# Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Structure of Semi-Positive-Definite Matrices and the Dilation Theory

#### 2.1. Outline of the Dilation Theory

- there exists a larger space from which the original function (or matrix) is deduced;
- we can choose the “dilated” function to be simpler. For instance, when dealing with matrices, each of its coefficients can be expressed as the projection of a larger unitary matrix. In this case, we obtain a unitary dilation. This approach has been for example developed in [34,35,36] for the stationary dilation of periodically-correlated processes.

#### 2.1.1. Dilation and Rotation of Contractions

#### 2.1.2. Dilation and Isometries

#### 2.1.3. Dilation and Measure

#### 2.2. Construction of Dilation Matrices

**Theorem**

**1**(Structure of a positive-definite block matrix)

**.**

- The operator $A=\left(\begin{array}{cc}X& Y\\ {Y}^{*}& Z\end{array}\right)$ is positive
- There exists a unique contraction Γ in $\mathcal{L}\left(\mathcal{R}\right(Z),\mathcal{R}(X\left)\right)$ such that$$Y={X}^{1/2}\Gamma {Z}^{1/2}$$

**Proof.**

**Theorem**

**2.**

- ${R}_{kk}\u2a7e0$ for all k
- there exists a family $\{{\Gamma}_{k,j}\mid k,j=1,\cdots n,k\u2a7dj\}$ of contraction such that$${R}_{k,j}={B}_{k,k}^{*}({L}_{k,j-1}{U}_{k+1,j-1}{C}_{k+1,j}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}{D}_{{\Gamma}_{k,k+l}^{*}}\cdots {D}_{{\Gamma}_{k,j-l}^{*}}{\Gamma}_{k,j}{D}_{{\Gamma}_{k+1,j}}\cdots {D}_{{\Gamma}_{j-1,j}}){B}_{j,j}$$

**Proof.**

- If ${R}^{\left(n\right)}$ is a semi-positive definite complex-valued Toeplitz kernel, then all the $\left\{{\Gamma}_{n}\right\}$ are complex-valued and respect $|{\Gamma}_{i}|<1$. The structure and the construction procedure for obtaining such a complex-valued parameter is identical whether the kernel is real or complex.
- The framework proposed by Constantinescu and recalled previously is quite general and can be extended to Matrix Orthogonal Polynomial on the Unit Circle (MOPUC) development. By referring to ([43], Section 3.1), matrix polynomials stem from a Szëgo recursion akin to the scalar case and thus provide a sequence of Verblunsky coefficients, or parcors that are matrices. Again in ([43], Section 3.11), a correspondence between MOPUC and CMV matrices [13] is provided, which are equivalent to dilation matrices. The construction procedure remains the same for the dilation matrices, but the parcors become in that case matrices (they are matrix-valued Verblunsky coefficients), which can be obtained by a matrix version of the Schur/Geronimus algorithm [44].

## 3. Analysis of Curves on a Manifold Induced by the Dilation

#### 3.1. Preliminaries on Lie Groups

- The geodesics through e (the identity element) are the integral curves $t\mapsto exp\left(tu\right),\phantom{\rule{4pt}{0ex}}u\in \mathfrak{g}$, that is, the one-parameter groups. In addition, because left and right are isometries and isometries maps geodesics to geodesics, the geodesics through any point $a\in G$ are the left (right) translates of the geodesics through e$$\gamma \left(t\right)={L}_{a}\left(exp\left(tu\right)\right),\phantom{\rule{4pt}{0ex}}u\in \mathfrak{g}.$$$${\gamma}^{\prime}\left(0\right)=\left(d{L}_{a}\right)e\left(u\right).$$
- The Levi-Civita connection is given by : ${\nabla}_{X}Y={\displaystyle \frac{1}{2}}[X,Y],\phantom{\rule{4pt}{0ex}}\forall X,Y\in \mathfrak{g}$

#### 3.2. Basic Outline of Geometry

#### 3.3. Metric and Distance over $\mathcal{M}$ and $\mathcal{S}$

#### 3.4. The Geodesic Equation

**Theorem**

**3.**

**Proof.**

**Input**: a set of rotation matrices ${\left\{{W}_{i}\right\}}_{i}$, seen as a partial observation of a closed trajectory on $SO\left(n\right)$.- Map the set of matrices $\left\{{W}_{i}\right\}$ into the a set of matrices in the Lie algebra $\left\{{V}_{i}\right\}$ using the inverse exponential map.
- Go back in the base manifold $SO\left(n\right)$ with the exponential map.
- Shift the interpolated curve in order to fulfill the condition $c\left(0\right)=e$ and compute the SRV transformation given by (41)
- Compute the distance defined by (38). The optimization is carried out by dynamic programming.
**Output**: distance between two curves in the manifold defined by the set of curves in $SO\left(n\right)$, and geodesic path between the curves.

#### 3.5. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Defect Operator, Elementary Rotation

**Theorem**

**A1.**

- There exists a contraction Γ in z such that $X=\Gamma Y$,
- ${X}^{*}X\u2a7d{Y}^{*}Y$.

**Proof.**

**Theorem**

**A2**(row contraction)

**.**

**Proof.**

**Theorem**

**A3**(Structure of row contraction)

**.**

- The operator ${T}^{n}=[{T}_{1}\phantom{\rule{1.em}{0ex}}{T}_{2}\phantom{\rule{1.em}{0ex}}\cdots {T}^{n}]$ in $\mathcal{L}({\oplus}_{k=1}^{n}{\mathcal{H}}_{k},{\mathcal{H}}^{\prime})$ is a contraction
- ${T}_{1}={\Gamma}_{1}$ is a contraction and, for $k>2$, there exists uniquely determined contractions ${\Gamma}_{k}\phantom{\rule{4pt}{0ex}}\in \mathcal{L}({\mathcal{H}}_{k},\mathcal{R}\left({\gamma}_{k}\right))$ such that ${T}_{k}={D}_{{\Gamma}_{1}^{*}}{D}_{{\Gamma}_{2}^{*}}\cdots {D}_{{\Gamma}_{k-1}^{*}}{\Gamma}_{k}$.

**Proof.**

## Appendix B. Geodesic Equation in the Space of Curve $\mathcal{M}$

**Theorem**

**A4.**

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**Figure 1.**Illustration of a sampled closed trajectory drawn in $SO\left(n\right)$ or $SU\left(n\right)$ that materializes the time varying of the Periodically Correlated (PC) measure for a stochastic process. Each ${W}_{i}$ is a dilation matrix built through the parcors. Recall that a PC process is a process such that ${R}_{s,t}={R}_{s+T,t+T}$ for a certain T, where ${R}_{\xb7,\xb7}$ stands for the correlation function of the process.

**Figure 2.**Example of a reparametrization of a curve. Here, it consists in changing the discretization with nonlinear time sample.

**Figure 3.**The tangent space ${T}_{\left[c\right]}\mathcal{M}$ at a point $\left[c\right]$ in the shape space $\mathcal{S}$ is isomorphic to the horizontal part ${\mathcal{H}}_{\mathcal{M}}$ of the tangent space at a point on the associated fiber.

**Figure 4.**1000 samples of PC processes generated by (

**a**) a modulated zero mean and unit variance stationary random process $a\left(t\right)$; (

**b**) a periodic AR(2) model with a period of 54 points; (

**c**) a periodic AR(2) model with a period of 20 points; and (

**d**) a periodic ARMA(2,1) model with a period of 20 points.

**Figure 5.**Representation inside the ball of radius $\pi $ of the four PC processes drawn in Figure 4, arranged in the same order with 200 interpolated points represented with green stars, the dashed black line is the theoretical curve and the red dots are the representation of dilation matrices. (

**a**) is the representation of Figure 4a, (

**b**) is the representation of Figure 4b, (

**c**) is the representation of Figure 4c and (

**d**) is the representation of Figure 4d.

**Figure 6.**Representation inside the ball of radius $\pi $ of the four PC processes drawn in Figure 4, arranged in the same order with 50 interpolated points except for plot (

**b**) which has been computed with 100 interpolated points represented with green stars; the dashed black line is the theoretical curve and the red dots are the representation of dilation matrices. (

**a**) is the representation of Figure 4a, (

**b**) is the representation of Figure 4b, (

**c**) is the representation of Figure 4c and (

**d**) is the representation of Figure 4d.

**Figure 7.**Geodesic interpolation with respect to (43) between the green dashed curve (gold standard signal of Figure 8) and the dashed red curve (signal of Figure 4c), first row for 200 interpolated points, second row for 50 interpolated points. For this scenario $s\in [1/4,\phantom{\rule{0.277778em}{0ex}}1/2,\phantom{\rule{0.277778em}{0ex}}3/4]$ which corresponds to [(

**a**–

**d**), (

**b**–

**e**), (

**c**–

**f**)] respectively.

**Figure 8.**A PAR(2) signal with a period of 20 points, 1000 samples were generated, and its corresponding SO(3) representation inside the ball of radius $\pi $.

**Figure A1.**we consider a beam of curves, which consists in a slight modification of the geodesic. The different curves are indexed by $\tau $. The idea is to find which of these curves gives the minimal energy to go from ${c}_{0}$ to ${c}_{1}$.

**Table 1.**Table of the distances between all the PC processes of Figure 4 to the gold standard PC process of Figure 8 through the distance of their curves’ shapes on $SO\left(3\right)$, $SO\left(4\right)$ and $SO\left(5\right)$. We have interpolated with roughly twice and four times the number of original points. We also applied here a DP to solve the optimization assignment problem.

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**MDPI and ACS Style**

Dugast, M.; Bouleux, G.; Marcon, E. Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds. *Entropy* **2018**, *20*, 717.
https://doi.org/10.3390/e20090717

**AMA Style**

Dugast M, Bouleux G, Marcon E. Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds. *Entropy*. 2018; 20(9):717.
https://doi.org/10.3390/e20090717

**Chicago/Turabian Style**

Dugast, Maël, Guillaume Bouleux, and Eric Marcon. 2018. "Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds" *Entropy* 20, no. 9: 717.
https://doi.org/10.3390/e20090717