| var {IntervalQuestionStat} | R Documentation |
Calculate the sample Fréchet variance of a random interval
Description
This function calculates the sample Fréchet variance of a single realization
of n nonempty compact real intervals drawn from an interval-valued
random set stored as an IntervalList object.
Usage
## S4 method for signature 'IntervalList'
var(x, theta = 1)
Arguments
x |
A list of intervals, that is, an |
theta |
A single positive real number saved as a |
Details
Let \mathcal{X} be an interval-valued random set
and let \left(x_{1},x_{2},\ldots,x_{n}\right) be a sample of n
independent observations drawn from \mathcal{X}. Then, the sample
Fréchet variance (see Fréchet, 1948) is defined as the following
non-negative real number given by
s_{\mathcal{X}}^{2} =
\frac{1}{n}\sum_{i=1}^{n}d_{\theta}^{2}\left(x_{i}, \overline{x}\right),
where \theta>0 and \overline{x} denotes the sample Aumann mean
of \left(x_{1},x_{2},\ldots,x_{n}\right). Due to \theta-distance
definition, this deviation measure can also be computed as follows,
s_{\mathcal{X}}^{2} =
s_{\mathrm{mid}~\mathcal{X}}^{2}+\theta\cdot
s_{\mathrm{spr}~\mathcal{X}}^{2},
where
s_{\mathrm{mid}~\mathcal{X}} = \frac{1}{n}\sum_{i=1}^{n}
(\mathrm{mid}~x_{i} - \mathrm{mid}~\overline{x})^{2},
s_{\mathrm{spr}~\mathcal{X}} = \frac{1}{n}\sum_{i=1}^{n}
(\mathrm{spr}~x_{i} - \mathrm{spr}~\overline{x})^{2}.
Value
This function returns the calculated sample Fréchet variance of the given
n interval, which is defined as a non-negative real number. Therefore,
the output of this function is a single numeric object.
Author(s)
José García-García garciagarjose@uniovi.es
References
Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l'institut Henri Poincaré, 10(4):215-310.
See Also
Other sample central tendency and covariance measures such as sample
Aumann mean and sample covariance can be calculated through
mean() and cov() functions, respectively.
Examples
## Some var() examples
list <- IntervalList(c(1, 3), c(2, 5))
var(list)
var(list, theta = 1/3)