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A Comparison between Successive Estimate of TVAR(1) and TVAR(2) and the Estimate of a TVAR(3) Process^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Successive Estimation Using TVAR(1) and TVAR(2) Processes

## 3. Restrictions of TVAR(3) Process with Linear Root Motion

#### 3.1. Calculation of the Roots from the Time-Stable Coefficients

#### 3.2. Restrictions for Linear Root Motion

## 4. Application: Two GNSS Time Series

- Via three TVAR(1) processes.
- Via a TVAR(1) process followed by a TVAR(2) process.
- Via a TVAR(2) process followed by a TVAR(1) estimate.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A GNSS time series ${\mathcal{S}}_{t}$ at the GNSS station in Pogum, Germany (TGPO) from end of 2009 to the begin of 2019.

**Figure 2.**Reduced time series ${\widehat{\mathcal{S}}}_{t}$. This is created from the time series ${\mathcal{S}}_{t}$ by removing the data jump, extrapolating the data gaps and eliminating the trend.

**Figure 3.**The roots of stationary AR(3) processes from each 100 consecutive observations. Here, the same-colored points correspond to the evaluation of a window, and the color gradient represents the temporal assignment.

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**MDPI and ACS Style**

Korte, J.; Brockmann, J.M.; Schuh, W.-D.
A Comparison between Successive Estimate of TVAR(1) and TVAR(2) and the Estimate of a TVAR(3) Process. *Eng. Proc.* **2023**, *39*, 90.
https://doi.org/10.3390/engproc2023039090

**AMA Style**

Korte J, Brockmann JM, Schuh W-D.
A Comparison between Successive Estimate of TVAR(1) and TVAR(2) and the Estimate of a TVAR(3) Process. *Engineering Proceedings*. 2023; 39(1):90.
https://doi.org/10.3390/engproc2023039090

**Chicago/Turabian Style**

Korte, Johannes, Jan Martin Brockmann, and Wolf-Dieter Schuh.
2023. "A Comparison between Successive Estimate of TVAR(1) and TVAR(2) and the Estimate of a TVAR(3) Process" *Engineering Proceedings* 39, no. 1: 90.
https://doi.org/10.3390/engproc2023039090