## Friday, 2 March 2018

### Hidden Markov Modelling of Synthetic Periodic Time Series Data

I am currently working on a method of predicting/projecting cyclic price action, based upon John Ehlers' sinewave indicator code, and to test it I am using Octave's implementation of a Hidden Markov model in the Octave statistics package hosted at Sourceforge.

Basically I measure the dominant cycle period ( using either the above linked sinewave indicator code or autocorrelation periodogram code ) and use the vector of measured dominant cycle periods as input to the hmmestimate function. Using the output from this, the hmmgenerate function is then used to generate a new period vector, these periods are converted to a slowly, periodically varying sine wave, and additive white Gaussian noise is added to the signal to produce a final signal upon which Monte Carlo testing of my proposed indicator can be conducted. A typical plot of the varying, dominant cycle periods looks like
whilst the noisy sine wave signal derived from this looks like this.
The relevant Octave code for all this is shown in the two code boxes below
clear all ;

% load all datafile names from Oanda
cd /home/dekalog/Documents/octave/oanda_data/daily ;
oanda_files_d = glob( "*_ohlc_daily" ) ; % cell with filenames matching *_ohlc_daily, e.g. eur_usd_ohlc_daily

all_transprobest = zeros( 75 , 75 ) ;

for ii = 1 : 124

filename = oanda_files_d{ ii } ;
current_data_d = load( "-binary" , filename ) ;
data = getfield( current_data_d , filename ) ;
midprice = ( data( : , 3 ) .+ data( : , 4 ) ) ./ 2 ;
period = autocorrelation_periodogram_2_5( midprice ) ;
period(1:49) = [] ;
min_val = min( period ) ; max_val = max( period ) ;
hmm_periods = period .- ( min_val - 1 ) ; hmm_states = hmm_periods ;

[ transprobest , outprobest ] = hmmestimate( hmm_periods , hmm_states ) ;
all_transprobest( min_val : max_val , min_val : max_val ) = all_transprobest( min_val : max_val , min_val : max_val ) .+ transprobest ;

endfor

all_transprobest = all_transprobest ./ 124 ;
[ i , j ] = find( all_transprobest ) ;

if ( min(i) == min(j) && max(i) == max(j) )
transprobest = all_transprobest( min(i):max(i) , min(j):max(j) ) ;
outprobest = eye( size(transprobest,1) ) ;
hmm_min_period_add = min(i) - 1 ;
endif

cd /home/dekalog/Documents/octave/period/hmm_period ;
save all_hmm_periods_daily transprobest outprobest hmm_min_period_add ;

and
clear all ;
cd /home/dekalog/Documents/octave/snr ;
cd /home/dekalog/Documents/octave/period/hmm_period ;

[ gen_period , gen_states ] = hmmgenerate( 2500 , transprobest , outprobest ) ;
gen_period = gen_period .+ hmm_min_period_add ;
gen_sine = sind( cumsum( 360 ./ gen_period ) ) ;
noise_val = mean( [ all_snr(:,1) ; all_snr(:,2) ] ) ;
noisy_sine = awgn( gen_sine , noise_val ) ;
[s,s1,s2,s3] = sinewave_indicator( noisy_sine ) ; s2(1:50) = s1(1:50) ;

figure(1) ; plot( gen_period , 'k' , 'linewidth' , 2 ) ;
figure(2) ; plot( gen_sine , 'k' , 'linewidth' , 2 , noisy_sine , 'b' , 'linewidth' , 2 , s , 'r' , 'linewidth' , 2 , ...
s1 , 'g' , 'linewidth' , 2 , s2 , 'm' , 'linewidth' , 2 ) ;
legend( "Gen Sine" , "Noisy Sine" , "Sine Ind" , "Sine Ind lead1" , "Sine Ind Lead2" ) ;

I hope readers find this useful if they need to generate synthetic, cyclic data for their own development/testing purposes too.