#
On the Family of Covariance Functions Based on ARMA Models^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Related Work

## 2. The Family of Non-Repeated Poles ARMA Models

#### Positive Definiteness in Higher Dimensions

## 3. Generalization to Repeated Poles ARMA Models

#### 3.1. The Half-Integer Matérn Covariance Function

#### 3.2. Repeated Poles ARMA Models

#### 3.3. Bounds for the Polynomial Coefficients of Markov Models

#### 3.4. Oscillatory Repeated Poles ARMA Models

## 4. Application to Altimetry Data: A Demonstration

`fmincon`, cf. [34] and implemented a constrained least squares fitting.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AR | Autoregressive |

ARMA | Autoregressive moving average |

MA | Moving average |

SOGM | Second-order Gauss–Markov |

SLA | Sea level anomalies |

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**Figure 1.**Permissible areas for parameters. For each dimension, the permissible area becomes a subset of that of the lower dimension. (

**a**) Permissible areas of the parameters ζ and η in different dimensions 1 to 3. (

**b**) Permissible areas of the weights c

_{1}and c

_{2}in different dimensions 1 to 3 shown for fixed parameter c = 1.

**Figure 3.**Functions of the type in Equation (11) with different orders fitted to the empirical covariances.

**Figure 4.**Functions of the type in Equation (16) with different orders fitted to the empirical covariances.

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**MDPI and ACS Style**

Schubert, T.; Brockmann, J.M.; Korte, J.; Schuh, W.-D.
On the Family of Covariance Functions Based on ARMA Models. *Eng. Proc.* **2021**, *5*, 37.
https://doi.org/10.3390/engproc2021005037

**AMA Style**

Schubert T, Brockmann JM, Korte J, Schuh W-D.
On the Family of Covariance Functions Based on ARMA Models. *Engineering Proceedings*. 2021; 5(1):37.
https://doi.org/10.3390/engproc2021005037

**Chicago/Turabian Style**

Schubert, Till, Jan Martin Brockmann, Johannes Korte, and Wolf-Dieter Schuh.
2021. "On the Family of Covariance Functions Based on ARMA Models" *Engineering Proceedings* 5, no. 1: 37.
https://doi.org/10.3390/engproc2021005037