Tics: amplitude and delay distributions - page 4
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The first figure at the beginning of the branch shows a typical noise exponent. Exactly the same
exponent is obtained if you calculate, for example, the number of points that the rate passes in
5 minutes and then create a histogram N from the number of points.
The second figure shows the change in market volatility over the course of a week - the variability there is apparent, its changes are also random in nature.
I am not saying that there is a strictly deterministic periodicity there, but the statistical regularity is there and has an objective character (the Asian lull). In my opinion, the "deterministic part" of the process can be modelled with acceptable accuracy by a periodic function.It is more advantageous to look for long-term patterns.
Thanks again for the reminder. I am doing the same thing, and have decided to analyse ticks not to directly profit from their behaviour, but to, shall we say, form sensible risk management tactics.Thanks for the valuable information, New. Please explain what a "typical noise exponent" is, i.e. which specific probability density function it corresponds to. You don't have to give the formula; just give its name as accepted in statistics.
Yes, it's more of a slang term. If for example the magnitude of the signal amplitude is randomly distributed,
then the spectrum would be similar to the first figure, i.e. the number (number) of signals with higher amplitude
would fall exponentially. If there were any anomalies (regularities), there would be "peaks and spikes on
this inverse exponent.
The Asian lull is of course an objective thing, unless the Japs go wild, but I think it is difficult to use
.
If for example the amplitude of the signal is randomly distributed,
then the spectrum will be similar to the first figure, i.e. the number (number) of signals with higher amplitude
will fall exponentially. If there were any anomalies (patterns), there would be "peaks" and "spikes" on
this inverse exponent.
And second: note that the first graph is not a histogram of amplitudes, but a histogram of lags. Almost everything is more or less clear with amplitudes.
P.S. I did not find the term "noise exponent" in the Internet.
I highlighted the critical words in your reply. How random is that?
And second: note that the first graph is not a histogram of amplitudes, but a histogram of lags. The amplitudes are more or less clear.
Actually, it does not matter which distribution is Gaussian or Poisson exponent here and there.
Suppose the lags are distributed according to Gauss. Suppose that the Gaussian maximum of lags lies in the 1-second region, then the number of lags with duration t will be N0*(1/exp(t-to)) with some kpf factor, where N0 is the number of lags at the maximum. To identify the specifics of the distribution one needs to carefully study it near the maximum (you have it near 1-second), but in practice this is usually not necessary, and often impossible due to errors and limitations - hence the generalized slang term noise exponent. In practice again it is more important to find deviations - if you had a lag number peak for example around 50 seconds with N of say 3000 then it would be interesting.
Of course, there is no special difference between Gauss and Poisson distributions in the end: there is a single peak in both cases, and the behavior of all curves near the maximum is the same (parabola), which makes it easy to ignore the 3rd and 4th moments of distributions (asymmetries and excesses). In general, the differences between all single-mode distributions are absolutely ephemeral - especially if they both have the same exponents. One can also forget about heavy tails, it is all nonsense, from the evil one...
P.S. of 31.10.2012: It was a joke, but I was not understood then...
Be so kind as not to quit halfway through.
P.S. They have. One only for now. I did so: on the second chart from the first page of the branch, to somehow smooth out the frantic differences in tick delays, I simply calculated their logarithms. Here's a pseudo-random delay logarithm process for a couple of weeks in April (1st and 2nd):
.......................
Both processes have become more "homogeneous" compared to the processes of the delay times themselves. The logarithms of the lags are now numbers in the intervals from about 0 (lag = 1 second) to 7 (lag greater than 1000 sec). ..............
I suspect that the "quasi-stationarity" of the lag logarithm process as a function of time did not appear here by accident. ..........
you will very likely get something close to Gaussian.
This is a general pattern from statistics - sort of a corollary to the central limit theorem (CLT).
If a random variable is unbounded (i.e. can take values from minus to plus infinity),
then according to the CPT many random factors will drive the distribution function of that variable to the normal law.
Assuming, of course, that all the assumptions of the TPT are met.
Similarly, if a random variable is strictly positive
(e.g. the time interval between the previous and subsequent events),
then that random variable will obey a lognormal distribution.
Or, likewise, the logarithm of that quantity will obey a normal distribution.
These statements are true for a great many random variables.
For example for prices, for the size of deposits in a bank, for the height of people, etc.
Similarly, if a random variable is strictly positive
Mak, read Peters, he's on Spider. He will quickly dispel your dreams of normality/lognormality in the market. Anyway, risk assessment based on the normal hypothesis is very much at odds with reality.(e.g. time interval between previous and subsequent events),
then this random variable will obey a lognormal distribution.
My daydreams on these topics went out about seven years ago.
Mak, maybe you're right about lognormality of ticks distribution on lags, but it can't be proved directly...
1. p.d.f. lags ticks:
2. lags as a function of time (a few very large lags had to be removed to see more clearly the areas of lag concentration; these large lags are actually very few):
3. pdf of amplitudes:
What do we see? No surprises in the third graph (as with EURUSD, there are two sharp peaks), but the first two make us think: pdf lags have clearly marked extremums in the area of even seconds and the lag versus time function confirms this. Perhaps, this is connected with peculiarities of quotation of this index.
It is interesting to note that similar charts/histograms for gold do not show anything too special compared to EURUSD, although they are admittedly much more "noisy".