Absolute courses - page 9
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No. There is a single solution that does not require the assumption of additional equations. That is, mathematically requiring some kind of addition to the system, but physically not. Say, such solution is possible (I have implemented it): the "principle of least action", i.e. reaching the known (realized) increments ED, PD, EP, for example, or another triangle, by minimal changes (minimizing the sum of modules) separately E, P, D. By minimal relative changes, so that there is something to compare and add up the modules. But the solution found from such assumption will not satisfy the lint test. Let's say, if we find dollar (separately from time in relation to itself in the past) from EURUSD, EURJPY, USDJPY, the result will be similar (this is generally speaking cool, for it means that this relationship - the principle of least action - is much closer to the truth than the equation zeroing the sum of currencies, however it is not exactly true - not exactly similar, not equal to the graph if we find D(t) from another triangle, for example GBPUSD, GBPJPY, USDJPY).
It is argued that the solution found from one triangle must coincide with the solution found from any other triangle, only then it can be considered true.
increments:
Explain how dED (second line, left-hand side) became eED (third line, left-hand side)
I divided the equation from the second line by ED[i-1], isn't it obvious? And dED[i-1,i]/ED[i-1] = eED[i-1,i], that is the relative change in EURUSD at the time interval between bars i-1 and i.
The most advantageous from the point of view of "least action" is just to aim k at zero. This, of course, makes the problem meaningless.
God be with you, colleague. I meant relative increments. Nothing depends on k at all. It simply reduces. And I am not saying that the solution {eED, ePD, eEP} corresponding to the minimal sum of moduli eE, eP, eD is true (e is epsilon). Nope. Not true. But it is at least a more reasonable "third relationship", for the general nature of the change in, say, D(t) will be similar when finding it from different "triangles". But similar doesn't mean equal, so we won't be able to use it. We need an exact solution. And without any additional assumptions, if only "least action".
Now, I hope you understand what I'm talking about.
I don't understand at all :-) Have you learned how to take derivatives?
And you still haven't learned how to take derivatives...
God be with you, colleague. I was referring to relative increments. Nothing depends on k at all.
It just reduces. And I am not saying that the solution {eED, ePD, eEP} corresponding to the minimal sum of moduli eE, eP, eD is true (e is epsilon). Nope. Not true. But it is at least a more reasonable "third relationship", for the general nature of the change in, say, D(t) will be similar when finding it from different "triangles". But similar doesn't mean equal, so we won't be able to use it. We need an exact solution. And without any additional assumptions, at least "least action".
For the reason stated above, the solution {eED, ePD, eEP} corresponding to the minimal sum of moduli or any other norm you specify is zero, or rather an infinitesimal value.
To dispel doubts, I'll explain on my fingers.
1. You introduce some norm N depending on eE, eP, eD, and it must have at least the following properties:
- symmetric with respect to currency substitution for each other
- Monotonicity: N1<N2 if and only if (other things being equal) eE1<eE2 (similarly for the other two currencies)
- equality to zero with eE, eP, eD=0
2. We want to minimise the norm, i.e. to find such a triplet eE, eP, eD for which N(eE, eP, eD)->min when the initial equations are solved.
Let us prove that this is impossible.
Suppose we succeeded, the vector {eE, eP, eD} is successfully matched. However, we may note that, for example, the vector {eE/2, eP/2, eD/2} also satisfies the original equations, hence it must provide a norm greater than {eE, eP, eD} (because it is the point of minimum!). However, the property of monotonicity tells us otherwise. We have arrived at a contradiction, the impossibility is proved.