## Monday, 28 September 2015

### Runge-Kutta Example and Code

Following on from my last post I thought I would, as a first step, code up a "straightforward" Runge-Kutta function and show how to deal with the fact that there is no "magic mathematical formula" to calculate the slopes that are an integral part of Runge-Kutta.

My approach is to fit a quadratic function to the last n_bars of price and take the slope of this via my Savitzky-Golay filter convolution code, and in doing so the Runge-Kutta value k1 can easily be obtained. The extrapolation beyond the k1 point to the points k2 and k3 is, in effect, trying to fit to points that have a half bar lead over the last available price. To accommodate this I use the last n_bar - 1 points of a 2 bar simple moving average plus the "position" of points k2  and k3 to calculate the slopes at k2 and k3. A 2 bar simple moving average is used because this has a half bar lag and is effectively an interpolation of the known prices at the "half bar intervals," and therefore points k2 and k3 are one h step ahead of the last half bar interval. The k4 point is again simply calculated directly from prices plus the half bar interval projection from point k3. If all this seems confusing, hopefully the Octave code below will clear things up.
clear all

%  create the raw price series
period = input( 'Period? ' ) ;
sideways = zeros( 1 , 2*period ) ;
uwr = ( 1 : 1 : 2*period ) .* 1.333 / period ; uwr_end = uwr(end) ;
unr = ( 1 : 1 : 2*period ) .* 4 / period ; unr_end = unr(end) ;
dwr = ( 1 : 1 : 2*period ) .* -1.333 / period ; dwr_end = dwr(end) ;
dnr = ( 1 : 1 : 2*period ) .* -4 / period ; dnr_end = dnr(end) ;

trends = [ sideways , uwr , unr.+uwr_end , sideways.+uwr_end.+unr_end , dnr.+uwr_end.+unr_end , dwr.+uwr_end.+unr_end.+dnr_end , sideways ] .+ 2 ;
noise = randn( 1 , length(trends) ) .* 0.0 ;
price = sinewave( length( trends ) , period ) .+ trends .+ noise ;
ma_2 = sma( price , 2 ) ;

%  regress over 'n_bar' bars
n_bar = 9 ;
%  and a 'p' order fit
p = 2 ;

%  get the relevant coefficients
slope_coeffs = generalised_sgolay_filter_coeffs( n_bar , p , 1 ) ;

%  container for 1 bar ahead projection
projection_1_bar = zeros( 1 , length( price ) ) ;

for ii = n_bar : length( price )

%  calculate k1 value i.e. values at price(ii), the most recent price
k1 = price( ii-(n_bar-1) : ii ) * slope_coeffs( : , end ) ;
projection_of_point_k2 = price(ii) + k1 / 2 ;

%  calculate k2 value
k2 = [ ma_2( ii-(n_bar-2) : ii ) ; projection_of_point_k2 ]' * slope_coeffs( : , end ) ;
projection_of_point_k3 = price(ii) + k2 / 2 ;

%  calculate k3 value
k3 = [ ma_2( ii-(n_bar-2) : ii ) ; projection_of_point_k3 ]' * slope_coeffs( : , end ) ;
projection_of_point_k4 = price(ii) + k3 / 2 ;

%  calculate k4 value
k4 = [ price( ii-(n_bar-2) : ii ) , projection_of_point_k4 ] * slope_coeffs( : , end ) ;

%  the runge-kutta weighted moving average
projection_1_bar(ii) = price(ii) + ( k1 + 2 * ( k2 + k3 ) + k4 ) / 6 ;

end

%  shift for plotting
projection_1_bar = shift( projection_1_bar , 1 ) ;
projection_1_bar( : , 1:n_bar ) = price( : , 1:n_bar ) ;

plot( price , 'c' , projection_1_bar , 'r' ) ;
This code produces a plot like this, without noise,

and this with noise ( line 12 of the code ).

The cyan line is the underlying price and the red line is the Runge-Kutta 1 bar ahead projection. As can be seen, when the price is moving in rather straight lines the projection is quite accurate, however, at turnings there is some overshoot, which is to be expected. I'm not unduly concerned about this overshoot as my intention is simply to get the various k values as features, but this overshoot does have some implications which I will discuss in a future post.