Volumes, volatility and Hearst index - page 9
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Temperatures do not flow from Brownian motion, nor do timeframes flow from ticks. On a neighbouring thread, to Prival, a known ticks supporter, I gave two pictures.
EURUSD30 - 7200 bars
EURUSD60 - 3600 bars
We can see that frequencies are different. It is an obvious fact that Open60[0] = Open30[0] and Close30[1] = Close60[0], while the result of the Fourier analysis is different! But this is only at first glance.
The ticks from which the corresponding timeframes are obtained are all different. Some ticks relate to a pipsqueak investor, other ticks relate to investors with other timeframes. In addition, each tick has different pose sizes behind it (which we don't get). On what basis are we combing all economically different ticks under the same heading? Of course, all timeframes are related. What is a trend on one, is a correction on the other.
It is nonsense to attribute ticks to investors, and even to classify them as pips or non-pips. This simple truth is beyond the grasp of many. Bars consist of ticks. You can slice bars any way you want with ticks, not just candlesticks, which are 2 centuries old.
Z.S., this is zombification ... take the blinders off.
Take your blinders off your eyes. Give me the formula, this is an economic tick and this is not an economic tick ...
1. What do you think "average mileage" is? A definition is desirable.
2. where did the formula 1) come from ? What is the k-factor ? Is it what you call the "Hurst coefficient" ?
4. The k coefficient does not appear anywhere in the table, and the fact that according to the results of this table h -> 1/2 is only a consequence of the fact that pure SB is considered. The asymptotic tendency to 1/2 can hardly be called a happy fact, since the case of SB is only a boundary case on which one can check the calibration. As a result of this check it turns out that we can only obtain 1/2 for the Hurst exponent asymptotically, in the limit of large N. Do you think it will work in practice?
I don't know where you got this formula from, but the Hurst exponent is not there.
And what I'm counting, unfortunately, you haven't understood at all. However, if it was a question (there was an unexpected question mark at the end of an affirmative sentence :-), I can assure you - it hadn't even occurred to me.
Formula 1) is taken from a probability theory textbook on random walk. The coefficient k relates the number of steps in a random walk to the average distance traveled in N steps and k is not the Hearst coefficient at all. I explicitly wrote that the Hurst coefficient is the sqrt, i.e. the degree to which N is raised, and for a random walk the Hurst coefficient is 1/2.
With the help of the formula about random walk I've given you a diagram of how your Hurst is asymptotically tending to 1/2 from above. If you didn't understand about random walk or think it doesn't apply to your calculation, forget what I wrote to you.
Just answer, don't you find your table odd in that your Hurst is never less than 1/2 for randomly generated numbers?
Just in case:
The first result of this study is the demonstration that when N is small, the Hearst exponent for random walk is significantly different from 1/2.
That is, when you read that the market is not random because the Hearst exponent for it is greater than 1/2, you must first of all ask yourself: On what statistics did the author draw this conclusion.
The second result of this study is the tabulation of the dependence of the Hearst exponent for random walk on N.
That is, if you have a time series with not too much N and want to use the Hearst exponent to determine its proximity to a random walk, you should calculate the Hearst exponent and compare it to the corresponding number from this table. Not with 1/2.
Formula 1) is taken from a probability theory textbook on random walk. The coefficient k relates the number of steps in a random walk to the average distance traveled in N steps and k is not the Hearst coefficient at all. I explicitly wrote that the Hearst coefficient lies in sqrt, i.e. the degree to which N is raised, and for random walk the Hearst coefficient is 1/2.
With the random walk formula I gave you a layout of how your Hurst tends asymptotically to 1/2 from above. If you didn't immediately understand about the random walk or think it doesn't apply to your calculation, then forget what I wrote to you.
Just answer, don't you find your table odd in that your Hurst is never less than 1/2 for randomly generated numbers?
Please provide a link to a textbook. The formula High - Low = k * sqrt(N) is a loose (and incorrect) transposition of Hurst formula R/S = k * N^h, where the average R is the average value (High - Low). The root arises only for SB, so it turns out that for SB it should be h = 1/2. It should, but it doesn't. Which is what my table shows.
So I don't find it odd that your Hearst score for SB doesn't happen to be less than 1/2. But I do find it odd that for SB it is greater than 1/2 all the time, and tends to that value only asymptotically as N increases.
Please provide a link to a textbook. The formula High - Low = k * sqrt(N) is a loose (and incorrect) transposition of a formula - it is not a Hearst transposition. It is a theorver theorem for SB. I used it to show why in your table the values for SB are >1/2 all the time. You see, the theorem for SB predicts the result of your calculation for SB, which you pass off as Hearst. It's you pleasing Hearst by the ears where it doesn't exist. The SB theorem is sufficient to explain your results. Hurst's R/S = k * N^h, where the average spread of R is the average value (High - Low) is not correct, it is not R/S analysis, it is self-referential. Hearst's R/S analysis has no R as an average value, this is your fiction. The root occurs only for SB, which is why it turns out that for SB it should be h = 1/2. It should, but it doesn't happen. - To clarify. It does not happen according to your NOT correct Hearst calculation formula - Which is what my table shows. - Your table shows the result predicted by probability theory, which is not surprising. What is surprising is your conclusion when your calculation does not match Hearst's theory for SB.
So I don't find it strange that for SB the Hearst exponent is never less than 1/2. But I do find it odd that for SB it is greater than 1/2 all the time, and tends to that value only asymptotically as N grows. - SB loving only persistence is nonsense.
The third column in Table 2a shows the value of K - the number of intervals that had to be generated to get the given accuracy acc=0.001. If we take into account that the total number of all possible trajectories is 2^N, then starting from N=32 the number K is a tiny fraction of this total number. And with increasing N this fraction rapidly decreases.
However, from the practical point of view this is of little joy. The interval N=16384, based on the density of ticks in 2009, corresponds to about one day. To calculate the average range R with an accuracy of 0.001 in a stationary market would take 2452000 trading days (i.e. 9430 years). It is unlikely to be of interest to anyone. However, if the accuracy is lowered significantly, it may be possible to reach adequate statistical data sets.
The sixth column(D) of Table 2a coincides very precisely in values with the second(N), and the ninth with the 10th(LOG(D)=LOG(N)), as it should be according to the previously given formula for the variance of increments. And the values of R at N=4, 8 and 16 coincide with the corresponding values from the previous table, where exact theoretical values of the mean spread are given. That is, the chosen level of accuracy and the corresponding sample sizes K do ensure the reliability of the resulting data.
The main interest is the last column, where the values of the Hurst index are given. The result in the n-th row was calculated using two points, the n-th and the previous one. Theoretically for the considered SB the Hurst index should have been equal to 0.5. However, as you can see, it is not the case. For small values of the interval N the index differs significantly from 0.5 and only with increasing N tends to 0.5, apparently asymptotically. I would like to underline the fundamental nature of this point: choosing different values of intervals into which we divide the series in order to calculate the Hurst exponent, we will get quite different values. Therefore, trying to evaluate the character of SR using the Hurst index, we should either have a tabulated curve for pure SB (this is the required calibration) with which to compare data from the experiment, or use very large intervals. Both options are practically unacceptable for real-world use.
I have bolded and underlined your words. After them, I would conclude that I do not calculate Hearst correctly, especially since this Hearst for SB in your table 2b, is always greater than 0.5. But here I am prompted that you have made a little discovery. It is suggested that you use your table as a normalisation, viz:
The second result of this study is to tabulate the dependence of the Hurst index For random walk on N.
That is, if you have a time series with not too much N and want to use the Hearst exponent to determine how close it is to a random walk, you should calculate the Hearst exponent and compare it to the corresponding number from this table. Not with 1/2.
To Candid: Yurixx calculates the Hearst ratio incorrectly. It does not agree with the theory for SB. Instead of pointing out his mistake, you propose to use this miscalculated coefficient for rationing? That's awful. If I have a time series with not too large N and wanting to use the Hurst coefficient to determine the degree of its closeness to a random walk , first of all I will use a mathematically sound estimate of the Hurst coefficient for my case, but not a table which records 1/2 + k/ln(N). Hearst's estimation for small N is expensive.
To me, what Yurixx reckons is not Hurst. Again, I've already shown why his Hurst in table 2b is greater than 1/2 all the time. All strictly by probability theory. No lyrics like "it should, but I want to call it Hurst".
No, the market certainly has a memory. Except that Peters' methods are questionable. Mainly on three counts: 1. There is no theoretical basis that provides a basis and calibration for comparing calculation results for different cases. 2. The data sets used are too small to provide the necessary level of confidence in the results. 3. In his calculations, Peters has piled up all the fractal levels and assumed implicit stationarity of the series. In our setting this has no value or meaning.
1. "grounds and calibration to compare results of calculations for different cases" - may I ask what this means? Which results need to be calibrated?
2. "The data sets used are too small to provide the necessary level of confidence in the results." - How did you assess this? Hurst, for example, got reliable results on quite a ridiculous number of samples. Can you hail your Hurst result with +/- error?
3. "proceeded on the implicit assumption of the stationarity of the series" - and it is correct that he did so, otherwise you wouldn't have written the book on Hearst in the markets. With non-stationary returns Hurst != 1/2 has nothing to do with persistence.
I think that pronouncing Hurst and kicking Peters would be a good place to start with the results to fit the theory.
to Candid: Yurixx calculates Hearst's coefficient incorrectly. It does not agree with the theory for SB. Instead of pointing out his mistake, you suggest that this miscalculated coefficient should be used for rationing? That's awful. If I have a time series with not too large N and wanting to use the Hurst index to determine the degree of its closeness to a random walk , first of all I will use a mathematically sound estimate of the Hurst index for my case, but not a table in which they are written 1/2 + k/ln(N). Hearst's estimation for small N is expensive.
To me, what Yurixx thinks is not Hearst.
If you are referring to a textbook, then give a specific reference. A textbook is not the same as a textbook. If you remember, the starting point here was exactly Feynman's textbook.
I finally realized what the main bug in Vita's conclusion is - the second assumption, h = log (k * sqrt(N)) / log (N), is also wrong.
The Hurst figure is defined as the slope of log(High - Low) versus log (N), and Vita wrote the slope of the ray from the origin to the point [log(High - Low), log (N)].
This is a standard error and this point was discussed here earlier too.
Finally I realized what the main bug in Vita's conclusion is - the second assumption, h = log (k * sqrt(N)) / log (N), is also wrong.
The Hurst figure is defined as the slope of log(High - Low) versus log (N), and Vita wrote the slope of the ray from the origin to the point [log(High - Low), log (N)].
This is a standard error and this point has also been discussed here before.
Once again, the Hurst exponent has nothing to do with it. Take the textbook "Introduction to Probability Theory" by Kolmogorov. There you will find the formula for the average run at random walk. High - Low is proportional to Open - Close, which is the average run in Yurixx's calculation, which is proportional to the root of the number of Kolmogorov steps. I substituted the formula from the textbook into Yurixx' s formula. Got the result, which agrees exactly with the tabulated calculation. You see, nowhere here Hearst is and has not been since the beginning. Someone may call the red-painted cart a ferrari to attribute properties of the ferrari to his cart, someone may call his home-grown calculation for the derived series Hearst to attribute properties of the Hearst to his calculation.
Ask Yurixx to calculate Hurst for series N*N from 0 to 1000 .
Hearst doesn't care what the series is measured in. For Hearst, substituting 1 pip = 38 parrots changes nothing. Yurixx's formula is killed by this substitution. Level of Nile and other series from everyday life, not to mention mathematical abstractions like N*N*N, cannot be measured by Yurixx'a formula , because the artificial limit imposed on the series has nothing to do with the real world and was written to make the truck red, i.e. "à la Hurst from Yurixx'a" was less than one and for SB tended to 1/2. There is no further resemblance.