Skip to contents

Sample Interval-valued Covariance

Source: R/int_cov.r
int_cov_z.Rd

Calculate the interval-valued covariance matrix in function of z

Usage

int_cov_z(z, data)

Arguments

z

A vector of 0 and 1, indicating which observations should be considered for the calculation

data

An intData object containing the macrodata/interval data

Value

The symbolic covariance matrix

Details

Let \(\boldsymbol{z}\in\{0,1\}^n\) be a vector indicating which \(m\) observations are “active”. This function calculates the sample interval-valued covariance matrix in function of \(\boldsymbol{z}\): \(\boldsymbol{S}_B(\boldsymbol{z})\). Let \(\boldsymbol{C}\), \(\boldsymbol{R}\) be the matrices of centers and ranges, respectively. Additionally, set: $$\overline{\boldsymbol{c}}_B(\boldsymbol{z})=\dfrac{1}{m}\boldsymbol{C}^{\top}\boldsymbol{z}, \qquad \overline{\boldsymbol{r}}_B(\boldsymbol{z})=\dfrac{1}{m}\boldsymbol{R}^{\top}\boldsymbol{z}.$$ The sample interval-valued covariance matrix is obtained according to the LatentCase:

  • "U_id_symmetric": The latent variables are identically distributed and symmetric: $$\boldsymbol{S}_B(\boldsymbol{z})=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top+\left(\dfrac{\delta}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\delta\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top,$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

  • "U_id": The latent variables are identically distributed: $$\boldsymbol{S}_B(\boldsymbol{z})=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top+\left(\dfrac{\delta}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\delta\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\\ +\left(\dfrac{\mathbb{E}(U)}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{r}_{h}^{\top}\right)-\dfrac{\mathbb{E}(U)}{2}\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top+\left(\dfrac{\mathbb{E}(U)}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{c}_{h}^{\top}\right)-\dfrac{\mathbb{E}(U)}{2}\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top,$$ where \(\delta=\mathbb{E}(U^2)/4\) and \(\mathbb{E}(U)\) are the parameters of the latent variables.

  • "General": The latent variables do not have any nice properties: $$\boldsymbol{S}_B(\boldsymbol{z})=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top+\left(\dfrac{1}{4m}\boldsymbol{\mathfrak{E}}_{UU}\bullet\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\dfrac{1}{4}\boldsymbol{\mathfrak{E}}_{UU}\bullet\left[\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\right]\\ +\left(\dfrac{1}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{r}_{h}^{\top}\right)\boldsymbol{\Psi}-\dfrac{1}{2}\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\boldsymbol{\Psi}+\boldsymbol{\Psi}\left(\dfrac{1}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{c}_{h}^{\top}\right)-\dfrac{1}{2}\boldsymbol{\Psi}\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top,$$ where:

    • \(\boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p))\),

    • \([\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j)\), \(i\neq j\), with \(\mathcal{E}(U_i,U_j)=\int_0^1 F_{U_i}^{-1}(t) F_{U_j}^{-1}(t) \, dt\),

    • \([\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2)\), \(i,j=1,\dots,p\),

    • \(\bullet\) denotes the Schur (or entrywise) product of matrices.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Examples

data(creditcard)
credit_card_int <- creditcard$intData

z <- rep(1, nrow(credit_card_int))
credit_card_cov<-int_cov_z(z,credit_card_int)