References and notes
- Called ‘demon’ by Greeks
- Stopford, Martin, 1997, Maritime Economics, 2nd ed. (London: Routledge) [3rd ed., 2009]
- See the weekly Greek newspaper Sunday Kathimerini, May 18, 2008
- The shipping loan tenure is normally four years
- This is comparably a relatively long period of time, but this is necessary for the nonlinear methods chosen, which require a minimum 500 of observations
- A method developed by the British hydrologist Hurst 53, building a dam in the Nile, to determine long-memory effects and fractional Brownian motion. It provides a measurement of how the distance that is covered by a particle increases over longer and longer time scales. This model is a generalization of Einstein's model of random walk 107 to include not only white noise but also black and pink noise. This is analytically presented in methodology section
- Lo , AW . 1991 . Long-term memory in stock market prices . Econometrica , 59 : 1279 – 1313 .
- This acronym is made of by the first letters of the surnames of the economists invented this test 108. With the BDS-statistic, the difference in the dispersion of the observations is tested, in spaces for a dimension from 2 to n, compared with white noise
- The integral indicates the probability that two points are within a certain distance of each other
- Goulielmos , AM and Psifia , M-E . 2007 . A study of trip and time charter freight rate indices: 1968–2003 . Maritime Policy and Management Int. Journal , 34 ( 1 ) : 55 – 67 .
- SIRIOPOULOS, C., 1998, Analysis and Tests of Univariate Financial Time Series, Typothito. (Athens: G. Dardanos Editions) [in Greek]
- Li , J and Parsons , MG . 1997 . Forecasting tanker freight rate using neural networks . Maritime Policy and Management , 24 ( 1 ) : 9 – 30 .
- A computer-based simulation considered as imitating the human nervous system, which can learn from experience
- Lyridis , DV , Zacharioudakis , P. Mitrou , P and Mylonas , A . 2004 . Forecasting tanker market using artificial neural networks . Maritime Economics and Logistics , 6 : 93 – 108 .
- Small , M . 2005 . Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance , Singapore : World Scientific .
- (1) AR (Yule 17) = Auto Regressive process = a stationary stochastic process where the current value of a time series is related to the past p values (p=any integer). An AR process has an infinite memory (2) ARMA = Autoregressive Moving Average process = a stationary stochastic process that can be a mixed model of AR and MA. MA = a stationary stochastic process in which the observed time series is the result of the moving average (MA) of an unobserved random time series. (3) ARIMA = Autoregressive Integrated Moving Average process = a non-stationary stochastic process related to ARMA. (4) ARCH (1982) = an AR with Conditional Heteroskedasticity = a nonlinear stochastic process where the variance is time-varying and conditional upon the past variance, with high peaks at the mean and fat tails. These are models for the analysis of the predictability of variance introduced by Engle, Bollerslev, Nelson and others. (5) The GARCH (1986) means the Generalized ARCH. It refers to a set of statistical tools to model data whose variability changes with time. The changes in variability are controlled by the data's own past behaviour, and it has been generalized to accommodate more circumstances as a further development of the initial ARCH. (6) EGARCH (1991) = Exponential GARCH
- Yule , GU . 1927 . On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers . Philosophical Transactions of the Royal Society , 226 : 267 – 298 .
- Engelen , S , Meersman , H and Van De Voorde , E . 2006 . Using system dynamics in maritime economics: An endogenous decision model for ship-owners in the dry bulk sector . Maritime Policy and Management , 33 ( 2 ) : 141 – 158 .
- The model started as a business game in 2005 between Antwerp and Delft Technical University and was presented at the 2005 IAME conference
- Sterman , JD . 2000 . Business Dynamics—Systems Thinking and Modeling for a Complex World , New York : McGraw-Hill .
- GOULIELMOS, A. M. and GOULIELMOS, MARCOS, 2009, How to determine timing in deciding to build or charter a vessel. International Journal of Transport Economics, 36(2), 74–97
- GOULIELMOS, A. M. and PSIFIA, M.-E., 2006, Variations in charter for a time series between 1971 and 2002: Can we model them as an effective tool in shipping finance? Transport Economics International Journal, June, 257–278
- Goulielmos , AM and Psifia , M-E . 2006 . Shipping finance: Time to follow a new track? . Maritime Policy and Management Int. Journal , 33 : 301 – 320 .
- Goulielmos , AM , Goulielmos and Matina . 2008 . Do the nonlinear tools make a difference in handling shipping derivative? . International Journal of Transport Economics , XXXV ( 3 ) : 345 – 371 .
- GOULIELMOS, A. M., GIZIAKIS, C. V. and GEORGANTZI, A., Forthcoming, the ‘price discovery’ role of freight derivatives (futures), 2006–2008. International Journal of Transport Economics
- GOULIELMOS, A. M. and SIROPOULOU, E., 2006, Determining the duration of cycles in the market of second hand tanker ships 1976–2001: Is prediction possible? International Journal of Bifurcation and Chaos, June, 2119–2127
- It is a system of nonlinear differential (or difference) equations with time as the main independent variable
- It is a nonlinear dynamic system producing results that seem to be random characterized by sensitive dependence on initial conditions and the existence of a ‘strange’ (chaotic) ‘attractor’ in its ‘phase space’. The concepts are defined below
- LILLEKJENDLIE, B., CHRISTOPHERSEN, N. and KUGIUMTZIS, D., 1995, Chaotic time series, Part II, System identification and prediction. Modeling, Identification and Control, Part II, 15(4), 225–243
- LILLEKJENDLIE, B., CHRISTOPHERSEN, N. and KUGIUMTZIS, D., 1995, Chaotic time series, Part II, Estimation of some invariant properties in state space. Modeling, Identification and Control, Part I, 15(4), 205–224
- Brockwell , PJ and Davis , RA . 1996 . Introduction to Time Series and Forecasting. Springer Texts in Statistics , New York : Springer-Verlag .
- Smale , S . 1963 . “ Diffeomorphisms with many periodic points ” . In Differential and Combinational Topology , Edited by: Cairns , SS . 63 – 80 . Princeton, NJ : Princeton University Press .
- Lorenz , EN . 1963 . Deterministic nonperiodic flow . Journal of Atmospheric Science , 20 : 130 – 141 .
- Eckman , JP and Ruelle , D . 1985 . Ergodic theory of chaos and strange attractors . Review of Modern Physics , 57 ( 3, part I ) : 617 – 656 .
- Takens , F . 1981 . Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics , 898 New York : Springer-Verlag .
- A non-stationary time series presents a trend or seasonality or a changing variability. In technical terms: all time series ‘moments’ do not change with time in case of stationarity. Nonlinear stationarity has a different meaning however 15: nonlinear stationarity is a situation unchanged by the system's dynamics over time. In this case the work is done with the so called ‘invariants’ (*), which are: the correlation dimension and entropy; the complexity and information; the divergence of nearby trajectories; the non linear prediction error or the Lyapunov exponent. For Small [15, p. 47 et seq.], (*) the ‘invariant’ is a quantity describing the dynamic behaviour of a system with the special property that its value does not depend on the coordinate system; or the value of a dynamic invariant obtained directly from the original system is the same in a particular transformation, such as a change from an original phase space to time-delay reconstruction (as this applies to this paper)
- Peters , EE . 1994 . Fractal Market Analysis: Applying Chaos Theory to Investment and Economics , New York : J. Wiley .
- For traditional linear, this was econometrics, stationarity exists when all moments (mean, standard deviation, kurtosis, skewness, and all similar statistics) remain unchanged with time. In contrast with this, in dynamical systems the evolution operator Φ does not change with time if we want to have a stationary dynamical system. In fact, we do, as otherwise non-stationary dynamical systems are exceedingly difficult to model from time series. Following Small [15, p. 4] we have to choose the system M⊆ Rk to be the smallest system (k lowest defined below) such that the corresponding Φ: Z × is stationary (i.e. the model includes all outside influences)
- Dickey , DA and Fuller , WA . 1979 . Distribution of the estimators for autoregressive time series without a unit root . Journal of the American Statistical Association , 74 : 427 – 431 .
- JARQUE, C. M. and BERA, A. K., 1980, An efficient large-sample test for normality of observations and regression residuals. Working Papers in Econometrics, No 40, Canberra: Australian National University
- Ljung , GM and Box , GEP . 1978 . On a measure of the lack of fit in time series models . Biometrika , 65 : 297 – 303 .
- Box , GEP and Pierce , DA . 1970 . Distribution of residual autocorrelations in ARMA time series models . Journal of the American Statistical Association , 65 : 1509 – 1526 .
- This means finding the dimension of the system. The dimension is a common subject in the nonlinear literature. We have dimensions giving us important information for: (1) capacity; (2) embedding; (3) information; (4) correlation; (5) point-wise; and (6) generalized. In this paper, only the embedding and the correlation dimensions are used
- A system is defined as an arrangement of parts where one influences the other in a uniform way. A dynamical system is defined as a system that evolves in time and creates one or more time series
- Thanks to certain scientists, theorems valid for continuous time have been modified to apply to discrete time
- SIRIOPOULOS, C. and LEONTITSIS, A., 2000, Chaos: Analysis and Forecasting Time Series [Including software NLTSA V.2.0]. (Thessalonica: Anikoula editions)
- CONT, R., 2007, Volatility clustering in financial markets: empirical facts and agent-based models, in Long Memory in Economics, edited by Gilles Teyssiere and Allan P. Kirman (Springer)
- The ratio of (R/S)n to the square root of a time index, n. R = range, S = standard deviation (local)
- This means that we divide the adjusted range by the local standard deviation. Rescaled range (R/S) is a name short for ‘Range (R) divided by standard deviation (S)’ [109, p. 202]. Dividing by standard deviation makes the term normalized. This work is titled as ‘normalized volatility’
- Adjusted to a mean of zero, and this is always non-negative
- This is one measure of the existence of trend in time series. It has been discovered also by L. O. Holder, a pure mathematician, mentioned by Mandelbrot and Hudson [109, p. 187]
- Hurst 53 anticipated the ‘renormalization group theory’ in Physics [37, p. 57]
- Hurst , HE . 1951 . Long term storage capacity of reservoirs . Transactions of the American Society of Civil Engineers , 116 : 770 – 799; 800–808 .
- Hampton , MJ . 1991 . Long and Short Shipping Cycles , 3rd , Cambridge : Cambridge Academy of Transport .
- This indirectly rejects the long held traditional view that shipping cycles are uniformly of four years’ duration, with two years boom and two years depression. This assumption led the Japanese company Sanco to almost catastrophe in 1981–87 110, 111
- A probability density function that is statistically self-similar, or in other words, in different increments of time the statistical characteristics remain the same
- PERETTI, DE C., 2007, Long memory and hysteresis, in Long Memory in Economics, edited by Gilles Teyssiere and Allan P. Kirman (Springer)
- It shows how the correlations change from one time-period to another (totalling 150 years in the case of Baillie)
- Baillie , RT . 1996 . Long memory processes and fractional integration in econometric . Journal of Econometrics , 73 : 5 – 59 .
- Steeb , WH . 2008 . The Nonlinear Workbook , 4th , River Edge, NJ : World Scientific .
- Proof: let Rn = R1 * n 1/α meaning that the sum of n values scales as n 1/α times the initial value. Taking log of both sides and solving for alpha, we have: α = log(n)/log(Rn) − log (R1), as H = log(R/S)/log(n) thus α = 1/H [37, pp. 212–3]. Mandelbrot-Hudson [109, p. 202] put it the other way round as: H = 1/alpha which however gives the same result
- Acronym for (1980–1, 1989) the ‘autoregressive fractionally integrated moving average process’
- Given by ϕ(L) (1 − L) d Xt = θ(L)ε t t∈{1, …, T} due to Hosking 112 and Granger and Joyeux 113, where the error term can follow a stationary GARCH process (non-Gaussian), ϕ and θ are polynomials with roots outside the unit circle, variance < infinity, L = a lag operator and d is the differentiating parameter, taking a real value, and absolute if d < 0.5 and d ± 0 is for memory for invertible series. Moreover when d = 0 indicates short term memory 114
- GOULIELMOS, A. M. and KASELIMI, V., 2009, Nonlinear Forecasting of Container Traffic as a Managerial Tool: A Case Study of the Port of Piraeus, 1973–2008. Paper presented at IAME conference, Copenhagen, June
- This is a measure of the ‘peakedness’ of the probability density function, where α = 2 for a normal distribution. See Mandelbrot [109, pp. 261–2, 295–6]. Alpha means volatility. It is an exponent measuring how wildly prices vary or how fat the tails of the price-change curve are
- Mandelbrot , BB . 1972 . Statistical methodology for nonperiodic cycles: from the covariance to R/S Analysis . Annals of Economic and Social Measurement , 1 ( 3 ) : 259 – 290 .
- Greene , MT and Fielitz , BD . 1977 . Long term dependence in Common Stock Returns . Journal of Financial Economics , 4 : 339 – 349 .
- The method as mentioned developed by Hurst 53 to determine long-memory effects and fractional Brownian motion. A measurement of how the distance covered by a particle increases over longer and longer time scales
- Goulielmos , AM . 2004 . A treatise of randomness tested also in marine accidents . Disaster Prevention and Management Int. Journal , 13 : 208 – 217 .
- Mandelbrot , BB and Wallis , JR . 1969 . Some long run properties of geophysical records . Water Resources Research , 5 : 321 – 340 .
- MANDELBROT, B. B. and TAQQU, M., 1979, Robust R/S analysis of long run serial correlation. Bulletin of the Int. Statistical Institute, 48, 59–104
- Press , W , Teukolsky , S , Vetterling , W and Flannery , B . 1996 . Numerical Recipes in C , Cambridge : Cambridge University Press .
- [(N − 1)/2 = (996 − 1)/2 = 497 (integer)]
- In relation to the random movement of molecules in a liquid, which according to Einstein is random (random walk model). Random collisions of the molecules between them produce random movements of molecules. Given N random steps, then the overall distance travelled is proportional to the quantity
- Basic feature of the test is that its distribution follows asymptotically the normal distribution with mean 0 and variance equal to 1. Significance tests can be carried out even when higher moments do not exist
- KANZLER, L., 1999, Very fast and correctly sized estimation of the BDS statistic (Oxford: Christ Church and Department of Economics, University of Oxford) [http://users.ox.ac.uk.uk/∼econlrk]
- An estimation of the fractal dimension measuring the probability that two points selected randomly in the phase space are to be found within a certain distance one from the other and shows the variation in this probability as distance changes
- BDS=
- The radius is better fixed as equal to one σ, but normally this range of values is used: ½ < e/σ < 2
- There is a restriction that N/m > 200 which is almost satisfied here at maximum m, as we have 996/5 = 199.2
- An attractor M belongs to Rk as a subset (or is included in it, i.e. M ⊆ Rk ). We have an evolution operator Φ such that (Zn, t) = Z n + t. The system state Zn exists within a manifold M and evolves according to Φ. This dynamic motion is observed in R by an observation function (time series)
- This is a number of points in the phase space towards which the trajectories tend asymptotically over time for a certain range of initial conditions. The repeated behavior is what one looks for
- If the attractor is strange, the points never repeat themselves and the orbits never intersect, but both points and orbits stay within the same region of the phase space. Strange attractors are non-periodic and generally have a fractal (non-integer) dimension. This is a configuration of a nonlinear ‘chaotic’ system
- Grassberger , P and Procaccia , I . 1983 . Measuring the strangeness of strange attractors . Physica D , 9 : 189 – 208 .
- The ‘correlation dimension’ defined above is the one usually used to show the system's dimension
- Selection of the time delay is required to make our observations independent amongst them when these are placed as vectors with independent coefficients in an m-dimensional system of co-ordinates
- It is an area of the phase space—a set of points—towards which trajectories are attracted. These are possible states of the system during the evolution of time. The attractor may reveal characteristics of the dynamics of the system within which it is observed
- This is a number that qualitatively describes how an object fills its space. Fractals, in contrast with Euclidean plane geometry with solid and continuous objects, are rough and often discontinuous, like a waffle ball
- When we have a dynamic system, both stationary and deterministic, we must examine it. Let us have an observation function g : providing us with a way to measure the current state of the system g (Zn) and observing Xn = g(Zn) at many successive times. According to Takens’ embedding theorem, if the embedding dimension de is sufficiently large, i.e. equal, for example to 10, or in our case: 2D + 1 (where D = 3.95) giving a result near 9 (8.9 = 7.9 + 1) ≤ de , then the evolution of Xn, Xn–1, Xn–2, …, Xn–de is the same as the evolution of Zn. Embedding Xn→Xn, Xn–1, Xn–2, …, Xn–de contains the same information as the original system, as de is large enough. The function g is twice differentiable and there is a sufficiently long data record available (996 weeks) and the data are sampled sufficiently often (each week)
- As fractality is one of the properties of strange attractors
- The system also requires theoretically four equations to describe it
- This is a measure of the dynamics of the attractor where each dimension has a Lyapunov exponent. A positive exponent measures sensitive dependence on initial conditions and how much a forecast can diverge when starting conditions differ
- Kugiumtzis , D , Lingjaerde , O and Christophersen , N . 1998 . Regularized local liner prediction of chaotic time series . Physica D , 122 : 344 – 360 .
- Sauer , T . 1993 . Time Series Prediction by Using Delay Coordinate Embedding , Reading, MA : Addison-Wesley .
- Sugihara , G and May , R . 1990 . Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series . Nature , 344 : 734 – 740 .
- The software is available in 46. Nonlinear software is also TISEAN freely downloaded from the internet. Some routines we have used were written in MATLAB 5.3
- This is a common method in forecasting
- Kugiumtzis , D , Lingjaerde , O and Christophersen , N . 1998 . Regularized local liner prediction of chaotic time series . Physica D , 122 : 344 – 360 .
- Farmer , DJ and Sidorowich , JJ . 1987 . Predicting chaotic time series . Physical Review Letters , 59 : 845 – 848 .
- This is a method belonging to the so-called ‘geometrical methods’ to select a system's dimension 115. This is based on the assumption that the points in phase-space with embedding dimension m are located at distances R between them. Then we test what happens if m = m + 1
- For details of the ‘Singular Value Decomposition’, see 102
- Broomhead , SD and King , GP . 1986 . Extracting qualitative dynamics from experimental data . Physica D , 20 : 217 – 236 .
- To reconstruct the dynamics of the original attractor that gives rise to the observed time series, we seek an embedding space where we may reconstruct an attractor from the scalar data so as to preserve the invariant characteristics of the original unknown attractor. The dEx1 phase space vector s(n) is constructed by assigning coordinates: s1(n) = s(n), s2(n) = s(n-τ), …, s dE(n) = s(n–(dE–1)τ and T = 1 for simplicity (time delay). dE is the embedding dimension. For more details see p. 35 of Feng and Tse 116
- Abarbanel , HD , Brown , R and Kadke , JB . 1990 . Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra . Physical Review A , 41 ( 4 ) : 1782 – 1806 .
- Abarbanel , HD , Brown , R , Sidorowich , J and Tsimring , LS . 1993 . The analysis of observed chaotic data in physical systems . Reviews of Modern Physics , 65 ( 4 ) : 1331 – 1392 .
- The program accepts up to 16 384 observations as mentioned, and it is contained in a diskette 46 and written in MS-DOS, with its code written in C/C++ and compiler the GNU CC 2.8.0 (1998) in an environment RHIDE 1.4 (1997). The program needs 3.8 MB zipped
- Einstein , A . 1905, Uber die von der molekularkinetischsen theorie der warme geforderte bewegung von in ruhenden flussigkeiten suspendierten teilchen, Annals of Physics, 322
- BROCK, W., DECHERT, W. and SHEINKMAN, J., 1986, A Test of Independence ased on the Correlation Dimension. SSRI report No 8702. Department of Economics, University of Wisconsin; University of Houston; University of Chicago. [In1987, the report appeared in Econometrics Reviews, 15, 197–235; in 1996, with LE BARON, B., it appeared in the same journal.]
- Mandelbrot , BB and Hudson , RL . 2004 . The (Mis) Behavior of Markets: A Fractal View of Risk, Ruin and Reward , New York : Basic Books .
- GOULIELMOS, A. M., 2007, Shipping Finance, 2nd ed. edited by A. Stamoulis. (Athens: Piraeus) [in Greek]
- GOULIELMOS, A. M., 2009, Risk analysis of the Aframax freight market and of its new building and second hand prices, 1976–2008 and 1984–2008. International Journal of Shipping and Transport Logistics, 1(1)
- Hosking , JRM . 1981 . Fractional differencing . Biometrica , 68 : 165 – 176 .
- Granger , CWJ and Joyeux , R . 1980 . An introduction to long-memory time series models and fractional differencing . Journal of Time Series Analysis , 1 ( 1 ) : 15 – 29 .
- Barkoulas , JT , Labys , WC and Onochie , J . 1997 . Fractional dynamics in international commodity prices . Journal of Futures Markets , 17 ( 2 ) : 161 – 189 .
- Kennel , M , Brown , R and Abarbanel , H . 1992 . Determining embedding dimension for phase-space reconstruction using a geometrical construction . Physical Review A , 45 : 3403 – 3411 .
- Feng , JC and Tse , CK . 2008 . Reconstruction of Chaotic Signals with Applications to Chaos-Based Communications , Singapore : World Scientific .