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

^{1}

^{2}

^{3}

^{*}

## 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

## References

- Absil, Pierre-Antoine, Robert Mahony, and Rodolphe Sepulchre. 2008. Optimization Algorithms on Matrix Manifolds. Princeton: Princeton University Press. [Google Scholar]
- Anderson, Theodore Wilbur. 1971. The Statistical Analysis of Time Series. New York: Wiley. [Google Scholar]
- Bierens, Herman J., and Luis F. Martins. 2010. Time-varying cointegration. Econometric Theory 26: 1453–90. [Google Scholar] [CrossRef]
- Breitung, Jörg, and Sandra Eickmeier. 2011. Testing for structural breaks in dynamic factor models. Journal of Econometrics 163: 71–84. [Google Scholar] [CrossRef] [Green Version]
- Casals, Jose, Alfredo Garcia-Hiernaux, Miguel Jerez, Sonia Sotoca, and A. Alexandre Trindade. 2016. State-Space Methods for Time Series Analysis: Theory, Applications and Software. Chapman & Hall/CRC, Monographs on Statistics & Applied Probability. New York: CRC Press. [Google Scholar]
- Chikuse, Yasuko. 2003. Statistics on Special Manifolds. New York: Springer. [Google Scholar]
- Chikuse, Yasuko. 2006. State space models on special manifolds. Journal of Multivariate Analysis 97: 1284–94. [Google Scholar] [CrossRef]
- Del Negro, Marco, and Christopher Otrok. 2008. Dynamic Factor Models with Time-Varying Parameters: Measuring Changes in International Business Cycles. Staff Report No. 326. New York: Federal Reserve Bank of New York. [Google Scholar]
- Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods, 2nd ed. Oxford: Oxford University Press. [Google Scholar]
- Eickmeier, Sandra, Wolfgang Lemke, and Massimiliano Marcellino. 2014. Classical time varying factor-augmented vector auto-regressive models—Estimation, forecasting and structural analysis. Journal of the Royal Statistical Society Series A (Statistics in Society) 178: 493–533. [Google Scholar] [CrossRef]
- Hamilton, James Douglas. 1994. Time Series Analysis. Princeton: Princeton University Press. [Google Scholar]
- Hannan, Edward J. 1970. Multiple Time Series. New York: Wiley. [Google Scholar]
- Herz, Carl S. 1955. Bessel functions of matrix argument. Annals of Mathematics 61: 474–523. [Google Scholar] [CrossRef]
- Hoff, Peter D. 2009. Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data. Journal of Computational and Graphical Statistics 18: 438–56. [Google Scholar] [CrossRef]
- Khatri, C. G., and Kanti V. Mardia. 1977. The von Mises-Fisher matrix distribution in orientation statistics. Journal of the Royal Statistical Society, Series B 39: 95–106. [Google Scholar] [CrossRef]
- Koopman, Lambert Herman. 1974. The Spectral Analysis of Time Series. New York: Academic Press. [Google Scholar]
- Mardia, Kanti V. 1975. Statistics of directional data (with discussion). Journal of the Royal Statistical Society, Series B 37: 349–93. [Google Scholar]
- Prentice, Michael J. 1982. Antipodally symmetric distributions for orientation statistics. Journal of Statistical Planning and Inference 6: 205–14. [Google Scholar] [CrossRef]
- Rothman, Philip, Dick van Dijk, and Philip Hans Franses. 2001. A Multivariate STAR analysis of the relationship between money and output. Macroeconomic Dynamics 5: 506–32. [Google Scholar]
- Stock, James, and Mark Watson. 2009. Forecasting in dynamic factor models subject to structural instability. In The Methodology and Practice of Econometrics. A Festschrift in Honour of David F. Hendry. Edited by David F. Hendry, Jennifer Castle and Neil Shephard. Oxford: Oxford University Press, pp. 173–205. [Google Scholar]
- Stock, James H., and Mark W. Watson. 2002. Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97: 1167–79. [Google Scholar] [CrossRef]
- Swanson, Norman Rasmus. 1998. Finite sample properties of a simple LM test for neglected nonlinearity in error correcting regression equations. Statistica Neerlandica 53: 76–95. [Google Scholar] [CrossRef]
- Wong, Roderick S. C. 2001. Asymptotic Approximations of Integrals. In Classics in Applied Mathematics. Philadelphia: Society for Industrial and Applied Mathematics. [Google Scholar]

**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 $-{\mathbf{\alpha}}_{0}$.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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