# Capturing a Change in the Covariance Structure of a Multivariate Process

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## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Description and Approach

#### 1.2. Outline of Paper

#### 1.3. Mathematical Toolbox

- (Ref. [16]) The multivariate gamma function, denoted ${\mathrm{\Gamma}}_{q}\left(\alpha \right),$ is defined as$$\begin{array}{ccc}\hfill {\mathrm{\Gamma}}_{q}\left(\alpha \right)& =& {\int}_{\mathbf{S}>\mathbf{0}}etr\left(\right)open="("\; close=")">-\mathbf{S}{\left|\mathbf{S}\right|}^{\alpha -\frac{1}{2}\left(\right)open="("\; close=")">q+1}\hfill & d\mathbf{S}\end{array}& =& {\pi}^{\frac{1}{4}q\left(\right)open="("\; close=")">q-1}\prod _{i=1}^{q}\mathrm{\Gamma}\left(\right)open="["\; close="]">\alpha -\frac{1}{2}\left(\right)open="("\; close=")">i-1\hfill $$$$\begin{array}{c}\hfill {\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\alpha ,\tau \\ =& {\pi}^{\frac{1}{4}q\left(\right)open="("\; close=")">q-1}\prod _{j=1}^{q}\mathrm{\Gamma}\left(\right)open="["\; close="]">\alpha +{t}_{j}-\frac{1}{2}\left(\right)open="("\; close=")">j-1\hfill \end{array}$$$${\int}_{\mathbf{S}>\mathbf{0}}etr\left(\right)open="("\; close=")">-\mathbf{SX}d\mathbf{S}={\mathrm{\Gamma}}_{q}\left(\alpha \right){\left|\mathbf{S}\right|}^{-\alpha}.$$
- (Ref. [16]) The multivariate beta function, denoted by ${\beta}_{q}\left(\right)open="("\; close=")">\alpha ,b$, is defined as$${\beta}_{q}\left(\right)open="("\; close=")">\alpha ,\beta {\left(\right)}^{{\mathbf{I}}_{q}}\beta -\frac{1}{2}\left(\right)open="("\; close=")">q+1$$
- (Ref. [15]) Meijer’s G-function with the parameters ${\alpha}_{1},\dots ,{\alpha}_{r}$ and ${\beta}_{1},\dots ,{\beta}_{s}$ is defined as$${G}_{r,s}^{m,n}\left(\right)open="("\; close=")">{x|}_{{\beta}_{1},\dots ,{\beta}_{s}}^{{\alpha}_{1},\dots ,{\alpha}_{r}}$$$$g\left(h\right)=\frac{{\prod}_{j=1}^{m}\mathrm{\Gamma}\left(\right)open="("\; close=")">{\beta}_{j}+h}{{\prod}_{j=1}^{n}}$$
- (Refs. [15,17,18]) The hypergeometric function of matrix argument is defined by$${}_{r}{F}_{s}\left(\right)open="("\; close=")">{\alpha}_{1},\dots ,{\alpha}_{r};{\beta}_{1},\dots ,{\beta}_{s};\mathbf{S}{\left(\right)}_{{\beta}_{1}}\tau \cdots {\left(\right)}_{{\beta}_{s}}\tau $$
- Two special cases of Equation (9) are of interest:
- 1.
- If $\mathbf{X}:\left(\right)open="("\; close=")">q\times q$ is a symmetric matrix where $\u2225\mathbf{X}\u2225<1,$ then$${}_{1}{F}_{0}\left(\right)open="("\; close=")">\alpha ;\mathbf{X}$$
- 2.
- If $\mathbf{X}:\left(\right)open="("\; close=")">q\times q$ is a symmetric matrix where $\u2225\mathbf{X}\u2225<1,$ then$$\begin{array}{ccc}& & {}_{2}{F}_{1}\left(\right)open="("\; close=")">\alpha ,\beta ;c;\mathbf{X}\hfill \end{array}{\left(\right)}^{{\mathbf{I}}_{q}}-\beta $$

- (Ref. [4]) Two particular results are of interest here.
- 1.
- If $\mathbf{S}:\left(\right)open="("\; close=")">q\times q$$>\mathbf{0},$ $\mathbf{B}:\left(\right)open="("\; close=")">q\times q$ free of elements of $\mathbf{S}$, then$$\begin{array}{ccc}& & {\int}_{\mathbf{S}>\mathbf{0}}{\left|\mathbf{S}\right|}^{\alpha -\frac{1}{2}\left(\right)open="("\; close=")">q+1}{\left(\right)}^{{\mathbf{I}}_{q}}-\beta \hfill & {\left(\right)}^{{\mathbf{I}}_{q}}-c\\ d\mathbf{S}\end{array}$$
- 2.
- The confluent hypergeometric function $\mathrm{\Psi}(\xb7)$ of symmetric matrix $\mathbf{R}:\left(\right)open="("\; close=")">q\times q$ is defined by$$\mathrm{\Psi}\left(\right)open="("\; close=")">\alpha ,c,\mathbf{R}{\left|\mathbf{S}\right|}^{\alpha -\frac{1}{2}\left(\right)open="("\; close=")">q+1}$$$$\begin{array}{ccc}& & {\int}_{\mathbf{Y}>\mathbf{0}}{\left|\mathbf{Y}\right|}^{\beta -\frac{1}{2}\left(\right)open="("\; close=")">q+1}etr\left(\right)open="("\; close=")">-\mathbf{XY}\hfill & \mathrm{\Psi}\left(\right)open="("\; close=")">\alpha ,c,\mathbf{Y}\\ d\mathbf{Y}\end{array}$$$$\begin{array}{ccc}& & {\int}_{\mathbf{Y}>\mathbf{0}}{\left|\mathbf{Y}\right|}^{\beta -\frac{1}{2}\left(\right)open="("\; close=")">q+1}etr\left(\right)open="("\; close=")">-\mathbf{XY}\hfill & \mathrm{\Psi}\left(\right)open="("\; close=")">\alpha ,c,{\mathbf{B}}^{\frac{1}{2}}{\mathbf{YB}}^{\frac{1}{2}}\\ d\mathbf{Y}\end{array}$$

## 2. Methodology

**Theorem**

**1.**

- 1.
- Equation (3) is given by$$\begin{array}{ccc}& & f({\mathbf{U}}_{0},{\mathbf{U}}_{1})\hfill \\ & =& C{\left(\right)}^{{\mathbf{U}}_{0}}\frac{1}{2}{v}_{2}-\frac{1}{2}\left(\right)open="("\; close=")">q+1\hfill & {\left(\right)}^{{\mathbf{I}}_{q}}\frac{1}{2}{v}_{3}\end{array}$$
- 2.
- ${\mathbf{U}}_{0}$ is given by$$f\left(\right)open="("\; close=")">{\mathbf{U}}_{0}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{2}$$
- 3.
- ${\mathbf{U}}_{1}$ is given by$$\begin{array}{ccc}& & f\left(\right)open="("\; close=")">{\mathbf{U}}_{1}\hfill \end{array}$$$$\left(\right)open="\parallel "\; close="\parallel ">{\mathbf{I}}_{q}-{\left(\right)}^{{\mathbf{I}}_{q}}\frac{1}{2}{\left(\right)}^{{\mathbf{I}}_{q}}\frac{1}{2}$$

**Proof.**

- 1.
- The joint pdf of $\mathbf{X},{\mathbf{W}}_{0},{\mathbf{W}}_{1}$ is given by$$\begin{array}{c}\hfill f\left(\right)open="("\; close=")">\mathbf{X},{\mathbf{W}}_{0},{\mathbf{W}}_{1}\\ =& C{\left|\mathbf{X}\right|}^{\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}-q-1}{\left(\right)}^{{\mathbf{W}}_{0}}\frac{1}{2}\left(\right)open="("\; close=")">{v}_{2}-q-1\hfill \end{array}$$$$\mathbf{U}=\mathbf{X},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\mathbf{U}}_{0}={\mathbf{X}}^{-\frac{1}{2}}{\mathbf{W}}_{0}{\mathbf{X}}^{-\frac{1}{2}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\mathbf{U}}_{1}={\left(\right)}^{\mathbf{X}}-\frac{1}{2},$$$$\mathbf{X}=\mathbf{U},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\mathbf{W}}_{0}={\mathbf{U}}^{\frac{1}{2}}{\mathbf{U}}_{0}{\mathbf{U}}^{\frac{1}{2}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\mathbf{W}}_{1}={\left(\right)}^{\mathbf{U}}\frac{1}{2}.$$$$\begin{array}{c}\hfill J\left(\right)open="("\; close=")">\mathbf{X},{\mathbf{W}}_{0},{\mathbf{W}}_{1}\to \mathbf{U},{\mathbf{U}}_{0},{\mathbf{U}}_{1}\\ =& J\left(\right)open="("\; close=")">\mathbf{X}\to \mathbf{U}J\left(\right)open="("\; close=")">{\mathbf{W}}_{0}\to {\mathbf{U}}_{0}\hfill & J\left(\right)open="("\; close=")">{\mathbf{W}}_{1}\to {\mathbf{U}}_{1}\end{array}$$$$\begin{array}{ccc}& & f\left(\right)open="("\; close=")">\mathbf{U},{\mathbf{U}}_{0},{\mathbf{U}}_{1}\hfill \end{array}{\left(\right)}^{{\mathbf{I}}_{q}}\frac{1}{2}{v}_{3}$$
- 2.
- The marginal pdf of ${\mathbf{U}}_{0}$ is obtained by integrating $f({\mathbf{U}}_{0},{\mathbf{U}}_{1})$ (see Equation (16)) with respect to ${\mathbf{U}}_{1}$ using Equation (6):$$\begin{array}{ccc}& & f\left(\right)open="("\; close=")">{\mathbf{U}}_{0}\hfill \end{array}$$
- 3.
- From Equations (16) and (13) and (14) it follows that$$\begin{array}{ccc}\hfill \phantom{\rule{-30.0pt}{0ex}}& & f\left(\right)open="("\; close=")">{\mathbf{U}}_{1}\hfill \end{array}etr\left(\right)open="("\; close=")">-\frac{1}{2}\left(\right)open="("\; close=")">{\mathbf{I}}_{q}+{\lambda}^{-1}{\mathbf{U}}_{1}$$

**Remark**

**1.**

**Corollary**

**1.**

- 1.
- The product moment $E\left(\right)open="("\; close=")">{\left(\right)}^{{\mathbf{U}}_{0}}{h}_{1}$ is given by$$E\left(\right)open="("\; close=")">{\left(\right)}^{{\mathbf{U}}_{0}}{h}_{1}=\frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}-{h}_{1}}{{\mathrm{\Gamma}}_{q}}$$
- 2.
- The product moment $E\left(\right)open="("\; close=")">{\left(\right)}^{{\mathbf{U}}_{1}}{h}_{2}$ is given by$$\begin{array}{c}\hfill E\left(\right)open="("\; close=")">{\left(\right)}^{{\mathbf{U}}_{1}}{h}_{2}\end{array}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}$$

**Theorem**

**2.**

- 1.
- the pdf of $\left(\right)$ is given by$$f\left(\left(\right),{\mathbf{U}}_{0}\right)$$
- 2.
- with cumulative distribution function (CDF)$$\begin{array}{c}\hfill {F}_{\left(\right)}\left(c\right)\\ =& Pr\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\mathbf{U}}_{0}\le c\hfill \end{array}$$
- 3.
- The pdf of $\left(\right)$ is given by$$\begin{array}{ccc}& & f\left(\left(\right),{\mathbf{U}}_{1}\right)\hfill \end{array}$$
- 4.
- with CDF$$\begin{array}{ccc}& & {F}_{\left(\right)}\left(c\right)\hfill \end{array}$$

**Remark**

**2.**

## 3. Numerical Example

`meijerG`; in our case, we used

`MeijerG`in the software Mathematica.

**Remark**

**3.**

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem 2.**

- 1.
- From Equation (23),$$E\left(\right)open="("\; close=")">{\left(\right)}^{{\mathbf{U}}_{0}}h-1=\frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}-h+1}{{\mathrm{\Gamma}}_{q}}$$$$E\left(\right)open="("\; close=")">{\left(\right)}^{{\lambda}^{-1}}h-1=\frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}-h+1}{{\mathrm{\Gamma}}_{q}}$$Using the well-known Mellin transform of $f\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\lambda}^{-1}{\mathbf{U}}_{0}$:$${M}_{f}\left(h\right)\equiv E\left(\right)open="("\; close=")">{\left(\right)}^{{\lambda}^{-1}}h-1.$$Expressing the multivariate gamma functions in Equation (A1) as a product of gamma functions and substituting it in the Mellin transform Equation (A2), gives$$\begin{array}{c}{M}_{f}\left(h\right)=\frac{{\pi}^{\frac{q\left(\right)open="("\; close=")">q-1}{}}}{{\displaystyle \sum _{j=1}^{q}}}{\displaystyle \sum _{j=1}^{q}}\mathrm{\Gamma}\left(\right)open="["\; close="]">{b}_{j}+h\hfill & {\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{2}\end{array}$$The pdf of $\left(\right)$ is uniquely obtained from the inverse Mellin transform ([15]) of Equation (A3) and using Equation (8) and is given by$$\begin{array}{ccc}& & f\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\lambda}^{-1}{\mathbf{U}}_{0}\hfill \end{array}& =& \frac{1}{2\pi i}{\int}_{\omega -i\infty}^{\omega +i\infty}{M}_{f}\left(h\right){\left(\right)}^{{\lambda}^{-1}}-hdh\hfill $$
- 2.
- Let $u=\left(\right)open="|"\; close="|">{\mathbf{U}}_{0}$$u>0$ then from Equation (25) the CDF is defined as$$\begin{array}{c}\hfill {F}_{\left(\right)}\left(c\right)\\ =& Pr\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\mathbf{U}}_{0}\le c\hfill \end{array}$$Applying [15] results from pages 142, 59, and 69, yields$$\begin{array}{ccc}& & {F}_{\left(\right)}\left(c\right)\hfill \end{array}\left(\right)open="("\; close=")">{a}_{1},1,\dots ,\left(\right)open="("\; close=")">{a}_{q},1$$
- 3.
- From Equation (24) the Mellin transform ([15]) of $f\left(\left(\right),{\mathbf{U}}_{1}\right)$ is$$\begin{array}{cc}\hfill {M}_{f}\left(h\right)=& \frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}}{-}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}+h-1{\lambda}^{\frac{1}{2}{v}_{1}q}\hfill & {\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}\end{array}$$Using Equations (5) and (9) the Gauss hypergeometric function of matrix argument in Equation (A5) can be written as$$\begin{array}{ccc}& & {}_{2}{F}_{1}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1},\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}-h+1;\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}\hfill & ;\left(\right)open="("\; close=")">1-\lambda \\ {\mathbf{I}}_{q}\end{array}$$This gives$$\begin{array}{ccc}\hfill {M}_{f}\left(h\right)& \equiv & \frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}+h-1}{{\lambda}^{\frac{1}{2}{v}_{1}q}}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}\hfill \end{array}$$The multivariate gamma function in Equation (A6) can be written as$$\begin{array}{ccc}& & \begin{array}{c}\hfill {\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{3}+h-1={\pi}^{\frac{q\left(\right)open="("\; close=")">q-1}{}}& {\displaystyle \sum _{j=1}^{q}}\mathrm{\Gamma}\left(\right)open="["\; close="]">{b}_{j}+h\\ ,\end{array}\hfill \end{array}$$$$\begin{array}{ccc}& & \begin{array}{c}\hfill {\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2}-h+1,\tau & ={\pi}^{\frac{q\left(\right)open="("\; close=")">q-1}{}}\\ {\displaystyle \sum _{j=1}^{q}}\mathrm{\Gamma}\left(\right)open="["\; close="]">1-{a}_{j}-h\end{array},\hfill \end{array}$$Substituting Equations (A7) and (A8) in Equation (A6) gives$$\begin{array}{ccc}\hfill {M}_{f}\left(h\right)& \equiv & \frac{{\lambda}^{\frac{1}{2}{v}_{1}q}}{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1}}\hfill & \sum _{t=0}^{\infty}\sum _{\tau}\frac{{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}{v}_{1},\tau}{}{\mathrm{\Gamma}}_{q}\left(\right)open="("\; close=")">\frac{1}{2}\left(\right)open="("\; close=")">{v}_{1}+{v}_{2},\tau \\ t!\end{array}{\pi}^{\frac{q\left(\right)open="("\; close=")">q-1}{}}$$The pdf of $\left(\right)$ is obtained from the inverse Mellin transform ([15]) of Equation (A9) and from the definition of the Meijer’s G-function Equation (8) as$$\begin{array}{ccc}& & f\left(\left(\right),{\mathbf{U}}_{1}\right)\hfill \end{array}$$
- 4.
- Let $u=\left(\right)open="|"\; close="|">{\mathbf{U}}_{1}$$u>0$ then from Equation (27) the CDF is defined as$$\begin{array}{c}\hfill {F}_{\left(\right)}\left(c\right)\\ =& Pr\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\mathbf{U}}_{1}\le c\hfill \end{array}$$Applying [15] results from page 142, 59, and 69, yields$$\begin{array}{ccc}& & {F}_{\left(\right)}\left(c\right)\hfill \end{array}$$

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$\mathit{\lambda}$ | q | $\mathit{\gamma}=0.01$ | $\mathit{\gamma}=0.025$ | $\mathit{\gamma}=0.05$ | $\mathit{\gamma}=0.1$ |
---|---|---|---|---|---|

2 | 1 | 7.00608 | 5.05263 | 3.83785 | 2.80643 |

1 | 1 | 3.50304 | 2.52632 | 1.91893 | 1.40321 |

0.5 | 1 | 1.75152 | 1.26316 | 0.95946 | 0.70161 |

2 | 2 | 7.29343 | 4.50351 | 2.97902 | 1.80558 |

1 | 2 | 3.64671 | 2.25176 | 1.48951 | 0.91746 |

0.5 | 2 | 1.82336 | 1.12588 | 0.74475 | 0.46006 |

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**MDPI and ACS Style**

Bekker, A.; Ferreira, J.T.; Human, S.W.; Adamski, K.
Capturing a Change in the Covariance Structure of a Multivariate Process. *Symmetry* **2022**, *14*, 156.
https://doi.org/10.3390/sym14010156

**AMA Style**

Bekker A, Ferreira JT, Human SW, Adamski K.
Capturing a Change in the Covariance Structure of a Multivariate Process. *Symmetry*. 2022; 14(1):156.
https://doi.org/10.3390/sym14010156

**Chicago/Turabian Style**

Bekker, Andriette, Johannes T. Ferreira, Schalk W. Human, and Karien Adamski.
2022. "Capturing a Change in the Covariance Structure of a Multivariate Process" *Symmetry* 14, no. 1: 156.
https://doi.org/10.3390/sym14010156