compositions (version 2.0-4)

mvar: Metric summary statistics of real, amount or compositional data

Description

Compute the metric variance, covariance, correlation or standard deviation.

Usage

mvar(x,...)
mcov(x,...)
mcor(x,...)
msd(x,...)
# S3 method for default
mvar(x,y=NULL,...)
# S3 method for default
mcov(x,y=x,...)
# S3 method for default
mcor(x,y,...)
# S3 method for default
msd(x,y=NULL,...)

Value

a scalar number, informing of the degree of variation/covariation of one/two datasets.

Arguments

x

a dataset, eventually of amounts or compositions

y

a second dataset, eventually of amounts or compositions

...

further arguments to stats::var or stats::cov. Typically a robust=TRUE argument. e.g. use

Author

K.Gerald v.d. Boogaart http://www.stat.boogaart.de, Raimon Tolosana-Delgado

Details

The metric variance (mvar) is defined by the trace of the variance in the natural geometry of the data, or also by the generalized variance in natural geometry. The natural geometry is equivalently given by the cdt or idt transforms.

The metric standard deviation (msd) is not the square root of the metric variance, but the square root of the mean of the eigenvalues of the variance matrix. In this way it can be interpreted in units of the original natural geometry, as the radius of a sperical ball around the mean with the same volume as the 1-sigma ellipsoid of the data set.

The metric covariance (mvar) is the sum over the absolute singular values of the covariance of two datasets in their respective geometries. It is always positive. The metric covariance of a dataset with itself is its metric variance. The interpretation of a metric covariance is quite difficult, but useful in regression problems.

The metric correlation (mcor) is the metric covariance of the datasets in their natural geometry normalized to unit variance matrix. It is a number between 0 and the smaller dimension of both natural spaces. A number of 1 means perfect correlation in 1 dimension, but only partial correlations in higher dimensions.

References

Daunis-i-Estadella, J., J.J. Egozcue, and V. Pawlowsky-Glahn (2002) Least squares regression in the Simplex on the simplex, Terra Nostra, Schriften der Alfred Wegener-Stiftung, 03/2003

Pawlowsky-Glahn, V. and J.J. Egozcue (2001) Geometric approach to statistical analysis on the simplex. SERRA 15(5), 384-398

See Also

var, cov, mean.acomp, acomp, rcomp, aplus, rplus

Examples

data(SimulatedAmounts)
mvar(acomp(sa.lognormals))
mvar(rcomp(sa.lognormals))
mvar(aplus(sa.lognormals))
mvar(rplus(sa.lognormals))

msd(acomp(sa.lognormals))
msd(rcomp(sa.lognormals))
msd(aplus(sa.lognormals))
msd(rplus(sa.lognormals))

mcov(acomp(sa.lognormals5[,1:3]),acomp(sa.lognormals5[,4:5]))
mcor(acomp(sa.lognormals5[,1:3]),acomp(sa.lognormals5[,4:5]))
mcov(rcomp(sa.lognormals5[,1:3]),rcomp(sa.lognormals5[,4:5]))
mcor(rcomp(sa.lognormals5[,1:3]),rcomp(sa.lognormals5[,4:5]))

mcov(aplus(sa.lognormals5[,1:3]),aplus(sa.lognormals5[,4:5]))
mcor(aplus(sa.lognormals5[,1:3]),aplus(sa.lognormals5[,4:5]))
mcov(rplus(sa.lognormals5[,1:3]),rplus(sa.lognormals5[,4:5]))
mcor(rplus(sa.lognormals5[,1:3]),rplus(sa.lognormals5[,4:5]))

mcov(acomp(sa.lognormals5[,1:3]),aplus(sa.lognormals5[,4:5]))
mcor(acomp(sa.lognormals5[,1:3]),aplus(sa.lognormals5[,4:5]))