# State-Space Models on the Stiefel Manifold with a New Approach to Nonlinear Filtering

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## Abstract

**:**

## 1. Introduction

## 2. Vector Models with Time-Varying Reduced Rank Parameters

## 3. Issues about the Specification of the Time-Varying Reduced Rank Parameter

**Proposition**

**1.**

**β**can be linearly normalized if the $r\times r$ upper block ${\mathit{b}}_{1}$ in

**Proof.**

**Proposition**

**2.**

**α**can be linearly normalized if the $r\times r$ upper block ${\mathit{a}}_{1}$ in

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

## 4. State-Space Models on the Stiefel Manifold

#### 4.1. The Stiefel Manifold and the Matrix Langevin Distribution

#### 4.2. Models

## 5. The Filtering Algorithms

**Proposition**

**3.**

**Proposition**

**4.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Proposition**

**5.**

**Proposition**

**6.**

## 6. Evaluation of the Filtering Algorithms by Simulation Experiments

- $T=100$, the sample size,
- $p\in \{2,3,10,20\}$, the dimension of the dependent variable ${\mathit{y}}_{t},$
- $r\in \{1,2\}$, the rank of the matrix ${\mathit{A}}_{t},$
- ${\mathit{x}}_{t}$, the explanatory variable vector has dimension ${q}_{1}=3$ ensuring that ${q}_{1}>r$ always holds, and each ${\mathit{x}}_{t}$ is sampled independently (over time) from a ${N}_{3}(\mathbf{0},{\mathit{I}}_{3})$,
- $\mathbf{\beta}={(1,-1,1)}^{\prime}/\sqrt{3},$
- ${\mathbf{\alpha}}_{0}={(1,-1,1,\dots )}^{\prime}/\sqrt{p}$, the initial value of ${\mathbf{\alpha}}_{t}$ sequence for the data generating process,
- $\mathsf{\Omega}=\rho {\mathit{I}}_{p}$, the covariance matrix of the errors is diagonal with $\rho \in \{0.1,0.5,1\},$
- $\mathit{D}=d{\mathit{I}}_{r}$, and $d\in \{5,50,500,800\}$.

- We sample from Model 1 by using the identified version in (18). First, simulate ${\mathbf{\alpha}}_{t}$ given ${\mathbf{\alpha}}_{t-1}$, and then ${\mathit{y}}_{t}$ given ${\mathbf{\alpha}}_{t}$. We save the sequence of the latent process ${\mathbf{\alpha}}_{t}$, $t=1,\dots ,T$.
- Then, we apply the filtering algorithm on the sampled data to obtain the filtered modal orientation ${\mathit{U}}_{t}$, $t=1,\dots ,T$.
- We compute the normalized distances ${\delta}_{t}({\mathbf{\alpha}}_{t},{\mathit{U}}_{t})$ and report by plotting them against the time t.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## Appendix B. Proof of Proposition 2

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**Figure 1.**Euclidean state space for $p=2$ and $r=1$. Points 1–3 are possible locations of the latent variable ${({\alpha}_{1t},\phantom{\rule{0.166667em}{0ex}}{\alpha}_{2t})}^{\prime}$. Circles are isodensity contours assuming ${({\alpha}_{1,t+1},\phantom{\rule{0.166667em}{0ex}}{\alpha}_{2,t+1})}^{\prime}|{({\alpha}_{1t},\phantom{\rule{0.166667em}{0ex}}{\alpha}_{2t})}^{\prime}\sim {N}_{2}({({\alpha}_{1t},\phantom{\rule{0.166667em}{0ex}}{\alpha}_{2t})}^{\prime},{I}_{2})$.

**Figure 4.**Normalized distances ${\delta}_{t}$ for the settings $p\in \{2,10,20\}$, $r=1$, $\rho =0.1$ and $d=50$.

**Figure 5.**Normalized distances ${\delta}_{t}$ for the settings $p\in \{2,10,20\}$, $r=1$, $\rho =0.1$ and $d=500$.

**Figure 6.**Normalized distances ${\delta}_{t}$ for the settings $p=2$, $r=1$, $\rho =1$ and $d\in \{5,50,500\}$.

**Figure 7.**Normalized distances ${\delta}_{t}$ for the settings $p=2$, $r=1$, $\rho =0.1$ and $d\in \{5,50,500\}$.

**Figure 8.**Normalized distances ${\delta}_{t}$ for the settings $p=3$, $r\in \{1,2\}$, $\rho =0.1$ and $d=\{500,800\}$.

**Figure 9.**Normalized distances ${\delta}_{t}$ for the settings $p\in \{2,10,20\}$, $r=1$, $\rho =0.1$ and $d=50$. The initial value of the filtering algorithm is $-{\mathbf{\alpha}}_{0}$.

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**MDPI and ACS Style**

Yang, Y.; Bauwens, L.
State-Space Models on the Stiefel Manifold with a New Approach to Nonlinear Filtering. *Econometrics* **2018**, *6*, 48.
https://doi.org/10.3390/econometrics6040048

**AMA Style**

Yang Y, Bauwens L.
State-Space Models on the Stiefel Manifold with a New Approach to Nonlinear Filtering. *Econometrics*. 2018; 6(4):48.
https://doi.org/10.3390/econometrics6040048

**Chicago/Turabian Style**

Yang, Yukai, and Luc Bauwens.
2018. "State-Space Models on the Stiefel Manifold with a New Approach to Nonlinear Filtering" *Econometrics* 6, no. 4: 48.
https://doi.org/10.3390/econometrics6040048