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**Freight forecasting method**

Relation between a freight rate and a time-charter equivalent in simplified form is as follows:

(1)

where - freight rate, - time-charter equivalent, - voyage costs (disbursements, canal dues, fuel costs and brokerage), - voyage duration, - lot.

ISM defines market situation in a certain region/sub-region with the integral freight index and the average time-charter equivalent. Calculation procedure is given in Chapter 2.

Dynamics of indices is cyclic in most cases. Capesize Global TCE Index is given as an example:

The above description is obviously cyclic. Peaks reiteration period is about 13-14 weeks.

The aim of the forecasting model is to find basic cyclical constituents of the description (signal) and to implement extrapolation, assuming that the numerical parameters of approximation model will be stable during a certain confidence interval of time.

A signal can be represented as series:

(2)

where and - coefficients at polynomial and cyclical parts of decomposition; - period and - shift (phase) of the corresponding harmonic.

In general, if a signal is precisely represented, the second sum has an infinite number of terms and is revealed as a decomposition of excess in Fourier series while signal is given as reduced Taylor series. When building an approximation model, reduced series are also used in a cyclical part where number of additives varies depending on the features of initial signal. Similarly, parameter of a model is a number of terms in polynomial part of equation.

The classic method of determination of frequencies of basic cyclical constituents is based on selection of typical peaks in Fourier spectrum of the signal. This method applies when an additive broadband noise is overlapped on a signal. In our case, noise has a multiplicative character expressed as variability of the period which may deviate by 15-20% of its value. This fact extends the spectrum of signal significantly and makes it impossible to expose main harmonics of signal amid the additive constituent of noise.

Determination of periods of main parts is based on integral wavelet transform. Signal decomposition can be written as follows:

(3)

where - chosen wavelet, and characterize scale and change during signal analysis. Thus, scale parameter relates to the corresponding period with the expression:

(4)

where - time step, - central frequency of the chosen wavelet.

The value of signal wavelet-coefficients given above is shown with colour in the chart below:

Minimums of function are marked with dark colour. It is obvious that the function has a recurrence on certain scale levels. The task is to define those levels, so we calculate correlation of energy distribution and scale. We assume the second central moment of shift parameter correlation as a measure of energy:

(5)

where - domain for in accordance with parameter ,

– average value of function on the corresponding scale cut.

Energy distribution of the signal under study can be shown as follows:

Description has the evident maximums for the values of 14.55 and 26.29 week periods. Moreover, harmonics corresponding to the periods 3.58, 5.93 and 52.14 that are critical points of function can be included into the sum.

Change of each harmonic is determined by approaching of each of them to the initial signal with the least square method. Then, the initial function is centered and rationed on its mean square deviation (in order to follow the condition of approximate equality of amplitudes between the initial signal and approximating function:

(6)

where – average value of signal. Integration is made along the time domain.

Having defined periods and phases included into a cyclical constituent of harmonics, we need to determinate coefficients and at the terms of sums. Their calculation is based on the least square method, taking into account additional weighting coefficients. Thus, the later values of description are more informative while building an approximation model. Due to discretization in time, initial description is specified as a vector.

Then formula (2) can be represented as the overdetermined system of linear algebraic equations (summation symbol is omitted):

(7)

where , – vector that contains desired coefficients. Matrix is formed by the horizontal concatenation of matrices - kernels of polynomial and cyclical sums: (8)

where , . It should be noted that lines of matrix are the function of time.

Vector is determined as per matrix expression:

(9)

where , - square matrix of weighting coefficients, which off-diagonal entries are equal to zero, exponentially increasing weighting coefficients are on the main diagonal:

(10)

Having built approximation model by adding the fixed value to the received function, we combine its chart with initial signal in a point that corresponds to the current moment of time.

This forecasting procedure is based on assumption that the value of vector remains unchanged during a certain confidence interval of time. Thus, the forecast values of integral index are calculated from correlation:

(11)

where – current moment of time, – forecast depth.

The smoothing exponential multipliers are entered in the sum (2) in order to weaken the effect of considerable increase of reduced Taylor series when cyclical constituent peaks are extrapolated and indemnified. Then, matrix can be determined with a horizontal concatenation:

(12)

where , .

Values of smoothing coefficients, as well as number of terms of polynomial constituent, interval of approximation, values of weighting coefficients and wavelet selection are parameters of the model while determinating basic harmonics of signal.

The above chart is an example of result of forecasting for the week . Actual values of average time-charter equivalent are marked with blue colour, values of corresponding approximation model – with green. The result of forecasting is marked with red colour. The chart shows that this procedure reflects real dynamics of the process and provides sufficient accuracy for the applied tasks.

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