a trading strategy based on Elliott Wave Theory - page 18
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I've read the literature and this is what I've come to:
Given: parabola y = A*x^2, point P = (Xp, Yp)
Find: the distance from P to the parabola.
From P to the parabola, draw a perpendicular (the normal to the parabola passing through P)
Denote by O = (Xo, Yo) the point of intersection of this normal with the parabola
Tangent to the parabola in point O has tangent angle tan(a) = 2*A*Xo (value of derivative in point O).
The tangent to the parabola in point O has to be perpendicular to vector OP.
From this we obtain a system of equations:
1. Yo = A*Xo^2 (the value of the parabola in point Xo)
2. tan(a) = 2*A*Xo (the angle of the tangent in point O)
3. cos(a)*sin(a) + (Xp - Xo)*(Yp - Yo) = 0 (the condition of perpendicularity of vectors)
now we have a system of three equations with three unknowns (Xo, Yo, a), so it can be solved.
rewrite Eq. 2 with sin and cos
substitute value Yo (from 1st equation) into 3rd equation, and we obtain a system:
1. sin(a) = 2*A*Xo*cos(a)
2. cos(a)*sin(a) + (Xp - Xo)*(Yp - A*Xo^2) = 0
we have got a system of 2 equations with 2 unknowns (Xo, a) which is better ;)
now express Xo from equation 1 and substitute this Xo into equation 2.
we obtain a trigonometric equation with one unknown (a)
Once you solve and find (a) you can reverse order to find Xo, then Yo
and then using Pythagoras we find the distance OP.
that's it :)
The only thing left - to solve the last equation, and it is not a small turn out.
Who wants to try it?
Thank you! Really simple geometry I'm a bit rusty :o)
There are even some ready-made algorithms on the web for solving cubic equations. Here is the first one with C code example:
http://algolist.manual.ru/maths/findroot/cubic.php
Thank you! Indeed, I've forgotten a bit about simple geometry :o)
There are even ready algorithms on the web for solving cubic equations. Here's the first one with C code example:
http://algolist.manual.ru/maths/findroot/cubic.php
Sorry for late reply. In general it is true that it is a parabola. Only you have not taken everything into account and risk falling to the level of "impossibility of approximation", let's call it that. My point is - you do not know exactly what the parabola itself is, but from the potentiality of the price field it follows that it is a parabola and if you incorrectly define or approximate the equation, it is unclear what you will get. Read carefully what I wrote above - you don't need a trajectory equation, you need a pivot zone. In mathematics you cannot always get an exact answer, but you can almost always estimate it - this is done by limiting transitions. And the integral methods I used work precisely because they are not related to the quality of approximation, but evaluate the solution, which is built on the above principles. Let me try to explain: most people try to identify the distribution of price in samples to construct confidence intervals. And due to their inability to do so precisely, they announce it as white noise, completely ignoring the existence and proof of the central limit theorem of statistics - any convergent distribution (which means that the area under the distribution curve is finite - more strictly: the non-integer integral converges) converges to normal with increasing degrees of freedom. So you don't really care about the shape of the curve to estimate the area - it is sufficient that the number is finite - then you can apply estimates. And here as well - you don't need the traetcurve itself - you need the zone of its extremum and this can be estimated using integral methods. So the whole task comes down to determining the convergence of samples and the use of mathematical estimations based on the principles mentioned above.
Good luck and good luck with trends.
That is as far as I understood the task is first to find such a sample of price series, in which when approximating it by any more or less true parabola the sum of squares of distances from price series points to this parabola will not change too much when varying the parabola coefficients? In other words, first we come up with an assumption about the existence of such an "optimal" sample, for which the sum of squares of distances does not change significantly (strictly within certain limits) when varying the parameters of the parabola? Actually since I haven't encountered such information anywhere, it is almost a discovery for me, if I may say so!:o) At first sight it is of course unbelievable, but if you have defined an extreme sample in such a way, this supposition must be true. Let's check it.
And then having such an "extreme" sample, we simply count the number of points situated at different intervals of this parabola. Further, knowing that the area under the curve of the price series and the parabola must be equal to the certain value, we determine the difference between what we have calculated using the available data and what should be in the interval according to the normal distribution. Then we sum up these differences separately to the left and to the right of the parabola. As a result, we get a ratio, e.g. the sum of the differences on the left refers to the sum of the differences on the right as 20/80% (probability of going up = 20%, probability of going down = 80%). Am I getting it right now or not really? Correct me then, please!
It's easier to solve with a distance function:
R = sqrt((Xp - Xo)^2 + (Yp - Yo)^2)
R = sqrt(Xp^2 - 2*Xp*Xo + Xo^2 + Yp^2 - 2*Yp*Yo + Yo^2)
substitute Yo = A*Xo^2:
R = sqrt(Xp^2 - 2*Xp*Xo + Xo^2 + Yp^2 - 2*Yp*A*Xo^2 + A^2*Xo^4)
it is easier to take dR^2/dXo instead of dR/dXo:
dR^2/dXo = -2*Xp + 2*Xo - 4*Yp*A*Xo + 4*A^2*Xo^3
By equating dR^2/dXo to zero, we obtain a cubic equation of the form a*X^3 + b*X + c = 0
a = 4*A^2
b = 2 - 4*Yp*A
c = -2*Xp
As far as I remember, both the central and the integral limit theorems refer to a sample where N -> infinity.
It's not clear how it can be relied upon when using a small sample size (number of bars)?
Moreover, they are formulated for equally distributed random variables, which is not the case with the market.
And finally all the theorems are based on the assumption that events are independent - one can argue much about it - whether market fluctuations are independent variables, but it seems to me that they are not.
Again due to "inertia" of the market, otherwise there would be no such thing as a "trend", which implies "dependence" of the market.
It would be interesting to hear comments...
Moreover, they are formulated for equally distributed random variables, and I think the market isn't.
And finally all the theorems are based on the assumption that events are independent - one can argue much about it - whether market fluctuations are independent variables, but it seems to me that they are not.
Again due to "inertia" of the market, otherwise there would be no such thing as a "trend", which implies "dependence" of the market.
Maybe the essence of the idea is that if we approximate this small sample, for example for a time period of 3-6 months by a parabola, then in terms of the parabola it is possible to apply this reasoning? That is, we end up with estimates in the plane perpendicular to the line of the parabola and not those estimates parallel to the price coordinate that everyone understands. I understand that Vladislav applies the same integral estimations to linear regression channels. That is, the probability of reversals for a linear regression channel can be determined using the same integral methods. And by simply analyzing information from different channels (linear regression and parabola) it obtains a more accurate estimation of market conditions (probability of reversal and continuation of movement).
However, I do not fully understand the question of estimating possible reversals in time. For example, Vladislav, do you use a simple postulate of the Murray's theory that if we take a period of time according to which levels are calculated and divide it into 8 parts, then in the areas of these parts there should be some critical points (reversal or breakout points)? That is, if we take the default parameters of the indicator P=64 (Period of 1440 - 1 day), then having divided by 8 we have an assumption that such crisis events must occur approximately every 8 trading days? Or something like that? Can you tell me please? Because if you use something else (e.g. somehow integral estimates of the probability of reversal), then at first glance the idea of time-based forecasting is not clear. Can you please tell me what the point is here?
Good luck and good luck with the trends.
Vladislav, can you describe the use of the standard deviation in your strategy in more detail in terms of estimating the position of the confidence interval we are in at the current moment in time? Suppose we have already found the optimal parabola(s) and linear regression channel(s) (based on the Hurst coefficient) through a head-to-head recalculation of all possible samples over the last six months, and know the current probability of reversal based on an integral estimation method. How can we now apply the standard deviation in this whole system as well? That is, what parameters should be chosen to calculate the standard deviation values? Maybe, in this case, we should just make the mouvings chart for which the standard deviation is calculated coincide as much as possible with the obtained optimum parabola or something else? That is, to begin with we simply plot, for example, a standard MA (or an unusual one - tell us which one?) and compare its divergence with an optimal parabola for the last week, for example, fitting this parabola with the value of the bar number parameter for which the MA is calculated. And then having obtained the value of МА parameters we bring it to the standard deviation indicator and thus find the deviation, by which we determine the confidence interval from the line of the optimal parabola? Or am I mistaken? Correct me, please!