Continuing with the work on the zero crossing function, for now I have decided to use a

"roofing filter" to detrend and smooth the raw price data and then apply some simple cycle extractor code

```
DEFUN_DLD ( cycle_extractor, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} cycle_extractor (@var{price})\n\
This function takes a single price vector input and outputs\n\
a vector of the cycle extracted from the input.\n\
@end deftypefn" )
{
octave_value_list retval_list ;
int nargin = args.length () ;
int vec_length = args(0).length () ;
// check the input argument
if ( nargin != 1 )
{
error ("Invalid argument. Input is a single price vector.") ;
return retval_list ;
}
if ( vec_length < 50 )
{
error ("Invalid argument. Input is a single price vector.") ;
return retval_list ;
}
if ( error_state )
{
error ("Invalid argument. Input is a single price vector.") ;
return retval_list ;
}
// end of input checking
// inputs
ColumnVector price = args(0).column_vector_value () ;
ColumnVector cycle = args(0).column_vector_value () ; cycle(0) = 0.0 ; cycle(1) = 0.0 ;
double alpha = 0.07 ;
for ( octave_idx_type ii (2) ; ii < vec_length ; ii ++ ) // Start the main loop
{
cycle(ii) = ( 1.0 - 0.5 * alpha ) * ( 1.0 - 0.5 * alpha ) * ( price(ii) - 2.0 * price(ii-1) + price(ii-2) ) + 2.0 * ( 1.0 - alpha ) * cycle(ii-1) - ( 1.0 - alpha ) * ( 1.0 - alpha ) * cycle(ii-2) ;
}
retval_list(0) = cycle ;
return retval_list ;
} // end of function
```

to create the inphase input for the zero cross function. For the imaginary or quadrature input I have decided to use the simple trigonometric identity of the derivative of a sine wave being the cosine (i.e. 90 degree phase lead), easily implemented using the bar to bar difference of the sine wave, or in our case the above simple cycle. I might yet change this, but for now it seems to work. The upper pane in the screen shot below shows the raw price in blue, the extracted cycle in black and the cycle derivative in red.

It can be seen that most of the time the cycle (inphase) either leads or is approximately in sync with the price, whilst the quadrature is nicely spaced between the zero crossings of the cycle.

The lower pane shows the measured periods as in the previous posts. The zero crossing measured periods are now a bit more erratic than before, but some simple tests show that, on average, the zero crossing measurement is consistently closer to the real period than the sine wave indicator period; however, this improvement cannot said to be statistically significant at any p-value.

Now I would like to introduce some other work I've been doing recently. For many years I have wanted to measure the instantaneous period using

Maximum entropy spectral estimation, which has been popularised over the years by

John Ehlers. Unfortunately I had never found any code in the public domain which I could use or adapt, until now this is. My discovery is

High Resolution Tools For Spectral Analysis. This might not actually be the same as Ehlers' MESA, but it certainly covers the same general area and, joy of joys, it has freely downloadable MATLAB code along with an accessible description of the theoretical background along with some academic papers.

As is usual in situations like this, I have had to refactor some of the MATLAB code so that it can run in

Octave without error messages; specifically this non Octave function

```
function [msg,A,B,C,D] = abcdchk(A,B,C,D)
%ABCDCHK Checks dimensional consistency of A,B,C,D matrices.
% ERROR(ABCDCHK(A,B,C,D)) checks that the dimensions of A,B,C,D
% are consistent for a linear, time-invariant system model.
% An error occurs if the nonzero dimensions are not consistent.
%
% [MSG,A,B,C,D] = ABCDCHK(A,B,C,D) also alters the dimensions
% any 0-by-0 empty matrices to make them consistent with the others.
% Copyright 1984-2001 The MathWorks, Inc.
% $Revision: 1.21 $ $Date: 2001/04/15 11:59:11 $
if nargin < 4, D = []; end
if nargin < 3, C = []; end
if nargin < 2, B = []; end
if nargin < 1, A = []; end
[ma,na] = size(A);
[mb,nb] = size(B);
[mc,nc] = size(C);
[md,nd] = size(D);
if mc==0 & nc==0 & (md==0 | na==0)
mc = md; nc = na; C = zeros(mc,nc);
end
if mb==0 & nb==0 & (ma==0 | nd==0)
mb = ma; nb = nd; B = zeros(mb,nb);
end
if md==0 & nd==0 & (mc==0 | nb==0)
md = mc; nd = nb; D = zeros(md,nd);
end
if ma==0 & na==0 & (mb==0 | nc==0)
ma = mb; na = nc; A = zeros(ma,na);
end
if ma~=na & nargin>=1
msg = 'The A matrix must be square';
elseif ma~=mb & nargin>=2
msg = 'The A and B matrices must have the same number of rows.';
elseif na~=nc & nargin>=3
msg = 'The A and C matrices must have the same number of columns.';
elseif md~=mc & nargin>=4
msg = 'The C and D matrices must have the same number of rows.';
elseif nd~=nb & nargin>=4
msg = 'The B and D matrices must have the same number of columns.';
else
msg = '';
end
```

should be in the loadpath, and in the function file dlsim.m, on line 71, the call to the internal MATLAB function

x = ltitr( a , b , u , x0 ) ;
should be replaced by
x = lsim ( ss (a , b , [ ] , [ ] , -1 ) , u , [ ] , x0 ) ;
(thanks to Lukas for this)
Having made these changes, this much simplified demo.m script

```
%% SIGNAL = 2 x sinusoids + noise
% Setting up the signal parameters and time history
N = 100 ;
mag0 = 1.8 ; mag1 = 1.5 ; o1 = 1.3 ; mag2 = 2 ; o2 = 1.35 ;
t = 0 : N - 1 ; t = t(:) ;
y0 = mag0 * randn(N,1) ;
y1 = mag1 * exp( 1i * ( o1 * t + 2 * pi * rand) ) ;
y2 = mag2 * exp( 1i * ( o2 * t + 2 * pi * rand )) ;
y = real( y0 .+ y1 .+ y2 ) ;
NN = 2048 ; th = linspace( 0, 2 * pi, NN ) ;
%% setting up filter parameters and the svd of the input-to-state response
thetamid = 1.325 ;
[ A , B ] = rjordan( 5, 0.88 * exp( thetamid * 1i ) ) ;
%% obtaining state statistics
R = dlsim_real( A, B, y' ) ;
%% maximum entropy
spectrum = me( R, A, B, th ) ;
spectrum = spectrum / max( spectrum ) * 1.2 ;
figure(1) ;
subplot(211) ; plot( real(y1), 'r', real(y2), 'm', y, 'b' ) ; legend( 'Underlying Sinewave 1', 'Underlying Sinewave 2', '2 Sinewaves Plus Noise' ) ; title( 'Noisy Price series' ) ;
subplot(212) ; plot( th(400:475), spectrum(400:475) ) ; title( 'The Spectrum - Peaks should be centred at 1.3 and 1.35' ) ;
```

produces this plot

The upper pane shows two sine waves (red and magenta), very close to each other in period and amplitude, combined with random noise to create price (in blue). Looking at just the blue price, it would seem to be almost impossible that there are in fact two sine waves hidden within noise in this short time series, yet the algorithm clearly picks these out as shown by the two peaks in the spectral plot in the lower pane. Powerful stuff!

It is my intention over the coming days to investigate using the zero crossing function to select the data length prior to using this spectral analysis to determine the instantaneous period of price.