Hearst index - page 26
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For some time I had to be distracted by other concerns - my daughter was 18 - I had no time for fractals ;))).
But such a switch - this is the first time I notice it - led to clear vision of the unsolved yet unsolved fractal problem.
Well, as soon as I come to my senses, we are going to solve this problem ;)
We'll be waiting:)
Despite the public's tepid interest in the topic, I am still following Peters' book. I've improved the methods and finally understood what is being calculated. So, here are the basic formulas:
The first thing we do is to convert the price series to logarithmic returns. The essence of the second formula is this: the average of logarithmic returns is the mathematical expectation of the series, if we calculate the average of the series from each value, then the sum of these values will always be equal to zero. This is the simplest way of detrending, but it has a huge impact on the result. Next, the minimum and maximum of the series are calculated. These calculations can be presented graphically in graphs:
The third formula, as everyone understands, is the calculation of the simple standard deviation.
H is therefore the simple ratio of the range to its logarithmic period.
Then the entire Peters series under analysis is divided into independent sub-periods. Each sub-period is calculated according to the methodology described above. As a result, there is an average value of RS, which is qualitatively different from Brownian motion. Since the dispersion of particles will be directly proportional to the logarithm of the period, the Hurst ratio, i.e. the ratio of timespan to period, must be a constant and be 0.5. In fact the formula is not perfect and tends to overestimate the result by 0.3, i.e. on obviously random series, Hurst will show 0.53, rather than 0.50. And it is not caused by the small sample, the more data we use, the more accurate indicator will be in the 0,53 area.
So, using the proposed methodology, I analyzed 500 000 independent values and compared the real RTS market to them. Then I purposely implemented a deterministic component in a random series: if two previous values were negative, to the current value 1/2 of the standard deviation is added (trend series) and vice versa: if two previous values were negative, to the current value 1/2 of the standard deviation is added (anti-trend series). Here are the obtained charts:
As can be seen, the RTS market is not qualitatively distinguishable from a random walk, while trend and anti-trend series show the expected characteristics.
Now let's look how this indicator looks in the dynamics:
As we can see, there are two main problems with the indicator: on sharp reversals, the MO will not be significant, while the swing will be high, which leads to an unreasonable overstatement of the indicator. On the contrary, in a clear uptrend the MO will be the main portion of the movement, but fluctuations around the MO will be small and thus the heurst will be again lower than it should be.
Thus, we can make a preliminary conclusion that the suggested method can not adequately describe the market price movement and effectively identify trend and anti-trend components.
Despite the public's tepid interest in the topic...
What is the distribution of "known random data"?
May I divulge the methodology of generation?
So, using the proposed methodology, I analyzed 500,000 independent values and compared the real RTS market to them. Then I purposely introduced a deterministic component into random series: if two previous values were negative, to the present value 1/2 of a standard deviation is added (trend series), and vice versa: if two previous values were negative, to the present value 1/2 of a standard deviation is added (anti-trend series). These are the graphs that came out:
And I don't understand the way to get an "anti-trend" series.
And the definition itself is a bit strange.
How is it a flat? and why only two negative ones are used and not three positive ones?
;)
Despite the public's tepid interest in the topic, I am still following Peters' book. I've improved the methods and finally understood what is being calculated. So, here are the basic formulas:
The first thing we do is convert the price series to logarithmic returns. The essence of the second formula is this: the average of logarithmic returns is the mathematical expectation of the series, if we calculate the average of the series from each value, then the sum of these values will always be equal to zero. This is the simplest way of detrending, but it has a huge impact on the result. Next, the minimum and maximum of the series are calculated. These calculations can be shown graphically in the graphs:
The third formula, as everyone understands, is the calculation of the simple standard deviation.
H is therefore the simple ratio of the range to its logarithmic period.
Then the entire Peters series under analysis is divided into independent sub-periods. Each sub-period is calculated according to the methodology described above. As a result, there is an average value of RS, which is qualitatively different from Brownian motion. Since the dispersion of particles will be directly proportional to the logarithm of the period, the Hurst ratio, i.e. the ratio of timespan to period, must be a constant and be 0.5. In fact the formula is not perfect and tends to overestimate the result by 0.3, i.e. on obviously random series, Hurst will show 0.53, rather than 0.50. And it is not caused by the small sample, the more data we use, the more accurate indicator will be in the 0,53 area.
So, using the proposed methodology, I analyzed 500 000 independent values and compared the real RTS market to them. Then I purposely implemented a deterministic component in a random series: if two previous values were negative, to the current value 1/2 of the standard deviation is added (trend series) and vice versa: if two previous values were negative, to the current value 1/2 of the standard deviation is added (anti-trend series). Here are the obtained charts:
As can be seen, the RTS market is not qualitatively distinguishable from a random walk, while trend and anti-trend series show the expected characteristics.
Now let's look how this indicator looks in the dynamics:
As we can see, there are two main problems with the indicator: on sharp reversals, the MO will not be significant, while the swing will be high, which leads to an unreasonable overstatement of the indicator. On the contrary, in a clear uptrend the MO will be the main portion of the movement, but fluctuations around the MO will be small and thus the heurst will be again lower than it should be.
Thus, we can make a preliminary conclusion that the suggested method can not adequately describe the market price movement and effectively identify trend and anti-trend components.
Proving the invalidity of any known theory is a great success. It is clearing the path to true knowledge. Congratulations!
Where's the proof? And of whose failure?
I didn't get it in my sleep...
Why does each conversion of source row to returns - Close[i] / Close[i - 1]?
Judging by the screenshots and the reference to detrending, it is incorrect to talk about MO (especially the returns-row). In this case you are referring to MO as a linear regression of a sample of price series. It is by subtracting it that you get detrending. And, in fact, the green line on your screenshot is not MO (it should be horizontal), it is a linear regression.
From the formula you can see that Hurst is the ratio of the maximum spread to the average spread of the detrended price series. Dividing by the logarithm of the sample size is just some formal fitting (normalisation). The point is the ratio of the maximum to the average.
Any such analysis is highly dependent on the condition of the original series. I.e. by what condition the i-th element is taken. You have the classic one - in equal intervals of time. But there are other methods that allow to take into account both High and Low prices for these time intervals. So there is much less loss of information.
Proving the invalidity of any known theory is a great success. It is clearing the way for true knowledge. Congratulations!
Is this sarcasm? I'm not trying to disprove anything, I just calculated the indicator using the suggested methodology - the output is indistinguishable from SB.
And I don't understand the way to get an "anti-trend" series.
and the definition itself is a bit strange.
Why are only two negative ones used, and not three positive ones?
The data was generated using an Excel add-in: "Random number generation".
The definition of "flat" is not quite correct. In this case it means the antipersistent series. The methodology has been specifically tailored to formula #2. As you can see, the formula is designed to "catch" just such perturbations. "two negative" is an arbitrary choice. The effect will be traceable for any number, as long as it is smaller than the sampling period (the so-called Peters memory effect).
The data were generated through an Excel add-in: "Random Number Generation".
The definition of "flat" is really not quite correct. What is meant here is an antipersistent series. The methodology was specifically tailored to formula #2. As you can see, the formula is designed to "catch" just such perturbations. "two negative" is an arbitrary choice. The effect will be seen for any number, as long as it is smaller than the sampling period (the so-called Peters memory effect).
So does this superstructure generate a uniformly distributed, normal or some other "random number"? Or don't you know?
Persistence and "trendiness" I take it you have the same thing?
ваЗачем каждый раз идет преобразование исходного ряда к returns - Close[i] / Close[i - 1]?
The conversion of the original series to returns only takes place once at the start of the calculation. Then, as you can see from formula #2 returns are reassembled into a consecutive series of increments.
Judging by the screenshots and mentioning of detrending, it is incorrect to talk about MO (especially about returns-row). In this case you are referring to MO as a linear regression of the sample price series. It is by subtracting it that you get detrending. And, in fact, the green line on your screenshot is not MO (it should be horizontal), it is a linear regression.
To avoid confusion let's look at the definition of IR: expected payoff is the average of a series of returns of a random variable. If a series of returns is cumulative, then for it the expectation is the sum of the increments of this series, or the simple difference between the final and the initial value. This is so because if the expectation is zero, then the difference between the ending and starting points of such an accumulated series will also always be zero, which can be clearly seen in the graph. Thus, subtracting the mean from the series is the easiest way to detrend. Basic statistical methods such as RMS do just that. Linear Regression, which you mentioned, is a bit different, it is searched through M.N.C. and generally more adequate to remove the trend component. But the figure shows exactly MO, but in the context of an accumulated series.
Any such analysis is highly dependent on the condition of the original series. I.e. by what condition the i-th element is taken. You have the classic one - after an equal interval of time.
Totally agree, my researches have shown, that proposed formulas do not work with returnees as such, but work with an accumulated series, but without taking into account its MO, which leads to loss of some information (value of MO proper), though visually the charts are almost the same as the original price ones.
But there are other methods, allowing taking into account both High and Low prices for these time intervals. I.e. there is much less loss of information.
I agree, the method is very crude and I don't think it's correct. Only two points are taken from the set, MO is discarded altogether. As a result irreversible information losses and incorrect work on initial series with non-stationary expectation. The way out is seen in application of ZigZag as a universal fractal ruler. For example, it can be the ratio of the traveled distance to the zig-zag knees.
So this superstructure generates a uniformly distributed, normal or some other "random number"? Or don't you know?
Persistence and "trendiness" I take it you have the same thing?
The distribution is normal, with zero MO and a given standard deviation. In this context, persistence and trending are the same thing. When I say "trend series" it means that the probability of coincidence of the sign of the increment with the sign of its previous returns is above 50%, anti-trendicity is the opposite, the probability of coincidence of the sign is less than 50%. This is not my definition, but exactly what is meant in the book.