Predicting the State of Synchronization of Financial Time Series using Cross Recurrence Plots Mostafa Shabani Martin Magris George Tzagkarakisyz Juho Kanniainenx Alexandros Iosiﬁdis

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Predicting the State of Synchronization of Financial

Time Series using Cross Recurrence Plots

Mostafa Shabani∗, Martin Magris∗, George Tzagkarakis†‡, Juho Kanniainen§, Alexandros Iosiﬁdis∗

∗Department of Electrical and Computer Engineering, Aarhus University, Denmark

†Foundation for Research and Technology - Hellas Institute of Computer Science, Greece

‡IRGO (EA 4190), University of Bordeaux, France

§Department of Computing Sciences, Tampere University, Finland

Abstract—Cross-correlation analysis is a powerful tool for

understanding the mutual dynamics of time series. This study

introduces a new method for predicting the future state of

synchronization of the dynamics of two ﬁnancial time series. To

this end, we use the cross-recurrence plot analysis as a nonlinear

method for quantifying the multidimensional coupling in the

time domain of two time series and for determining their state

of synchronization. We adopt a deep learning framework for

methodologically addressing the prediction of the synchronization

state based on features extracted from dynamically sub-sampled

cross-recurrence plots. We provide extensive experiments on

several stocks, major constituents of the S&P100 index, to

empirically validate our approach. We ﬁnd that the task of

predicting the state of synchronization of two time series is in

general rather difﬁcult, but for certain pairs of stocks attainable

with very satisfactory performance1.

Index Terms—Cross Recurrence Plot, Synchronization, Con-

volutional Neural Network, Financial Time Series

I. INTRODUCTION

Time series prediction and classiﬁcation in ﬁnance is signif-

icantly challenging due to the complexity, multivariate nature,

and non-stationary nature of time series in this domain [Mur-

phy, 1999]. Security trading and price dynamics in ﬁnancial

markets are particularly complex due to the interacting nature

and inter-connectedness of their underlying driving forces and

determinants leading to signiﬁcant co-movements in stocks’

prices. The characterization and modeling of multivariate time

series dynamics have long been discussed in the ﬁnancial liter-

ature, where the prevailing approach is that based on classical

econometric theory. Among the multivariate linear models,

the most widespread ones are vector autoregressive (VAR)

models [L¨

utkepohl, 1999], [L¨

utkepohl, 1991], vector moving

averages and ARMA (autoregressive moving average) models

[Reinsel, 1993] and cointegrated VAR models [Juselius, 2006].

Widespread is the use of multivariate conditional heteroskedas-

ticity GARCH-type, see e.g. [Bauwens et al., 2006] for a

review, multivariate stochastic volatility models [Harvey et al.,

1994], and more methods based on the realized volatility

[Chiriac and Voev, 2011, e.g.].

Among the non-linear models the threshold autoregressive

model [Tong, 1978], smooth transition autoregressive [Dijk

et al., 2002] and Markov switching models [Krolzig, 2013]

1Paper submitted to and under consideration at Pattern and Recognition

Letters.

are nowadays standard approaches. Alternatives include non-

parametric methods, functional coefﬁcient [Chen and Tsay,

1993a] and nonlinear additive AR models [Chen and Tsay,

1993b], recurrence analysis, and neural networks. The com-

plexity of modern ﬁnancial markets running over the so-called

limit-order book mechanism is, however, characterized by

typical non-linear, noisy, and often non-stationary dynamics. In

addition, the high-dimensional nature of the limit-order book

ﬂow and complexity of the interactions within it constitute

severe limits in the applicability of classic econometric meth-

ods for its modeling and forecasting. Besides a very limited

number of analytical and tractable models for the order ﬂow

and price dynamics in limit-order books [Cont et al., 2010],

[Huang and Kercheval, 2012], [Hawkes, 2018, e.g.], machine

learning methods have received much attention [Heaton et al.,

2017], [Dixon et al., 2020], as they are naturally appealing in

this context.

By considering the stock market as a complex system,

it is natural to apply such methods for addressing those

prediction problems where the application setting and assump-

tions beneath standard econometric techniques are stringent

or inadequate. Indeed, it has extensively been shown that, in

ﬁnancial applications, deep learning (DL) models are often

capable of outperforming traditional approaches due to their

ability to learn complex data representations based on end-to-

end data-driven training, see e.g. [Sezer et al., 2017], [Zhang

et al., 2019], [Tran et al., 2019], [Passalis et al., 2020],

[Shabani et al., 2022a], [Haselbeck et al., 2022]. DL models

have been adopted for a variety of problems ranging from

price prediction [Khare et al., 2017], [Fons et al., 2021],

[Bhandari et al., 2022], [Basher and Sadorsky, 2022], [Xu

and Zhang, 2021], limit-order book-based mid-price prediction

[Zhang et al., 2019], [Tran et al., 2019], [Shabani et al.,

2022a], [Shabani and Iosiﬁdis, 2020], [Shabani et al., 2022b],

and volatility prediction [Kyoung-Sook and Hongjoong, 2019],

[Liu, 2019], [Christensen et al., 2021].

Whereas the target of the above literature is generally the

analysis and prediction of single time series, this paper focuses

on an analysis of stock pair co-movements. Several trading

strategies can be put into play to take advantage of co-

movements and exploit statistical arbitrages, including pair-

trading, portfolio management, or relative and convergence

trading strategies applied at an intraday level, e.g., see [Guo

arXiv:2210.14605v2 [q-fin.ST] 2 Nov 2022

et al., 2017] for an overview. While DL provides a basis

for prediction given a set of descriptive features, the issue

of how to detect and quantify co-movements remains to be

addressed. This paper suggests the use of recurrence analysis

based on Cross Recurrence Plots (CRP) for detecting and

extracting features indicative of stocks’ shared dynamics or

co-movements, along with a deep learning framework for

predicting whether certain pairs of stocks will exhibit a shared

dynamics in the future (in the sense speciﬁed in Section

II). Not only in the view of extending the ML and applied

econometrics literature in this direction, but the possibility of

forecasting epochs of time series synchronization is likewise

relevant for practitioners.

For detecting and quantifying co-movements or more gen-

erally shared dynamical features in time series, the standard

econometric approach is that of cross-correlation analysis,

e.g., [Tsay, 2005, Chapter 8]. This intuitive linear approach,

based on the estimation and perhaps forecasting of cross-

correlation matrices, appears to be an element of a much wider

theory and methodological approach that has been explored

and developed in the last years within a broader generic non-

ﬁnancial setting. Simple cross-correlation analysis has been

remarkably extended and generalized towards methods that

help explore co-movements between time series within non-

linear, noisy, and non-stationary systems of very complex

dynamics, either ﬁnancial [Ma et al., 2013], [Bonanno et al.,

2001], [Ramchand and Susmel, 1998] or not [Webber Jr. and

Zbilut, 1994], [Marwan and Kurths, 2002], [Lancia et al.,

2014].

Recurrent analysis [Webber and Marwan, 2015] explores the

reconstruction of a phase-space using time-delay embedding

for quantifying characteristics of nonlinear patterns in a time

series over time [Takens, 1981]. This is done by calculating

the so-called Recurrence Plot [Eckmann et al., 1995], the core

concept of which is to identify all points in time that the

phase-space trajectory of a single time series visits roughly

the same area in the phase-space. Recurrence plot analysis has

no assumptions or limitations on dimensionality, distribution,

stationarity, and size of data [Webber and Marwan, 2015].

These characteristics make it suitable for multidimensional

and non-stationary ﬁnancial time series data analysis. The

CRP [Marwan, 1999] is an extension of the recurrence plot,

introduced to analyze the co-movement and synchronization

of two different time series. The CRP indicates points in

time that a time series visits the state of another time series,

with possibly different lengths in the same phase-space. These

concepts are discussed in further detail in Section II.

In this paper, we propose a method for predicting the state of

synchronization over time of two multidimensional 2ﬁnancial

time series based on their CRP. In particular, we use the

2Throughout the paper, with uni- or multi- variate we refer to the nature of

the analyses (RP as opposed to CRP), and with one- or multi- dimensional we

refer to the nature of the time series. That is, the RP (as presented in equation

(2)) provides a univariate analysis of a single one-dimensional time series,

while the CRP (as presented in latter equation (3)) a multivariate analysis of

two one-dimensional or multi-dimensional time series.

CRP to quantify the co-movements and extract the binary

representation of its diagonal elements as the prediction targets

for a DL model. For predicting the state of synchronization

at the next epoch we employ a Convolutional Neural Network

(CNN) that uses as inputs CRPs independently calculated from

data-crops obtained by applying ﬁxed-size sliding windows on

the time series. Our extensive experiments on 12 stocks of

the S&P100 index selected from different sectors show that

the proposed method can predict the synchronization of stock

pairs with satisfactory performance.

The remainder of the paper is organized as follows. Section

II introduces in detail the concepts and theory behind the CRP,

with an outlook on its applications in ﬁnancial and economic

problems. Our proposed approach for predicting time instances

of time series’ synchronization is presented in Section III.

Empirical results on real market data are provided in Section

IV, whilst Section V provides conclusions.

II. FINANCIAL TIME SERIES RECURRENCE ANALYSIS

Recurrence in the analysis of time series, seen as a nonlinear

dynamic system, is the repetition of a pattern over time. The

visualization of recurrences in the dynamics of a time series

can be expressed via a RP or recurrence matrix [Webber

and Marwan, 2015]. In other words, the RP represents the

recurrence of the phase-space trajectory to a state. The phase-

space of a d-dimensional time series Nwith Tobservations

N={n>

1,n>

2,...,n>

T}>, with nibeing the row-vector

representing a generic observation at time i,i= 1, . . . , T is

calculated using the time-delay embedding method. State Ni

in the phase-space is obtained by

Ni= [ni,ni+τ,...,ni+(k−1)τ], i = 1, . . . , T 0, (1)

where τdenotes the delay and kis the embedding dimension,

T0=T−τ(k−1),τand kcan, respectively, be determined

with the Average Mutual Information Function (AMI) method

[Fraser and Swinney, 1986] and the False Nearest Neighbors

(FNN) method of [Kennel et al., 1992]. For a uni-dimensional

time series Niis a row vector of size (1 ×k), for a d-

dimensional times series Niis a row vector of size (1 ×kd).

The recurrence state matrix of the reconstructed phase-space,

known as Recurrence Plot (RP), at times iand j, is deﬁned

Ri,j (ε) = H(ε− kNi−Njk), i, j = 1, . . . , T 0, (2)

where εis a threshold distance value, H(·)is the Heaviside

function, and k·k is the euclidean distance. Due to the

underlying embedding (1), Ri,j is deﬁned for i i= 1 up

to T0=T−τ(k−1). If two states Niand Njare in an

ε-neighbourhood the value of Ri,j is equal to 1, otherwise is

0.The value of εhighly affects the output of RP. When εis too

small or too large, the RP cannot identify the true recurrence

of states. There are different approaches for ﬁnding the best

value for εin the literature [Webber and Marwan, 2015]. We

follow the guidelines provided in [Schinkel et al., 2008] for

selecting ε. The values on the diagonal line of RP are equal

to one (i.e., Ri,i = 1) because in that case the two states

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PredictingtheStateofSynchronizationofFinancialTimeSeriesusingCrossRecurrencePlotsMostafaShabani,MartinMagris,GeorgeTzagkarakisyz,JuhoKanniainenx,AlexandrosIosidisDepartmentofElectricalandComputerEngineering,AarhusUniversity,DenmarkyFoundationforResearchandTechnology-HellasInstituteofComputerSci...

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Predicting the State of Synchronization of Financial Time Series using Cross Recurrence Plots Mostafa Shabani Martin Magris George Tzagkarakisyz Juho Kanniainenx Alexandros Iosiﬁdis

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