# Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Spatiotemporal Model Using ICA and VC Regression

- (1)
- Centralize and whiten the observed data, obtain the whitened data $\mathit{Z}$;
- (2)
- Choose an initial weight vector of unit norm (random) $\mathit{w}$;
- (3)
- Update ${\mathit{w}}^{+}$ by ${\mathit{w}}^{+}=\mathrm{E}\left[\mathit{Z}\mathrm{g}\left({\mathit{w}}^{T}\mathit{z}\right)\right]-\mathrm{E}\left[{\mathrm{g}}^{\prime}\left({\mathit{w}}^{T}\mathit{Z}\right)\right]\mathit{w}$, where $\mathrm{g}(\xb7)$ is a nonlinear function, such as ${\mathrm{g}}_{1}\left(y\right)=\mathrm{tan}\mathrm{h}\left({a}_{1}y\right)$, ${\mathrm{g}}_{2}\left(y\right)=y\mathrm{exp}\left(-{y}^{2}/2\right)$ and ${\mathrm{g}}_{3}\left(y\right)={y}^{3}$;
- (4)
- Normalize $\mathit{w}$ by $\mathit{w}={\mathit{w}}^{+}/\Vert {\mathit{w}}^{+}\Vert $;
- (5)
- Go back to step “3” if not converged.

#### 2.2. Varying Coefficient Regression Method

**is**called the pseudo-hat matrix.

## 3. Case Study

#### 3.1. GPS Data

#### 3.2. Spatiotemporal Analysis Using ICA

#### 3.3. Modeling Common Seasonal Signals

#### 3.4. Spatiotemporal Modeling and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Top 4 Scaled independent components (ICs) and the spatial responses (SRs). (

**a**) scaled IC1 and SR1. (

**b**) scaled IC2 and SR2. (

**c**) scaled IC3 and SR3. (

**d**) scaled IC4 and SR4.

**Figure 4.**Fitting results of (

**a**) IC1 and (

**b**) IC2 using least squares (LS) regression and varying coefficient (VC) regression.

**Figure 5.**Histogram of Residual in the fitting model of least squares regression (LS) and varying coefficient regression (VC) for (

**a**) IC1 and (

**b**) IC2, the std of residual are marked.

**Figure 6.**The root mean square (RMS) reduction in coordinate time series after the model correction for each site, the sites “tela” and “suth” are marked.

**Figure 7.**Vertical coordinate time series and the fitting lines of spatiotemporal model for sites (

**a**) “tela” and (

**b**) “suth”.

**Figure 8.**Comparison between ICs derived displacements (GPS(ICs)) and mass loading displacements at site “tela”.

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**MDPI and ACS Style**

Liu, B.; Xing, X.; Tan, J.; Xia, Q.
Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression. *Sensors* **2020**, *20*, 5627.
https://doi.org/10.3390/s20195627

**AMA Style**

Liu B, Xing X, Tan J, Xia Q.
Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression. *Sensors*. 2020; 20(19):5627.
https://doi.org/10.3390/s20195627

**Chicago/Turabian Style**

Liu, Bin, Xuemin Xing, Jianbo Tan, and Qing Xia.
2020. "Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression" *Sensors* 20, no. 19: 5627.
https://doi.org/10.3390/s20195627