Trend‐cycle Estimation Using Fuzzy Transform and Its Application for Identifying Bull and Bear Phases in Markets

• Author: Vilém Novák,Linh Nguyen,Soheyla Mirshahi
• Published date: 01 July 2020
• Date: 01 July 2020
• DOI: http://doi.org/10.1002/isaf.1473

Received: 19 September 2019Revised: 19 February 2020Accepted:25April 2020

DOI: 10.1002/isaf.1473

RESEARCH ARTICLE

Trend-cycle Estimation Using Fuzzy Transform and Its

Application for Identifying Bull andBear Phases inMarkets

Linh NguyenVilém NovákSoheyla Mirshahi

Institute for Rese arch and Applicatio ns of

Fuzzy Modelling, NSC IT4Innovations,

Universityo f Ostrava,30. dubna 22, 701 03

Ostrava 1, Czech Republic

Correspondence

Linh Nguyen, Ins titute for Research and

Applications ofFuzzy Modelling, N SC

IT4Innovations, U niversity of Ostrava, 30.

dubna 22, 701 03 Ostrava 1, Czech Republic.

Email: linh.nguyen@osu.cz

Funding information

Czech Ministry of Educ ation, Youth and

Sports , Grant/ Award Numbe r:

CZ.02.1.01/0.0/0.0/17_049/0008414;

Grantová Agenturǎ

Ceské Republiky,

Grant/Award Number: 18-13951S

Summary

This paper is focused on one of the fundamental problems in financial time-series

analysis; namely,the identification ofthe historical bull and bear phases.We start with

the proof that the trend-cycle can be well estimated using the technique of a higher

degree fuzzy transform. Then, we suggest a mathematical definition of the bull and

bear phases and provide a novel technique for their identification. As a consequence,

the turning points (i.e. the points where the market changes its phase) are detected.

We illustrate our methodology on several examples.

KEYWORDS

bull and bear phases, decomposition model, financial time series, fuzzy transform, trend-cycle,

market analysis

1INTRODUCTION

Financial crises in the 19th and early 20th centuries put the world

economy in to complicated situatio ns that initiated big interest in e co-

nomic and financial cycles prediction. In thissituation, any information

giving a better understanding of the behaviour of markets, such as

their trends,cyc les,structure of breaks, and so on, can be useful. The

data of m arkets are usual ly represent ed in the form of t ime series. T o

unde rstand the m better, p eople pu rsuing ma rkets traditi onally de com-

pose the corresponding data into several interpretable components

thatprovide essentialinformationaboutthe market. Notethat the idea

of time-series decomposition is not new. Persons (1919) described

four typesoffluctuationsintimeseries; namely,thesecular trend,

cyclical movement, seasonal movement, and residualvariations. These

are nowadays taken as standard compon ents of (financial) time series

and called trend,cycle,seasonalco mponent,andirregular fluctua tions.In

many situations , however, it is dif ficult to distinguish c learly the trend

and cycle, and so they are often joined into one component called

trend-cycle.Hence,the decompositionis reduced into the trend-cycle,

seasonal compo nent, and irregular fluctuations.

There are two main approaches to time-series decomposition: the

model-based approach (e.g. X11-ARIMA model, Dagum, 1980; the

state-space model, Godolphin & Triantafyllopoulos, 2006) and the

non-parametric approach (e.g. seasonal trend decomposition based

on loess (STL) method, Theodosiou,2011; and singular spectrum

analysis; Alexandrov et al. 2008). The estimationofthe trend-cycle is

an essentialissue in the aforementioned methods. A new technique of

fuzzy transform (F-transform)—which can be taken as a non- parametric

technique—has recentlyshown benefit not only in the solution of this

task but also in the forecasting of the time series (Novák et al. 2008;

Novák et al. 2010; Novák et al. 2014; Di Martino et al. 2011;̌

Sť

epnǐ

cka

et al. 2011).

The F-transform is a special techniquefor approximation of func-

tions that was introduced by Perfilieva (2006). The fundamental

conceptofitisthatofafuzzy partition,which is a set offuzzy

sets having a special shape. After the fuzzy partition is specified, the

F-transform has two phases: direct and inverse.Theformertransforms

a given functioninto a vector of componentsmade up of polynomials

defined over the supports of the fuzzy sets forming the fuzzy parti-

tion. The inve rseF -transform provides an approximatio n of the given

function from the components obtained in the direct phase. We dis-

tinguish the F0-transform and the higher degree one (Fm-transform,

m∈N) that was introduced by Perfilievaet al. (2011).

One of the tasks in the the ory of the F-transform is compu tation of

the F-transfo rm components. Re cently, a novel ap proach to this issue

has been introduced using monomial bases that make it possible to

compute the c omponents in a way that is simple rtha n the original one

(seeNguyenet al. 2017; Hoľ

capek & Nguyen, 2018; Hol ̌

capek et al.

Intell Sys A cc Fin Mgmt. 2020;27:111– 124.wileyonlinelibrary.com/journal/isaf© 2020 John Wiley & Sons, Ltd.111