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Forecasting Abrupt Changes in Foreign Exchange Markets: Method Using Dynamical Network Marker

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Forecasting Abrupt Changes in Foreign Exchange Markets: Method Using Dynamical Network Marker Forecasting abrupt changes in foreign exchange markets: method using dynamical network marker Shunsuke Oya1, Kazuyuki Aihara1,2 and Yoshito Hirata1,2 1 Deparment of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan 2 Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan E-mail: [email protected] Received 4 June 2014, revised 10 August 2014 Accepted for publication 2 October 2014 Published 20 November 2014 New Journal of Physics 16 (2014) 115015 doi:10.1088/1367-2630/16/11/115015 Abstract We apply the idea of dynamical network markers (Chen et al 2012 Sci. Rep. 2 342) to foreign exchange markets so that early warning signals can be provided for any abrupt changes. The dynamical network marker constructed achieves a high odds ratio for forecasting these sudden changes. In addition, we also extend the notion of the dynamical network marker by using recurrence plots so that the notion can be applied to delay coordinates and point processes. Thus, the dynamical network marker is useful in a variety of contexts in science, tech- nology, and society. Keywords: early warning signals, econophysics, foreign exchange markets, delay coordinates, recurrence plots 1. Introduction There are many situations where it is desirable to forecast when sudden changes in foreign exchange markets might occur because their changes influence the fundamentals of economies for many countries. Although methods of providing early warning signals for the markets have been proposed [1, 2], these methods have some limitations because they assume that their models remain valid even when we face sudden changes. However, these assumptions are too Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 16 (2014) 115015 1367-2630/14/115015+26$33.00 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft New J. Phys. 16 (2014) 115015 S Oya et al strong because sudden changes might be due to bifurcations that change the qualitative characteristics of the underlying economic systems. Chen et al [3] proposed an index, called a dynamical network marker, for providing early warning signals that forecast qualitative changes accompanying bifurcations. In this paper, we modify the construction of the dynamical network marker to create an index for a dynamical network marker from a single time series. Then we apply the dynamical network marker to a time series of foreign exchange markets. Moreover, we extend the dynamical network marker using recurrence plots [4, 5] so that the dynamical network marker can be applied to a set of delay coordinates and point processes. The rest of this paper is organized in as follows. In section 2, we briefly review the dynamical network marker. In section 3, we modify the construction of the dynamical network marker so that we can apply the index for a dynamical network marker to a single time series. In section 4, we extend the dynamical network marker using recurrence plots with delay coordinates [6, 7] or distances for point processes [8, 9]. In section 5, we discuss the results and conclude this paper. 2. Dynamical network marker The dynamical network marker proposed by Chen et al [3] provides an early warning signal for a change in a network system within which a bifurcation is involved. Let G =(V, E) represent a graph with a set of nodes V and a set of edges E, for which we now consider all-to-all connections between every two nodes of V. We the divide nodes into two groups: Vl ⊂ V is a subset of nodes which we call the leading network; the remaining subset Vol= VV\ are the nodes that are outside of the leading network. If a network system is getting close to a bifurcation point, then the standard deviations for the temporal changes of the nodes in Vl increase; the correlation coefficients between the two nodes in Vl also increase; each correlation coefficient between a node in Vl and another node in Vo decreases to 0. Let σ be the mean for the standard deviations for the temporal changes of the nodes in the leading network. Denote by Cl the mean of the correlation coefficients between the two nodes in Vl. We also use Co for the mean of the correlation coefficients between the nodes in Vl and the nodes in Vo. Then, an index for the dynamical network marker is defined as σC I = l .(1) Co In the paper by Chen et al [3], the authors calculated σ, Cl, and Co for a set of samples for each of the three conditions, normal, pre-disease, and disease states, and showed that I is significantly largest for the pre-disease state, where a transition related to a bifurcation is going on. 3. Modification for a single time series We modify the index I so that we can obtain an index for a dynamical network maker from a single time series generated from a network system, especially foreign exchange markets. Suppose that a g-dimensional time series xs( ) of the observed part of the state is given and that each variable xsm ( ) of time series xs( ) corresponds to a price for a currency against another at 2 New J. Phys. 16 (2014) 115015 S Oya et al time s. In addition, we suppose that there is the unobserved part xs¯ ( ) of the state. Then, the system we consider can be written as ⎫ xs(1)(),+= f() xs xs¯ () + Ws (),⎪ ⎬ (2) xs¯ (1)+= f¯ () xs(), xs¯ () + Ws¯ (),⎭⎪ where Ws( ) and Ws¯ ( ) are dynamical noise, either of which might be zero. We regard the unobserved part xs¯ ( ) as a bifurcation parameter and apply the idea of the dynamical network biomarker. Let w be the window size. We split a time series {xs(): s=… 1,2, ,2 wt }of observables into two: the first part {xs(): s=… 1,2, , wt }is used for defining the leading network that can detect the early warning signals, and the second part {xs(): s=+ wt 1, wt +… 2, ,2 wt }is used for forecasting abrupt changes. Then, we define the size ct of changes for the observables during the time period of ((wt− 1), wt] as ⎧ ⎪ maxxsm ( )− min xsm ( ) ⎨ swt∈−(( 1), wt] swt∈−( ( 2), wt ( − 1)] ct = max max , m ⎪ minxsm ( ) ⎩ swt∈−( ( 2), wt ( − 1)] ⎫ minxsm ( )− max xsm ( ) ⎪ swt∈−(( 1), wt swt wt ] ∈−( ( 2), ( − 1)] ⎬. maxxsm ( ) ⎪ swt∈−( ( 2), wt ( − 1)] ⎭ Given a leading network Vl, the index I is also calculated for the window ((wt− 1), wt], which we denote by IVtl( ). Namely, for a leading network Vl,wedefine IVtl( ) mathematically as follows: σtl()VC ltl, () V IVtl()= , CVot, () l 1 xt¯d ()= ∑ xsd (), w swt∈−(( 1), wt] 1 2 στ(,td )=−∑ ()xxtd () ¯d () , w τ∈−((wt 1), wt] 1 σσtl()V = ∑ (,td ), Vl dV∈ l 11 CVlt, () l = ∑ 2 wtdteσσ(, ) (, ) VVll− (,)de∈× Vll Vd , ≠ e ×−−∑ ()()xxtxxtddee()ττ¯ () () ¯ () , τ∈−((wt 1), wt] 3 New J. Phys. 16 (2014) 115015 S Oya et al and 11 CVot, () l = ∑ VVlo wtdteσσ(, ) (, ) (,)de∈× Vlo V ×−−∑ ()()xxtxxtddee()ττ¯ () () ¯ () . (3) τ∈−((wt 1), wt] Then we use threshold θI(Vl) for IVtl( ) and threshold θc for ct to forecast whether or not the change ct is larger, depending on whether or not IVtl( ) is larger. We restrict θI(Vl) so that it is greater than or equal to the median for the values for the first half of each dataset. We count the numbers of incidences for (IVtl()< θ Il ( V) and ctc+1 < θ ), (IVtl()< θ Il ( V) and ctc+1 ⩾ θ ), (IVtl()⩾ θ Il ( V) and ctc+1 < θ ), and (IVtl()⩾ θ Il ( V) and ctc+1 ⩾ θ ) for the first half of each dataset. Let these numbers be b1,1(VVlIl ,θθ ( ), c ), bVV 1,2 ( lIl , θθ ( ), c ), bVV 2,1 ( lIl , θθ ( ), c ), and b2,2 (,VVlIlθθ (), c), respectively. The leading network is selected in this paper so that the 2 following chi-squared value χθ(,VVlIl (),) θ cfor the first half of the dataset is maximized in terms of parameters θIl(V ) and θc over a set of leading networks which consist of all the sub- networks containing two or three nodes and the edges connecting them: ( pVi ()lIl,,θθ() V c × qVj ()lIl,,θθ() V c × nV()lIl,,θθ() V c 2 2 − bVijlIl, ,,θθ() V c 2 ()) χθ()VVlIl,,() θ c= ∑ , ij,1= ( pVi ()lIl,,θθ() V c × qVj ()lIl,,θθ() V c × nV()lIl,,θθ() V c) 2 ∑bVij, () l,,θθ I() V l c j=1 pVi ()lIl,,θθ() V c= , nV()lIl,,θθ() V c 2 ∑bVij, () l,,θθ I() V l c i=1 qVj ()l = , nV()lIl,,θθ() V c 2 nV()()lIl,,θθ() V c= ∑ bij, V l ,,, θθ I () V l c ij,1= 2 2 χχθθ()VVVl = max()lIl , () , c , (4) θθIl()V , c 4 New J. Phys. 16 (2014) 115015 S Oya et al Figure 1. Time series data for the coupled map lattice. ˆ 2 VVl = arg maxχ ()l , Vl∈×{, VVVVV ×× } ˆ ˆ 2 ˆ θχθθIl()VVV= argmax max (lIl,,,() c ) θθIl()V c ˆ ˆ 2 ˆ θχθθcl()VVV= argmax max (lIl,,,() c ) θθcI()Vl ˆ IIVttl= ().(5) We use the most significant leading network Vˆl and the threshold values θˆIl(Vˆ ) and IVˆcl( ˆ ) for forecasting the abrupt changes that happen in the second part of the time series. We write the chi-squared value obtained for the second part of the time series using Vˆl, θˆIl(Vˆ ) and IVˆcl( ˆ ) by χ 2.
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