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The question simplifies to ... how do I calculate standard deviation
at the current bar for all past values of some array without using a
loop, thereby eliminating the innermost loop and leaving only the
outer one.
When looking at most problems like this where the solution may not be
immediately obvious, the simplest way is to break the problem down
into its individual components and use EXPLORE to see that each
calculation is doing what it's supposed to and from the perspective
of speed it won't be any slower to do it this way, in some cases it
may actually be faster i.e. here's the way most people write a
stochastic calc ...
Sto = (C - LLV(C, Length)) / (HHV(C, Length) - LLV(C, Length));
The problem of course is that one has done the calc LLV(C, Length)
twice ... Simpler and of course faster is ...
LLVX = LLV(C, Length)
Sto = (C - LLVX) / (HHV(C, Length) - LLVX);
Back to your problem ...
StDevX = Sqrt(Cum ((X - Average(X)) ^ 2) / n)
Let's assume we want to see how to get the calculations correct at
BarIndex() == 10 ( The 11th Bar ) without using a loop and for the
moment we won't care if the calc is correct at BI() = 9 or 11 because
we know we can always write a loop to go around all of this if we
need to ...
// Let's generate some simple dummy data "X" to
// use where we can easily eyeball the results
// "X" can always be replaced by something real
X = BarIndex() + 100;
// The components
n = BarIndex() + 1;
CumX = Cum(X);
MeanX = CumX / n;
XMean = X - MeanX;
Mean2 = XMean ^ 2;
CumM2 = Cum(Mean2);
nCumM = CumM2 / n;
StDvX = sqrt(nCumM);
Filter = BarIndex() <= 10;
AddColumn(X, "X", 1.0);
AddColumn(n, "n", 1.0);
AddColumn(CumX, "CumX", 1.0);
AddColumn(MeanX, "MeanX", 1.2);
AddColumn(XMean, "X-MeanX", 1.2);
AddColumn(Mean2, "Mean2", 1.2);
AddColumn(CumM2, "CumM2", 1.2);
AddColumn(nCumM, "nCumX", 1.2);
AddColumn(StDvX, "StDevX", 1.2);
Try taking what's above and running it as an EXPLORE ... see the
columns (below "hopefully") it shows i.e. one for each component
including the data "X" ... It would appear that StDevX is correct not
only for BI() == 10 but for ALL the other bars as well without ANY
loops.
X n CumX MeanX X-MeanX Mean2 CumM2 nCumX StDevX
100 1 100 100.00 0.00 0.00 0.00 0.00 0.00
101 2 201 100.50 0.50 0.25 0.25 0.13 0.35
102 3 303 101.00 1.00 1.00 1.25 0.42 0.65
103 4 406 101.50 1.50 2.25 3.50 0.88 0.94
104 5 510 102.00 2.00 4.00 7.50 1.50 1.22
105 6 615 102.50 2.50 6.25 13.75 2.29 1.51
106 7 721 103.00 3.00 9.00 22.75 3.25 1.80
107 8 828 103.50 3.50 12.25 35.00 4.38 2.09
108 9 936 104.00 4.00 16.00 51.00 5.67 2.38
109 10 1045 104.50 4.50 20.25 71.25 7.13 2.67
110 11 1155 105.00 5.00 25.00 96.25 8.75 2.96
Since the rest of your AFL doesn't require any loops, one would
conclude that your AFL really needs NO loops at all.
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