Forecasting Financial Indices: The Baltic Dry Indices

Transcription

1 International Journal of Maritie, Trade & Econoic Issues pp Volue I, Issue (1), 2013 Forecasting Financial Indices: The Baltic Dry Indices Eleftherios I. Thalassinos 1, Mike P. Hanias 2, Panayiotis G. Curtis 3, John E. Thalassinos 4 Abstract: The ain ai of this paper is to use chaos ethodology in an attept to predict the Baltic Dry Indices (BDI, BCI, BPI) using the invariant paraeters of the reconstructed strange attractor that governs the syste s evolution. This is the result of the new eerging field in econo-physics which ainly consists of autonoous physic-atheatical odels that have been already applied to financial analysis. The proposed ethodology is estiating the optial delay tie and the iniu ebedding diension with the ethod of False Nearest Neighbors (FNN). Monitoring the trajectories of the corresponding strange attractor we achieved a 30, 60, 90 and 120 tie steps out of saple prediction. Key Words: Chaos ethodology, False Nearest Neighbors, Baltic Dry Indices JEL Classification: C02, C14, C22 1 Corresponding Author: Professor, University of Piraeus. Departent of Maritie Studies, e-ail: thalassi@unipi.gr, thalassinos@ersj.eu. 2 TEI Chalkidas, Athens,Greece, e-ail: hanias@teihal.gr 3 TEI Chalkidas, Athens, Greece, e-ail: pcurtis@teihal.gr 4 PhD University of Piraeus, Piraeus, Greece

2 110 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos 1. Introduction Predicting Baltic Dry Indices is possible applying algoriths used in physical sciences. This article cobines finance and non linear ethods fro chaos theory to exaine the predictability of the Baltic Dry Indices. The proposed ethodology applies non linear tie series analysis for the BCI, BDI and BPI indices covering the period fro until In particular the ethod of the False Nearest Neighbor (FNN) is used in the first step to evaluate the invariant paraeter of the syste as the iniu ebedding diension. In a second stage using the reconstructed state space the article achieves an out of saple ulti step tie series prediction. The ethodology is very dynaic copared to traditional ones (Thalassinos et al., 2009). The ost iportant property of the proposed ethodology is the fact that there is no need to know fro trial to error soe constants to fit and extrapolate the tie series. The ethodology calculates the invariant paraeters of the syste itself. The fact that shipping indices are sensitive in irregular shocks and crises which are innate eleents in chaotic systes, their predictability is uch easier using chaotic ethodology than any other forecasting ethod. Other benefits as they are pointed out in Thalassinos et al., (2009) are: The possibility to extract inforation about a coplex dynaic syste, which generates several observed tie series by using only one of the. To analyze the iage-syste with the sae topology that preserves the ain characteristics of the genuine syste. The article cobines aritie tie series daily data for the period of to (total nuber of observations 2000) with chaos ethodology to exaine the predictability of the three Baltic Dry Indices as the ost characteristic indices in the aritie industry naely the BCI, BDI and BPI. 2. BCI Tie Series According to the theory of observed chaotic data (Abarbanel, 1996; Sprott, 2003; Hanias et al., 2007a, 2007b; Thalassinos et al., 2009) any non-linear tie series can be presented as a set of signals x=x(t) as it shown at Figure 1. The sapling rate is Δt=1 day and the nuber of data N=2000. As it is presented in Figure 1 the corresponding index had a significant nuber of structural shifts in the period under study with a ax of 16,000 and a in of 1,000 points. Fro the tie series plots of the BCI in Figure 1, 1700 data points has been selected as the training data set, in other words the data that are used for the state space reconstruction and the other 300 data points as the test data set for out of saple period prediction.

3 Forecasting Financial Indices: The Baltic Dry Indices 111 Figure 1: Tie Series plots of the BCI BCI State space reconstruction Tie units (days) 3.1 Tie delay τ X r Fro the given data a vector i, i=1, 2, N, where N=1700, in the order diensional phase space given by the following relation (Kantz et al., 1997; Takens, 1981; Hanias et al., 2007a, 2007b; Thalassinos et al., 2009) is constructed as shown in equation (1): X r i = {xi, xi-τ, xi-2τ,., xi+(-1)τ} (1) X r where i represents a point in the diensional phase space in which the attractor is ebedded each tie, where τ is the tie delay τ = iδt. The eleent xi represents a value of the exained scalar tie series in tie, corresponding to the i-th coponent of the tie series. By using this ethod the phase space reconstruction to the proble of proper deterining suitable values of values of and τ is constructed. The next step is to find the tie delay (τ) and the ebedding diension () without using any other inforation apart fro the historical values of the indices. Equation (2) is used to calculate the tie delay by using the tie-delayed utual inforation proposed by Fraser et al., 1986 and Abarbanel, p( xi, xi+ τ ) I( τ ) = p( x ) i, xi+ τ log 2 ( ) ( ) x x p x + τ i p x i, i i+ τ (2) In equation (2), p(xi) is the probability of value xi and p(xi, xi+τ) denotes joint probability. I(τ) shows the inforation (in bits) being extracted fro the value in tie xi about the value in tie xi+ττ. The function I can be thought of as a nonlinear generalization of the autocorrelation function. For a rando process x(t) the I(t), is a

4 I 112 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos easure of the inforation about x(t+τ) contained in x(t). The first nadir of I(τ) gives the delay, tau0, such that x(t+tau0) adds axial inforation to that already known fro x(t). This tau0 is returned as an estiate of the proper tie lag for a delay ebedding of the given tie series. The tie delay is calculated as the first iniu of the utual inforation (Kantz et al., 1997; Fraser et al., 1986; Casdagli et al., 1991). Mutual inforation against the tie delays (with a iniu at 54 tie steps) for our tie series is presented in Figure 2. Figure 2: Mutual inforation I vs. tie delay τ Ebedding diension After obtaining the satisfactory value of τ, the ebedding diension is to be deterined in order to finish the phase space reconstruction. For this purpose the ethod of False Nearest Neighbors (Kantz et al., 1997 Kennel et al., 1992) is used. More specifically, the ethod is based on a fact that when ebedding diension is too low, the trajectory in the phase space will cross itself. If we are able to detect these crossings, we ay decide whether the used is large enough for correct reconstruction of the original phase space, when no intersections occur or not. When intersections are present for a given, the ebedding diension is too low and we have to increase it at least by one. Then, we test the eventual presence of selfcrossings again (Kennel et al., 1992; Abarbanel 1996). The practical realization of the described ethod is based on testing of the neighboring points in the - diensional phase space. Typically, we take a certain aount of points in the phase space and find the nearest neighbor to each of the. Then we copute distances for all these pairs and also their distances in the (+1) diensional phase space. The rate of these distances is given by equation (3) as follows: y i ( + 1) y n( i) ( + 1) P = y i ( ) y n( i) ( ) (3) where yi() represents the reconstructed vector, belonging to the i-th point in the - diensional phase space and index the n(i) denotes the nearest neighbour to the i-th point. If P is greater than soe value Pax, we call this pair of points False Nearest Neighbors (i.e. neighbors, which arise fro trajectory self-intersection and not fro τ

5 Forecasting Financial Indices: The Baltic Dry Indices 113 the closeness in the original phase space). In the ideal case, when the nuber of false neighbors falls to zero, then the value of is found. For this purpose we copute the rate of false nearest neighbours in the reconstructed phase space using the forula in equation (4): xi+ τ xn( i) + τ RA (4) where RA is the radius of the attractor, N 1 R A = x i x ( 5 ) N i = 1 and N 1 x = xi N i= 1 (6) is the average value of tie series. P P When the following criterion ax, (7) is satisfied then it can be used to distinguish between true and false neighbours (Kennel et al., 1992). The diension is found when the percent of false nearest neighbors decreases below soe liit, (Kugiutzis et al., 1994), so we choose Pax=10 as used typically. Before we apply the above procedure, we ust deterine the Theiler window for out tie series and exclude all pairs of points in the original series which are teporally correlated and are closer than this value of Theiler window. For this purpose we produce space tie separation plots (Kantz et al., 1997) as shown in Figure 3(a). Figure 3: Space tie separation plot Spatial Separation p = 1 p = 0.8 p = 0.6 p = 0.4 p = 0.2 Density Median Spatial Separation Median Spatial Separation olag = 53 to Orbital Lag Orbital Lag (a) (b) (a) Density function estiate of the edian contour (upper graph) in addition to a suggested range of suitable orbital lags. (b) The ost populous values of the edian contour are highlighted by a cross-hatched area that covers a plot of the edian curve (lower).

6 114 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos The space tie separation plots where produced with a desired ebedding diension 5, τ=54, and the probability associated with the first contour equal to 0.2 In Figure 3 we had plot a suary of the space-tie contours including a density function estiate of the edian contour in addition to a suggested range of suitable orbital lags. In the latter case, the ost populous values of the edian contour are highlighted by a cross-hatched area that covers a plot of the edian curve. The suggested range for a suitable orbital lag is based on the range of values that first escape this cross-hatched region, so we can choose a Theiler window fro 53 to 117. In our case we choose the Theiler window to be 65. Then we used atlab code to calculate the quantity P. The rate of false neighbors that is under the above liit Pax=10 is achieved for = 4, thus this value should be suitable for the purpose of phase space reconstruction. This is shown in Figure 4 for our tie series. Figure 4: Percent of False Nearest Neighbors nuber vs FNN% Tie Series Prediction of BCI The next step is to predict the evolution of BCI by coputing weighted average of evolution of close neighbors of the predicted state in the reconstructed phase space (Miksovsky et al., 2007; Sta et al., 1998). The reconstructed -diensional signal projected into the state space can exhibit a range of trajectories, soe of which have structures or patterns that can be used for syste prediction and odeling. Essentially, in order to predict k steps into the future fro the last -diensional vector point } N }, we have to find all the nearest neighbors NN in the B ( ε-neighborhood of this point. To be ore specific, let s set ε xn ) to be the set of points within ε of } N B ( ) (i.e. the ε-ball). Thus any point in ε xn is closer to

7 Forecasting Financial Indices: The Baltic Dry Indices 115 the } N than ε. All these points NN } coe fro the previous trajectories of the syste and hence we can follow their evolution k-steps into the future NN + k}. The final prediction for the point } N is obtained by averaging over all neighbors projections k-steps into the future. The algorith can be written as in equation (8) as follows: 1 N + k } = x NN + k B ( x ) NN X NN B ( xnn ) (8) B ( x NN ) where denotes the nuber of nearest neighbors in the neighborhood of the point } N ( Kantz et al., 1997). As an exaple we suppose that we want tο predict k=2 steps ahead. The basic principle of the prediction odel is visualized in Figure 5. The blue dot N} represents the last known saple, fro which we want to predict one and two steps into the future. The blue circles represent ε- neighborhoods in which three nearest neighbors were found. Figure 5: Basic prediction principle of the siple deterinistic odel } N N + 2 } N 2 } 1} N N } { } 1 x N + The next step in the algorith is to check that the projections, one and two steps into the past, of the points in NN } are also nearest neighbors of the two previous

8 116 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos { readings } xn 1 and N 2} respectively. This criterion excludes unrelated trajectories that enter and leave the ε-neighborhood of N} but do not track back to ε- neighborhoods of } N 1 and N 2}, thus aking the unsuitable for prediction. Assuing that any nearest neighbors have been found and checked using the criterion detailed previously, we project their trajectories into the future and average the to get results for } N + 1 and N + 2}. We used the values of τ and fro our previous analysis so the appropriate tie delay τ was chosen to be τ=54. We use as ebedding diension the 2* = 8 (Sprott J. C) and for the optiu nuber of nearest neighbors we used the value of ebedding diension = 4. These values of ebedding diension and nuber of nearest neighbors gave the better results for k=30 tie steps ahead. We apply the procedure for in saple forecasting until data point 1700 as shown at Figure 6 then we applied the procedure for out of saple prediction fro data point 1700 to data point 1730 as shown in Figure 7. Figure 6: and in saple predicted tie series for k=30 days ahead Predicted k=30 days ahead 6000 BCI Tie steps (days)

9 Forecasting Financial Indices: The Baltic Dry Indices 117 Figure7: and out of saple predicted tie series for k=30 days ahead Predicted k=30 days ahead BCI Tie steps (days) and predicted tie series for k=60, 90, 120 tie steps ahead are presented in Figures 8, 9, 10 respectively for out of saple period. Figure 8: and out of saple predicted tie series for k=60 days ahead Predicted k=60 days ahead BCI Tie steps (days)

10 118 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Figure 9: and out of saple predicted tie series for k=90 tie steps ahead Predicted k=90 days ahead BCI Tie steps (days) Figure 10: and out of saple predicted tie series for k=120 tie steps ahead BCI Predicted k=120 days ahead Tie Index (days)

11 Forecasting Financial Indices: The Baltic Dry Indices 119 The prediction error for establishing the quality of the fit was chosen to be the classical root ean square error (RMSE) and found to be 8.99x10-2, 9.54x10-2, 2.17x10-1, 2.26x10-1 for k=30, 60, 90, 120 respectively. 5. Tie Series Prediction of BDI The BDI tie series is presented as a signal x=x(t) as it shown in Figure 11. It covers data fro to The sapling rate was Δt=1 day and the nuber of data are N=2000. Figure 11: BDI Tie Series BDI Tie units (days) Fro this tie series we have choose 1700 data as the training data set, in other words the data that we used for the state space reconstruction and the other 300 data as the test data set for our out of saple prediction. We used the sae procedure as before and we had estiated the delay tie, Theiler window and ebedding diension. Mutual inforation against the tie delays (with a iniu at 55 tie steps) for BDI tie series is presented in Figure 12.

12 I 120 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Figure 12: Mutual inforation I vs. tie delay τ The space tie separation plot and the density function estiation plots are shown in Figure 13. Fro this plot we estiate the Theiler window to be 100. Figure 13: Space tie separation plot τ Spatial Separation p = 1 p = 0.8 p = 0.6 p = 0.4 p = 0.2 Density Median Spatial Separation Median Spatial Separation olag = 48 to Orbital Lag Orbital Lag (a) (b) (a) Density function estiate of the edian contour (upper graph) in addition to a suggested range of suitable orbital lags. (b) The ost populous values of the edian contour are highlighted by a cross-hatched area that covers a plot of the edian curve (lower). The percent of false nearest neighbors nuber FNN vs. is shown in Figure 14. Fro this it is clear that the ebedding diension is 4.

13 Forecasting Financial Indices: The Baltic Dry Indices 121 Figure 14: Percent of false nearest neighbors nuber FNN vs FNN% The next step is to predict 30, 60, 90 and 120 tie steps ahead. Figures 15, 16, 17, 18 and 19 are shown the in saple and out of saple period predictions. Figure 15: and in saple predicted tie series for k=30 days ahead Predicted k=30 days ahead 8000 BDI Tie units (days)

14 122 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Figure 16: and out of saple predicted tie series for k=30 days ahead Predicted k=30 days ahead BDI Tie units (days) Figure 17: and out of saple predicted tie series for k=60 days ahead Predicted k=60 days ahead BDI Tie units (days)

15 Forecasting Financial Indices: The Baltic Dry Indices 123 Figure 18: and out of saple predicted tie series for k=90 tie steps ahead Predicted k=90 days ahead BDI Tie units (days) Figure 19: and out of saple predicted tie series for k=120 tie steps ahead Predicted k=120 days ahead BDI Tie units(days) The prediction error for establishing the quality of the fit was chosen to be the classical root ean square error (RMSE) and found to be 5.68x10-2, 5.99x10-2, 6.53x10-2, 6.80x10-2 for k=30, 60, 90 and 120 respectively.

16 124 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos 5. Tie Series Prediction of BPI The BPI tie series is presented as a signal x=x(t) as it shown in Figure 20. It covers data fro to The sapling rate was Δt=1 day and the nuber of data are N=2000. Figure 20: BPI tie series BPI Tie units (days) Fro this tie series we have choose 1700 data as the training data set, in other words the data that we used for the state space reconstruction and the other 300 data as the test data set for our out of saple prediction. We used the sae procedure as before and we have estiated the delay tie, Theiler window and ebedding diension. Mutual inforation against the tie delays (with a iniu at 55 tie steps) for BDI tie series is presented in Figure 21.

17 I Forecasting Financial Indices: The Baltic Dry Indices 125 Figure 21: Mutual inforation I vs. tie delay τ The space tie separation plot and density function estiation plots are shown in Figure 22. Fro this plot we estiate the Theiler window to be 100. Figure 22: Space tie separation plot τ Spatial Separation p = 1 p = 0.8 p = 0.6 p = 0.4 p = 0.2 Density Median Spatial Separation Median Spatial Separation olag = 59 to Orbital Lag Orbital Lag (a) (b) (a) Density function estiate of the edian contour (upper graph) in addition to a suggested range of suitable orbital lags. (b) The ost populous values of the edian contour are highlighted by a cross-hatched area that covers a plot of the edian curve (lower). The percent of false nearest neighbors nuber FNN vs. is shown in Figure 23. Fro this fig it s clear that the ebedding diension is 4.

18 126 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Figure 23: Percent of false nearest neighbors nuber FNN vs FNN% The next step is to predict 30, 60, 90 and 120 tie steps ahead. Figures 24, 25, 26, 27 and 28 are shown the in saple and out of saple period predictions. Figure 24: and in saple predicted tie series for k=30 days ahead Predicted k=30 days ahead 6000 BPI Tie units (days)

19 Forecasting Financial Indices: The Baltic Dry Indices 127 Figure 25: and out of saple predicted tie series for k=30 days ahead Predicted k=30 days ahead BPI Tie units (days) Figure 26: and out of saple predicted tie series for k=60 days ahead Predicted k=60 days ahead BPI Tie units (days)

20 128 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Figure 27: and out of saple predicted tie series for k=90 days ahead Predicted k=90 days ahead BPI Tie units (days) Figure 28: and out of saple predicted tie series for k=120 days ahead. BPI Predicted k=120 days ahead Tie units (days)

21 Forecasting Financial Indices: The Baltic Dry Indices 129 The prediction error for establishing the quality of the fit was chosen to be the classical root ean square error (RMSE) and found to be 7.10x10-2, 7.56x10-2, 1.24x10-1, 2.36x10-1 for k=30, 60, 90 and 120 respectively. 6. Conclusion In this paper chaotic analysis has been used to predict Baltic Dry Indices tie series. After estiating the iniu ebedding diension, the proposed ethodology has pointed out that the syste is characterized as a high diension chaotic. Fro reconstruction of the systes strange attractors, it has been achieved a 30, 60, 90 and 120 out of saple tie steps prediction. In Table 1 the RMS values of the corresponding prediction are shown. Table 1: RMS values RMS 30 DAYS 60 DAYS 90 DAYS 120 DAYS BDI 5.68x x x x10-2 BCI 8.99x x x x10-1 BPI 7.10x x x x10-1 It is clear that the BDI has the iniu error and gives the best prediction copared to the other two indices. The prediction power of the suggested ethod is liited by the properties of the original syste and the series alone. Because the series represents the only source of inforation for the syste it is iportant to be as long as possible. Too short tie series will not worsen the prediction only but it ay also ake the proper reconstruction of the phase space ipossible. Another iportant liitation factor is the eventual containation by noise evidently, the ore noise is included in the series, the less accurate result can be gained. However the prediction horizon of 120 days would be useful for econoical analysis as it can reveal the trend in a very good anner. References Abarbanel, H.D.I. (1996), Analysis of Observed Chaotic Data, Springer, New York. Casdagli, M., Eubank, S., Farer, J.D. and Gibson, J. (1991), State Space Reconstruction in the Presence of Noise, Physica D, 51, pp Fraser, A.M., Swinney, H.L. (1986), Independent coordinates for strange attractors fro utual inforation, Physics Review A, pp. 33, Hanias, M.P., Curtis, G.P. and Thalassinos, E.J. (2007), Non-Linear Dynaics and Chaos: The Case of the Price Indicator at the Athens Stock Exchange, International Research Journal of Finance and Econoics, Issue 11, pp Hanias, M.P., Curtis, G.P. and Thalassinos, E.J. (2007), Prediction with Neural Networks: The Athens Stock Exchange Price Indicator, European Journal of Econoics, Finance And Adinistrative Sciences, Issue 9.

22 130 International Journal of Maritie, Trade & Econoic Issues, I (1) 2013 Eleftherios Thalassinos Mike Hanias Panayiotis Curtis John Thalassinos Hanias, M.P., Karras, A.D. (2007), Efficient Non Linear Tie Series Prediction Using Non Linear Signal Analysis and Neural Networks in Chaotic Diode Resonator Circuits, Springer Berlin/Heidelberg, Lecture Notes in Coputer Science, Advances in Data Mining. Theoretical Aspects and Applications, Vol. 4597, pp Kantz, H., Schreiber, T. (1997), Nonlinear Tie Series Analysis, Cabridge University Press, Cabridge. Kennel, M.B., Brown, R., Abarbanel, H.D.I. (1992). Deterining ebedding diension for phase-space reconstruction using a geoetrical construction, Physics Review A, 45, Kugiutzis, D., Lillekjendlie, B., Christophersen, N. (1994) Chaotic Tie Series, Modelling Identification and Control, 15, 205. Miksovsky, J., Raidl, A. (2007), On Soe Nonlinear Methods of Meteorological Tie Series Analysis, Proceedings of WDS Conference. Sta, C.J., Pijn, J.P.N., Pritchard, W.S. (1998), Reliable Detection of Nonlinearity in Experiental Tie Series with Strong Periodic Coponent, Physica D, Vol Sprott, J.C. (2003), Chaos and Tie series Analysis, Oxford University Press. Takens, F. (1981), Detecting Strange Attractors in Turbulence, Lecture notes in Matheatics, Vol. 898, Springer, New York. Thalassinos, I.E., Hanias M.P., Curtis, G.P., and Thalassinos, E.J. (2009), Chaos Theory: Forecasting the Freight Rate of an Oil Tanker, International Journal of Coputational Econoics and Econoetrics, Vol. X, Issue 1.