Following on from the previous post, the test I outlined in that post wasn't very satisfactory, which I put down to the fact that the Sigmoid transformation of the raw MFE/MAE indicator values is not amenable to the application of standard deviation as a meaningful measure. Instead, I changed the test to one based on the standard error of the mean, an example screen shot of which is shown below:-

The top pane shows the the long version of the indicator and the bottom pane the short version. In each there are upper and lower limits of the sample standard error of the mean above and below the population mean (mean of all values of the indicator) along with the cumulative mean value of the top N matches as shown on the x-axis. In this particular example it can be seen that around the 170-180 samples mark the cumulative mean moves inside the standard error limits, never to leave them again. The meaning I ascribe to this is that there is no value to be gained from using more than approximately 180 samples for machine learning purposes, for this example, as to use more samples would be akin to training on all available data, which makes the use of my Cauchy Schwarz matching algo superfluous. I repeated the above on all instances of sigmoid transformed and untransformed MFE/MAE indicator values to get an average of 325 samples for transformed, and an average of 446 samples for the untransformed indicator values across the 4 major forex pairs. Based on this, I have decided to use the top 450 Cauchy Schwarz matches for training purposes, which has ramifications for model complexity will be discussed shortly.

Returning to the above screen shot, the figure 2 inset shows the price bars that immediately follow the price bar for which the main screen shows the top N matches. Looking at the extreme left of the main screen it can be seen that the lower pane, short indicator has an almost maximum reading of 1 whilst the upper pane, long indicator shows a value of approx. 2.7, which is not much above the global minimum for this indicator and well below the 0.5 neutral level. This strongly suggests a short position, and looking at the inset figure it can be seen that over the 3 days following the extreme left matched bar a short position was indeed the best position to hold. This is a pattern that seems to frequently present itself during visual inspection of charts, although I am unable to quantify this in any way.

On the matter of model complexity alluded to above, I found the Learning From Data course I have recently completed on the edX platform to be very enlightening, particularly the concept of the VC dimension, which is nicely explained in the Learning From Data Video library. I'll leave it to interested readers to follow the links, but the big take away for me is that using 450 samples as described above implies that my final machine learning model must have an upper bound of approximately 45 on the VC dimension, which in turn implies a maximum of 45 weights in the neural net. This is a design constraint that I will discuss in a future post.

# Dekalog Blog

"Trading is statistics and time series analysis." This blog details my progress in developing a systematic trading system for use on the futures and forex markets, with discussion of the various indicators and other inputs used in the creation of the system. Also discussed are some of the issues/problems encountered during this development process. Within the blog posts there are links to other web pages that are/have been useful to me.

## Thursday, 11 December 2014

## Wednesday, 26 November 2014

### Test of the MFE/MAE Indicator

Continuing from my last post, wherein I stated I was going to conduct a more pertinent statistical test of the returns of the bars(s) immediately following the N best, Cauchy Schwarz matching algorithm matched bars in the price history, readers may recall that the basic premise behind this algorithm is that by matching current price action to the N best matches, the price action after these matches can be used to infer what will occur after the current price action. However, rather than test the price action directly I have decided to apply the test to the MFE/MAE indicator. There are several reasons for this, which are enumerated below.

This shows two sampling distributions of the mean for Long MFE/MAE indicator values > 0.5, the upper pane for sample sizes of 20 and the lower pane for 75. For simplicity I shall only discuss the Long > 0.5 version of the indicator, but everything that follows applies equally to the Short version. As expected the upper pane shows greater variance, and for the envisioned test a whole series of these sampling distributions will be produced for different sampling rates. The way I intend it to work is as follows:

where the cyan and red lines are the +/- 2 standard deviations above/below a notional mean value for the whole distribution of approximately 0.85, and the chart can be considered to be a type of control chart. The upper and lower control lines converge towards the right, reflecting the decreasing variance of increasingly large N sample means, as shown in the first chart above. The green line represents the cumulative N sample mean of the best N historical matches' future values. I have shown it as decreasing as it is to be expected that as more N matches are included, the greater the chance that incorrect matches, unexpected price reversals etc. will be caught up in this mean calculation, resulting in the mean value moving into the left tail of the sampling distribution. This effect combines with the shrinking variance to reach a critical point (rejection of the null hypothesis) at which the green line exits below the lower control line.

The purpose of all the above is provide a principled manner to choose the number N matches from the Cauchy-Schwarz matching algorithm to supply instances of training data to the envisioned neural net training. An incidental benefit of this approach is that it is indirectly a hypothesis test of the fundamental assumption underlying the matching algorithm; namely that past price action has predictive ability for future price action, and furthermore, it is a test of the MFE/MAE indicator. Discussion of the results of these tests in a future post.

- I intend to use the indicator as the target function for future Neural net training
- the indicator represents a reward to risk ratio, which indirectly reflects price action itself, but without the noise of said action
- this reward to risk ratio is of much more direct concern, from a trading perspective, than accurately predicting price
- since the indicator is now included as a feature in the matching algorithm, testing the indicator is, very indirectly, a test of the matching algorithm too

This shows two sampling distributions of the mean for Long MFE/MAE indicator values > 0.5, the upper pane for sample sizes of 20 and the lower pane for 75. For simplicity I shall only discuss the Long > 0.5 version of the indicator, but everything that follows applies equally to the Short version. As expected the upper pane shows greater variance, and for the envisioned test a whole series of these sampling distributions will be produced for different sampling rates. The way I intend it to work is as follows:

- take a single bar in the history and see what the value of the MFE/MAE indicator value is 3 bars later (assume > 0.5 for this exposition, so we compare to long sampling distributions only)
- get the top 20 matched bars for the above selected bar and the corresponding 20 indicator values for 3 bars later and take the mean of these 20 indicator values
- check if this mean falls within the sampling distribution of the mean of 20, as shown in the upper pane above by the vertical black line at 0.8 on the x axis. If it does fall with the sampling distribution, we accept the null hypothesis that the 20 best matches in history future indicator values and the value of the indicator after the bar to be matched come from the same distribution
- repeat the immediately preceding step for means of 21, 22, ... etc until such time as the null hypothesis can be rejected, shown in the lower pane above. At this point, we then then declare an upper bound on the historical number of matches for the bar to be predicted

where the cyan and red lines are the +/- 2 standard deviations above/below a notional mean value for the whole distribution of approximately 0.85, and the chart can be considered to be a type of control chart. The upper and lower control lines converge towards the right, reflecting the decreasing variance of increasingly large N sample means, as shown in the first chart above. The green line represents the cumulative N sample mean of the best N historical matches' future values. I have shown it as decreasing as it is to be expected that as more N matches are included, the greater the chance that incorrect matches, unexpected price reversals etc. will be caught up in this mean calculation, resulting in the mean value moving into the left tail of the sampling distribution. This effect combines with the shrinking variance to reach a critical point (rejection of the null hypothesis) at which the green line exits below the lower control line.

The purpose of all the above is provide a principled manner to choose the number N matches from the Cauchy-Schwarz matching algorithm to supply instances of training data to the envisioned neural net training. An incidental benefit of this approach is that it is indirectly a hypothesis test of the fundamental assumption underlying the matching algorithm; namely that past price action has predictive ability for future price action, and furthermore, it is a test of the MFE/MAE indicator. Discussion of the results of these tests in a future post.

## Wednesday, 12 November 2014

### First Use for the MFE/MAE Indicator

This first use is as an input to my Cauchy-Schwarz matching algorithm, previous posts about which can be read here, here and here. The screen shot below shows what I would characterise as a "good" set of matches:

The top left pane shows the original section of the price series to be matched, and the panes labelled #1, #5, etc. are the best match, 5th best match and so on respectively. The last 3 rightmost bars in each pane are "future" price bars, i.e. the 4th bar in from the right is the target bar that is being matched, matched over all the bars to the left or in the past of this target bar.I consider the above to be a set of "good" matches because, for the #1 through #25 matches for "future" bars:

- if one considers the logic of the mfe/mae indicator each pane gives indicator readings of "long," which all agree with the original "future" bars
- similarly the mae (maximum adverse excursion) occurs on the day immediately following the matched day
- the mfe (maximum favourable excursion) occurs on the 3rd "future" bar, with the slight exception of pane #10
- the marked to market returns of an entry at the open of the 1st "future" bar to the close of the 3rd "future" bar all show a profit, as does the original pane

In the above linked posts the test statistic used to judge the predictive efficacy of the matching algorithm was effect size. However, I think a more pertinent test statistic to use would be the average bar return over the bars immediately following a matched bar, and a discussion of this will be the subject of my next post.

## Wednesday, 5 November 2014

### A New MFE/MAE Indicator.

After stopping my investigation of tools for spectral analysis over the last few weeks I have been doing another mooc, this time Learning from Data, and also working on the idea of one of my earlier posts.

In the above linked post there is a video showing the idea as a "paint bar" study. However, I thought it would be a good idea to render it as an indicator, the C++ Octave .oct code for which is shown in the code box below.

The yellow equity curve is the cumulative, close to close, tick returns of a buy and hold strategy, the blue is the return going flat when indicated, and the red maintaining the previous position when flat is indicated. Not much to write home about. However, this second chart shows the return when one has the benefit of the "peek into the future" as discussed in my earlier post.

The colour of the curves are as before except for the addition of the green equity curve, which is the cumulative, vwap value to vwap value tick returns, a simple representation of what an equity curve with realistic slippage might look like. This second set of equity curves shows the promise of what could be achievable if a neural net to accurately predict future values of the above indicator can be trained. More in an upcoming post.

In the above linked post there is a video showing the idea as a "paint bar" study. However, I thought it would be a good idea to render it as an indicator, the C++ Octave .oct code for which is shown in the code box below.

```
DEFUN_DLD ( adjustable_mfe_mae_from_open_indicator, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} adjustable_mfe_mae_from_open_indicator (@var{open,high,low,close,lookback_length})\n\
This function takes four input series for the OHLC and a value for lookback length. The main outputs are\n\
two indicators, long and short, that show the ratio of the MFE over the MAE from the open of the specified\n\
lookback in the past. The indicators are normalised to the range 0 to 1 by a sigmoid function and a MFE/MAE\n\
ratio of 1:1 is shifted in the sigmoid function to give a 'neutral' indicator reading of 0.5. A third output\n\
is the max high - min low range over the lookback_length normalised by the range of the daily support and\n\
resistance levels S1 and R1 calculated for the first bar of the lookback period. This is also normalised to\n\
give a reading of 0.5 in the sigmoid function if the ratio is 1:1. The point of this third output is to give\n\
some relative scale to the unitless MFE/MAE ratio and to act as a measure of strength or importance of the\n\
MFE/MAE ratio.\n\
@end deftypefn" )
{
octave_value_list retval_list ;
int nargin = args.length () ;
// check the input arguments
if ( nargin != 5 )
{
error ( "Invalid arguments. Arguments are price series for open, high, low and close and value for lookback length." ) ;
return retval_list ;
}
if ( args(4).length () != 1 )
{
error ( "Invalid argument. Argument 5 is a scalar value for the lookback length." ) ;
return retval_list ;
}
int lookback_length = args(4).int_value() ;
if ( args(0).length () < lookback_length )
{
error ( "Invalid argument lengths. Argument lengths for open, high, low and close vectors should be >= lookback length." ) ;
return retval_list ;
}
if ( args(1).length () != args(0).length () )
{
error ( "Invalid argument lengths. Argument lengths for open, high, low and close vectors should be equal." ) ;
return retval_list ;
}
if ( args(2).length () != args(0).length () )
{
error ( "Invalid argument lengths. Argument lengths for open, high, low and close vectors should be equal." ) ;
return retval_list ;
}
if ( args(3).length () != args(0).length () )
{
error ( "Invalid argument lengths. Argument lengths for open, high, low and close vectors should be equal." ) ;
return retval_list ;
}
if (error_state)
{
error ( "Invalid arguments. Arguments are price series for open, high, low and close and value for lookback length." ) ;
return retval_list ;
}
// end of input checking
// inputs
ColumnVector open = args(0).column_vector_value () ;
ColumnVector high = args(1).column_vector_value () ;
ColumnVector low = args(2).column_vector_value () ;
ColumnVector close = args(3).column_vector_value () ;
// outputs
ColumnVector long_mfe_mae = args(0).column_vector_value () ;
ColumnVector short_mfe_mae = args(0).column_vector_value () ;
ColumnVector range = args(0).column_vector_value () ;
// variables
double max_high = *std::max_element( &high(0), &high( lookback_length ) ) ;
double min_low = *std::min_element( &low(0), &low( lookback_length ) ) ;
double pivot_point = ( high(0) + low(0) + close(0) ) / 3.0 ;
double s1 = 2.0 * pivot_point - high(0) ;
double r1 = 2.0 * pivot_point - low(0) ;
for ( octave_idx_type ii (0) ; ii < lookback_length ; ii++ ) // initial ii loop
{
// long_mfe_mae
if ( open(0) > min_low ) // the "normal" situation
{
long_mfe_mae(ii) = 1.0 / ( 1.0 + exp( -( ( max_high - open(0) ) / ( open(0) - min_low ) - 1.0 ) ) ) ;
}
else if ( open(0) == min_low )
{
long_mfe_mae(ii) = 1.0 ;
}
else
{
long_mfe_mae(ii) = 0.5 ;
}
// short_mfe_mae
if ( open(0) < max_high ) // the "normal" situation
{
short_mfe_mae(ii) = 1.0 / ( 1.0 + exp( -( ( open(0) - min_low ) / ( max_high - open(0) ) - 1.0 ) ) ) ;
}
else if ( open(0) == max_high )
{
short_mfe_mae(ii) = 1.0 ;
}
else
{
short_mfe_mae(ii) = 0.5 ;
}
range(ii) = 1.0 / ( 1.0 + exp( -( ( max_high - min_low ) / ( r1 - s1 ) - 1.0 ) ) ) ;
} // end of initial ii loop
for ( octave_idx_type ii ( lookback_length ) ; ii < args(0).length() ; ii++ ) // main ii loop
{
// assign variable values
max_high = *std::max_element( &high( ii - lookback_length + 1 ), &high( ii + 1 ) ) ;
min_low = *std::min_element( &low( ii - lookback_length + 1 ), &low( ii + 1 ) ) ;
pivot_point = ( high(ii-lookback_length) + low(ii-lookback_length) + close(ii-lookback_length) ) / 3.0 ;
s1 = 2.0 * pivot_point - high(ii-lookback_length) ;
r1 = 2.0 * pivot_point - low(ii-lookback_length) ;
// long_mfe_mae
if ( open( ii - lookback_length + 1 ) > min_low && open( ii - lookback_length + 1 ) < max_high ) // the "normal" situation
{
long_mfe_mae(ii) = 1.0 / ( 1.0 + exp( -( ( max_high - open( ii - lookback_length + 1 ) ) / ( open( ii - lookback_length + 1 ) - min_low ) - 1.0 ) ) ) ;
}
else if ( open( ii - lookback_length + 1 ) == min_low )
{
long_mfe_mae(ii) = 1.0 ;
}
else
{
long_mfe_mae(ii) = 0.0 ;
}
// short_mfe_mae
if ( open( ii - lookback_length + 1 ) > min_low && open( ii - lookback_length + 1 ) < max_high ) // the "normal" situation
{
short_mfe_mae(ii) = 1.0 / ( 1.0 + exp( -( ( open( ii - lookback_length + 1 ) - min_low ) / ( max_high - open( ii - lookback_length + 1 ) ) - 1.0 ) ) ) ;
}
else if ( open( ii - lookback_length + 1 ) == max_high )
{
short_mfe_mae(ii) = 1.0 ;
}
else
{
short_mfe_mae(ii) = 0.0 ;
}
range(ii) = 1.0 / ( 1.0 + exp( -( ( max_high - min_low ) / ( r1 - s1 ) - 1.0 ) ) ) ;
} // end of main ii loop
retval_list(2) = range ;
retval_list(1) = short_mfe_mae ;
retval_list(0) = long_mfe_mae ;
return retval_list ;
} // end of function
```

The way to interpret this is as follows:- if the "long" indicator reading is above 0.5, go long
- if the "short" is above 0.5, go short
- if both are below 0.5, go flat

The yellow equity curve is the cumulative, close to close, tick returns of a buy and hold strategy, the blue is the return going flat when indicated, and the red maintaining the previous position when flat is indicated. Not much to write home about. However, this second chart shows the return when one has the benefit of the "peek into the future" as discussed in my earlier post.

The colour of the curves are as before except for the addition of the green equity curve, which is the cumulative, vwap value to vwap value tick returns, a simple representation of what an equity curve with realistic slippage might look like. This second set of equity curves shows the promise of what could be achievable if a neural net to accurately predict future values of the above indicator can be trained. More in an upcoming post.

## Tuesday, 23 September 2014

### High Resolution Tools for Spectral Analysis - Update

Following on from my initial enthusiasm for the code on the High Resolution Tools for Spectral Analysis page, I have say that I have been unable to get the code performing as I would like it for my intended application to price time series.

My original intent was to use the zero crossing period estimation function, the subject of my last few posts, to get a rough idea of the dominant cycle period and then use the most recent data in a rolling window of this length as input to the high resolution code. This approach, however, ran into problems.

Firstly, windows of just the dominant cycle length (approximately 10 to 30 data points only) would lead to all sorts of errors being thrown from the toolkit functions as well as core Octave functions, such as divide by zero warnings and cryptic error messages that even now I don't understand. My best guess here is that the amount of data available in such short windows is simply insufficient for the algorithm to work, in much the same way as the Fast Fourier transform may fail to work if given too little data that is not a power of 2 in length. It might be possible to substantially rewrite the relevant functions, but my understanding of the algorithm and the inner workings of Octave means this is well beyond my pay grade.

My second approach was to simply extend the amount of data available by using the Octave repmat function on the windowed data so that all the above errors disappeared. This had very hit and miss results - sometimes they were very accurate, other times just fair to middling, and on occasion just way off target. I suspect here that the problem is the introduction of signal artifacts via the use of the repmat function, which results in Aliasing of the underlying signal.

As a result I shall not continue with investigating this toolbox for now. Although I only investigated the use of the me.m function (there are other functions available) I feel that my time at the moment can be used more productively.

My original intent was to use the zero crossing period estimation function, the subject of my last few posts, to get a rough idea of the dominant cycle period and then use the most recent data in a rolling window of this length as input to the high resolution code. This approach, however, ran into problems.

Firstly, windows of just the dominant cycle length (approximately 10 to 30 data points only) would lead to all sorts of errors being thrown from the toolkit functions as well as core Octave functions, such as divide by zero warnings and cryptic error messages that even now I don't understand. My best guess here is that the amount of data available in such short windows is simply insufficient for the algorithm to work, in much the same way as the Fast Fourier transform may fail to work if given too little data that is not a power of 2 in length. It might be possible to substantially rewrite the relevant functions, but my understanding of the algorithm and the inner workings of Octave means this is well beyond my pay grade.

My second approach was to simply extend the amount of data available by using the Octave repmat function on the windowed data so that all the above errors disappeared. This had very hit and miss results - sometimes they were very accurate, other times just fair to middling, and on occasion just way off target. I suspect here that the problem is the introduction of signal artifacts via the use of the repmat function, which results in Aliasing of the underlying signal.

As a result I shall not continue with investigating this toolbox for now. Although I only investigated the use of the me.m function (there are other functions available) I feel that my time at the moment can be used more productively.

## Friday, 19 September 2014

### Inputs For Zero Crossing Function

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

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

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

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.

```
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 functionx = 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 plotThe 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.

Labels:
Cycle Period,
Digital Signal Processing,
Octave

## Wednesday, 17 September 2014

### Zero Crossing Frequency/Period Measuring Function

Following on from my last post I have now finished coding the first two versions of this function, which are shown in the code boxes below.

It can be seen that they both perform admirably within their respective theoretical constraints on this ideal input. The subject of my next post will concentrate on the inputs that can be realistically extracted from live price data.

```
DEFUN_DLD ( iq_zc_period_indicator_v1, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} iq_zc_period_indicator_v1 (@var{Realpart,Quadrature})\n\
This function takes two input vectors of the real part and quadrature of price,\n\
the analytic signal, and outputs the period calculated from the zero crossings of\n\
the two input vectors.\n\
@end deftypefn" ) {
octave_value_list retval_list ;
int nargin = args.length () ;
int vec_length = args(0).length () ;
// check the input arguments
if ( nargin != 2 )
{
error ( "Invalid input argument(s). Inputs are two vectors, the realpart and quadrature." ) ;
return retval_list ;
}
if ( vec_length < 50 )
{
error ( "Invalid input argument length(s). Input vectors must be > 50 in length." ) ;
return retval_list ;
}
if ( args(1).length () != vec_length )
{
error ( "Invalid input argument length(s). Input vectors differ in length." ) ;
return retval_list ;
}
if ( error_state )
{
error ( "Invalid input argument(s). Inputs are two vectors, the realpart and quadrature." ) ;
return retval_list ;
}
// end of input checking
// inputs
ColumnVector inphase_1 = args(0).column_vector_value () ;
ColumnVector quadrature_1 = args(1).column_vector_value () ;
ColumnVector f_hat ( vec_length ) ;
ColumnVector inphase_1_state ( vec_length ) ;
ColumnVector quadrature_1_state ( vec_length ) ;
for ( octave_idx_type ii (0) ; ii < 50 ; ii ++ ) // Start the initialising loop
{
f_hat(ii) = 20.0 ; // a default period length
inphase_1_state(ii) = 0.0 ; // assuming exactly on zero cross line
quadrature_1_state(ii) = 0.0 ; // assuming exactly on zero cross line
} // end of initialising ii loop
double latest_i_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
double previous_i_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
double latest_q_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
double previous_q_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
// get the initial states of the inphase_1 and quadrature_1 vectors. A value of
// 1 is assigned if > 0, -1 for < 0 & 0.0 if == 0
inphase_1_state(49) = inphase_1(49) > 0.0 ? 1.0 : ( inphase_1(49) < 0.0 ? -1.0 : 0.0 ) ;
quadrature_1_state(49) = quadrature_1(49) > 0.0 ? 1.0 : ( quadrature_1(49) < 0.0 ? -1.0 : 0.0 ) ;
for ( octave_idx_type ii (50) ; ii < vec_length ; ii ++ ) // Start the main loop
{
// first, assign current states for the inphase_1 and quadrature_1 vectors on this
// new (ii) bar
inphase_1_state(ii) = inphase_1(ii) >= 0.0 ? 1.0 : ( inphase_1(ii) < 0.0 ? -1.0 : 0.0 ) ;
quadrature_1_state(ii) = quadrature_1(ii) >= 0.0 ? 1.0 : ( quadrature_1(ii) < 0.0 ? -1.0 : 0.0 ) ;
// check for a change in state, which indicates a zero crossing
if ( inphase_1_state(ii) == 1.0 && inphase_1_state(ii-1) == -1.0 )
{ // inphase_1 has crossed up over the zero line
latest_i_1_zc_time = double(ii-1) - inphase_1(ii-1) / ( inphase_1(ii) - inphase_1(ii-1) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_i_1_zc_time - previous_i_1_zc_time ) * 2.0 ; // calculate the period from times of zero crosses
previous_i_1_zc_time = latest_i_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( inphase_1_state(ii) == -1.0 && inphase_1_state(ii-1) == 1.0 )
{ // inphase_1 has crossed down over the zero line
latest_i_1_zc_time = double(ii-1) + inphase_1(ii-1) / ( inphase_1(ii-1) - inphase_1(ii) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_i_1_zc_time - previous_i_1_zc_time ) * 2.0 ; // calculate the period from times of zero crosses
previous_i_1_zc_time = latest_i_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( quadrature_1_state(ii) == 1.0 && quadrature_1_state(ii-1) == -1.0 )
{ // quadrature_1 has crossed up over the zero line
latest_q_1_zc_time = double(ii-1) - quadrature_1(ii-1) / ( quadrature_1(ii) - quadrature_1(ii-1) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_q_1_zc_time - previous_q_1_zc_time ) * 2.0 ; // calculate the period from times of zero crosses
previous_q_1_zc_time = latest_q_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( quadrature_1_state(ii) == -1.0 && quadrature_1_state(ii-1) == 1.0 )
{ // quadrature_1 has crossed down over the zero line
latest_q_1_zc_time = double(ii-1) + quadrature_1(ii-1) / ( quadrature_1(ii-1) - quadrature_1(ii) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_q_1_zc_time - previous_q_1_zc_time ) * 2.0 ; // calculate the period from times of zero crosses
previous_q_1_zc_time = latest_q_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else
{ // neither the inphase_1 or quadrature_1 have crossed the zeroline on this bar
f_hat(ii) = f_hat(ii-1) ; // set to most recently calcualted zero crossing period length
}
} // end of main ii loop
retval_list(0) = f_hat ;
return retval_list ;
} // end of function
```

and, almost identical to the above
```
DEFUN_DLD ( iq_zc_period_indicator_v1_1, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} iq_zc_period_indicator_v1_1 (@var{Realpart,Quadrature})\n\
This function takes two input vectors of the real part and quadrature of price,\n\
the analytic signal, and outputs the period calculated from the zero crossings of\n\
the two input vectors.\n\
@end deftypefn" )
{
octave_value_list retval_list ;
int nargin = args.length () ;
int vec_length = args(0).length () ;
// check the input arguments
if ( nargin != 2 )
{
error ( "Invalid input argument(s). Inputs are two vectors, the realpart and quadrature." ) ;
return retval_list ;
}
if ( vec_length < 50 )
{
error ( "Invalid input argument length(s). Input vectors must be > 50 in length." ) ;
return retval_list ;
}
if ( args(1).length () != vec_length )
{
error ( "Invalid input argument length(s). Input vectors differ in length." ) ;
return retval_list ;
}
if ( error_state )
{
error ( "Invalid input argument(s). Inputs are two vectors, the realpart and quadrature." ) ;
return retval_list ;
}
// end of input checking
// inputs
ColumnVector inphase_1 = args(0).column_vector_value () ;
ColumnVector quadrature_1 = args(1).column_vector_value () ;
ColumnVector f_hat ( vec_length ) ;
ColumnVector inphase_1_state ( vec_length ) ;
ColumnVector quadrature_1_state ( vec_length ) ;
for ( octave_idx_type ii (0) ; ii < 50 ; ii ++ ) // Start the initialising loop
{
f_hat(ii) = 20.0 ; // a default period length
inphase_1_state(ii) = 0.0 ; // assuming exactly on zero cross line
quadrature_1_state(ii) = 0.0 ; // assuming exactly on zero cross line
} // end of initialising ii loop
double latest_iq_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
double previous_iq_1_zc_time = 49.0 ; // as the "current" bar is bar no. 49
// get the initial states of the inphase_1 and quadrature_1 vectors. A value of
// 1 is assigned if > 0, -1 for < 0 & 0.0 if == 0
inphase_1_state(49) = inphase_1(49) > 0.0 ? 1.0 : ( inphase_1(49) < 0.0 ? -1.0 : 0.0 ) ;
quadrature_1_state(49) = quadrature_1(49) > 0.0 ? 1.0 : ( quadrature_1(49) < 0.0 ? -1.0 : 0.0 ) ;
for ( octave_idx_type ii (50) ; ii < vec_length ; ii ++ ) // Start the main loop
{
// first, assign current states for the inphase_1 and quadrature_1 vectors on this
// new (ii) bar
inphase_1_state(ii) = inphase_1(ii) >= 0.0 ? 1.0 : ( inphase_1(ii) < 0.0 ? -1.0 : 0.0 ) ;
quadrature_1_state(ii) = quadrature_1(ii) >= 0.0 ? 1.0 : ( quadrature_1(ii) < 0.0 ? -1.0 : 0.0 ) ;
// check for a change in state, which indicates a zero crossing
if ( inphase_1_state(ii) == 1.0 && inphase_1_state(ii-1) == -1.0 )
{ // inphase_1 has crossed up over the zero line
latest_iq_1_zc_time = double(ii-1) - inphase_1(ii-1) / ( inphase_1(ii) - inphase_1(ii-1) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_iq_1_zc_time - previous_iq_1_zc_time ) * 4.0 ; // calculate the period from times of zero crosses
previous_iq_1_zc_time = latest_iq_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( inphase_1_state(ii) == -1.0 && inphase_1_state(ii-1) == 1.0 )
{ // inphase_1 has crossed down over the zero line
latest_iq_1_zc_time = double(ii-1) + inphase_1(ii-1) / ( inphase_1(ii-1) - inphase_1(ii) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_iq_1_zc_time - previous_iq_1_zc_time ) * 4.0 ; // calculate the period from times of zero crosses
previous_iq_1_zc_time = latest_iq_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( quadrature_1_state(ii) == 1.0 && quadrature_1_state(ii-1) == -1.0 )
{ // quadrature_1 has crossed up over the zero line
latest_iq_1_zc_time = double(ii-1) - quadrature_1(ii-1) / ( quadrature_1(ii) - quadrature_1(ii-1) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_iq_1_zc_time - previous_iq_1_zc_time ) * 4.0 ; // calculate the period from times of zero crosses
previous_iq_1_zc_time = latest_iq_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else if ( quadrature_1_state(ii) == -1.0 && quadrature_1_state(ii-1) == 1.0 )
{ // quadrature_1 has crossed down over the zero line
latest_iq_1_zc_time = double(ii-1) + quadrature_1(ii-1) / ( quadrature_1(ii-1) - quadrature_1(ii) ) ; // calculate time of zero cross
f_hat(ii) = ( latest_iq_1_zc_time - previous_iq_1_zc_time ) * 4.0 ; // calculate the period from times of zero crosses
previous_iq_1_zc_time = latest_iq_1_zc_time ; // update the previous_zc_time to equal the latest_zc_time
}
else
{ // neither the inphase_1 or quadrature_1 have crossed the zeroline on this bar
f_hat(ii) = f_hat(ii-1) ; // set to most recently calcualted zero crossing period length
}
} // end of main ii loop
retval_list(0) = f_hat ;
return retval_list ;
} // end of function
```

The first measures the period every quarter of a cycle, based on a half period measure of zero crossings, and the second every quarter of a cycle based on quarter cycle measurements. The screen shot below shows typical performance on ideal sine and cosine inputs (top pane) with the period measurements shown in the lower pane.It can be seen that they both perform admirably within their respective theoretical constraints on this ideal input. The subject of my next post will concentrate on the inputs that can be realistically extracted from live price data.

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