a trading strategy based on Elliott Wave Theory - page 186
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The Hurst exponent is an integral characteristic of time series and describes the diffusion rate (the amount of deviation from time) of the quantity of interest. As a consequence, a lot of interesting points are simply not taken into account. Much more informative is the construction of correlogram of residual time series. As a special case, we can get an estimate of the Hearst exponent from it, but in addition, we have in our hands a powerful tool to determine more subtle and important indicators of the time series.
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Interesting interpretation of Hearst 's index, I haven't encountered such an understanding yet. The explanation "the value of the deviation from time" I admittedly did not quite understand.
Much more informative is the construction of correlogram of residual time series. As a special case, the Hurst exponent can be estimated from it
I'm currently finishing a working version (more accurate) of the indicator calculation, but using wavelet analysis. If you don't mind, tell me or give me some links how to get the Hurst index from the correlogram.
There are a lot of variants of its calculation. :о)
And there are many variants of its calculation, indeed. :o)
The volatility of an instrument s as a function of the number of bars n (or timeframe t) is calculated as the volatility determined on the minimum timeframe s0 multiplied by the ratio of the timeframe of interest t related to the minimum t0 and all this in the power of the Hurst index:
s=s0*(t/t0)^M where M is the Hurst index. Usually, for an integral time series based on a stationary normally distributed random variable, the Hurst exponent is 1/2 and indicates the unpredictable nature of the price formation. In this case the price after time t with the probability of 63% will be situated in the price corridor with the width s. Actually, I tried to call it a diffusion rate, perhaps too hastily :-) If Hearst's value is more than 1/2, then we may speak about the trend market, if it is less - about the backward price behaviour. Perhaps, this is all that can be extracted from the Hurst index analysis.
Not much, for the sophisticated researcher. The same, and much more detailed information about the price formation mechanism can be obtained from the analysis of the sample analogue of the autocorrelation function.
I can't remember it off the top of my head. If I remember, I'll give you the link.
As for volatility, how s0 is defined. If you can, give me a link or tell me more about it. I don't really understand. By timeframe, what do we mean by this formula?
The spectral density p(omega) of a stationary time series is defined by its autocorrelation function:
p(omega)=SUM(r(k)*exp{i*omega*k}), where the summation is from -infinity, to +infinity.
Since r(-k) = r(k), the spectral density can be written as:
p(omega)=1+2*SUM(r(k)*cos{omega*k}), where the summation is from 1, to +infinity.
Hence, the function p(omega) is harmonic with period 2Pi. The graph of the spectral density, called the spectrum, is symmetric with respect to omega = Pi. Therefore when analysing the behaviour of
p(omega) is restricted to values 0<=omega<=Pi/dt or by f from 0 to 1/(2*dt). It has the dimensionality of the square of the amplitude referred to a frequency unit.
The use of properties of this function in applied time series analysis is defined as "spectral time series analysis". A reasonably complete description of this approach is given, for example, in [Jenkins, Wats (1971, 1972)] and [Lloyd, Lederman (1990)].
As a rule, in the frequency analysis of filters, the value dt of the sampling interval is taken as 1, which respectively defines the frequency response on the interval (0...Pi) by frequency or (0...1/2) by f. When the fast Fourier transform (FFT) is used, the spectra are computed in the one-sided variant of positive frequencies in the frequency interval from 0 to 2Pi (from 0 to 1 Hz), where the complexly conjugate part of the main band spectrum (from -Pi to 0) takes the interval from Pi to 2Pi (to accelerate computation the periodicity principle of discrete spectra is used).
It is important for the meaningful analysis that the spectral density value characterises the strength of the relation that exists between the time series xt and the harmonic with the period 2Pi/omega. This makes it possible to use the spectrum as a means of capturing periodicities in the analyzed time series: the set of spectrum peaks determines the set of harmonic components in the expansion. If the series contains a hidden harmonic of the omega frequency, it also contains periodic terms with frequencies omega/2, omega/3, etc. This is the so called "echo", repeated by the spectrum at low frequencies.
Grasn, about the volatility.
Its calculation doesn't differ from the standard deviation estimation :
s0=SQRT(|SUM{High[i+1+k]-low[i+k]}^2|/{k-1}) where the summation is performed over all k from 0, to n. For the statistical reliability n should be greater than 100. s0 using this formula is calculated for the minimum timeframe, it usually is minutes. Knowing how the Hurst index depends on the timeframe you can find the value of volatility at any timeframe using the formula that is given in the post above. The reverse is also true: if you build the dependence of volatility on timeframe using the above formula after processing the statistical data, it will not be difficult to calculate the Hurst index.
Grasn, about volatility.
Calculating it is no different than estimating the standard deviation:
s0=SQRT(|SUM{High[i+1+k]-low[i+k]}^2|/{k-1}), where the summation is done over all k from 0, to n. For the statistical reliability n should be greater than 100. s0 using this formula is calculated for the minimum timeframe, it usually is minutes. Knowing how the Hurst index depends on the timeframe you can find the value of volatility at any timeframe using the formula that is given in the post above. The reverse is also true: if you build the dependence of volatility on timeframe using the above formula after processing the statistical data, it will not be difficult to calculate the Hurst index.
This is the point I do not understand.
I've got to get serious about DSP.
Neutron, in the above formula s0=SQRT(|SUM{High[i+1+k]-low[i+k]}^2|/{k-1})
there is something unclear. Perhaps the problem is that writing formulas in text format does not show all the subtleties. Could you please explain
1. Why do we need modulus of sum of squares of differences, if it is already a positive value
2. Why {k-1} in denominator is behind sum sign, if summing is done by
3. Why high and low refer to adjacent, not one, bars
By the way, grasn, remember our discussion about volatility ? Neutron, as you can see, states the same as I do: volatility is measured by standard deviation.
What's not clear? How the formula is derived, how one thing is expressed from another, or just, nothing is clear?
Just kidding!
I guess I have to get serious about DSP.
By the way grasn, remember our discussion about volatility? Neutron, as you can see, states the same as I do: volatility is estimated by the value of standard deviation.
I understood that, although I haven't come across such a definition of volatility. I am interested in this parameter as a qualifying criterion for choosing a reliable channel. I will have to see what I will get. Especially since there is a link with the Hurst index.
PS: DSP is indeed an interesting field and I remind you that you have already joined the ranks of "digitizers".