# Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters

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## Abstract

**:**

## 1. Introduction

## 2. A Class of Locally Asymptotically Self-Similar Processes

**Definition**

**1**

**.**A stochastic process ${\left\{{Y}_{t}^{\left(H\right)}\right\}}_{t\ge 0}$ (here the time indexes set is not necessarily continuous) is self-similar to self-similarity index $H\in (0,1)$ if, for all $n\in \mathbb{N}:=\{1,2,\dots \}$, all ${t}_{1},\dots ,{t}_{n}\ge 0$ and all $c>0$,

**Theorem**

**1.**

**Proof.**

**Definition**

**2**

**.**A continuous-time stochastic process ${\left\{{Z}_{t}^{\left(H\right(t\left)\right)}\right\}}_{t\ge 0}$ with its index $H(\u2022)$ being a continuous function valued in $(0,1)$, is called locally asymptotically self-similar, if for each $t\ge 0$, there exists a non-degenerate self-similar process ${\left\{{Y}_{u}^{\left(H\right(t\left)\right)}\right\}}_{u\ge 0}$ with self-similarity index $H\left(t\right)$, such that

**Proposition**

**1.**

**Proof.**

**Assumption**

**1.**

**Definition**

**3**

**.**A multifractional Brownian motion ${\left\{{W}_{H\left(t\right)}\left(t\right)\right\}}_{t\ge 0}$ is a continuous-time Gaussian process defined by:

- $\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tilde{W}\left(\xi \right)$ denotes a complex-valued Gaussian measure (see Proposition 2.1 in [43]) satisfying$${\int}_{\mathbb{R}}\tilde{f}\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tilde{W}\left(\xi \right)={\int}_{\mathbb{R}}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}W\left(t\right)\phantom{\rule{3.33333pt}{0ex}}a.s.$$for any$$f\in {L}^{2}\left(\mathbb{R}\right):=\left\{\left\{f:\phantom{\rule{3.33333pt}{0ex}}\mathbb{R}\to \mathbb{R}\right\}:\phantom{\rule{3.33333pt}{0ex}}{\int}_{\mathbb{R}}{\left|f\left(u\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}u<+\infty \right\},$$with $\tilde{f}\left(\xi \right):={\left(2\pi \right)}^{-1/2}{\int}_{\mathbb{R}}{e}^{i\xi u}f\left(u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}u$ being the Fourier transform of f and ${\left\{W\left(t\right)\right\}}_{t\in \mathbb{R}}$ being a standard Brownian motion.
- The Hurst functional parameter $H:\phantom{\rule{3.33333pt}{0ex}}[0,+\infty )\u27f6(0,1)$ is a Hölder function with exponent $\beta >\underset{t\in [0,+\infty )}{sup}H\left(t\right)$. Subject to this constraint the paths of mBm are almost surely continuous functions.

## 3. Clustering Stochastic Processes

#### 3.1. Covariance-Based Dissimilarity Measure between Autocovariance Ergodic Processes

**Definition**

**4.**

- For any integers $l\ge 1$, $m\ge 0$, ${X}_{l\dots l+m-1}^{\left(1\right)}$ is the shortcut notation of the row vector $\left({X}_{l}^{\left(1\right)},\dots ,{X}_{l+m-1}^{\left(1\right)}\right)$.
- The distance ρ between two equal-sized covariance matrixes ${M}_{1},{M}_{2}$ denotes the Frobenius norm of ${M}_{1}-{M}_{2}$. Recall that for a matrix ${A}_{M\times N}$, its Frobenius norm is defined by $\parallel {A}_{M\times N}{\parallel}_{F}:=\sqrt{{\sum}_{i=1}^{M}{\sum}_{j=1}^{N}{a}_{ij}^{2}},$ where for each $(i,j)\in \{1,\dots ,M\}\times \{1,\dots ,N\}$, ${a}_{ij}$ denotes the $(i,j)$-coefficient of ${A}_{M\times N}$.
- The sequence of positive weights ${\left\{{w}_{j}\right\}}_{j\ge 1}$ should be chosen such that $d\left({X}^{\left(1\right)},{X}^{\left(2\right)}\right)<+\infty $, i.e., the series on the right-hand side of Equation (19) is convergent. The choice of ${\left\{{w}_{j}\right\}}_{j}$ will be discussed in the forthcoming simulation study in Section 5.

**Remark**

**1.**

**Definition**

**5.**

- For $j=1,2$, $1\le l\le n$ and $m\le n-l+1$, $\nu \left({X}_{l\dots l+m-1}^{\left(j\right)}\right)$ denotes the empirical covariance matrix of the process ${X}^{\left(j\right)}$s path $({X}_{l}^{\left(j\right)},\dots ,{X}_{l+m-1}^{\left(j\right)})$, which is given below:$$\nu \left({X}_{l\dots l+m-1}^{\left(j\right)}\right):=\frac{{\sum}_{i=l}^{n-m+1}{({X}_{i}^{\left(j\right)}\phantom{\rule{3.33333pt}{0ex}}\dots \phantom{\rule{3.33333pt}{0ex}}{X}_{i+m-1}^{\left(j\right)})}^{T}({X}_{i}^{\left(j\right)}\phantom{\rule{3.33333pt}{0ex}}\dots \phantom{\rule{3.33333pt}{0ex}}{X}_{i+m-1}^{\left(j\right)})}{n-m-l+2},$$where ${(\u2022)}^{T}$ denotes the transpose of a matrix.

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.2. Covariance-Based Dissimilarity Measure between Locally Asymptotically Self-Similar Processes

**Proposition**

**2.**

**Proof.**

- L is chosen from $\{1,\dots ,n-K-1\}$.
- ${\mathbf{z}}_{1}^{\left(i\right)}$, ${\mathbf{z}}_{2}^{\left(i\right)}$ are the localized increment paths defined as in (29). Heuristically speaking, for $i=1,\dots ,n-K-1$, $\widehat{d}({\mathbf{z}}_{1}^{\left(i\right)},{\mathbf{z}}_{2}^{\left(i\right)})$ computes the “distance” between the two covariance structures (of the increments of ${\left\{{Z}_{t}^{H\left(t\right)}\right\}}_{t}$) indexed by the time in the neighborhood of ${t}_{i}$, and $\widehat{{d}^{*}}({\mathbf{z}}_{1},{\mathbf{z}}_{2})$ averages the above distances. It is worth noting that the value K describes the “sample size” used to approximate each local distance $\widehat{d}$. Therefore, its value should be picked neither too large nor too small and it can depend on n. It is suggested that $K\ge 5$ in order that the result of estimating the dissimilarity measure $\widehat{d}$ is acceptable. The largest value one can set for K is $n-2$ (correspondingly, $L=1$).

**Remark**

**5.**

## 4. Approximately Asymptotically Consistent Algorithms

#### 4.1. Offline and Online Algorithms

**Definition**

**6**

**.**Let $G=\left\{{G}_{1},\dots ,{G}_{\kappa}\right\}$ be a partitioning of $\mathbb{N}$ into κ disjoint sets ${G}_{k}$, $k=1,\dots ,\kappa $, such that the means and covariance structures of ${\mathbf{x}}_{i}$, $i\in \mathbb{N}$ are identical, if and only if $i\in {G}_{k}$ for some $k=1,\dots ,\kappa $. Such G is called ground truth of covariance structures. For $N\ge 1$, we denote by ${G|}_{N}$ the restriction of G to the first N sequences:

Algorithm 1: Offline clustering. |

Algorithm 2: Online clustering. |

#### 4.2. Computational Complexity and Consistency of the Algorithms

**Proposition**

**3.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Upper bound of**${\gamma}_{j}^{t}$: Similar to how (43) is derived, we use the triangle inequalities (Remark 5) and (39) to obtain:$$\begin{array}{ccc}\hfill & & \underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{min}{\gamma}_{j}^{t}<\frac{{\delta}_{min}}{2}\right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,N(t\left)\right\}\\ k,{k}^{\prime}\in \{1,\dots ,\kappa \}\\ k\ne {k}^{\prime}\end{array}}{min}\left(d\left({X}^{\left(k\right)},{X}^{\left({k}^{\prime}\right)}\right)-2\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)\right)<\frac{{\delta}_{min}}{2}\right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,N(t\left)\right\}\\ k\in \{1,\dots ,\kappa \}\end{array}}{max}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\frac{{\delta}_{min}}{4}\right).\hfill \end{array}$$$$\underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{min}{\gamma}_{j}^{t}<\frac{{\delta}_{min}}{2}\right)\le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,m\}\\ k\in \{1,\dots ,\kappa \}\end{array}}{max}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)<m\delta .$$$$\begin{array}{ccc}\hfill & & \underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{max}{\gamma}_{j}^{t}>{\delta}_{max}+2\epsilon \right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,N(t\left)\right\}\\ k,{k}^{\prime}\in \{1,\dots ,\kappa \}\\ k\ne {k}^{\prime}\end{array}}{max}\left(d\left({X}^{\left(k\right)},{X}^{\left({k}^{\prime}\right)}\right)+2\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)\right)>{\delta}_{max}+2\epsilon \right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,m\}\\ k\in \{1,\dots ,\kappa \}\end{array}}{max}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)<m\delta .\hfill \end{array}$$**Upper bound of**${\eta}^{t}$: By (56) and the fact that ${\sum}_{j=1}^{N\left(t\right)}{w}_{j}\ge {w}_{m},$ we have$$\begin{array}{ccc}\hfill & & \underset{t\ge T}{sup}\mathbb{P}\left({\eta}^{t}<\frac{{w}_{m}{\delta}_{min}}{2}\right)\le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{min}{\gamma}_{j}^{t}\sum _{j=1}^{N\left(t\right)}{w}_{j}<\frac{{w}_{m}{\delta}_{min}}{2}\right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{min}{\gamma}_{j}^{t}<\frac{{\delta}_{min}}{2}\right)<m\delta .\hfill \end{array}$$$$\begin{array}{ccc}\hfill & & \frac{1}{{\eta}^{t}}\sum _{j=1}^{N\left(t\right)}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)=\frac{1}{{\eta}^{t}}\sum _{j=1}^{m-1}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)\hfill \\ & & \phantom{\rule{56.9055pt}{0ex}}+\frac{1}{{\eta}^{t}}\sum _{j=m}^{J}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)+\frac{1}{{\eta}^{t}}\sum _{j=J+1}^{N\left(t\right)}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right).\hfill \end{array}$$

**Upper bound of the first term:**Using (53) and the fact that ${\left({\eta}^{t}\right)}^{-1}{\sum}_{j=1}^{m-1}{w}_{j}{\gamma}_{j}^{t}\le 1,$ we obtain$$\begin{array}{ccc}\hfill & & \underset{t\ge T}{sup}\mathbb{P}\left(\frac{1}{{\eta}^{t}}\sum _{j=1}^{m-1}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)\le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,m-1\}\\ k\in \{1,\dots ,\kappa \}\end{array}}{max}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)\hfill \\ & & \le (m-1)\delta .\hfill \end{array}$$**Upper bound of the second term:**Recall that ${\mathbf{x}}_{{c}_{k}^{j}}^{t}={\mathbf{x}}_{{s}_{k}}^{t}$ for all $j\in \{m,\dots ,J\}$ and $k\in \{1,\dots ,\kappa \}$. Therefore, by (52) and the fact that ${\left({\eta}^{t}\right)}^{-1}{\sum}_{j=m}^{J}{w}_{j}{\gamma}_{j}^{t}\le 1,$ for every $k\in \{1,\dots ,\kappa \}$ we have$$\begin{array}{ccc}\hfill & & {\displaystyle \underset{t\ge T}{sup}\mathbb{P}\left(\frac{1}{{\eta}^{t}}\sum _{j=m}^{J}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)=\underset{t\ge T}{sup}\mathbb{P}\left(\widehat{{d}^{*}}\left({\mathbf{x}}_{{s}_{k}}^{t},{X}^{\left(k\right)}\right)\frac{1}{{\eta}^{t}}\sum _{j=m}^{J}{w}_{j}{\gamma}_{j}^{t}>\epsilon \right)}\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\widehat{{d}^{*}}\left({\mathbf{x}}_{{s}_{k}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)<\delta .\hfill \end{array}$$**Upper bound of the third term:**By (52), (58) and (57),$$\begin{array}{ccc}\hfill & & \underset{t\ge T}{sup}\mathbb{P}\left(\frac{1}{{\eta}^{t}}\sum _{j=J+1}^{N\left(t\right)}{w}_{j}{\gamma}_{j}^{t}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\frac{2{\epsilon}^{2}({\delta}_{max}+2\epsilon )}{{w}_{m}{\delta}_{min}}\right)\hfill \\ & & \le \underset{t\ge T}{sup}\mathbb{P}\left(\underset{\begin{array}{c}j\in \{1,\dots ,N(t\left)\right\}\\ k\in \{1,\dots ,\kappa \}\end{array}}{max}\widehat{{d}^{*}}\left({\mathbf{x}}_{{c}_{k}^{j}}^{t},{X}^{\left(k\right)}\right)>\epsilon \right)\hfill \\ & & \phantom{\rule{56.9055pt}{0ex}}+\underset{t\ge T}{sup}\mathbb{P}\left(\underset{j\in \{1,\dots ,N(t\left)\right\}}{max}{\gamma}_{j}^{t}>{\delta}_{max}+2\epsilon \right)\hfill \\ & & <2m\delta .\hfill \end{array}$$

## 5. Tests on Simulated Data: Clustering Multifractional Brownian Motions

#### 5.1. Efficiency Improvement: ${log}^{*}$-Transformation

#### 5.2. Simulation Methodology

#### 5.3. Synthetic Datasets

**Case 1 (Monotonic function)**: The general form is taken to be$$H\left(t\right)=0.5+h\xb7t/Q,\phantom{\rule{1.em}{0ex}}t\in [0,Q],$$**Case 2 (Periodic function)**: The general form is taken to be$$H\left(t\right)=0.5+h\xb7sin(\pi t/Q),\phantom{\rule{1.em}{0ex}}t\in [0,Q],$$**Case 3 (Small turbulence on $\mathit{H}$)**: We proceed to examine if the proposed algorithm has the capacity of distinguishing processes with very similar behaviors. To this end we take consider clustering an fBm with Hurst parameter ${H}_{f}$, and an mBm with the Hurst functional parameter$$H\left(t\right)={H}_{f}+0.1\xb7sin(\pi t/Q),\phantom{\rule{1.em}{0ex}}t\in [0,Q],$$We perform the tests using two values of ${H}_{f}$: (i) ${H}_{f}=0.2$ and (ii) ${H}_{f}=0.8$. In both cases, the index $H\left(t\right)$ of the corresponding mBm is regarded to be ${H}_{f}$ plus some noise.

- 1.
- For $i=1,\dots ,5$, simulate 20 mBm paths in group i (corresponding to ${h}_{i}$), each path is with length of 305. Then the total number of paths $N=100$. To be more explicit we denote by$$S:=\left(\begin{array}{cccc}{x}_{1,1}& {x}_{1,2}& \cdots & {x}_{1,305}\\ {x}_{2,1}& {x}_{2,2}& \cdots & {x}_{2,305}\\ \vdots & \vdots & \ddots & \vdots \\ {x}_{100,1}& {x}_{100,2}& \cdots & {x}_{100,305}\end{array}\right),$$$${S}^{\left(i\right)}:=\left(\begin{array}{cccc}{x}_{20(i-1)+1,1}& {x}_{20(i-1)+1,2}& \cdots & {x}_{20(i-1)+1,305}\\ \vdots & \vdots & \ddots & \vdots \\ {x}_{20i,1}& {x}_{20i,2}& \cdots & {x}_{20i,305}\end{array}\right).$$
- 2.
- At each $t=1,\dots ,100$, we observe the first $n\left(t\right)=3t+5$ values of each path, i.e.,$${S}_{\mathrm{offline}}\left(t\right):=\left(\begin{array}{cccc}{x}_{1,1}& {x}_{1,2}& \cdots & {x}_{1,3t+5}\\ {x}_{2,1}& {x}_{2,2}& \cdots & {x}_{2,3t+5}\\ \vdots & \vdots & \ddots & \vdots \\ {x}_{100,1}& {x}_{100,2}& \cdots & {x}_{100,3t+5}\end{array}\right).$$

- 1.
- 2.
- At each $t=1,\dots ,100$ and $i=1,\dots ,5$, we observe the following dataset in the ith group:$${S}_{\mathrm{online}}^{\left(i\right)}\left(t\right):=\left(\begin{array}{ccccccc}{\tilde{x}}_{1,1}& {\tilde{x}}_{1,2}& \cdots & \cdots & \cdots & \cdots & {\tilde{x}}_{1,{n}_{1}\left(t\right)}\\ {\tilde{x}}_{2,1}& {\tilde{x}}_{2,2}& \cdots & \cdots & \cdots & {\tilde{x}}_{2,{n}_{2}\left(t\right)}\\ \vdots & \vdots & \ddots & \vdots \\ {\tilde{x}}_{{N}_{i}\left(t\right),1}& {\tilde{x}}_{{N}_{i}\left(t\right),2}& \cdots & {\tilde{x}}_{{N}_{i}\left(t\right),{n}_{{N}_{i}\left(t\right)}\left(t\right)}\end{array}\right),$$
- ${N}_{i}\left(t\right):=6+\lfloor (t-1)/10\rfloor $ denotes the number of paths in the ith group. Here $\lfloor \u2022\rfloor $ denotes the floor number. That is, starting from 6 paths in each group, 1 new path will be added into each group as t increases by 10.
- ${n}_{l}\left(t\right):=3{\left(t-{(l-6)}^{+}\right)}^{+}+5$, with ${(\u2022)}^{+}:=max(\u2022,0)$. This means each path observes three new values as t increases by 1.

#### 5.4. Experimental Results

**Case 1 (Monotonic function):**

**(1)**- Both algorithms attempt to be consistent in their circumstances, as the time t increases, in the sense that the corresponding misclassification rates decrease to 0.
**(2)**- Clustering mBms are asymptotically equivalent to clustering their tangent processes’ increments.
**(3)**- The online algorithm seems to have an overall better performance: its misclassification rates are 5–$10\%$ lower than that of offline algorithm. The reason may be that at early time steps the differences among the $H(\u2022)$s are not significant. Unlike the offline clustering algorithm, the online one is flexible enough to catch these small differences.

**Case 2 (Periodic function):**

**(1)**- Both misclassification rates of the clustering algorithms have generally a declining trend as time increases.
**(2)**- As the differences among the periodic function $H(\u2022)$s values go up and down, the misclassification rates go down and up accordingly.
**(3)**- The online clustering algorithm has an overall worse performance than the offline one. This may be because starting from $t=20$ the differences among $H(\u2022)$s become significantly large. In this situation, the offline clustering algorithm can better catch these differences, since it has a larger sample size (20 paths in each group) than the online one.

**Case 3 (Small turbulence on $\mathit{H}$):**

#### 5.5. Comparison to Traditional Approaches Designed for Clustering Finite-Dimensional Vectors

## 6. Real World Application: Clustering Global Financial Markets

#### 6.1. Motivation

#### 6.2. Data and Methodology

**Equity indexes returns**: We cluster the global stock indexes based on their empirical time-varying covariance structure. We use Algorithms 1 and 2 as the clustering approach. We select the index constituents of MSCI ACWI (All Country World Index) as the underlying stochastic processes in the datasets for clustering analysis. Each of the indexes is a realized path representing the historical monthly returns of the underlying economic entities. MSCI ACWI is the leading global equity market index and covers more than 85% of the market capitalization of the global stock market (As of December 2018, as reported on https://www.msci.com/acwi, accessed on 1 January 2022).**Sovereign CDS spreads**: We cluster the sovereign credit default swap (CDS) spreads of global economic entities. The sovereign CDS is an insurance-like financial product that provides default protection of treasury bonds for the economic entity (e.g., the government). The CDS spread reflects the cost to insurer of the exposure on a sovereign entity’s default. We select a five-year sovereign CDS spread as the indicator of sovereign credit risk, as the five-year product has the best liquidity on the CDS market. We overlap the sample of economic entities between the stock and CDS datasets, and the same set of underlying economics entities are present in the clustering analysis. Our CDS data source is Bloomberg.

#### 6.3. Clustering Results

## 7. Conclusions and Future Prospects

**(1)**- Given their flexibility, our algorithms are applicable to clustering any distribution stationary ergodic processes with finite variances, any autocovariance ergodic processes, and locally asymptotically self-similar processes whose tangent processes have autocovariance ergodic increments. Multifractional Brownian motion (mBm) is an excellent representative of the latter class of processes.
**(2)**- Our algorithms are efficient enough in terms of their computational complexity. A simulation study is performed on clustering mBm. The results show that both offline and online algorithms are approximately asymptotically consistent.
**(3)**- Our algorithms are successfully applied to cluster the real world financial time series (equity returns and sovereign CDS spreads) via the development level and via regions. The outcomes are self-consistent with the financial markets behavior and they reveal the level of impact between the economic development and regions on equity returns.

**(1)**- The clustering framework proposed in our paper only focuses on the cases where the true number of clusters $\kappa $ is known. The problem for which $\kappa $ is supposed to be unknown remains open.
**(2)**- If we drop the Gaussianity assumption, the class of stationary incremental self-similar processes becomes much larger. This will yield an introduction to a more general class of locally asymptotically self-similar processes, whose autocovariances do not exist. This class includes linear multifractional stable motion [55,56] as a paradigmatic example. Cluster analysis of such stable processes will no doubt lead to a wide range of applications, especially when the process distributions exhibit heavy-tailed phenomena. Neither the distribution dissimilarity measure introduced in [12] nor the covariance-based dissimilarity measures used in this paper would work in this case, hence new techniques are required to cluster such processes, such as considering replacing the covariances with covariations [35] or symmetric covariations [57] in the dissimilarity measures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

fBm | fractional Brownian motion |

mBm | multifractional Brownian motion |

gBm | geometric Brownian motion |

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**Figure 1.**The functional form of $H(\u2022)$ follows Equation (73): $H\left(t\right)=0.5+{h}_{i}\xb7t/100$ with $t=0,1,\dots ,100$. The left graph plots $H(\u2022)$ correspond to 5 different clusters. The right graph illustrates the misclassification rates output by (i) offline algorithm on offline dataset (solid line) and (ii) online algorithm on online dataset (dashed line). Both algorithms are performed based on the ${log}^{*}$- transformed covariance-based dissimilarity measure.

**Figure 2.**The functional form of $H(\u2022)$ follows Equation (74): $H\left(t\right)=0.5+{h}_{i}\xb7sin(\pi t/100)$ with $t=0,1,\dots ,100$. The left graph plots $H(\u2022)$ corresponding to 5 different clusters. The right graph illustrates the misclassification rates output by (i) offline algorithm on offline dataset (solid line) and (ii) online algorithm on online dataset (dashed line). Both algorithms are performed based on the ${log}^{*}$- transformed covariance-based dissimilarity measure.

**Figure 3.**The clustering performance using fBm with ${H}_{f}$ and mBm with $H(\u2022)$ given in Equation (74): $H\left(t\right)={H}_{f}+0.1\xb7sin(\pi t/100)$ with $t=0,1,\dots ,100$. The left graph is the case where ${H}_{f}=0.2$, and the right graph is the case where ${H}_{f}=0.8$. Both graphs illustrate the misclassification rates output by $\left(i\right)$ offline algorithm on offline dataset (solid line) and $\left(ii\right)$ online algorithm on online dataset (dashed line). Both algorithms are performed based on the ${log}^{*}$- transformed covariance-based dissimilarity measure.

**Figure 4.**The functional form of $H(\u2022)$ follows Equation (73): $H\left(t\right)=0.5+{h}_{i}\xb7t/100$ with $t=0,1,\dots ,100$. The left graph plots the misclassification rates output by K-means and hierarchical clustering methods using the offline dataset. The right graph shows the misclassification rates using the online dataset.

**Figure 5.**The functional form of $H(\u2022)$ follows Equation (74): $H\left(t\right)=0.5+{h}_{i}\xb7sin(\pi t/100)$ with $t=0,1,\dots ,100$. The left graph plots the misclassification rates output by K-means and hierarchical clustering methods using the offline dataset. The right graph shows the misclassification rates using the online dataset.

**Table 1.**The categories of major stock and sovereign CDS markets in the MSCI ACWI (All Country World Index). There are 23 developed economic entities and 24 emerging countries or areas. The geographical regions contain Americas, EMEA (Europe, Middle East and Africa), Pacific and Asia. Markets with * have missing sovereign CDS data in our data sample.

Developed Markets | Emerging Markets | ||||
---|---|---|---|---|---|

Americas | Europe & Middle East | Pacific | Americas | Europe & Middle East & Africa | Asia |

Canada | Austria | Australia | Brazil | Czech Republic | China (Mainland) |

USA | Belgium | Hong Kong | Chile | Greece * | India |

Denmark | Japan | Colombia | Hungary | Indonesia | |

Finland | New Zealand | Mexico | Poland | Korea | |

France | Singapore * | Peru | Russia | Malaysia | |

Germany | Turkey | Pakistan | |||

Ireland | Egypt | Philippines | |||

Israel | South Africa | Taiwan | |||

Italy | Qatar * | Thailand | |||

The Netherlands * | United Arab Emirates * | ||||

Norway | |||||

Portugal | |||||

Spain | |||||

Sweden | |||||

Switzerland | |||||

United Kingdom |

**Source:**MSCI ACWI (All Country World Index) market allocation. https://www.msci.com/acwi, accessed on 1 January 2022.

**Table 2.**The misclassification rates of clustering algorithms on different datasets. The clustering facts are geographical region and development level, respectively. Panel A presents the results from clustering global stock indexes, and Panel B presents the results from clustering sovereign CDS spreads.

Panel A | Offline Algorithm | Online Algorithm | ||
---|---|---|---|---|

Stock Returns | Regions | Emerging/Developed | Regions | Emerging/Developed |

offline dataset | 61.70% | 29.79% | 55.32% | 36.17% |

online dataset | 53.19% | 44.68% | 51.06% | 14.89% |

Panel B | Offline Algorithm | Online Algorithm | ||

CDS Spreads | Regions | Emerging/Developed | Regions | Emerging/Developed |

offline dataset | 64.29% | 28.57% | 71.43% | 26.19% |

online dataset | 54.76% | 47.62% | 59.52% | 26.19% |

**Table 3.**The misclassification cases when using offline algorithm on offline dataset and online algorithm on online dataset. Panel A reports the mis-categorized economics entities in the stock index clustering case, and Panel B reports the mis-categorized economics entities in the sovereign CDS spreads clustering case. The algorithm clusters the dataset into two groups: emerging market group and developed market group. The mis-categorized outcome are reported, where (i) entities from developed markets incorrectly clusters in emerging market, or (ii) vice versa.

Panel A: Equity Indexes Returns | |||
---|---|---|---|

Group 1 (Emerging Markets) | Group 2 (Developed Markets) | ||

Incorrect-Offline | Incorrect-Online | Incorrect-Offline | Incorrect-Online |

Austria | Austria | Korea | Czech Republic |

Finland | Finland | Chile | Qatar |

Germany | Portugal | Philippines | Peru |

Ireland | Malaysia | South Africa | |

Italy | Mexico | ||

Norway | |||

Portugal | |||

Spain | |||

New Zealand | |||

Panel B: Sovereign CDS Spreads | |||

Group 1 (Emerging Markets) | Group 2 (Developed Markets) | ||

Incorrect-Offline | Incorrect-Online | Incorrect-Offline | Incorrect-Online |

Ireland | Ireland | Chile | Chile |

Italy | Portugal | China (Mainland) | China (Mainland) |

Portugal | Czech Republic | Czech Republic | |

Spain | Korea | Hungary | |

Malaysia | Korea | ||

Mexico | Malaysia | ||

Poland | Mexico | ||

Thailand | Poland | ||

Thailand |

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## Share and Cite

**MDPI and ACS Style**

Rao, N.; Peng, Q.; Zhao, R.
Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters. *Fractal Fract.* **2022**, *6*, 222.
https://doi.org/10.3390/fractalfract6040222

**AMA Style**

Rao N, Peng Q, Zhao R.
Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters. *Fractal and Fractional*. 2022; 6(4):222.
https://doi.org/10.3390/fractalfract6040222

**Chicago/Turabian Style**

Rao, Nan, Qidi Peng, and Ran Zhao.
2022. "Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters" *Fractal and Fractional* 6, no. 4: 222.
https://doi.org/10.3390/fractalfract6040222