Example 1-Economical data

Fig.5. Here we show the realization of the Step 2 described in Section 2.2. The initial fit was realized with the help of the ECs method and initial hypothesis H0(t) and hypothesis H1(t)

Example 1. Economical Data

1 < months < 360

Ack

Fig.6. Here we show the optimal fit (the solid red line) that is realized with the help

of the function H(t,vopt). It minimizes the value of the relative error (9.291%) and

helps to obtain the optimal values of the power-law exponent < > = 3.07938 and < >=1.89072. The final fit (solid green line) is realized with the help of the function Hf(t). All fitting parameters are

collected in Table 1.

Fig.7. Besides the parameters that restore the structure of the scaling equation (2) we show here the distribution of the amplitudes Ack, Ask that define the

unknown log-periodic function in expression (19). In order to decrease the number of the fitting parameters and in the same time to keep the acceptable accuracy for the final fitting function in the interval (3-5%) we choose the value of the final mode (K) from the condition (2K+4)=104 < N=360.

22

a 2, 1 0, a 1, 1 0.

Economical Data

Fig.8. The final fit of the distribution of the quarter prices. The corresponding parameters are collected in Table 1. Other two curves (shown on Fig.2) are very close to each other and so their fit is not shown.

Fig.9. In comparison with Fig.7 we decrease in two times the number of corresponding amplitudes in order to keep the value of the relative error in the same limits (3-5%) and satisfy to the requirement (2K+4=54 < N=120).

Economical Data

1 < years < 30

Fig.10. The final fit of the distribution of the half-year prices. The corresponding parameters are collected in Table 1. Other two curves (shown on Fig.2) (the upper and down distribution of prices) are given also. The distribution of the amplitudes for the fitting curve corresponding to mean price is given in the small frame above. Other two distributions of amplitudes for upper and down fitting curves are similar and they are not shown.

Fig.11. The final fit of the distribution of the year prices. The corresponding parameters are collected in Table 1.

The distributions of amplitudes are shown below. For this compressed curve only 5 amplitudes are necessary.

Economical Data

Fig.12. In comparison with Figs.7 and 9 we keep the number of modes equaled K=5. This number is sufficient for keeping the value of the relative error in the limits (3-5%) and for satisfaction of the requirement (2K+4=14 < N=60).

Possible predictions for the next 5 months

meanPr_(1-360)

2.0predPr_(5-365)

Fig.13. What is happened if we continue the fitting functions calculated for 1 month and half-year (given in the small frame) for the 5 points up?

It is interesting to note that these curves demonstrate the essential decreasing of prices on gold in the nearest future.

Economical Data

Fig.14. It is instructive to see that the distribution of the annual prices also has a tendency to decrease these prices on the nearest 5 years. For the first two curves depicted on the previous figure one can explain the "decreasing" tendency by existence of a specific bend that is clearly seen on the right-hand of these curves. But the natural bend for the curve depicted here is clearly absent, however, the tendency to decreasing of the prices for the nearest 5 years is conserved.

Comments to Table 1. The fitting parameters collected in this table reflect the basic parameters of the final fitting function and the parameters that figuring in the scaling equation (2) and its solution

(3). Strictly speaking it is necessary to take the mean values of the columns 2,3,4,6 but we keep them as they are in order to see the variations of these values with respect to compression procedure. The last column shows the number of modes that can restore the unknown log-periodic function with accuracy given in column 7.

Fig.15. The distribution of mean temperature (measured in C0) for both Earth's Hemispheres. The data cover 130 year period of registration of mean temperature.

The key parameters shown on the previous table

ln 2 , 1 exp ln , ( 1 0), s0 A0 1 1 , ln , 1 exp ln , ( 1 0).

2 k 2k 1 .

Meteo-Data

Fig.16. If one can realize the reduction to three incident points then we obtain the curve similar to the curve depicted on the previous figure. This reduction is realized for the mean temperature belonging to the North hemisphere (NH). The temperatures Tup and Tdn show the limits of deviation of the central curve from its mean position.

Fig.17. These plots demonstrate the reduction to three points for the mean temperature belonging to the South Hemisphere (SH). As before the temperatures the curves Tup and Tdn demonstrate the limits of deviation of the central curve from its mean position.