Calculate the probability of reversal - page 6
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So it's smoothing.
)
It's a Gaussian mix approximation...
)
It's a Gaussian mix approximation...
Approximation by mix is a little different.
)
It's an approximation by a mix of Gaussians...
Well, I'm interested in the final result, not the individual Gaussians.
Then you need to find "applied statistics" by Kobzar and look there the second chapter)
Looked, but even a word such "approximation" did not find)
In general, it is strange. There are a million complicated ways of estimation. Except for one, the most understandable, simple and accurate - approximation.
Maybe I do not understand something?
I looked it up, but didn't even find the word "approximation".)
It's weird. There are a million clever ways of estimation out there. Except for one, which is the clearest, simplest and most accurate - approximation.
Maybe I do not understand something?
It is possible to come up with an enormous number of various estimates. You only have to then establish their consistency and unbiasedness, at the very least. This is not true for all "precise, simple and straightforward" estimators, a typical example is the denominator n-1 in an unbiased variance estimator.
If the existing estimate is also efficient and sufficient, then the invention of new estimates is either pointless at all, or must have some additional basis. Usually these are robustness considerations, dealing with small samples, outliers, missing values, etc.
What is meant is not a time series, but a histogram close to normal.
How deep into probabilistic approaches people go here and even forget that a histogram is just "a way of graphically representing tabular data" (Wiki), which says nothing about its content. As far as I can guess you are talking about a table of relative (relative to their total sum) sample frequencies Hi of some event x>xi, presumably close to a normal probability distribution. And about replacing it with values of normal distribution probabilities so that the error is minimal in some sense. What don't you like about formulas for direct calculation of these parameters, expectation and variance by their definition?
If xi in the table are equidistant, then:
- expectation is fitted simply as the arithmetic mean of all realisations = the weighted average of table values with weights equal to Íi from that table;
- dispersion - as the square root of the standard deviation (weights are the same Hi), or, if you want greater accuracy of estimation, not the standard deviation, but the standard deviation (the only difference is whether to divide by n or by n-1). Standard deviation estimation is unbiased.
...
Here, on the x-axis, you can see how many steps the person took away from the starting point, from -10 (to the left) to +10 (to the right) and the probability with which they did it in %. How do you find what the probability of turning around was at each step?
Your example is most likely the result of a simulation of the venerable Galton board with reflectors.
At any rate, it's very similar.
is very plausible at 10
iterations (i.e. a 'house' type board) for a Markov chain with transition probability matrix -
0.75 0.25 0 0 ... 0
0.25 0.5 0.25 0 ... 0
...
0 ... 0.25 0.5 0.25
0 ... 0 0.25 0.75
initial state 0 0 0 0 0 1000 0 0 0 0 0. i.e. from relative zero)
Why are you not happy with the formulas for directly calculating these parameters, the expectation and variance by their definition?
For example, they do not show fat-tailedness.
For example, they don't show fat-tailedness. And the Gaussians, based on them, are likely not to converge with the data in either the tails or the centre.
But nevertheless the significant difference is only in the outermost bunkers. As I said above, this is because of the reflective walls).
For example, they do not show thick-tailedness.
It is not the ways of estimating the parameters of a normal distribution (fitting, approximation) that show it at all. It is the normal distribution itself that does not have thick tails. Ask Alexander_K2, he was looking for those tails. Just look at the table with unit parameters. There are tables in every TV textbook, I think, and in every maths reference book. No matter how you adjust, you have to change the variant distribution to catch fat tails. And why exactly do you need a probability distribution? Exactly the probability distribution? Why these stamps for "some data"? Or is it not some data after all, but rather sample relative frequencies, as I guessed?
Maybe the point is that the probabilistic representation does not describe your data at all? Remember, how the expectation dances on Yuriy Asaulenko's pictures https://www.mql5.com/ru/forum/221552/page162#comment_6399653 on Forex rates. Don't you want to use the probabilistic representation for them? Then it is clear where the heavy tails come from.