Elementary Fourier analysis

Here is the comparison of my Elementary Fourier AFL analysis and FFT usual procedure.
The sample was 921 bars of the Nikkei smoothed RSI.
The error is decreased and, the most important, the fundamental period is more expressive.
 

Attachments

Quote from TSOKAKIS:

An oscillator array y may have a linear trend yL.
After the linear detrending, the new y-yL may be analysed into a sum of sinusoidal terms.
y-yL=g1*sin(2*pi*f1*(t-t1)+phi1)+g2*sin(2*pi*f2*(t-t1)+phi2)+...
We shall examine here the result of an elementary Fourier analysis fundamental frequency and its 2nd and 3rd harmonics
y=yL+g1*sin(2*pi*f1*(t-t1)+phi1)+g2*sin(2*pi*f2*(t-t1)+phi2)+g3*sin(2*pi*f2*(t-t1)+phi3)
f1 is the fundamental frequency
f2=2*f1
f3=2*f2
The amplitudes g1, g2 and g3 are positive since the phases phi1, phi2 and phi3 vary from 0 to 2*pi.
The fundamental period will be calculated in two steps with a final accuracy of 10 days.
A smoothed RSI [bold black line] is analysed in the following example:
Paste in an Indicator Builder the

// Elementary Fourier analysis, by D. Tsokakis, May 2004
t=Cum(1)-1;
C1=MA(RSI(50),100);
start=Cum(IsTrue(C1))==1;
t1=ValueWhen(start,t);
PlotShapes(shapeDownTriangle*start,colorYellow);
C10=ValueWhen(start,C1);Plot(C1,"C1",colorBlack,8);
GraphXSpace=2;
x = Cum(1);
lastx = LastValue( x );
Daysback = LastValue(Cum(IsTrue(C1)));
aa = LastValue( LinRegIntercept( C1, Daysback) );
bb = LastValue( LinRegSlope( C1, Daysback ) );
yy = Aa + bb * ( x - (Lastx - DaysBack) );
yy=IIf( x >= (lastx - Daysback), yy, -1e10 );
Plot( yy, "yy", colorRed );
detrend=C1-yy;
new1=detrend;Hor=LastValue(Cum(new1)/Cum(IsTrue(C1)));
pi=4*atan(1);n=12;
// Fundamental period, crude approximation
error00=10000;per01=0;g01=0;phi01=0;stg0=0.5;stp0=100;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0.5;g<=8;g=g+stg0)
{
for(per=300;per<=1000;per=per+stp0)
{f=1/per;
y=Hor+g*sin(2*pi*f*(t-t1)+phi);
error=LastValue(Cum(abs(y-new1)));
if(error<error00)
{error00=error;per01=per;g01=g;phi01=phi;}
}}}
f01=1/per01;y01=Hor+g01*sin(2*pi*f01*(t-t1)+phi01);
Plot(y01+yy,"y01",colorBlack,1);
Title=Name()+" [ Sample="+WriteVal(Daysback,1.0)+" bars ]"+"\nyS0="+WriteVal(Hor,1.2)+
"\nyS01="+
WriteVal(g01,1.1)+"*sin(2*pi*(1/"+
WriteVal(per01,1.0)+")*(t-t1)+"+
WriteVal(12*phi01/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error1 ="+
WriteVal(LastValue(Cum(abs(y01-new1))),1.0)+", Error1/bar ="+
WriteVal(2*LastValue(Cum(abs(y01-new1)))/Daysback,1.2)+" %";;
// Fundamental period, detailed approximation
error0=10000;per1=0;g1=0;phi1=0;stg=0.5;stp=10;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0.5;g<=8;g=g+stg)
{
for(per=per01-stp0;per<=per01+stp0;per=per+stp)
{f=1/per;
y=Hor+g*sin(2*pi*f*(t-t1)+phi);
error=LastValue(Cum(abs(y-new1)));
if(error<error0)
{error0=error;per1=per;g1=g;phi1=phi;}
}}}
f1=1/per1;y1=Hor+g1*sin(2*pi*f1*(t-t1)+phi1);
Plot(y1+yy,"y1",colorYellow,1);
Title=Title+
"\nyS1="+
WriteVal(g1,1.1)+"*sin(2*pi*(1/"+
WriteVal(per1,1.0)+")*(t-t1)+"+
WriteVal(12*phi1/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error1 ="+
WriteVal(LastValue(Cum(abs(y1-new1))),1.0)+", Error1/bar ="+
WriteVal(2*LastValue(Cum(abs(y1-new1)))/Daysback,1.2)+" %";;
// 2nd Harmonic
error0=10000;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0;g<=8;g=g+0.1)
{
per2=per1/2;f=1/per2;
y2=y1+g*sin(2*pi*f*(t-t1)+phi);
error2=LastValue(Cum(abs(y2-new1)));
if(error2<error0)
{error0=error2;g2=g;phi2=phi;}
}}
f2=1/per2;y2=y1+g2*sin(2*pi*f2*(t-t1)+phi2);
Plot(y2+yy,"y1",colorBlue,1);
Title=Title+
"\nyS2="+
WriteVal(g2,1.1)+"*sin(2*pi*(1/"+
WriteVal(per2,1.0)+")*(t-t1)+"+
WriteVal(12*phi2/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error2 ="+
WriteVal(LastValue(Cum(abs(y2-new1))),1.0)+", Error2/bar ="+
WriteVal(2*LastValue(Cum(abs(y2-new1)))/Daysback,1.2)+" %";;
// 3rd Harmonic
error0=10000;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0;g<=8;g=g+0.1)
{
per3=per2/2;f=1/per3;
y3=y2+g*sin(2*pi*f*(t-t1)+phi);
error3=LastValue(Cum(abs(y3-new1)));
if(error3<error0)
{error0=error3;g3=g;phi3=phi;}
}}
f3=1/per3;y3=y2+g3*sin(2*pi*f3*(t-t1)+phi3);
Plot(y3+yy,"y1",colorWhite,8);
Title=Title+
"\nyS3="+
WriteVal(g3,1.1)+"*sin(2*pi*(1/"+
WriteVal(per3,1.0)+")*(t-t1)+"+
WriteVal(12*phi3/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error3 ="+
WriteVal(LastValue(Cum(abs(y3-new1))),1.0)+", Error3/bar ="+
WriteVal(2*LastValue(Cum(abs(y3-new1)))/Daysback,1.2)+" %";

The bold white line is the final result.
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Well, it seems a little bit complicated?
 
Quote from Baruch:

Well, it seems a little bit complicated?
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What is complicated, the concept or the code ?
The concept, ie the sinusoidal analysis of a non-periodic function is quite common in theoretical and applied maths/engineering etc
The code is not bad, some friend may give a shorter one via C++ translation [if he comes to some result, I will let you know...]
Using Amibroker 4.50 in a PIII/800, 64MB RAM, it takes ~8sec for a 1000bar history.
FFT is quite faster but not that accurate.
 
Quote from TSOKAKIS:

What is complicated, the concept or the code ?
The concept, ie the sinusoidal analysis of a non-periodic function is quite common in theoretical and applied maths/engineering etc
The code is not bad, some friend may give a shorter one via C++ translation [if he comes to some result, I will let you know...]
Using Amibroker 4.50 in a PIII/800, 64MB RAM, it takes ~8sec for a 1000bar history.
FFT is quite faster but not that accurate.
More...

Oh, so simple?
 
Quote from Baruch:

Oh, so simple?
More...

Note also that the described procedure is innovative and unique in the T/A literature, as far as I know.
The statistical softwares usually include FFT [or DFT] but their use is limited, inaccurate, work with 2^n series [512, 1024, 2048, ... bars], suffer from the end-point effect and, practically, are not designed for T/A applications.
 
Quote from TSOKAKIS:

You may see now the FFT comparison and the results. The concept of my Elementary Fourier analysis is a bit different.
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Thanks for your contribution -- keep it coming.
 
Here is another comparison of the EFA [Elementary Fourier Analysis] versus the FFT [Fast Fourier Transformation].
In FFT I used the 7 most significant terms, filtering out the amplitude>0.05.
In EFA I used the fundamental period P and the harmonics P/2, P/3, P/4 and P/5.
The visual [and the detailed] comparison is obvious.
Since FFT prefers 2^n bars [256, 512, 1024 etc] it has a lot of problems after the first 512 bars.
[my sample was an 921-bar smoothed Nikkei RSI]
Dimitris Tsokakis
[I used DaDisp_SE2000 for FFT and Amibroker 4.50 for EFA ]
 

Attachments

Quote from TSOKAKIS:

An oscillator array y may have a linear trend yL.
After the linear detrending, the new y-yL may be analysed into a sum of sinusoidal terms.
y-yL=g1*sin(2*pi*f1*(t-t1)+phi1)+g2*sin(2*pi*f2*(t-t1)+phi2)+...
We shall examine here the result of an elementary Fourier analysis fundamental frequency and its 2nd and 3rd harmonics
y=yL+g1*sin(2*pi*f1*(t-t1)+phi1)+g2*sin(2*pi*f2*(t-t1)+phi2)+g3*sin(2*pi*f2*(t-t1)+phi3)
f1 is the fundamental frequency
f2=2*f1
f3=2*f2
The amplitudes g1, g2 and g3 are positive since the phases phi1, phi2 and phi3 vary from 0 to 2*pi.
The fundamental period will be calculated in two steps with a final accuracy of 10 days.
A smoothed RSI [bold black line] is analysed in the following example:
Paste in an Indicator Builder the

// Elementary Fourier analysis, by D. Tsokakis, May 2004
t=Cum(1)-1;
C1=MA(RSI(50),100);
start=Cum(IsTrue(C1))==1;
t1=ValueWhen(start,t);
PlotShapes(shapeDownTriangle*start,colorYellow);
C10=ValueWhen(start,C1);Plot(C1,"C1",colorBlack,8);
GraphXSpace=2;
x = Cum(1);
lastx = LastValue( x );
Daysback = LastValue(Cum(IsTrue(C1)));
aa = LastValue( LinRegIntercept( C1, Daysback) );
bb = LastValue( LinRegSlope( C1, Daysback ) );
yy = Aa + bb * ( x - (Lastx - DaysBack) );
yy=IIf( x >= (lastx - Daysback), yy, -1e10 );
Plot( yy, "yy", colorRed );
detrend=C1-yy;
new1=detrend;Hor=LastValue(Cum(new1)/Cum(IsTrue(C1)));
pi=4*atan(1);n=12;
// Fundamental period, crude approximation
error00=10000;per01=0;g01=0;phi01=0;stg0=0.5;stp0=100;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0.5;g<=8;g=g+stg0)
{
for(per=300;per<=1000;per=per+stp0)
{f=1/per;
y=Hor+g*sin(2*pi*f*(t-t1)+phi);
error=LastValue(Cum(abs(y-new1)));
if(error<error00)
{error00=error;per01=per;g01=g;phi01=phi;}
}}}
f01=1/per01;y01=Hor+g01*sin(2*pi*f01*(t-t1)+phi01);
Plot(y01+yy,"y01",colorBlack,1);
Title=Name()+" [ Sample="+WriteVal(Daysback,1.0)+" bars ]"+"\nyS0="+WriteVal(Hor,1.2)+
"\nyS01="+
WriteVal(g01,1.1)+"*sin(2*pi*(1/"+
WriteVal(per01,1.0)+")*(t-t1)+"+
WriteVal(12*phi01/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error1 ="+
WriteVal(LastValue(Cum(abs(y01-new1))),1.0)+", Error1/bar ="+
WriteVal(2*LastValue(Cum(abs(y01-new1)))/Daysback,1.2)+" %";;
// Fundamental period, detailed approximation
error0=10000;per1=0;g1=0;phi1=0;stg=0.5;stp=10;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0.5;g<=8;g=g+stg)
{
for(per=per01-stp0;per<=per01+stp0;per=per+stp)
{f=1/per;
y=Hor+g*sin(2*pi*f*(t-t1)+phi);
error=LastValue(Cum(abs(y-new1)));
if(error<error0)
{error0=error;per1=per;g1=g;phi1=phi;}
}}}
f1=1/per1;y1=Hor+g1*sin(2*pi*f1*(t-t1)+phi1);
Plot(y1+yy,"y1",colorYellow,1);
Title=Title+
"\nyS1="+
WriteVal(g1,1.1)+"*sin(2*pi*(1/"+
WriteVal(per1,1.0)+")*(t-t1)+"+
WriteVal(12*phi1/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error1 ="+
WriteVal(LastValue(Cum(abs(y1-new1))),1.0)+", Error1/bar ="+
WriteVal(2*LastValue(Cum(abs(y1-new1)))/Daysback,1.2)+" %";;
// 2nd Harmonic
error0=10000;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0;g<=8;g=g+0.1)
{
per2=per1/2;f=1/per2;
y2=y1+g*sin(2*pi*f*(t-t1)+phi);
error2=LastValue(Cum(abs(y2-new1)));
if(error2<error0)
{error0=error2;g2=g;phi2=phi;}
}}
f2=1/per2;y2=y1+g2*sin(2*pi*f2*(t-t1)+phi2);
Plot(y2+yy,"y1",colorBlue,1);
Title=Title+
"\nyS2="+
WriteVal(g2,1.1)+"*sin(2*pi*(1/"+
WriteVal(per2,1.0)+")*(t-t1)+"+
WriteVal(12*phi2/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error2 ="+
WriteVal(LastValue(Cum(abs(y2-new1))),1.0)+", Error2/bar ="+
WriteVal(2*LastValue(Cum(abs(y2-new1)))/Daysback,1.2)+" %";;
// 3rd Harmonic
error0=10000;
for(phi=0;phi<2*pi;phi=phi+pi/n)
{
for(g=0;g<=8;g=g+0.1)
{
per3=per2/2;f=1/per3;
y3=y2+g*sin(2*pi*f*(t-t1)+phi);
error3=LastValue(Cum(abs(y3-new1)));
if(error3<error0)
{error0=error3;g3=g;phi3=phi;}
}}
f3=1/per3;y3=y2+g3*sin(2*pi*f3*(t-t1)+phi3);
Plot(y3+yy,"y1",colorWhite,8);
Title=Title+
"\nyS3="+
WriteVal(g3,1.1)+"*sin(2*pi*(1/"+
WriteVal(per3,1.0)+")*(t-t1)+"+
WriteVal(12*phi3/pi,1.0)+"*pi/"+WriteVal(n,1.0)+"), Error3 ="+
WriteVal(LastValue(Cum(abs(y3-new1))),1.0)+", Error3/bar ="+
WriteVal(2*LastValue(Cum(abs(y3-new1)))/Daysback,1.2)+" %";

The bold white line is the final result.
More...

Are you sure about 3rd Harmonic?
 
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