# Homology Groups of Embedded Fractional Brownian Motion

H. Masoomy<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, 1983969411, Tehran, Iran  S. Tajik<EMAIL_ADDRESS>Department
of Physics, Brock University, St. Catharines, ON L2S 3A1, Canada  S. M. S.
Movahed<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, 1983969411, Tehran, Iran

###### Abstract

A well-known class of non-stationary self-similar time series is the
fractional Brownian motion (fBm) considered to model ubiquitous stochastic
processes in nature. In this paper, we study the homology groups of high-
dimensional point cloud data (PCD) constructed from synthetic fBm data. We
covert the simulated fBm series to a PCD, a subset of unit $D$-dimensional
cube, employing the time delay embedding method for a given embedding
dimension and a time-delay parameter. In the context of persistent homology
(PH), we compute topological measures for embedded PCD as a function of
associated Hurst exponent, $H$, for different embedding dimensions, time-
delays and amount of irregularity existed in the dataset in various scales.
Our results show that for a regular synthetic fBm, the higher value of the
embedding dimension leads to increasing the $H$-dependency of topological
measures based on zeroth and first homology groups. To achieve a reliable
classification of fBm, we should consider the small value of time-delay
irrespective of the irregularity presented in the data. More interestingly,
the value of scale for which the PCD to be path-connected and the post-
loopless regime scale are more robust concerning irregularity for
distinguishing the fBm signal. Such robustness becomes less for the higher
value of embedding dimension.

Persistent Homology, Fractional Brownian Motion, Time Delay Embedding

##  I Introduction

Data generation from a wide range of experiments and observations leads to
great opportunities to use modeling in the context of complex systems.
Assigning a shape to various types of data and accordingly computing their
persistence in terms of so-called resolution has recently become the focus of
many studies. Furthermore, the spatio-temporal distribution (shape) of
datasets has been beneficial in various branches of science Vuran  _et al._
(2004); Diggle (2006); Chaplain  _et al._ (2001). Among different criteria
such as linear algebra Strang  _et al._ (1993); Defranza and Gagliardi (2015)
and common statistical properties Jaynes (1957); Goulden (1939); Mandel
(2012); Albert and Barabási (2002); Costa  _et al._ (2007); Barthélemy (2011);
Wu  _et al._ (2015); Boguna  _et al._ (2021), the geometrical and topological
features has provided complementary evaluations Adler (1981); Carlsson (2009).
Depending on what type of information is needed and based on limitations in
computational resources, one- and/or $n$-point statistics of geometrical and
topological properties Rice (1944, 1945, 1954); Bardeen  _et al._ (1986);
Matsubara (2003); Pogosyan  _et al._ (2009); Gay  _et al._ (2012); Matsubara
and Codis (2020); Vafaei Sadr and Movahed (2021), and complex network based
analysis (Albert and Barabási, 2002; Costa  _et al._ , 2007; Barthélemy, 2011;
Wu  _et al._ , 2015; Boguna  _et al._ , 2021, and references therein), can be
taken into account.

Inspired by geometrical fractals Mandelbrot (1982), the notion of self-
similarity and self-affinity is employed to quantify the statistical
properties of different fields of generally $(N+D)$-dimensionAdler (1981) 111A
typical $(N+D)$-dimensional stochastic field (process), is a measurable
mapping from probability space into a $\mathbb{R}^{N}$-valued function on
$\mathbb{R}^{D}$ Euclidean space, in which the labels $N$ and $D$ are
respectively devoted to $N$ dependent and $D$ independent parameters,
generating a $(N+D)$-dimensional random field or a stochastic process. A
pioneer method to quantify a self-similar process based on Hurst exponent
Hurst (1951) which is known as fractional Brownian motion (fBm) (its
associated increment is called fractional Gaussian noise (fGn)) Mandelbrot and
Van Ness (1968), is called Multifractal Detrended Fluctuation Analysis (MFDFA)
Peng  _et al._ (1994, 1995); Kantelhardt  _et al._ (2002) (see also (Jiang
_et al._ , 2019; Eghdami  _et al._ , 2018, and references therein)). Besides
the noted methods for classification, prediction, and examination of datasets,
one can focus on topological features of data to study it under the banner of
Topological Data Analysis (TDA) Dey and Wang (2022); Edelsbrunner and Harer
(2022); Zomorodian (2005); Ghrist (2008); Wasserman (2018) and accordingly
construct a powerful algebra-topological-based approach Nakahara (2003);
Munkres (2018); Hatcher (2005). To this end, Persistent Homology (PH) as a
shape-based tool that examines the evolution of global features of the data
related to topological invariants is utilized Otter  _et al._ (2017); Speidel
_et al._ (2018); Maletić  _et al._ (2016); Donato  _et al._ (2016); Pereira
and de Mello (2015); Masoomy  _et al._ (2021a).

There are many methods to construct higher dimensional sets from a
$(1+1)$-dimensional series and then implement TDA. The visibility graph method
Lacasa  _et al._ (2008), correlation network method Yang and Yang (2008) and
time delay embedding (TDE) method Takens (1981); Packard  _et al._ (1980) have
been used as the pre-processing part on the input data. The visibility graph
and correlation network methods constructe a complex network from the time
series by considering the visibility condition between data points of
underlying data and correlation between sub-time series of the dataset,
respectively, while the TDE method maps the time series into a $D$-dimensional
point cloud data (PCD) by a time-delay parameter, $\tau$. There are also some
undirected methods for converting time series to the complex network, in which
the reconstructed PCD (by TDE method) is converted to the network. The
recurrence network method Gao and Jin (2009); Marwan  _et al._ (2009); Donner
_et al._ (2010) and $k$-nearest neighbor (kNN) network method Shimada  _et
al._ (2008) are the examples. Briefly, in the recurrence network method, the
connections are considered based on the proximity of the embedded data points
(state vectors), while in the kNN method all state vectors (nodes) are
connected with their $k$ nearest neighbors.

Quantifying the scaling exponent of fBm and fGn has been done by various
methods Peng  _et al._ (1994, 1995); Kantelhardt  _et al._ (2002); Jiang  _et
al._ (2019); Eghdami  _et al._ (2018), but the specious effect of complicated
trends, the impact of non-stationarities, the finite size effect and
irregularities have remained as the main challenges in the mentioned
algorithms Hu  _et al._ (2001); Chen  _et al._ (2002); Nagarajan and Kavasseri
(2005); Ma  _et al._ (2010); Eghdami  _et al._ (2018). Subsequently, some
efforts have been devoted to removing or at least tuning down the above
discrepancies from different points of view ranging from network analysis
Lacasa  _et al._ (2009); Ni  _et al._ (2009); Xie and Zhou (2011); Masoomy and
Najafi (2021), reconstructed phase space of fBm series by TDE method in terms
of recurrence network analysis Liu  _et al._ (2014) by taking into account the
false nearest neighbors algorithm Kennel  _et al._ (1992) and mutual
information method Fraser and Swinney (1986), recurrence network of fBms Zou
_et al._ (2015), to performing TDA on the weighted natural visibility graph
constructed from fGn series Masoomy  _et al._ (2021b).

The main purpose of this work is to figure out how the reconstructed phase
space (embedded PCD) of the fBm time series changes by varying the intrinsic
parameter of the signal, i.e. the Hurst exponent, $H$, and the algorithmic
parameters of the TDE method, i.e. the embedding dimension, $D$, and the time-
delay, $\tau$. Furthermore, the influence of irregularity adjusted by a
parameter, $q$, the fraction of the number of missing data points, $T_{\rm
missing}$, in the time series to its total number of data points, $T$, as a
measure for the amount of irregularity of underlying dataset will be examined
in this work. More precisely, we utilize the PH method to get deep insight
into how the state vectors constructed by the TDE method are distributed from
a topological viewpoint. One of the advantages of this idea is that we can
study the global properties of the phase space measured by the population of
the homology groups of the weighted topological space (called weighted
simplicial complex) mapped from the corresponding reconstructed phase space.
And more importantly, we can also capture the evolution of these topological
invariants by varying the proximity parameter (threshold), $\epsilon$,
continuously. Subsequently, we examined the behavior of the topological
measures computed from the $d$th persistent diagram, $\mathcal{D}_{d}$, and
$d$th Betti number, $\beta_{d}$, as a function of the proximity parameter
(called $d$th Betti curve) for $d=0,1$. We also aim to find the optimal choice
for the parameters $D$ and $\tau$ to distinguish the fBms of various $H$ and
also the results would be robust against the irregularity.

Our results indicate that the computed topological measures of embedded fBm
are sensitive to the value of the Hurst exponent. Generally, the statistics of
$d$th homology classes have strong Hurst-dependency. The evolution of
$d$-dimensional topological holes ($d$-holes) occurs in low (high) scales for
fBms with $H\gtrsim 0.5$ ($H\lesssim 0.5$). This $H$-dependency grows by
increasing the dimension of constructed PCD, i.e. for a good estimation for
$H$ one can deal with high-dimensional PCD. The situation is more noticeable
when the time series contains some irregularity. For a signal with a high
value of irregularity, the proposed measures become more $D$-dependent and the
accuracy of estimating the Hurst exponent decays for high-dimensional PCD. In
other words, the topological features extracted from low-dimensional PCDs have
the minimum $q$-dependency, i.e. the features of 2-dimensional reconstructed
phase space are more robust than the $D$-dimensional PCD for $D>2$ in the
presence of irregularity, suggesting the best value $D=2$ for the estimation
of Hurst exponent of irregular fBms.

The rest of this paper is organized as follows: In the next section, our
methodology and pipeline for analysis of the fBm signal are introduced. The
numerical results of synthetic fBm time series, via reconstructed phase-space
distribution of the state vectors from the topological viewpoint, are given in
section III. Summary and concluding remarks will be presented in section IV.

Figure 1: A synthetic regular ($q=0$) fBm time series of length $T=10^{4}$
with $H=0.5$ (top panel) and associated phase space of size $N=9\times 10^{3}$
reconstructed by the TDE method for embedding dimension $D=2$ and time-delay
$\tau=10^{3}$ (bottom panel). Figure 2: Left panel: low-dimensional
simplicies, 0-simplex, 1-simplex, 2-simplex and 3-simplex (sorted from top to
bottom). Right panel: a 3-dimensional simplicial complex containing
$|\Sigma_{0}|=39$ 0-simplicies, $|\Sigma_{1}|=63$ 1-simplicies,
$|\Sigma_{2}|=25$ 2-simplicies, $|\Sigma_{3}|=3$ 3-simplicies and determined
by the Betti numbers $\vec{\beta}=(2,4,0,0)$. The subspaces determined by
different colors illustrate example for a 1-chain (red), 1-cycle (pink),
1-boundary (green) and 1-hole (orange).

Figure 3: The persistent homology (PH) of a random PCD. This figure shows how
the PH method extracts the topological features of a random 2-dimensional PCD
of size $N=15$. Top panel: growing VR simplicial complex by increasing the
proximity parameter $\epsilon$ (from left to right). Through the filtration,
topological properties of the PCD change (connected components represented by
black points and blue lines, while topological loops represented by orange
lines). Bottom panel: the PB (PD in inset) associated with the filtration
shows the persistence (evolution) of extracted topological features by
persistence bars (pairs), filled blue diamond and empty orange square symboles
are for the zeroth and first homology generators, respectively.

##  II Methodology and our Pipeline

For the sake of clarity, we will give a brief about the computational methods
utilized in this paper to assess synthetic fractional Brownian motion (fBm)
signal by paying attention to the mathematical preliminaries. More precisely,
the PH of reconstructed phase space (embedded PCD) from fBm for different
noticeable specifications and our pipeline is explained in this section.

###  II.1 Time Delay Embedding

The reconstruction of phase space from a typical time series has been
implemented to capture the evolution of the deterministic and chaotic
dynamical systems and determine the correlations between associated quantities
Packard  _et al._ (1980); Myers  _et al._ (2019). Time delay embedding (TDE)
is a mathematically well-defined method to map a time series into high-
dimensional Euclidean space to make such a finite-dimensional phase space
Takens (1981). For a given discrete time series, $x=\\{x_{t}\\}_{t=1}^{T}$, of
length $T$, we make a set of $N$ state vectors of dimension $D$ in
$D$-dimensional Euclidean space, $\mathbb{R}^{D}$, which is called
$D$-dimensional PCD, for a given time-delay, $\tau$, as follows:

$X(x,D,\tau)=\Bigr{\\{}\vec{x}_{t}\in\mathbb{R}^{D}~{}\Bigr{|}~{}\vec{x}_{t}\equiv(x_{t},x_{t+\tau},...,x_{t+(D-1)\tau})\Bigr{\\}}_{t=1}^{N}$
(1)

Notice that $X(x,D=1,\tau)=x$ and the PCD size $N=T-(D-1)\tau$. According to
the Takens’ theorem Takens (1981), we can recreate a topologically equivalent
$D$-dimensional phase space from a time series by means of this method. Figure
1 shows a synthetic fBm time series with $H=0.5$ of length $T=10^{4}$ (top
panel) and associated 2-dimensional PCD of size $N=9\times 10^{3}$
reconstructed by the TDE method for $\tau=10^{3}$ (bottom panel). As noticed
before, for any value of $D$ and $\tau$, one can construct PCD from fBm,
$x(H,q,T)$, and therefore, the associated homology groups are examined.

###  II.2 Persistent Homology

Topology is a branch of pure mathematics dealing with the abstract objects
living in high-dimensional topological spaces (e.g. the real line, sphere,
torus, and more complicated spaces) to classify them in terms of their global
properties (e.g. connectedness, genus, the Euler characteristics, etc.)
Nakahara (2003). These global features of the space usually are topologically
invariant, which means that they do not change under continuous deformations
and are not dependent on the way the corresponding space is made
(triangulated, mathematically). By the term continuous deformation, we mean
any continuous changes (elastic motions) of the space like shrinking,
stretching, rotating, reflecting, etc., but not cutting or gloving. These
continuous deformations are defined by the concept of homeomorphism in
algebraic topology Arnold (2011).

Here we introduce the required objects and definitions in homology theory. In
algebraic topology, to determine the topological properties of a space, the
associated space is triangulated by mapping it into a collection of
$d$-dimensional simplicies ($d$-simplicies), called simplicial complex. A
$d$-simplex $\sigma_{d}=[\vec{x}_{0},...,\vec{x}_{d}]$ is a convex-hull subset
of $d$-dimensional Euclidean space, $\mathbb{R}^{d}$, determined by its
$(d+1)$ geometrically independent points $\\{\vec{x}_{0},...,\vec{x}_{d}\\}$
in $\mathbb{R}^{d}$. For instance, a point is a 0-simplex, a line segment is a
1-simplex, a triangle is a 2-simplex, a tetrahedron is a 3-simplex etc (see
the left panel of Fig. 2). A simplicial complex, $\mathcal{K}$, is a
collection of simplices such that any subsimplex of any simplex in the
simplicial complex is in the simplicial complex. Also, any pair of simplices
are either disjoint or they intersect in a lower-dimensional simplex existing
in the simplicial complex. The dimension of a simplicial complex is defined as
the dimension of the largest simplex in it (see the right panel of Fig. 2)
Nakahara (2003). For a simplicial complex, $\mathcal{K}$, containing
$|\Sigma_{d}|$ $d$-simplices, one can create a $|\Sigma_{d}|$-dimensional
vector space, $\mathcal{C}_{d}(\mathcal{K})$, called $d$-chain group of the
simplicial complex $\mathcal{K}$, by the basis considered as the set of all
$d$-simplicies
$\Sigma_{d}(\mathcal{K})=\\{\sigma_{d}^{i}\\}_{i=1}^{|\Sigma_{d}|}$ of the
$\mathcal{K}$ and the vectors, called $d$-chains, as fallows:

$c_{d}\equiv\sum_{i=1}^{|\Sigma_{d}|}a_{i}\sigma_{d}^{i}~{}~{};~{}~{}a_{i}~{}\in~{}\mathbb{Z}_{2}\equiv\\{0,1\\}$
(2)

The boundary of a $d$-simplex is the union of all its $(d-1)$-subsimplices
which is obtained by applying the boundary operator $\partial_{d}$ on the
simplex:

$\partial_{d}(\sigma_{d})=\sum_{i=0}^{d}(-1)^{i}~{}[\vec{x}_{0},...,\vec{x}_{i-1},\vec{x}_{i+1},...,\vec{x}_{d}]$
(3)

The $d$-cycle group
$\mathcal{Z}_{d}(\mathcal{K})\equiv\\{c_{d}\in\mathcal{C}_{d}(\mathcal{K})~{}|~{}\partial_{d}(c_{d})=\emptyset\\}$
of $\mathcal{K}$ is defined the set of all boundaryless $d$-chains of the
$\mathcal{K}$, i.e. any $d$-chain mapped to the empty space (set) is a
$d$-cycle $z_{d}\in\mathcal{Z}_{d}(\mathcal{K})$. Another subspace of
$\mathcal{C}_{d}(\mathcal{K})$ is called $d$-boundary group
$\mathcal{B}_{d}(\mathcal{K})\equiv\\{c_{d}\in\mathcal{C}_{d}(\mathcal{K})~{}|~{}\partial_{d+1}(c_{d+1})=c_{d}\\}$
containing all $d$-chain of $\mathcal{K}$ which is the boundary of a
$(d+1)$-chain in $\mathcal{C}_{d+1}(\mathcal{K})$. The elements of
$\mathcal{B}_{d}(\mathcal{K})$ are called $d$-boundary. Since the boundary
operator satisfies the property
$\partial_{d-1}(\partial_{d}(c_{d}))=\emptyset$ for any
$c_{d}\in\mathcal{C}_{d}(\mathcal{K})$, we can conclude that
$\mathcal{B}_{d}(\mathcal{K})\subseteq\mathcal{Z}_{d}(\mathcal{K})$. In order
to ignore $d$-cycles of the simplicial complex $\mathcal{K}$ that are also
boundary, one can consider a topological equivalence relation on
$\mathcal{Z}_{d}(\mathcal{K})$ such that, any pair of cycles
$z_{d}^{i},z_{d}^{j}\in\mathcal{Z}_{d}(\mathcal{K})$ are equivalent
(homologous), if $z_{d}^{i}-z_{d}^{j}\in\mathcal{B}_{d}(\mathcal{K})$. This
equivalence relation partitions $\mathcal{Z}_{d}(\mathcal{K})$ into a union of
disjoint subsets, called $d$th homology classes. The $d$-homology group of the
simplicial complex $\mathcal{K}$ is defined as
$\mathcal{H}_{d}(\mathcal{K})\equiv\\{[z_{d}]~{}|~{}z_{d}\in\mathcal{Z}_{d}(\mathcal{K})\\}$,
where $[z_{d}]$ represents the homology class of $z_{d}$. The $d$th Betti
number of simplicial complex $\mathcal{K}$, denoted by
$\beta_{d}(\mathcal{K})$, as a topological invariant of $\mathcal{K}$, is the
dimension of $d$-homology group of $\mathcal{K}$. Intuitively,
$\beta_{d}(\mathcal{K})$ indicates the number of $d$-dimensional topological
holes ($d$-holes) of the topological space triangulated by the simplicial
complex $\mathcal{K}$.

Figure 4: The proposed pipeline in this paper: A synthetic fractional Brownian
motion (fBm) time series is converted to a point cloud data (PCD) by the time-
delay embedding (TDE) method. The evolution of homology groups of simplicial
complex mapped from the constructed point cloud data by the Vitoris-Rips (VR)
method is calculated by the persistent homology (PH) technique and summarized
as persistence pairs (PPs) in persistence diagrams (PDs). Finally, topological
measures based on the persistence diagrams and Betti curves are computed.

Figure 5: Upper panels: the PDs of $d$th homology groups (filled blue diamond
for $d=0$ and empty orange square for $d=1$) for 2-dimensional PCD (inset
plots) converted from regular fBm series with $H=0.2$ (left), $H=0.5$ (middle)
and $H=0.8$ (right) by the TDE method for time-delay $\tau=100$. The
population and distribution of PPs varies by $H$ which cause the
$H$-dependency of topological measures. Lower panels: the $D$-dependency of
$0$th (filled blue diamond) and $1$st (empty orange square) PD of regular fbm
with $H=0.5$ embedded to unit $D$-cube ($D=2,3,4$ from left to right) for
$\tau=1000$. The inset plots show the visualization of reconstructed PCD
projected to the standard planes of the space.

Consider a $D$-dimensional PCD
$X=\\{\vec{x}_{i}\in\mathbb{R}^{D}~{}|~{}\vec{x}_{i}\equiv(x_{i}^{\delta})_{\delta=1}^{D}\\}_{i=1}^{N}$
of size $N$ as a finite ($N\neq\infty$) discrete subset of $D$-dimensional
Euclidean space $\mathbb{R}^{D}$. The first step to study this type of dataset
from topological viewpoint, i.e. to understand the topological space
underlying the PCD, is finding a triangulation to tessellate the dataset
Edelsbrunner and Harer (2022). We construct a simplicial complex $\mathcal{K}$
from the PCD $X$. The obvious triangulation of $X$ is creating a simplicial
complex containing only $N$ 0-simplices,
$\mathcal{K}(X)=\\{\sigma_{0}^{i}\equiv[\vec{x}_{i}]~{}|~{}\vec{x}_{i}\in
X\\}_{i=1}^{N}$, with trivial topology
($\beta_{0}(\mathcal{K}(X))=N,\beta_{d}(\mathcal{K}(X))=0$ for $d>0$). To go
beyond this simple structure, one can build the simplicial complex of the PCD
in larger scale, i.e. higher value of proximity of the vectors in $X$. To this
end, for a fixed value of the proximity parameter $\epsilon\geq 0$, we can
create a simplicial complex $\mathcal{K}(X,\epsilon)$ associated with the PCD
$X$, as a collection of simplcies such that any $d$-simplex
$\sigma_{d}=[\vec{x}_{0},...,\vec{x}_{d}]$ in $\mathcal{K}(X,\epsilon)$
corresponds to $(d+1)$ vectors $\\{\vec{x}_{0},...,\vec{x}_{d}\\}$ in $X$ and
the Euclidean distance between any pair of the vectors
$\\{\vec{x}_{0},...,\vec{x}_{d}\\}$ is less than the threshold. The
constructed simplicial complex $\mathcal{K}(X,\epsilon)$ is called Vietoris-
Rips (VR) simplicial complex Zomorodian (2005). After building the VR
simplicial complex $\mathcal{K}(X,\epsilon)$ from the dataset, one can
calculate the topological objects defined before. But the structural
properties of $\mathcal{K}(X,\epsilon)$ are highly dependent on the chosen
scale $\epsilon$. To overcome this issue, the persistent homology (PH) method,
from topological data analysis (TDA), computes the evolution of the extracted
topological invariants ($d$th Betti numbers) by varying the proximity
parameter $\epsilon$, continuously Munch (2017). Precisely, PH builds a
growing sequence of VR simplicial complexes, called filtration, by increasing
the proximity parameter and captures the structural changes of the simplicial
complex in terms of Betti numbers $\beta_{d}(\epsilon)$, called Betti curve.
Therefore, one can get the evolution of any $d$th homology class $[z_{d}]$ in
filtered simplicial complex expressed by persistence pair (PP)
$\epsilon^{[z_{d}]}=(\epsilon_{\rm appear}^{[z_{d}]},\epsilon_{\rm
disappear}^{[z_{d}]})$ and summarize all PPs in a multiset
$\mathcal{D}_{d}=\\{\epsilon_{i}^{[z_{d}]}\\}_{i}$, known as $d$th persistence
diagram (PD), where $\epsilon_{\rm appear}^{[z_{d}]}$ and $\epsilon_{\rm
disappear}^{[z_{d}]}$ are the scales in which the homology class $[z_{d}]$
appears and disappears, respectively. Figure 3 illustrates the mechanism of
homology class extraction from a typical random 2-dimensional PCD of size
$N=15$ by the PH method. The top row shows the filtration process in which by
increasing the proximity parameter (diameter of the gray circles centered by
the state vectors) the topology of the VR simplicial complex varies. The
zeroth (first) homology classes are represented by black points and blue lines
(orange lines). The bottom panel indicates the persistence barcode (PB) and
persistence diagram (PD) (inset plot) of the zeroth (blue bars in PB and
filled blue points in PD) and first (orange bars in PB and orange empty points
in PD) homology classes associated with the filtration.

The number of PPs in $d$th PD is called $n_{d}=|\mathcal{D}_{d}|$ and for
$d=0$, this quantity has trivial value $n_{0}=N$. The reason is correspondence
between the state vectors and PPs in 0th PD. To quantify the distribution of
PPs in $d$th PD, one can calculate the Shannon entropy of the lifespans of the
$d$th homology classes in $\mathcal{D}_{d}$. By the lifespan of a $d$th
homology class $[z_{d}]$, we mean the positive quantity
$\ell^{[z_{d}]}\equiv\epsilon_{\rm disappear}^{[z_{d}]}-\epsilon_{\rm
appear}^{[z_{d}]}$. This measure for the entropy, so-called persistence
entropy (PE), is formulated as follows:

$E_{d}=-\displaystyle\sum_{i=1}^{n_{d}}\frac{\ell_{i}^{[z_{d}]}}{\sum_{i=1}^{n_{d}}\ell_{i}^{[z_{d}]}}~{}\log\left(\frac{\ell_{i}^{[z_{d}]}}{\sum_{i=1}^{n_{d}}\ell_{i}^{[z_{d}]}}\right)$
(4)

Betti curves are common visualization of PDs which indicate that how the
population of homology classes (Betti numbers) evolve through the scale,
$\epsilon$. In fact, the number of $d$-holes of a filtered simplicial complex
for a fixed value of $\epsilon$, denoted by $\beta_{d}(\epsilon)$, is the
number of PPs in $d$th PD for which $\epsilon_{(i)\rm
appear}^{[z_{d}]}\leq\epsilon$ and $\epsilon_{(i)\rm
disappear}^{[z_{d}]}>\epsilon~{},~{}i=1,...,n_{d}$. Therefore, the calculation
of $d$th Betti curve can be written as:

$\beta_{d}(\epsilon)=\sum_{i=1}^{n_{d}}\Theta\left(\epsilon-\epsilon_{(i)\rm
appear}^{[z_{d}]}\right)~{}\Theta\left(\epsilon_{(i)\rm
disappear}^{[z_{d}]}-\epsilon\right)$ (5)

It is possible to define other relevant measures based on the behavior of
Betti curves e.g. $\epsilon_{d}^{\rm appear}$, $\epsilon_{d}^{\rm diappear}$
and $\epsilon_{d}^{\rm maximize}$ which are known as critical scales for $d$th
Betti curve and they read as:

$\epsilon_{d}^{\rm appear}\equiv{\rm
max}\Bigr{(}\epsilon~{}\Bigr{|}~{}\int_{0}^{\epsilon}\beta_{d}(\epsilon^{\prime})~{}{\rm
d}\epsilon^{\prime}=0\Bigr{)}$ (6)

and

$\epsilon_{d}^{\rm diappear}\equiv{\rm
min}\Bigr{(}\epsilon~{}\Bigr{|}~{}\int_{\epsilon}^{+\infty}[\beta_{d}(\epsilon^{\prime})-c_{d}]~{}{\rm
d}\epsilon^{\prime}=0\Bigr{)}$ (7)

where $c_{d=0}=1$ and $c_{d>0}=0$, and

$\epsilon_{d}^{\rm maximize}\equiv{\rm
min}\Bigr{(}\epsilon~{}\Bigr{|}~{}\frac{\partial\beta_{d}(\epsilon^{\prime})}{\partial\epsilon^{\prime}}|_{\epsilon^{\prime}=\epsilon}=0~{},~{}\frac{\partial^{2}\beta_{d}(\epsilon^{\prime})}{\partial\epsilon^{\prime
2}}|_{\epsilon^{\prime}=\epsilon}\leq 0\Bigr{)}$ (8)

According to the structure of VR simplicial complex at $\epsilon=0$, we obtain
$\epsilon_{0}^{\rm appear}=\epsilon_{0}^{\rm maximize}=0$. Furthermore, for
$d=0$ the critical scale $\epsilon_{0}^{\rm disappear}$ separates the
connected regime ($\epsilon\geq\epsilon_{0}^{\rm disappear}$) from the
disconnected regime ($\epsilon<\epsilon_{0}^{\rm disappear}$). For $d=1$, the
critical scales $\epsilon_{1}^{\rm appear}$ and $\epsilon_{1}^{\rm disappear}$
determine the loopless regimes (pre-loopless regime
$\epsilon<\epsilon_{1}^{\rm appear}$ and post-loopless regime
$\epsilon>\epsilon_{1}^{\rm disappear}$) and $\epsilon_{1}^{\rm maximize}$ is
the scale at which the simplicial complex becomes loopful. The integration of
the Betti curves is the last proposed measure:

$B_{d}\equiv\int_{\epsilon_{d}^{\rm appear}}^{\epsilon_{d}^{\rm
disappear}}\beta_{d}(\epsilon^{\prime})~{}{\rm d}\epsilon^{\prime}$ (9)

Figure 6: Upper panels: the zeroth (left) and first (lright) Betti curve of
reconstructed 4-dimensional phase space for time-delay $\tau=1$ from regular
fBms of various Hurst exponent. Lower panels: the Betti curves of
reconstructed phase space of various dimensions by time-delay $\tau=10$ from
regular fBm series with $H=0.1$. These curves indicate the strong
$D$-dependency of topological properties of the embedded PCD.

Now we are interested in analyzing the effect of the parameters $D$, $\tau$
and $q$ on the statistics of the $d$th homology group ($d=0,1$) and the
$H$-dependency of the mentioned nontrivial topological measures, namely
$\epsilon_{0}^{\rm disappear}$, $B_{0}$, $E_{0}$, $\epsilon_{1}^{\rm appear}$,
$\epsilon_{1}^{\rm disappear}$, $\epsilon_{1}^{\rm maximize}$, $B_{1}$,
$E_{1}$ and $n_{1}$ and to look for which ones are more appropriate for
estimating the Hurst exponent of the fBm series in the presence of
irregularity in details in next section.

Figure 7: The $D$-dependency of topological measures introduced in the text.
Here we set the parameters as $\tau=1$ and $q=0$.

###  II.3 Pipeline

Our proposed pipeline includes the following steps (see Fig. 4):

(i) The fBm series $x(H,q,T)$ of distinct Hurst exponent $H$ with some
irregularity $q$ with size $T$ is simulated and embedded to a high-dimensional
Euclidean space by TDE method for embedding dimension $D$ and time-delay
$\tau$ to construct a phase space $X(H,q,N,D,\tau)$ of size $N=T-(D-1)\tau$.

(ii) The reconstructed phase space is mapped to simplicial complex
$\mathcal{K}(H,q,N,D,\tau,\epsilon)$ by the Vietoris-Rips (VR) method for
various scales $\epsilon$.

(iii) PH method is applied to extract the statistics and evolution of
$d$-dimensional topological holes ($d$-holes) of the scale-dependent VR
simplicial complex, then the associated generators are visualized as PPs in
$d$th PD $\mathcal{D}_{d}(H,q,N,D,\tau)$ for zeroth and first homology groups.

(iv) Some topological measures are directly computed from the PDs and some
others can be defined by the criticality of the behavior of $d$th Betti curve
$\beta_{d}(H,q,N,D,\tau,\epsilon)$ calculated from PDs.

It is worth noticing that, we define the normalized quantities
$\tilde{\beta}_{d}\equiv\beta_{d}/N$ and $\eta\equiv N\epsilon$ and use them
to calculate the introduced topological measurements instead of the quantities
$\beta_{d}$ and $\epsilon$ in the rest of this paper.

Figure 8: Upper panel: the 2-dimensional PCD of regular fBm with $H=0.9$ for
time-delay $\tau=10$ (blue), $\tau=100$ (red) and $\tau=1000$ (orange) and
associated PDs (inset plots). Lower panel: the PDs of 2-dimensional PCDs
(insets) for $q=0$ (orange), $q=0.05$ (red) and $q=0.09$ (blue). Here we took
$H=0.1$. In these plots, the associated zeroth and first PDs are represented
by filled blue diamond and empty orange square symboles, respectively.

Figure 9: Upper panels: the evolution of $0$th (left) and $1$st (right) Betti
number of reconstructed 10-dimensional phase space from regular fBm of $H=0.1$
for different time-delay parameter. Middel panels: the number of connected
components (left) and topological loops (right) for $D=2$ and $H=0.1$ as a
function of threshold for various value of irregularity $q$. The lower panel
is the same as middle panel except for $D=10$.

##  III Results

According to our pipeline, at first, we generate synthetic fBm for any given
Hurst exponent based on the Holmgren-Riemann-Liouville fractional integral
Mandelbrot and Van Ness (1968); Kahane (1993); Reed  _et al._ (1995)

$\displaystyle
x(H,t)=\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}\int_{0}^{t}(t-s)^{H-\frac{1}{2}}{\rm
d}x(s)$ (10)

here $\Gamma$ is the Gamma function and ${\rm d}x(s)=x(s+{\rm d}s)-x(s)$ is
the increment of $(1+1)$-dimensional Brownian motion (see Masoomy  _et al._
(2021b) for more details). Then we convert and rescale our mock fBm time
series of various Hurst exponent $H\in(0,1)$ with $\Delta H=0.1$ to the unit
$D$-dimensional cube ($D$-cube) $[0,1]^{D}$, a subspace of $D$-dimensional
Euclidean space, from $D=2$ to $D=10$ ($\Delta D=1$), by using TDE method for
$\tau=1,10,100,1000$ and irregularity value, defined as the fraction of
missing data points in time series to the time series length $q\equiv T_{\rm
missing}/T$, from $q=0$ (regular) to $q=0.09$ by step $\Delta q=0.01$. Notice
that we have valued the length of time series $T$, such that the PCD size
$N=T-(D-1)\tau$ be fixed $N=2^{10}$ and the missing data points are selected
uniformly randomly. In PH part, the proximity parameter varies continuously
from $\epsilon_{\rm min}=0$ to $\epsilon_{\rm max}=0.2\sqrt{D}$, where
$\sqrt{D}$ is the maximum possible distance between any two points in unit
$D$-cube (diameter). All proposed topological measurements in this work are
computed and averaged over $10^{3}$ realizations. For computational part, we
utilize ”Ripser” Python package Tralie  _et al._ (2018).

Now, we are going to evaluate topological measures for reconstructed phase
space (embedded PCD) from generated fBm. We are interested in examining the
effect of relevant parameters, namely $(H,q)$ (so-called intrinsic parameters)
and $(D,\tau)$ (algorithmic parameters) on $d$th homology group ($d=0,1$). The
upper panels of Fig. 5 show the $d$th PD ($d=0$ filled blue diamond and $d=1$
empty orange square) for Hurst exponents of $H=0.2,0.5,0.8$ (from left to
right) for fixed parameters $D=2$, $\tau=100$ and $q=0$. The insets are the
visualization of the reconstructed PCD. The $d$th Betti curves of various $H$
are also shown in the upper panels of Fig. 6. In this part, the parameters are
$D=4$, $\tau=1$ and $q=0$. Mentioned plots reveal how the topological
distribution of state vectors in $D$-cube and consequently the corresponding
PDs and Betti curves change for various values of Hurst exponent of the fBm
series. The amount of memory encoded in the correlation quantity of the fBm
signal impacts the pattern of reconstructed PCD, such that the higher value of
$H$ corresponding to a higher value of correlation in fBm leads to the
continuous trajectory (slow changes) than the lower value of correlation. In
fact, by decreasing $H$, the $N$ state vectors become randomly distributed in
$D$-cube. Almost all the proposed topological measures decrease by increasing
the Hurst exponent for the various value of parameters $D$, $\tau$, and $q$.

Figure 10: The time-delay parameter $\tau$ impacts on the $H$-dependency of
topological measures. Figure 11: The proposed measures as a function of the
Hurst exponent for various amounts of irregularity controlled by $q$.

###  III.1 $D$-dependency

The statistics and evolution of homology classes of PCD are strongly dependent
on the dimension of the Euclidean space as the embedding dimension. The zeroth
(filled blue diamond) and first (empty orange square) PD of a regular ($q=0$)
embedded fBm with $H=0.5$ by time-delay $\tau=1000$ to 2-, 3- and 4-cube is
shown in the lower panels of Fig. 5. The inset plots are the projections of
the reconstructed PCD into the standard 2-dimensional planes
$P_{ij}\equiv\\{c_{i}\hat{e}_{i}+c_{j}\hat{e}_{j}~{}|~{}c_{i},c_{j}\in\mathbb{R}\\}$,
where $\\{\hat{e}_{i}\\}_{i=1}^{D}$ is the standard basis of the space. The
Betti curves for $d=0$ and $d=1$ of various $D$ are also shown in the lower
panels of Fig. 6. Since the typical distance between state vectors increases
by increasing embedding dimension for any given value of Hurst exponent, the
homology groups evolve (appeared, disappeared and maximized) in higher value
of threshold (see also the lower panels of Fig. 5 as a consistent
representation). The $H$-dependency of the measures increases by $D$ as shown
in Fig. 7. This means that the topological differences between the
distribution of state vectors for various $H$ are more significant in higher
dimensions which can be captured by the homology classes. However, the
sensitivity of $E_{0}$ for the fBm signal with $H\lesssim 0.5$ is less than
other defined measures irrespective of the embedding dimension.

The interesting thing about the $D$-dependency of the pattern of state vectors
is that the topological distribution of these vectors for signals with
$H\gtrsim 0.5$ are more robust versus the embedding dimension $D$. It is worth
noting that, the $D$-dependency of the homology groups evolution is
considerable when the embedded time series is irregular (we will discuss in
III.3).

###  III.2 $\tau$-dependency

We consider a special case $\tau$=0 for which the embedded PCD,
$X(x,D,\tau=0)=\\{\vec{x}_{t}=(x_{t},...,x_{t})\\}_{t=1}^{L}$, is a subset of
a line along $\vec{e}\equiv\sum_{i=1}^{D}\hat{e}_{i}$ with trivial topology,
$\tilde{\beta}_{d}(H,q,N,D,\tau=0,\eta)=0$, $d>0$, where
$\\{\hat{e}_{i}\\}_{i=1}^{D}$ is the standard basis of $D$-dimensional
Euclidean space. The upper panel of Fig. 8 illustrates 2-dimensional PCD
reconstructed from highly correlated ($H=0.9$) fBm for various time-delay
values $\tau=10,100,1000$, blue, red and orange, respectively (inset plots)
and associated PDs (filled diamond for $d=0$ and empty square for $d=1$). The
higher value of $\tau$ leads to construct more extended PCD as shown in the
insets plots of upper panel of Fig. 8 and consequently, the amount of loops on
the higher thresholds grows.

The Betti curves of 10-dimensional PCD of fBm with $H=0.1$ for various value
of $\tau$ also are shown in the upper panel of Fig. 9. Figure 10 reveals the
behavior of our proposed topological measures as a function of the Hurst
exponent for various values of $\tau$. Since the rate of the autocorrelation
function decaying for the fBm time series decreases by increasing $H$
suggesting the small values of $\tau$ are more proper for estimating the Hurst
exponent of the fBm series especially for $H\lesssim 0.5$ regime which is
consistent with the statement in Zou  _et al._ (2015) from the network
analysis point of view. For almost $H\lesssim 0.5$ regime, the autocorrelation
of the fBm signal goes down rapidly compared to that for $H\gtrsim 0.5$ and
therefore as represented in different panels of Fig. 10, we find that for
small $\tau$ our measures are as sensitive as enough to estimate Hurst
exponent for the whole range $H\in(0,1)$ of underlying fBm signal. In
addition, our results demonstrate that particularly the $E_{0}$, $E_{1}$ and
$n_{1}$ for large enough $\tau$ and small value of Hurst exponent loose their
$H$-dependency.

###  III.3 $q$-dependency

The lower panel of Fig. 8 shows the zeroth (filled diamond) and first (empty
square) PD of 2-dimensional PCDs (inset plots) converted from irregular ($q=0$
orange, $q=0.05$ red and $q=0.09$ blue) fBms with $H=0.1$ by $\tau=10$. The
zeroth (middle left panel) and first (middle right panel) Betti curve of
2-dimensional PCDs mapped from the fBm signal with $H=0.1$ imposed by various
irregularity $q$ are illustrated in Fig. (9). The lower panels of Fig. (9) are
for 10-dimensional PCDs and the rest parameters are the same as that for
middle panels. This plot reveals the robustness of topological measures when
we consider low-dimensional PCDs against the irregularity in the time series.
To explain this fact, by imposing irregularity to the time series $x$ of
length $T$ makes the time series and embedded PCD $X$ loose $T_{\rm
missing}(q)=qT$ data points and $N_{\rm missing}(D,q)\geq T_{\rm missing}$
state vectors, respectively. The number of missing state vectors $N_{\rm
missing}$ strongly depends on the embedding dimension, such that $N_{\rm
missing}(D=1,q)=T_{\rm missing}$ and $N_{\rm missing}$ increases by $D$.
Precisely, for a given irregular time series with irregularity equates to $q$,
the number of missing data points in the series is $T_{\rm missing}=qT$, and
according to $N=T-(D-1)\tau$, the influence of missing data point in time
series grows for higher value of $D$ in state vectors of $X$. Mentioned effect
can be recognized in the zeroth Betti number of $X$ (see the beginning of
zeroth Betti curve in the middle and lower left panels of Fig. 9) as
$N\tilde{\beta}_{0}(\eta=0)=N(q)\leq T-(D-1)\tau$. This phenomenon affects the
behavior of topological measurements versus the Hurst exponent. Such that some
measures increase by $q$ but some others decrease. This effect dramatically is
high in higher dimensions (see Fig. 11).

##  IV Summary and Concluding remarks

In this work, we studied the topological signatures based on the homology
groups of a point cloud data (PCD) constructed from the synthetic fractional
Brownian motion (fBm) series to classify these series using the corresponding
Hurst exponent. We simulated the mock fBms for different $H$ and converted
them to a $D$-dimensional PCD (a discrete subset of unit $D$-cube) according
to the time delay embedding (TDE) method for various embedding dimensions and
time delays. Then, by using the Vietoris-Rips (VR) method the embedded PCD is
mapped into the simplicial complex for continuously varying proximity
parameter $\epsilon$ and filtered by the persistent homology (PH) technique to
capture the evolution of homology groups of triangulated PCD through the
scales. For the zeroth and first homology group the homology classes are
stored as pairs, so-called persistence pairs (PPs), in the zeroth and first
persistence diagram (PD). Any PP contains appear and disappear scale of the
homology class.

Relying on the population and distribution of PPs we defined the number of PPs
and persistent entropy revealing the global properties of the corresponding
PCD. Also, the zeroth and first Betti curves, $\beta_{d}(\epsilon)$, were
computed to determine some transition scales for connectivity and loop
structures in PCD. As we expected, all topological measures depend on the
parameters ($H,q,D,\tau,\epsilon$) explained in our pipeline (subsection
II.3). We assessed the $H$-dependency of topological measures considered in
this paper.

Our results demonstrated that the $H$-dependency of our measures
($\eta_{0}^{\rm disappear},B_{0},E_{0}$ for zeroth homology group,
$\eta_{1}^{\rm appear},\eta_{1}^{\rm disappear},\eta_{1}^{\rm
maximize},B_{1},E_{1},n_{1}$ for first homology group) grows by increasing the
embedding dimension (Fig. 7). The $D$-dependency goes down for the higher
value of the Hurst exponent since for this range of $H$, the amount of
autocorrelation becomes high for a small time-delay. The time-delay imprint on
the topological measures has been illustrated in Fig. 10 demonstrating that
the small value of $\tau$ is more reliable for determining the corresponding
Hurst exponent in the whole range $H\in(0,1)$ of synthetic fBm.

Motivated by irregularity existed in the realistic dataset in the universality
class of fBm or fGn (Ma  _et al._ (2010); Eghdami  _et al._ (2018)), irregular
mock series was simulated and quantified by a single irregular quantity ($q$).
For the higher value of $H$, we expected the statistical properties of
irregular fBm series remain almost unchanged. Our analysis showed that almost
all topological criteria have weak dependency on $q$, meanwhile the $E_{0}$ ,
$E_{1}$ and $n_{1}$ depended on irregularity existed in data (Fig. 11). This
means that those measures which are more related to the size and ordering of
data are more sensitive to $q$, consequently, we propose that taking into
account $\eta_{0}^{\rm disappear}$, $\eta_{1}^{\rm disappear}$ are more robust
concerning data loss. In addition, the $q$-dependency increases by increasing
the embedding dimension. Also, we find that selecting the lower value of time-
delay makes a proper measure to estimate the value of the Hurst exponent for
various types of fBm irrespective of irregularity.

For regular fBm series, the higher value of $D$, the more sensitive behavior
of topological measures is. While the influence of irregularity would be
magnified for the higher value of the embedding dimension yielding the more
$H$-dependency happens for lower $D$.

The final remarks are as follows: the size dependency of our proposed measures
for any given value of Hurst exponent is important to examine. Incorporating
the embedding approach enables us to evaluate the higher dimensional
topological holes which are nontrivial hidden shapes in the underlying series,
particularly in the absence of irregularity. Both mentioned tasks will be left
for our next research.

## Acknowledgment

The authors are very grateful to B. Askari for his constructive comments. A
part of the numerical simulations were carried out on the computing cluster of
the Brock University.

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