var_scale {ecostatscale}

Variance scaling function

Description

Extrapolate variance observed at spatial or ecological scale b to a different scale, B. Equivalent to Eq.7a in the main text.

Usage

var_scale(mvar_b, murho_b, mucov_b = NULL, b = 1, B)

Arguments

mvar_b

Mean variance of abundance values observed at scale b.

murho_b

Mean Pearson correlation coefficient of abundance values observed at scale b, calculated as mucov_b/mvar_b. If NULL, mucov_b is used instead.

mucov_b

Mean covariance of abundances observed at scale b. Ignored if mrho_b is not NULL. Defaults to NULL.

b

Size of observed scale. Defaults to 1.

B

Size of desired scale for extrapolation.

Value

Extrapolated variance at scale B.

Examples

# extrapolate from scale of 1 to 10 - e.g. from a 1m2 patch to a 10m2 patch
var_scale(mvar_b = 1, murho_b = 0.5, b = 1, B = 10)

# example with 100 simulated species
nsp<-100 # number of species
var_b<-1 # species-level abundance variance
cov_b<-(-0.01) # between-specie abundance covariance
# note - if nsp is large, cov_b must be near zero
# this is because, e.g. many variables cannot all be
# simultaneously negatively correlated

# make a covariance matrix based on var_b and cov_b
sigmamat<-diag(nsp)*var_b+(1-diag(nsp))*cov_b
# simulate 1000 observations of 100 species
sim_x<-mvtnorm::rmvnorm(n=1e3, mean = rep(0,100), sigma = sigmamat)

# calculate mean variance, covariance, and correlation from sim_x
cvmat<-cov(sim_x)
mvar_b<-mean(diag(cvmat))
mucov_b<-mean(cvmat[row(cvmat)!=col(cvmat)])
murho_b<-mucov_b/mvar_b

# test function vs. observation
# note - answers match exactly
var(rowSums(sim_x))
var_scale(mvar_b, murho_b = murho_b, b=1, B=100)