## Monday, 9 December 2013

### Preliminary Tests of Walk Forward NN Training Idea.

In my suspending work on Brownian bands post I suggested that I wanted to do some preliminary tests of the idea of training a neural net on a walk forward basis. The test I had in mind was a Monte Carlo permutation test as described in David Aronson's book  "Evidence Based Technical Analysis."  Before I go to the trouble of refactoring all my NN code I would like to be sure that, if a NN can be successfully trained in this way, the end result will have been worth the effort. With this aim in mind I used the below implementation of the MC test routine to test the above linked post's idea. The code, as usual, is Octave .oct function C++ code, but taking advantage of Octave's in-built libraries to avoid loops where possible.
DEFUN_DLD ( position_vector_permutation_test, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} position_vector_permutation_test (@var{returns,position_vector_matrix,iters})\n\
This function takes as inputs a vector of returns, a position vector matrix the columns of which are\n\
the individual position vectors that are to be part of the test and the number of iterations for the test.\n\
If the matrix consists only of one position vector the test performed is the single position vector permuation\n\
test. If there are two or more position vectors the test performed is the data mining bias adjusted permutation\n\
test. The null hypothesis that is tested is that the position vector(s) give results that are no better than those\n\
a random ordering of equivalent net positions would give. The alternative hypothesis is that the position vector(s) give\n\
results that are better than a random ordering would give. The function returns a p-value(s) for the position vector(s)\n\
being tested. If small enough, we can reject the null hypothesis and accept the alternative hypothesis. The returned\n\
p-values are expressed as percentages, i.e 0.05 means a 5% significance value. The test statistic is the sum of returns.\n\
*IMPORTANT NOTE* Apart from basic input argument checks to avoid error messages and seg faults this function does not\n\
check that the returns vector and position vector matrix are properly alligned etc. with regard to accuracy of test results\n\
and makes no assumptions about this. It is the user's responsibility to ensure that the position vector(s) are offset to\n\
match the returns for the particular test in question.\n\
@end deftypefn" )

{
octave_value_list retval_list ; // the return value(s) are p value(s)
int nargin = args.length () ;

// check the input arguments
if ( nargin != 3 )
{
error ("Insufficient arguments. Inputs are a returns vector, a position vector matrix and a scalar value for iters.") ;
return retval_list ;
}

if ( args(0).length () != args(1).rows () )
{
error ("Dimensions mismatch. Length of returns vector should == No.rows of position_vector_matrix.") ;
return retval_list ;
}

if ( args(2).rows() > 1 )
{
error ("Invalid 3rd argument. This should be a scalar value for the number of permutations to perform.") ;
return retval_list ;
}

if ( args(2).length () != args(2).rows () )
{
error ("Invalid 3rd argument. This should be a scalar value for the number of permutations to perform.") ;
return retval_list ;
}

if ( error_state )
{
error ("Error state. See usage") ;
return retval_list ;
}
// end of input checking

set_warning_state ( "Octave:broadcast" , "off" ) ;

RowVector returns = args(0).row_vector_value () ; // Returns vector input
Matrix position_vector = args(1).matrix_value () ; // Positions vector input

// compute the return(s) of the candidate model(s)
Matrix model_returns ( 1 , position_vector.cols() ) ;
model_returns = returns * position_vector ; // matrix multiplication

// normalised returns multiplier
Matrix normalised_multiplier ( 1 , position_vector.cols() ) ;
normalised_multiplier.fill ( 0.0 ) ;

if ( position_vector.cols() > 1 ) // more than one position vector
{
for ( octave_idx_type ii (0) ; ii &lt; position_vector.cols() ; ii++ )
{
for ( octave_idx_type jj (0) ; jj &lt; position_vector.rows() ; jj++ )
{
normalised_multiplier( 0 , ii ) += fabs( position_vector( jj , ii ) ) ;
} // end of jj loop
normalised_multiplier( 0 , ii ) = sqrt( position_vector.rows() ) / sqrt( normalised_multiplier( 0 , ii ) ) ;
} // end of ii loop
} // end of if ( position_vector.cols() > 1 )
else // only one position vector
{
normalised_multiplier( 0 , 0 ) = 1.0 ;
}

// use this normalised_multiplier to adjust the model returns
model_returns = product( model_returns , normalised_multiplier ) ;

// Now do the Monte-Carlo replications
int temp ;
int k1 ;
int k2 ;
Matrix trial_returns ( 1 , position_vector.cols() ) ;
double max_trial_return ;
Matrix counts ( 1 , position_vector.cols() ) ;
counts.fill ( 0.0 ) ;
MTRand mtrand1 ; // Declare the Mersenne Twister Class - will seed from system time

for ( octave_idx_type ii (0) ; ii &lt; args(2).int_value() ; ii++ ) // args(2).int_value() is the no. of permutaions to perform
{
trial_returns.fill( 0.0 ) ; // set trial returns to zero

k1 = args(0).length () - 1 ; // initialise prior to shuffling the returns vector

while ( k1 > 0 ) // While at least 2 left to shuffle
{
k2 = mtrand1.randInt ( k1 ) ; // Pick an int from 0 through k1

if (k2 > k1)     // check if random vector index no. k2 is > than max vector index - should never happen
{
k2 = k1 - 1 ; // But this is cheap insurance against disaster if it does happen
}

temp = returns ( k1 ) ;           // allocate the last vector index content to temp
returns ( k1 ) = returns ( k2 ) ; // allocate random pick to the last vector index
returns ( k2 ) = temp ;           // allocate temp content to old vector index position of random pick
k1 = k1 - 1 ;                     // count down
}                                 // Shuffling is complete when this while loop exits

// Compute returns for this random shuffle
trial_returns = returns * position_vector ;

// and now normalise these returns
trial_returns = product( trial_returns , normalised_multiplier ) ;

max_trial_return = *std::max_element( &trial_returns( 0,0 ), &trial_returns( 0,trial_returns.cols() ) ) ;

for ( octave_idx_type jj (0) ; jj &lt; trial_returns.cols() ; jj++ )
{
if ( max_trial_return >= model_returns(0,jj) ) // If the best random system(s) beats the candidate system(s)
{
counts(0,jj) += 1 ; // Count it
}
}

} // end of main ii loop

// now write out the results
Matrix p_values ( 1 , position_vector.cols() ) ;
Matrix iters = args(2).matrix_value () ;
p_values = quotient( counts , iters ) ; // count divided by number of permutations

retval_list(0) = p_values ;

set_warning_state ( "Octave:broadcast" , "on" ) ;

return retval_list ; // Return the output to Octave

} // end of function

I am happy to report that the tests were passed with flying colours, although of course this shouldn't really be surprising as there is the benefit of peeking a few days into the future in the historical record.

My next task will be to implement the walk forward version of the NN, which I anticipate will take multiple weeks at a bare minimum to get satisfactorily working Octave code. More in due course.

## Friday, 6 December 2013

### Brownian Oscillator

In my previous post I said I was thinking about an oscillator based on Brownian bands and the lower pane in the video below shows said oscillator in action. The upper pane simply shows the prices with super imposed Brownian bands. The oscillator ranges between 0 and 1 and is effectively a measurement of the randomness of price movement. The calculation is very simple: for consecutive look back periods of 1 to the dominant cycle period a counter is incremented by 1 for each look back length in which price is between the Brownian bands for that look back length, and the sum of this counter is then divided by the dominant cycle period. The logic should be easily discernible from the code given in the code box below. According to this logic the higher the oscillator value the more random price movement is, and the lower the value the higher the "trendiness" of the price series. The value of the oscillator could also be interpreted as the percentage of look back periods which indicate random movement. Nb. I have changed the calculation of the bands to use absolute price differences rather than log differences - I find that this change leads to better behaved bands.

I have not done any testing of this oscillator as an indicator but I think it might have some value as a filter for a trend following system, or as a way of switching between trend following and mean reversion types of systems. The fact that the oscillator values lie in the range 0 to 1 also makes it suitable as an input to a neural net.

The Octave .oct code
DEFUN_DLD ( brownian_oscillator, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {} brownian_oscillator (@var{price,period})\n\
This function takes price and period vector inputs and outputs a value\n\
normalised by the max look back period for the number of look back periods\n\
from 1 to max look back period inclusive for which the price is between\n\
Brownian bands.\n\
@end deftypefn" )

{
octave_value_list retval_list ;
int nargin = args.length () ;

// check the input arguments
if ( nargin != 2 )
{
error ( "Invalid arguments. Type help to see usage." ) ;
return retval_list ;
}

if ( args(0).length () < 50 ) // check length of price vector input
{
error ( "Invalid arguments. Type help to see usage." ) ;
return retval_list ;
}

if ( args(1).length () != args(0).length () ) // lookback period length argument
{
error ( "Invalid arguments. Type help to see usage." ) ;
return retval_list ;
}

if ( error_state )
{
error ( "Invalid arguments. Type help to see usage." ) ;
return retval_list ;
}
// end of input checking

ColumnVector price = args(0).column_vector_value () ;
ColumnVector period = args(1).column_vector_value () ;
ColumnVector abs_price_diff = args(0).column_vector_value () ;
ColumnVector osc = args(0).column_vector_value () ; osc(0) = 0.0 ;
double sum ;
double osc_sum ;
double up_bb ;
double low_bb ;
int jj ;

for ( octave_idx_type ii (1) ; ii < 50 ; ii++ ) // initialising loop
{
abs_price_diff(ii) = fabs( price(ii) - price(ii-1) ) ;
osc(ii) = 0.0 ;
}

for ( octave_idx_type ii (50) ; ii < args(0).length () ; ii++ ) // main loop
{
// initialise calculation values
sum = 0.0 ;
osc_sum = 0.0 ;
up_bb = 0.0 ;
low_bb = 0.0 ;

for ( jj = 1 ; jj < period(ii) + 1 ; jj++ ) // nested jj loop
{

abs_price_diff(ii) = fabs( price(ii) - price(ii-1) ) ;
sum += abs_price_diff(ii-jj+1) ;
up_bb = price(ii-jj) + (sum/double(jj)) * sqrt(double(jj)) ;
low_bb = price(ii-jj) - (sum/double(jj)) * sqrt(double(jj)) ;

if ( price(ii) <= up_bb && price(ii) >= low_bb )
{
osc_sum += 1.0 ;
}

} // end of nested jj loop

osc(ii) = osc_sum / period(ii) ;

} // end of main loop

retval_list(0) = osc ;

return retval_list ;

} // end of function