| cov {IntervalQuestionStat} | R Documentation |
Calculate the sample covariance between two random intervals
Description
This function calculates the sample covariance between two realizations of
n nonempty compact real intervals drawn from two random intervals
saved as two different IntervalList objects.
Usage
## S4 method for signature 'IntervalList,IntervalList'
cov(x, y, theta = 1)
Arguments
x |
A list of intervals, that is, an |
y |
A list of intervals, that is, an |
theta |
A single positive real number saved as a |
Details
Let \mathcal{X} and \mathcal{Y} be two interval-valued random
sets and let \left((x_{1},y_{1}),(x_{2},y_{2}), \ldots,
(x_{n},y_{n})\right) be a sample of n independent observations drawn
from \left(\mathcal{X},\mathcal{Y}\right). Then, the sample covariance
between \mathcal{X} and \mathcal{Y} is defined as the following
real number given by
s_{\mathcal{X}~\mathcal{Y}} =
s_{\mathrm{mid}~\mathcal{X}~\mathrm{mid}~\mathcal{Y}}+\theta\cdot
s_{\mathrm{spr}~\mathcal{X}~\mathrm{spr}~\mathcal{Y}},
where \theta>0 and
s_{\mathrm{mid}~\mathcal{X}~\mathrm{mid}~\mathcal{Y}} =
\frac{1}{n}\sum_{i=1}^{n}(\mathrm{mid}~x_{i} -
\mathrm{mid}~\overline{x})(\mathrm{mid}~y_{i}-\mathrm{mid}~\overline{y}),
s_{\mathrm{spr}~\mathcal{X}~\mathrm{spr}~\mathcal{Y}} =
\frac{1}{n}\sum_{i=1}^{n}(\mathrm{spr}~x_{i} -
\mathrm{spr}~\overline{x})(\mathrm{spr}~y_{i}-\mathrm{spr}~\overline{y}),
with \overline{x} and \overline{y} being the sample Aumann means
of the given one-dimensional random samples.
Value
This function returns the calculated sample covariance of two samples
of n interval-valued data, which is defined as a real number.
Therefore, the output of this function is a single numeric value.
Author(s)
José García-García garciagarjose@uniovi.es
See Also
Other sample central tendency and dispersion measures such as sample
Aumann mean and sample Fréchet variance can be calculated through
mean() and var() functions, respectively.
Examples
## Some cov() examples changing theta
list1 <- IntervalList(c(0, 3, 2, 5, 6), c(4, 5, 4, 8, 7))
list2 <- IntervalList(c(3, 0, 3, 1, 4), c(7, 4, 6, 2, 6))
cov(list1, list2)
cov(list1, list2, 1/3)
## Note that cov(X, X) = var(X)
cov(list1, list1)
var(list1)
cov(list1, list1, 1/3)
var(list1, 1/3)