cov-methods: Calculate the sample covariance between two random intervals

In IntervalQuestionStat: Tools to Deal with Interval-Valued Responses in Questionnaires

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 IntervalList object.
y	A list of intervals, that is, an IntervalList object with the same length as x.
theta	A single positive real number saved as a numeric object. By default, theta = 1.

Details

Let \mathcal{X} and \mathcal{Y} be two interval-valued random sets and let ≤ft((x_{1},y_{1}),(x_{2},y_{2}), …, (x_{n},y_{n})\right) be a sample of n independent observations drawn from ≤ft(\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}}+θ\cdot s_{\mathrm{spr}~\mathcal{X}~\mathrm{spr}~\mathcal{Y}},

where θ>0 and

s_{\mathrm{mid}~\mathcal{X}~\mathrm{mid}~\mathcal{Y}} = \frac{1}{n}∑_{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}∑_{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 [e-mail]

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)

IntervalQuestionStat documentation built on Nov. 1, 2022, 5:06 p.m.