEff <- computeEffectiveNumObs(res, na.rm = TRUE))# generate autocorrelated time series res <- stats::filter(rnorm(1000), filter = rep(1,5), circular = TRUE) res[100:120] <- NA # plot the series of autocorrelated random variables plot(res) # plot their empirical autocorrelation function acf(res, na.action = na.pass) #effAcf <- computeEffectiveAutoCorr(res) # the effective number of parameters is less than number of 1000 samples (nEff <- computeEffectiveNumObs(res, na.rm = TRUE))
The distribution of y = a - b + s, where a and b are two lognormal random variables and s is a constant to be estimated, can be approximated by a lognormal distribution.
estimateDiffLognormal(mu_a, mu_b, sigma_a, sigma_b, corr = 0) pDiffLognormalSample( mu_a, mu_b, sigma_a, sigma_b, corr = 0, q = 0, nSample = 1e+05 )estimateDiffLognormal(mu_a, mu_b, sigma_a, sigma_b, corr = 0) pDiffLognormalSample( mu_a, mu_b, sigma_a, sigma_b, corr = 0, q = 0, nSample = 1e+05 )
mu_a |
center parameter of the first term |
mu_b |
center parameter of the second term |
sigma_a |
scale parameter of the first term |
sigma_b |
scale parameter of the second term |
corr |
correlation between the two random variables |
q |
vector of quantiles |
nSample |
number of samples |
estimateDiffLognormal: numeric vector with components mu, sigma, and shift, the components of the shifted lognormal distribution.
pDiffLognormalSample: vector of probabilities
estimateDiffLognormal: Estimate the shifted-lognormal approximation to difference of two lognormals
pDiffLognormalSample: Distribution function for the difference of two lognormals based on sampling.
Default provides the probability that the difference is significantly larger
than zero.
Estimate lognormal distribution parameters from a sample
estimateParmsLognormFromSample(x, na.rm = FALSE) estimateStdErrParms(x, na.rm = FALSE)estimateParmsLognormFromSample(x, na.rm = FALSE) estimateStdErrParms(x, na.rm = FALSE)
x |
numeric vector of sampled values |
na.rm |
a logical value indicating whether NA values should be stripped before the computation proceeds. |
The expected value of a can be determined with higher accuracy the larger the sample. Here, the uncorrelated assumption is applied at the log scale and distribution parameters are returned with the same expected value as the sample, but with uncertainty (sigma) decreased by sqrt(nfin - 1).
Since with low relative error, the lognormal becomes very close to the normal distribution, the distribution of the mean can be well approximated by a normal with sd(mean(x)) ~ sd(x)/sqrt(n-1).
numeric vector with components mu and sigma,
i.e., the center parameter (mean at log scale, log(median)) and
the scale parameter (standard deviation at log scale)
estimateParmsLognormFromSample: Estimate lognormal distribution parameters from a sample
estimateStdErrParms: Estimate parameters of the lognormal distribution of the mean from an uncorrelated sample
.mu <- log(1) .sigma <- log(2) n = 200 x <- exp(rnorm(n, mean = .mu, sd = .sigma)) exp(pL <- estimateParmsLognormFromSample(x)) # median and multiplicative stddev c(mean(x), meanx <- getLognormMoments(pL["mu"],pL["sigma"])[,"mean"]) c(sd(x), sdx <- sqrt(getLognormMoments(pL["mu"],pL["sigma"])[,"var"])) # stddev decreases (each sample about 0.9) to about 0.07 # for the mean with n replicated samples se <- estimateStdErrParms(x) sqrt(getLognormMoments(se["mu"],se["sigma"])[,"var"]) sd(x)/sqrt(n-1) # well approximated by normal # expected value stays the same c(meanx, getLognormMoments(se["mu"],se["sigma"])[,"mean"]).mu <- log(1) .sigma <- log(2) n = 200 x <- exp(rnorm(n, mean = .mu, sd = .sigma)) exp(pL <- estimateParmsLognormFromSample(x)) # median and multiplicative stddev c(mean(x), meanx <- getLognormMoments(pL["mu"],pL["sigma"])[,"mean"]) c(sd(x), sdx <- sqrt(getLognormMoments(pL["mu"],pL["sigma"])[,"var"])) # stddev decreases (each sample about 0.9) to about 0.07 # for the mean with n replicated samples se <- estimateStdErrParms(x) sqrt(getLognormMoments(se["mu"],se["sigma"])[,"var"]) sd(x)/sqrt(n-1) # well approximated by normal # expected value stays the same c(meanx, getLognormMoments(se["mu"],se["sigma"])[,"mean"])
Estimate the parameters of the lognormal approximation to the sum
Estimate the parameters of the lognormal approximation to the sum
estimateSumLognormalSample( mu, sigma, resLog, effAcf = computeEffectiveAutoCorr(resLog), isGapFilled = logical(0), na.rm = TRUE ) estimateSumLognormalSampleExpScale(mean, sigmaOrig, ...) estimateSumLognormal( mu, sigma, effAcf = c(), corr = Diagonal(length(mu)), corrLength = if (inherits(corr, "ddiMatrix")) 0 else nTerm, sigmaSum = numeric(0), isStopOnNoTerm = FALSE, na.rm = isStopOnNoTerm )estimateSumLognormalSample( mu, sigma, resLog, effAcf = computeEffectiveAutoCorr(resLog), isGapFilled = logical(0), na.rm = TRUE ) estimateSumLognormalSampleExpScale(mean, sigmaOrig, ...) estimateSumLognormal( mu, sigma, effAcf = c(), corr = Diagonal(length(mu)), corrLength = if (inherits(corr, "ddiMatrix")) 0 else nTerm, sigmaSum = numeric(0), isStopOnNoTerm = FALSE, na.rm = isStopOnNoTerm )
mu |
numeric vector of center parameters of terms at log scale |
sigma |
numeric vector of scale parameter of terms at log scale |
resLog |
time series of model-residuals at log scale to estimate correlation |
effAcf |
numeric vector of effective autocorrelation
This overrides arguments |
isGapFilled |
logical vector whether entry is gap-filled rather than an original measurement, see details |
na.rm |
neglect terms with NA values in mu or sigma |
mean |
numeric vector of expected values |
sigmaOrig |
numeric vector of standard deviation at original scale |
... |
further arguments passed to |
corr |
numeric matrix of correlations between the random variables |
corrLength |
integer scalar: set correlation length to smaller values to speed up computation by neglecting correlations among terms further apart. Set to zero to omit correlations. |
sigmaSum |
numeric scalar: possibility to specify a precomputed scale parameter instead of computing it. |
isStopOnNoTerm |
if no finite estimate is provided then by default NA is returned for the sum. Set this to TRUE to issue an error instead. |
If there are no gap-filled values, i.e. all(!isGapFilled) or
!length(isGapFilled) (the default), distribution parameters
are estimated using all the samples. Otherwise, the scale parameter
(uncertainty) is first estimated using only the non-gapfilled records.
Also use isGapFilled == TRUE for records, where sigma cannot be trusted. When setting sigma to missing, this is also affecting the expected value.
If there are only gap-filled records, assume uncertainty to be (before v0.1.5: the largest uncertainty of given gap-filled records.) the mean of the given multiplicative standard deviation
numeric vector with components mu, sigma,
and nEff,
i.e. the parameters of the lognormal distribution at log scale
and the number of effective observations.
estimateSumLognormalSample: In addition to estimateSumLognormal take care of missing values
and estimate correlation terms.
estimateSumLognormalSampleExpScale: Before calling estimateSumLognormalSample estimate
lognormal parameters from value and its uncertainty given
on original scale.
estimateSumLognormal: Estimate the parameters of the lognormal approximation to the sum
Lo C (2013) WKB approximation for the sum of two
correlated lognormal random variables.
Applied Mathematical Sciences, Hikari, Ltd., 7 , 6355-6367
10.12988/ams.2013.39511
# distribution of the sum of two lognormally distributed random variables mu1 = log(110) mu2 = log(100) sigma1 = log(1.2) sigma2 = log(1.6) (coefSum <- estimateSumLognormal( c(mu1,mu2), c(sigma1,sigma2) )) # repeat with correlation (coefSumCor <- estimateSumLognormal( c(mu1,mu2), c(sigma1,sigma2), effAcf = c(1,0.9) )) # expected value is equal, but variance with correlated variables is larger getLognormMoments(coefSum["mu"],coefSum["sigma"]) getLognormMoments(coefSumCor["mu"],coefSumCor["sigma"])# distribution of the sum of two lognormally distributed random variables mu1 = log(110) mu2 = log(100) sigma1 = log(1.2) sigma2 = log(1.6) (coefSum <- estimateSumLognormal( c(mu1,mu2), c(sigma1,sigma2) )) # repeat with correlation (coefSumCor <- estimateSumLognormal( c(mu1,mu2), c(sigma1,sigma2), effAcf = c(1,0.9) )) # expected value is equal, but variance with correlated variables is larger getLognormMoments(coefSum["mu"],coefSum["sigma"]) getLognormMoments(coefSumCor["mu"],coefSumCor["sigma"])
Construct the full correlation matrix from autocorrelation components.
getCorrMatFromAcf(nRow, effAcf)getCorrMatFromAcf(nRow, effAcf)
nRow |
number of rows in correlation matrix |
effAcf |
numeric vector of effective autocorrelation components . The first entry, which is defined as 1, is not used. |
Compute summary statistics of a log-normal distribution
getLognormMoments(mu, sigma, m = exp(mu + sigma2/2) - shift, shift = 0) getLognormMedian(mu, sigma, shift = 0) getLognormMode(mu, sigma, shift = 0)getLognormMoments(mu, sigma, m = exp(mu + sigma2/2) - shift, shift = 0) getLognormMedian(mu, sigma, shift = 0) getLognormMode(mu, sigma, shift = 0)
mu |
numeric vector: location parameter |
sigma |
numeric vector: scale parameter |
m |
mean at original scale, may override default based on mu |
shift |
shift for the shifted lognormal distribution |
for getLognormMoments a numeric matrix with columns
mean (expected value at original scale)
, var (variance at original scale)
, and cv (coefficient of variation: sqrt(var)/mean).
For the other functions a numeric vector of the required summary.
getLognormMoments: get the expected value, variance, and coefficient of variation
getLognormMedian: get the median
getLognormMode: get the mode
Limpert E, Stahel W & Abbt M (2001)
Log-normal Distributions across the Sciences: Keys and Clues.
Oxford University Press (OUP) 51, 341,
10.1641/0006-3568(2001)051[0341:lndats]2.0.co;2
scaleLogToOrig
# start by estimating lognormal parameters from moments .mean <- 1 .var <- c(1.3,2)^2 parms <- getParmsLognormForMoments(.mean, .var) # # computed moments must equal previous ones (ans <- getLognormMoments(parms[,"mu"], parms[,"sigma"])) cbind(.var, ans[,"var"]) # getLognormMedian(mu = log(1), sigma = log(2)) getLognormMode(mu = log(1), sigma = c(log(1.2),log(2)))# start by estimating lognormal parameters from moments .mean <- 1 .var <- c(1.3,2)^2 parms <- getParmsLognormForMoments(.mean, .var) # # computed moments must equal previous ones (ans <- getLognormMoments(parms[,"mu"], parms[,"sigma"])) cbind(.var, ans[,"var"]) # getLognormMedian(mu = log(1), sigma = log(2)) getLognormMode(mu = log(1), sigma = c(log(1.2),log(2)))
Calculate mu and sigma of lognormal from summary statistics.
getParmsLognormForMedianAndUpper(median, upper, sigmaFac = qnorm(0.99)) getParmsLognormForMeanAndUpper(mean, upper, sigmaFac = qnorm(0.99)) getParmsLognormForLowerAndUpper(lower, upper, sigmaFac = qnorm(0.99)) getParmsLognormForLowerAndUpperLog(lowerLog, upperLog, sigmaFac = qnorm(0.99)) getParmsLognormForModeAndUpper(mle, upper, sigmaFac = qnorm(0.99)) getParmsLognormForMoments(mean, var, sigmaOrig = sqrt(var)) getParmsLognormForExpval(mean, sigmaStar)getParmsLognormForMedianAndUpper(median, upper, sigmaFac = qnorm(0.99)) getParmsLognormForMeanAndUpper(mean, upper, sigmaFac = qnorm(0.99)) getParmsLognormForLowerAndUpper(lower, upper, sigmaFac = qnorm(0.99)) getParmsLognormForLowerAndUpperLog(lowerLog, upperLog, sigmaFac = qnorm(0.99)) getParmsLognormForModeAndUpper(mle, upper, sigmaFac = qnorm(0.99)) getParmsLognormForMoments(mean, var, sigmaOrig = sqrt(var)) getParmsLognormForExpval(mean, sigmaStar)
median |
geometric mu (median at the original exponential scale) |
upper |
numeric vector: value at the upper quantile, i.e. practical maximum |
sigmaFac |
sigmaFac=2 is 95% sigmaFac=2.6 is 99% interval. |
mean |
expected value at original scale |
lower |
value at the lower quantile, i.e. practical minimum |
lowerLog |
value at the lower quantile, i.e. practical minimum at log scale |
upperLog |
value at the upper quantile, i.e. practical maximum at log scale |
mle |
numeric vector: mode at the original scale |
var |
variance at original scale |
sigmaOrig |
standard deviation at original scale |
sigmaStar |
multiplicative standard deviation |
For getParmsLognormForMeanAndUpper
there are two valid solutions, and the one with lower sigma
, i.e. the not so strongly skewed solution is returned.
numeric matrix with columns 'mu' and 'sigma', the parameter of the lognormal distribution. Rows correspond to rows of inputs.
getParmsLognormForMedianAndUpper: Calculates mu and sigma of lognormal from median and upper quantile.
getParmsLognormForMeanAndUpper: Calculates mu and sigma of lognormal from mean and upper quantile.
getParmsLognormForLowerAndUpper: Calculates mu and sigma of lognormal from lower and upper quantile.
getParmsLognormForLowerAndUpperLog: Calculates mu and sigma of lognormal from lower and upper quantile at log scale.
getParmsLognormForModeAndUpper: Calculates mu and sigma of lognormal from mode and upper quantile.
getParmsLognormForMoments: Calculate mu and sigma from moments (mean anc variance)
getParmsLognormForExpval: Calculate mu and sigma from expected value and geometric standard deviation
Limpert E, Stahel W & Abbt M (2001)
Log-normal Distributions across the Sciences: Keys and Clues.
Oxford University Press (OUP) 51, 341,
10.1641/0006-3568(2001)051[0341:lndats]2.0.co;2
# example 1: a distribution with mode 1 and upper bound 5 (thetaEst <- getParmsLognormForModeAndUpper(1,5)) mle <- exp(thetaEst[1] - thetaEst[2]^2) all.equal(mle, 1, check.attributes = FALSE) # plot the distributions xGrid = seq(0,8, length.out = 81)[-1] dxEst <- dlnorm(xGrid, meanlog = thetaEst[1], sdlog = thetaEst[2]) plot( dxEst~xGrid, type = "l",xlab = "x",ylab = "density") abline(v = c(1,5),col = "gray") # example 2: true parameters, which should be rediscovered theta0 <- c(mu = 1, sigma = 0.4) mle <- exp(theta0[1] - theta0[2]^2) perc <- 0.975 # some upper percentile, proxy for an upper bound upper <- qlnorm(perc, meanlog = theta0[1], sdlog = theta0[2]) (thetaEst <- getParmsLognormForModeAndUpper( mle,upper = upper,sigmaFac = qnorm(perc)) ) #plot the true and the rediscovered distributions xGrid = seq(0,10, length.out = 81)[-1] dx <- dlnorm(xGrid, meanlog = theta0[1], sdlog = theta0[2]) dxEst <- dlnorm(xGrid, meanlog = thetaEst[1], sdlog = thetaEst[2]) plot( dx~xGrid, type = "l") #plot( dx~xGrid, type = "n") #overplots the original, coincide lines( dxEst ~ xGrid, col = "red", lty = "dashed") # example 3: explore varying the uncertainty (the upper quantile) x <- seq(0.01,1.2,by = 0.01) mle = 0.2 dx <- sapply(mle*2:8,function(q99){ theta = getParmsLognormForModeAndUpper(mle,q99,qnorm(0.99)) #dx <- dDistr(x,theta[,"mu"],theta[,"sigma"],trans = "lognorm") dx <- dlnorm(x,theta[,"mu"],theta[,"sigma"]) }) matplot(x,dx,type = "l") # Calculate mu and sigma from expected value and geometric standard deviation .mean <- 1 .sigmaStar <- c(1.3,2) (parms <- getParmsLognormForExpval(.mean, .sigmaStar)) # multiplicative standard deviation must equal the specified value cbind(exp(parms[,"sigma"]), .sigmaStar)# example 1: a distribution with mode 1 and upper bound 5 (thetaEst <- getParmsLognormForModeAndUpper(1,5)) mle <- exp(thetaEst[1] - thetaEst[2]^2) all.equal(mle, 1, check.attributes = FALSE) # plot the distributions xGrid = seq(0,8, length.out = 81)[-1] dxEst <- dlnorm(xGrid, meanlog = thetaEst[1], sdlog = thetaEst[2]) plot( dxEst~xGrid, type = "l",xlab = "x",ylab = "density") abline(v = c(1,5),col = "gray") # example 2: true parameters, which should be rediscovered theta0 <- c(mu = 1, sigma = 0.4) mle <- exp(theta0[1] - theta0[2]^2) perc <- 0.975 # some upper percentile, proxy for an upper bound upper <- qlnorm(perc, meanlog = theta0[1], sdlog = theta0[2]) (thetaEst <- getParmsLognormForModeAndUpper( mle,upper = upper,sigmaFac = qnorm(perc)) ) #plot the true and the rediscovered distributions xGrid = seq(0,10, length.out = 81)[-1] dx <- dlnorm(xGrid, meanlog = theta0[1], sdlog = theta0[2]) dxEst <- dlnorm(xGrid, meanlog = thetaEst[1], sdlog = thetaEst[2]) plot( dx~xGrid, type = "l") #plot( dx~xGrid, type = "n") #overplots the original, coincide lines( dxEst ~ xGrid, col = "red", lty = "dashed") # example 3: explore varying the uncertainty (the upper quantile) x <- seq(0.01,1.2,by = 0.01) mle = 0.2 dx <- sapply(mle*2:8,function(q99){ theta = getParmsLognormForModeAndUpper(mle,q99,qnorm(0.99)) #dx <- dDistr(x,theta[,"mu"],theta[,"sigma"],trans = "lognorm") dx <- dlnorm(x,theta[,"mu"],theta[,"sigma"]) }) matplot(x,dx,type = "l") # Calculate mu and sigma from expected value and geometric standard deviation .mean <- 1 .sigmaStar <- c(1.3,2) (parms <- getParmsLognormForExpval(.mean, .sigmaStar)) # multiplicative standard deviation must equal the specified value cbind(exp(parms[,"sigma"]), .sigmaStar)
When comparing values at log scale that have different sd at original scale, better compare log(mean) instead of mu.
scaleLogToOrig(logmean, sigma) scaleOrigToLog(mean, sd)scaleLogToOrig(logmean, sigma) scaleOrigToLog(mean, sd)
logmean |
log of the expected value |
sigma |
standard deviation at log scale |
mean |
expected value at original scale |
sd |
standard deviation at original scale |
numeric matrix with columns mean, and sd at original scale
scaleLogToOrig: get logmean and sigma at log scale
scaleOrigToLog: get mean and sd at original scale
xLog <- data.frame(logmean = c(0.8, 0.8), sigma = c(0.2, 0.3)) xOrig <- as.data.frame(scaleLogToOrig(xLog$logmean, xLog$sigma)) xLog2 <- as.data.frame(scaleOrigToLog(xOrig$mean, xOrig$sd)) all.equal(xLog, xLog2) xLog3 <- as.data.frame(getParmsLognormForMoments(xOrig$mean, xOrig$sd^2)) all.equal(xLog$sigma, xLog3$sigma) # but mu < logmeanxLog <- data.frame(logmean = c(0.8, 0.8), sigma = c(0.2, 0.3)) xOrig <- as.data.frame(scaleLogToOrig(xLog$logmean, xLog$sigma)) xLog2 <- as.data.frame(scaleOrigToLog(xOrig$mean, xOrig$sd)) all.equal(xLog, xLog2) xLog3 <- as.data.frame(getParmsLognormForMoments(xOrig$mean, xOrig$sd^2)) all.equal(xLog$sigma, xLog3$sigma) # but mu < logmean
Compute the standard error accounting for empirical autocorrelations
seCor( x, effCor = if (missing(effCov)) computeEffectiveAutoCorr(x) else effCov/var(x, na.rm = TRUE), na.rm = FALSE, effCov, nEff = computeEffectiveNumObs(x, effCor, na.rm = na.rm) )seCor( x, effCor = if (missing(effCov)) computeEffectiveAutoCorr(x) else effCov/var(x, na.rm = TRUE), na.rm = FALSE, effCov, nEff = computeEffectiveNumObs(x, effCor, na.rm = na.rm) )
x |
numeric vector |
effCor |
numeric vector of effective correlation components
first entry at zero lag equals one. See |
na.rm |
logical. Should missing values be removed? |
effCov |
alternative to specifying effCor: numeric vector of
effective covariance components
first entry is the variance. See |
nEff |
possibility to specify precomputed number of effective observations for speedup. |
The default uses empirical autocorrelation
estimates from the supplied data up to first negative component.
For short series of x it is strongly recommended to to
provide effCov that was estimated on a longer time series.
numeric scalar of standard error of the mean of x
set off-diagonal values of a matrix
setMatrixOffDiagonals(x, diag = 1:length(value), value, isSymmetric = FALSE)setMatrixOffDiagonals(x, diag = 1:length(value), value, isSymmetric = FALSE)
x |
numeric square matrix |
diag |
integer vector specifying the diagonals 0 is the center +1 the first row to upper and -2 the second row to lower |
value |
numeric vector of values to fill in |
isSymmetric |
set to TRUE to to only specify the upper diagonal element but also set the lower in the mirrored diagonal |
matrix with modified diagonal elements
Compute the unbiased variance accounting for empirical autocorrelations
varCor( x, effCor = computeEffectiveAutoCorr(x), na.rm = FALSE, nEff = computeEffectiveNumObs(x, effAcf = effCor) )varCor( x, effCor = computeEffectiveAutoCorr(x), na.rm = FALSE, nEff = computeEffectiveNumObs(x, effAcf = effCor) )
x |
numeric vector |
effCor |
numeric vector of effective correlation components
first entry at zero lag equals one. See |
na.rm |
logical. Should missing values be removed? |
nEff |
possibility to specify precomputed number of effective observations for speedup. |
The default uses empirical autocorrelation
estimates from the supplied data up to first negative component.
For short series of x it is strongly recommended to to
provide effCov that was estimated on a longer time series.
numeric scalar of unbiased variation of x