Firstly, I create my "ideal" market types by using an Octave .oct function, the code for which is given below. In this code can also be seen my implementation of the code for the Trend Vigor indicator, which I think is slightly different from Elher's. The code is commented, so no further description is required here.

.oct function code

```
// This function takes as arguments a single value input for a sinewave period and a single value input for a degrees increment value. A sinewave of the
// given period is created and then trends are added to the sinewave such that 5 hypothesised "ideal" market types are created: these are
// 1) a perfectly cyclic sideways market with no trend i.e. just the sinewave component
// 2) an uptrending market with cyclic retracements (uwr) such that retracements are down to the 50% Fibonacci ratio level
// 3) an uptrending market with no retracemnets (unr) i.e. the uptrend completely swamps the downward cyclic component counter to the trend
// 4) a downtrending market with cyclic retracements (dwr) such that retracements are up to the 50% Fibonacci ratio level
// 5) a downtrending market with no retracements (dnr) i.e. the downtrend completely swamps the upward cyclic component counter to the trend
// The cyclic component of these markets is then extracted using the bandpass indicator. These vector lengths are 500 in order to allow time for the
// bandpass calculations to settle down. The peak to peak amplitude is then recovered from the bandpass at the end of each market vector. The trend slope
// is also calculated at the end of each market vector. The trend vigor indicator value is then calculated thus
// trend_slope/p_to_p. The bandpass delta is set to 0.2.
// The idea is that
// for a sideways market the value should be about 0
// for a trending with retracement market the value should be > 0 && < 1 or < 0 && > -1, depending on whether an up trend or down trend
// for a trending with no retracement market the ratio should be > 1 or > -1, depending on whether an up trend or down trend
// The original sinewave is then repeatedly phase shifted by the degrees increment value, the above process repeated and new values are calculated.
// All the calculated values for each market type are the vector outputs of the function.
#include
```
#include
#include
#define PI 3.14159265
DEFUN_DLD (trend_vigor_dist, args, , "Inputs are period & degrees increment value, outputs are vectors of repeated median slope values")
{
octave_value_list retval_list;
if (args(0).length () < 1 | args(0).length () > 1)
{
error ("Invalid arguments. Inputs are single value period length and single value degrees increment value");
return retval_list;
}
if (args(1).length () < 1 | args(1).length () > 1)
{
error ("Invalid arguments. Inputs are single value period length and single value degrees increment value");
return retval_list;
}
if (error_state)
{
error ("Invalid arguments. Inputs are single value period length and single value degrees increment value");
return retval_list;
}
// read inputs
int period = args(0).int_value ();
double degrees_inc = args(1).double_value ();
double period_inc = 360.0 / double(period);
int length = period + 1; // the length of the "lookback". Is equal to period + 1 to encompass a full period
// vectors to hold created market values
ColumnVector sideways_vec(500); // vector to hold sideways market values
ColumnVector sideways_bandpass_vec(500); // vector to hold bandpass of sideways market values
ColumnVector uwr_vec(500); // vector to hold uwr market values
ColumnVector uwr_bandpass_vec(500); // vector to hold bandpass of uwr market values
ColumnVector unr_vec(500); // vector to hold unr market values
ColumnVector unr_bandpass_vec(500); // vector to hold bandpass of unr market values
ColumnVector dwr_vec(500); // vector to hold dwr market values
ColumnVector dwr_bandpass_vec(500); // vector to hold bandpass of dwr market values
ColumnVector dnr_vec(500); // vector to hold dnr market values
ColumnVector dnr_bandpass_vec(500); // vector to hold bandpass of dnr market values
// calculate the market trend_incs
double uwr_trend_inc = 12 / ( 5 * double(period) );
double unr_trend_inc = 4 / double(period);
double dwr_trend_inc = -( 12 / ( 5 * double(period) ) );
double dnr_trend_inc = -( 4 / double(period) );
// declare variables for bandpass and trend vigor calculations
double delta = 0.2;
double beta = cos( (360.0/period)*PI/180.0 );
double gamma = 1.0 / cos( (720.0*delta/period)*PI/180.0 );
double alpha = gamma - sqrt(gamma*gamma - 1.0);
double power_side;
double power_uwr;
double power_unr;
double power_dwr;
double power_dnr;
double rms;
double p_to_p;
// create output vectors
int output_vec_length = int ( 360 / degrees_inc );
ColumnVector sideways_dist ( output_vec_length ); // create output column of correct length for sideways market
ColumnVector uwr_dist ( output_vec_length ); // create output column of correct length for uwr market
ColumnVector unr_dist ( output_vec_length ); // create output column of correct length for unr market
ColumnVector dwr_dist ( output_vec_length ); // create output column of correct length for dwr market
ColumnVector dnr_dist ( output_vec_length ); // create output column of correct length for dnr market
for (octave_idx_type ii (0); ii < output_vec_length; ii++)
{
// Create the market types and their bandpasses for this ii iteration
for (octave_idx_type jj (0); jj < 500; jj++)
{
// First create the sideways market type
sideways_vec(jj) = sin( (degrees_inc*ii + period_inc*jj) * PI / 180 );
if ( jj < 2 )
{
sideways_bandpass_vec(jj) = 0;
}
else
{
sideways_bandpass_vec(jj) = 0.5*(1.0 - alpha)*(sideways_vec(jj) - sideways_vec(jj-2)) + beta*(1.0 + alpha)*sideways_bandpass_vec(jj-1) - alpha*sideways_bandpass_vec(jj-2);
}
// next, the uwr retracement market (uwr)
uwr_vec(jj) = sideways_vec(jj) + jj*uwr_trend_inc;
if ( jj < 2 )
{
uwr_bandpass_vec(jj) = 0;
}
else
{
uwr_bandpass_vec(jj) = 0.5*(1.0 - alpha)*(uwr_vec(jj) - uwr_vec(jj-2)) + beta*(1.0 + alpha)*uwr_bandpass_vec(jj-1) - alpha*uwr_bandpass_vec(jj-2);
}
// next, the unr retracement market (unr)
unr_vec(jj) = sideways_vec(jj) + jj*unr_trend_inc;
if ( jj < 2 )
{
unr_bandpass_vec(jj) = 0;
}
else
{
unr_bandpass_vec(jj) = 0.5*(1.0 - alpha)*(unr_vec(jj) - unr_vec(jj-2)) + beta*(1.0 + alpha)*unr_bandpass_vec(jj-1) - alpha*unr_bandpass_vec(jj-2);
}
// next, the dwr retracement market (dwr)
dwr_vec(jj) = sideways_vec(jj) + jj*dwr_trend_inc;
if ( jj < 2 )
{
dwr_bandpass_vec(jj) = 0;
}
else
{
dwr_bandpass_vec(jj) = 0.5*(1.0 - alpha)*(dwr_vec(jj) - dwr_vec(jj-2)) + beta*(1.0 + alpha)*dwr_bandpass_vec(jj-1) - alpha*dwr_bandpass_vec(jj-2);
}
// next, the dnr retracement market (dnr)
dnr_vec(jj) = sideways_vec(jj) + jj*dnr_trend_inc;
if ( jj < 2 )
{
dnr_bandpass_vec(jj) = 0;
}
else
{
dnr_bandpass_vec(jj) = 0.5*(1.0 - alpha)*(dnr_vec(jj) - dnr_vec(jj-2)) + beta*(1.0 + alpha)*dnr_bandpass_vec(jj-1) - alpha*dnr_bandpass_vec(jj-2);
}
} // end of jj loop to create the different markets and their bandpasses
// now loop over end of each market vector to create the distributions
power_side = 0.0;
power_uwr = 0.0;
power_unr = 0.0;
power_dwr = 0.0;
power_dnr = 0.0;
for (octave_idx_type jj (0); jj < length; jj++)
{
power_side = power_side + sideways_bandpass_vec(499-jj)*sideways_bandpass_vec(499-jj) + sideways_bandpass_vec(499-jj-int(period/4.0))*sideways_bandpass_vec(499-jj-int(period/4.0)) ;
power_uwr = power_uwr + uwr_bandpass_vec(499-jj)*uwr_bandpass_vec(499-jj) + uwr_bandpass_vec(499-jj-int(period/4.0))*uwr_bandpass_vec(499-jj-int(period/4.0)) ;
power_unr = power_unr + unr_bandpass_vec(499-jj)*unr_bandpass_vec(499-jj) + unr_bandpass_vec(499-jj-int(period/4.0))*unr_bandpass_vec(499-jj-int(period/4.0)) ;
power_dwr = power_dwr + dwr_bandpass_vec(499-jj)*dwr_bandpass_vec(499-jj) + dwr_bandpass_vec(499-jj-int(period/4.0))*dwr_bandpass_vec(499-jj-int(period/4.0)) ;
power_dnr = power_dnr + dnr_bandpass_vec(499-jj)*dnr_bandpass_vec(499-jj) + dnr_bandpass_vec(499-jj-int(period/4.0))*dnr_bandpass_vec(499-jj-int(period/4.0)) ;
}
// fill the distribution vectors
rms = sqrt( power_side / (period+1) ) ;
p_to_p = 2.0 * 1.414 * rms ;
sideways_dist(ii) = ( sideways_vec(499) - sideways_vec(499-period) ) / p_to_p ;
rms = sqrt( power_uwr / (period+1) ) ;
p_to_p = 2.0 * 1.414 * rms ;
uwr_dist(ii) = ( uwr_vec(499) - uwr_vec(499-period) ) / p_to_p ;
rms = sqrt( power_unr / (period+1) ) ;
p_to_p = 2.0 * 1.414 * rms ;
unr_dist(ii) = ( unr_vec(499) - unr_vec(499-period) ) / p_to_p ;
rms = sqrt( power_dwr / (period+1) ) ;
p_to_p = 2.0 * 1.414 * rms ;
dwr_dist(ii) = ( dwr_vec(499) - dwr_vec(499-period) ) / p_to_p ;
rms = sqrt( power_dnr / (period+1) ) ;
p_to_p = 2.0 * 1.414 * rms ;
dnr_dist(ii) = ( dnr_vec(499) - dnr_vec(499-period) ) / p_to_p ;
} // end of main ii loop
retval_list(4) = dnr_dist;
retval_list(3) = dwr_dist;
retval_list(2) = unr_dist;
retval_list(1) = uwr_dist;
retval_list(0) = sideways_dist;
return retval_list;
}

This function is called by this simple Octave script```
clear all
inc = input( "Enter phase increment: ");
for ii = 6:50
[sideways_dist,uwr_dist,unr_dist,dwr_dist,dnr_dist] = trend_vigor_dist(ii,inc);
A=[sideways_dist,uwr_dist,unr_dist,dwr_dist,dnr_dist];
file = strcat( int2str(ii),"_period_dist" );
dlmwrite(file,A)
endfor
```

which writes the output of the tests to named files which are to be used for further analysis in R.Firstly, using R, I wanted to see what the distribution of the results looks like, so this R script

```
rm(list=ls())
data <- as.matrix(read.csv(file="20_period_dist",head=FALSE,sep=,))
side <- density(data[,1])
uwr <- density(data[,2])
max_uwr_y <- max(uwr$y)
max_uwr_x <- max(uwr$x)
min_uwr_x <- min(uwr$x)
unr <- density(data[,3])
max_unr_y <- max(unr$y)
max_unr_x <- max(unr$x)
min_unr_x <- min(unr$x)
dwr <- density(data[,4])
max_dwr_y <- max(dwr$y)
max_dwr_x <- max(dwr$x)
min_dwr_x <- min(dwr$x)
dnr <- density(data[,5])
max_dnr_y <- max(dnr$y)
max_dnr_x <- max(dnr$x)
min_dnr_x <- min(dnr$x)
plot_max_y <- max(max_uwr_y,max_unr_y,max_dwr_y,max_dnr_y)
plot_max_x <- max(max_uwr_x,max_unr_x,max_dwr_x,max_dnr_x)
plot_min_x <- min(min_uwr_x,min_unr_x,min_dwr_x,min_dnr_x)
par(mfrow=c(2,1))
plot(uwr,xlim=c(plot_min_x,plot_max_x),ylim=c(0,plot_max_y),col="red")
par(new=TRUE)
plot(unr,xlim=c(plot_min_x,plot_max_x),ylim=c(0,plot_max_y),col="blue")
par(new=TRUE)
plot(dwr,xlim=c(plot_min_x,plot_max_x),ylim=c(0,plot_max_y),col="green")
par(new=TRUE)
plot(dnr,xlim=c(plot_min_x,plot_max_x),ylim=c(0,plot_max_y),col="black")
plot(side)
```

produces plots such as this outputwhere the top plot shows the distributions of the uwr, unr, dwr and dnr markets, and the lower plot the sideways market. In this particular case, the spread of each distribution is so narrow ( measured differences of the order of thousandths of a decimal place ) that I consider that for practical purposes the distributions can be treated as single values. This simple R boot strap script gets the average value of the distributions to be used as this single point value.

```
rm(list=ls())
data <- as.matrix(read.csv(file="trend_vigor_dist_results",head=FALSE,sep=,))
side <- data[,1]
uwr <- data[,2]
unr <- data[,3]
dwr <- data[,4]
dnr <- data[,5]
side_samples <- matrix(0,50000)
uwr_samples <- matrix(0,50000)
unr_samples <- matrix(0,50000)
dwr_samples <- matrix(0,50000)
dnr_samples <- matrix(0,50000)
for(ii in 1:50000) {
side_samples[ii] <- mean(sample(side, replace=T))
uwr_samples[ii] <- mean(sample(uwr, replace=T))
unr_samples[ii] <- mean(sample(unr, replace=T))
dwr_samples[ii] <- mean(sample(dwr, replace=T))
dnr_samples[ii] <- mean(sample(dnr, replace=T))
}
side_mean <- mean(side_samples)
uwr_mean <- mean(uwr_samples)
unr_mean <- mean(unr_samples)
dwr_mean <- mean(dwr_samples)
dnr_mean <- mean(dnr_samples)
```

For readers' interest, the actual values are 0.829, 1.329, -0.829 and -1.329 with 0 for the sideways market.This final plot is the same Natural Gas plot as in my previous post, but with the above values substituted for Ehler's default values of 1 and -1.

What I intend to do now is use these values as the means of normal distributions with varying standard deviations as inputs for my Naive Bayes classifier. Further Monte Carlo testing will be done such that values for the standard deviations are obtained that result in the classifier giving false classifications, when tested using the "ideal" markets code above, within acceptable limits, most probably a 5% classification error rate.