Algorithm- Step 1
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mean price |
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in KiloUSD |
2.0 |
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up price |
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down price |
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1.5 |
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Troy Ounce |
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1.0 |
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for one |
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of Prices |
0.5 |
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Corridor |
0.0 |
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-10 |
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1< quarters <120 |
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mean price |
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in KiloUSD |
2.0 |
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up price |
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down price |
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1.5 |
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Troy Ounce |
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1.0 |
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for one |
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of prices |
0.5 |
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Corridor |
0.0 |
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0 |
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60 |
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1< half years < 60 |
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Fig.2. In the result of application of procedure described as Step 1 (reduction to three incident points) one can take the self-similar curve showing the quarter distribution of prices.
The degree of compression is equaled 3.
Fig. 3. The property of self-similarity of the initial curve is conserved if one takes the half- year period intervals. The degree of compression is 6 (in comparison with the Fig. 1).
11
Algorithm-Step 2
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mean price |
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2.0 |
up price |
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down price |
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09.2012 |
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1.5 |
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Ounce in |
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for Troy |
1.0 |
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09.1983 |
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prices |
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0.5 |
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Corridor of |
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1< years < 30 |
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Fig.4. The self-similarity is still conserved if we realize the reduction of twelve months points to three points characterizing a year. We note that the interval of deviation of prices from its mean value (a specific measure of uncertainty) is increased with narrowing of the initial interval. We compressed 12 months to one year (the degree of compression is 12).
Requirements:
R1. Any external factor (time, frequency, coordinate and etc.,) measured in general in number of the measured points it is desirable to reduce to minimal number of 100-150 points in order to provide the value of the fitting error less than 10%.
R2. For economical and other data associated with human activity it is natural to choose the conventional intervals as years, half-years, months, weeks, days and etc.
R3. The scaling factor should lies in acceptable limits in order to keep the calculated value of the power-law parameter (associated with fractal dimension) in the interval 0 < < 3(4).
12
Algorithm-Step 2
Step-2. Criterion of selection of the initial hypothesis yc1(t)
H0 (t) A0 A1t 0 Ac1 yc1 (t) As1 ys1 (t),
yc1 (t) t 1 cos 1 ln t , ys1 (t) t 1 sin 1 ln t
1, 2 . 1
D3 H |
(t) b D2 H |
(t) b DH |
(t) b H |
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(t) K, D t |
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r3 b r2 |
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Y (t) H0 (t) |
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t |
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X (t) H (u) du |
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t0 |
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t |
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X2 (t) |
ln t ln u H0 (u) du |
... , C2 |
b2 , |
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t0 |
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t |
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H0 (u) du |
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X3 |
(t) |
ln t ln u 2 |
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... , C3 b3 , |
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u |
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t0 |
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Here and below we keep the same notations for 4 linear unknown amplitudes A0, A1, Ac1, As1.
Other three unknown nonlinear parameters entering in (6) can be found by the eigen- coordinates (ECs)
method |
b1 |
0 |
2 1 |
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Y (t) Cs X s (t). |
b2 2 0 1 12 2 |
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s 1 |
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The pair of brackets in the last expressions <…> = defines the arithmetic mean of the neighboring expression located on the left that should be subtracted from it.
s 4,5, 6 |
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X |
(t) ln7 s (t) |
... , C |
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(K, D3H |
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13 |
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Application of the ECs method for evaluation of the constants and the desired function
14
Step-2. More complex case
If the initial hypothesis is supposed to be correct, then we can expect that the exponents 0 and 1 should have at least the same sign and cannot be strongly deviated from each other.
In the opposite case it is necessary to consider another hypothesis. When the value of the relative error is remained rather large the initial hypothesis should be replaced by an alternative hypothesis that looks not as simple as the initial expression H0(t) . In order to
have the justified criterion that shows the conditions of replacement of H0(t) by another
hypothesis it is necessary to consider a more complicated initial expression: |
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H0 (t) A0 A1t |
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B1t |
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Acp ycp (t) Asp ysp (t), |
0 0 1,2 |
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p 1 |
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6 nonlinear |
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ycp (t) exp p ln(t) cos p ln t |
, ysp |
(t) exp p ln(t) sin p ln t , p 1, 2. |
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parameters! |
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H0 (t) h0 (t) y2 (t), h0 (t) A1t |
Ac1 yc1 (t) As1 ys1 (t), |
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y |
(t) A B t 0 |
Ac yc (t) As ys |
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ycp (t) exp p |
ln(t) cos p ln t , ysp (t) exp p ln(t) sin p ln t , p 1, 2. |
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One can apply the ECs method for the function y2(t) (having the same structure as the previous function H0(t)) and consider the nonlinear parameters 0 , 1, 1 as known. The corresponding program is shown in the previous slide
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Step 2. More complex case
Taking into account the invariance of the expressions relatively procedure of the n-fold integration Jn:
Jn exp ax cos(bx) K1,n (a,b) exp ax cos(bx) K2,n (a,b) exp ax sin(bx),
Jn exp ax sin(bx) Q1,n (a,b) exp ax cos(bx) Q2,n (a,b) exp ax sin(bx),
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Y (t) Cs X s (t).
s 1
X7 (t) t 1 
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X8 (t) t 1 cos( 1 ln(t)) 
...
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X9 (t) t 1 sin( 1 ln(t)) 
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Now we are ready to formulate the criterion of the applicability of the initial hypothesis.
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r |
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1 rp1, rp2 1.45, 1 r 1.45.
The upper limit (1.45) of this inequality is chosen from the following condition. Being rounded off the integer value it should give again the unit value. If this condition is violated, then it is necessary to consider new hypotheses.
16
Algorithm-Step 3
Step 3. The optimization procedure
The third step is related to optimization of the power-law exponent and inoculating frequency that are located in the intervals:
min( 0 , 0 , 1, 2 ) min max( 0 , 0 , 1, 2 ) max , min( 1, 2 ) min max( 1, 2 ) max .
If some nonlinear fitting parameter p is located in the given limits [pmin, pmax] then one can introduce the function:
p(v) tv e, |
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H (t, v) A0 A1 exp( (v) ln t) Ac yc(t, v) As ys(t,v) |
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pmax pmin |
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pminvmax |
pmax vmin |
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yc(t, v) exp( (v) ln t) cos( (v) ln t), |
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ys(t,v) exp( (v) ln t) sin( (v) ln t), |
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(v) t0v e0 , (v) t1v e1 |
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min RelErr v v |
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17
Optimization programs
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Algorithm-Step 4
Step 4. The final fit of the initial function
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K |
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H f (t) A0 A1t |
Ack yck (t) Ask ysk (t) , |
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k 1 |
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ln t |
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yck (t) exp |
ln t cos 2 k |
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ln t |
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ysk (t) exp |
ln t sin 2 k |
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min RelErr K |
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y(t) H f |
t, K |
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mean |
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y(t) |
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ln |
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exp |
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figures in the |
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ln |
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1 exp |
ln , ( 1 |
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Table 1. |
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2 k |
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2k 1 . |
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Idea of data reduction
One important comment: It is obvious that this parameter K should satisfy to the following requirement 2K+4 <N, where N denotes the total number of the measured points figuring in the initial function y(tj) (j=1,2,…,N). In the opposite case, when the number of unknown
modes K exceeds the number of data points N (2K+4 > N), the fitting procedure becomes useless and so it should be rejected.
19
Mini-programs for the final fitting
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