Heterogeneity Statistics {RFA}R Documentation

Compute 3 heterogeneity statistics.

Description

Compute 3 heterogeneity statistics based on the sample L-moments. It is usefull to test if a region could be considered as homogeneous or not, that is if all events from each site could came from only one distribution once normalized with the index flood.

Usage

heterogeneity(N.sim,N.site,size,param,Vsite)

Arguments

N.sim Numeric. The number of stochastic region generated. It is recommended to set N.sim at list 500.
N.site Numeric. The number of site within the region.
size Numeric vector of length equal to N.site. The sample size of each site within the region.
param Numeric vector of length 4. The parameter vector of the fitted regional Kappa distribution.
Vsite Numeric vector of length 3. The statistic values for the region.

Details

The code first fits a 4-parameter Kappa distribution from the weigthed mean of the first 4 sample L-moments of each site. Then, it generates a large number of stochastics region with the same properties as the real one. Observations are sampled in the Kappa distribution fitted earlier.

For each stochastic region, 3 statistics are computed and the behaviour of these values are directly compared with the 'real' statistics. Of course, if they have the same behaviour, then the region should be considered as homogeneous.

The general form of the test statistics is

H = (V - m) / sigma

where V is one of the 3 statistics V1, V2, V3 defined later evaluated of the 'real' region, m and sigma are the mean and standart deviation of the N statistics V evaluated on each stochastic region.

The 3 statistics V1, V2 and V3 are:

V = { sum_{i=1}^N n_i ( tau_i - tau_R )^2 / sum_{i=1}^N n_i }^(1/2)

sum_{i=1}^N n_i { (tau_i - tau_R)^2 + (tau_(3,i) - tau_(3,R))^2 }^(1/2) / sum_{i=1}^N n_i

sum_{i=1}^N n_i { (tau_(4,i) - tau_(4,R))^2 + (tau_(3,i) - tau_(3,R))^2 }^(1/2) / sum_{i=1}^N n_i

where n_i is the sample size of site i, tau_i, tau_R are the at-site i and regional sample L-CV respectively, tau_{3,i}, tau_{3,R} are the at-site i and regional sample L-Skewness respectively, tau_{4,i}, tau_{4,R} are the at-site i and regional sample L-Kurtosis respectively and N is the number of stochastic region generated.

The last 2 statistics lack power to discriminate between homogeneous and heterogeneous region. They rarely yield values larger than 2 even for grossly heterogeneous regions.

For the first statistic, it is suggest that the region be regarded as acceptably homogeneous if H1 < 1, possibly heterogeneous if 1 <= H1 < 2, definitely heterogeneous if H1 <= 2.

Value

The program print value of the three statistics H1, H2 and H3 and return a list of several not so usefull objects.

Author(s)

Ribatet Mathieu

References

Hosking, J. R. M. and Wallis, J. R. (1997) Regional Frequency Analysis. Cambridge University Press.

Examples

data(northCascades)
lmom <- c(1, 0.1103, 0.0279, 0.1366)
kappaParam <- kappalmom(lmom)
heterogeneity(500, 19, size = northCascades[,1],
kappaParam, c(0.0104, .0339, .0405))
##The heterogeneity statistics given by Hosking for this case
##study are H1 = 0.62, H2 = -1.49 and H3 = -2.37
##Taking into account sample variability, results should be
##consistent

[Package RFA version 0.0-8 Index]