Determining the Noise Covariance Matrix R for a Kalman Filter

An important part of getting a Kalman filter to work well is tuning the process noise covariance matrix Q and the measurement noise covariance matrix R. This post is about obtaining the R matrix, with a post about the Q matrix to come in due course.

In my last post about the alternative version extended kalman filter to model a sine wave I explained that the 4 measurements are the sine wave value itself, and the phase, angular frequency and amplitude of the sine wave. The phase and angular frequencies are "measured" using outputs of the sinewave indicator and the amplitude is calculated as an RMS value over a measured period. The "problem" with this is that the accuracies of such measurements on real data are unknown, and therefore so is the R matrix, because the underlying cycle properties are unknown. My solution to this is offline Monte Carlo simulation.

As a first step the Octave code in the box immediately below

clear all ;
pkg load statistics ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load data

raw_data = dlmread( 'raw_data_for_indices_and_strengths' ) ;
delete_hkd_crosses = [ 3 9 12 18 26 31 34 39 43 51 61 ] .+ 1 ; % +1 to account for date column
raw_data( : , delete_hkd_crosses ) = [] ;
raw_data( : , 1 ) = [] ; % delete data vector 

all_g_c = dlmread( "all_g_mults_c" ) ; % the currency g mults
all_g_c( : , 7 ) = [] ; % delete hkd index

all_g_s = dlmread( "all_g_sv" ) ;      % the gold and silver g mults   
all_g_c = [ all_g_c all_g_s(:,2:3) ] ; % a combination of the above 2
all_g_c( : , 2 : end ) = cumprod( all_g_c( : , 2 : end) , 1 ) ;
all_g_c( : , 1 ) = [] ; % delete date vector

all_raw_data = [ raw_data all_g_c ] ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% set up recording vectors
amplitude = zeros( size( all_raw_data , 1 ) , 1 ) ;
all_amplitudes = zeros( size( all_raw_data ) ) ;
all_periods = all_amplitudes ;

for ii = 1 : size( all_raw_data , 2 )
  
  price = all_raw_data( : , ii ) ;
  period = autocorrelation_periodogram( price ) ;
  all_periods( : , ii ) = period ;
  [ hp , ~ ] = highpass_filter_basic( price ) ; hp( 1 : 99 ) = 0 ;

  for jj = 100 : size( all_raw_data , 1 )
    amplitude( jj ) = sqrt( 2 ) * sqrt( mean( hp( round( jj - period( jj ) / 2 ) : jj , 1 ).^2 ) ) ;
  endfor
  
  % store information about amplitude changes as a multiplicative constant
  all_amplitudes( : , ii ) = amplitude ./ shift( amplitude , 1 ) ;

endfor

% truncate
all_amplitudes( 1 : 101 , : ) = [] ;
all_periods( 1 : 101 , : ) = [] ;

max_period = max( max( all_periods ) ) ;
min_period = min( min( all_periods ) ) ;

% unroll
all_amplitudes = all_amplitudes( : ) ; all_periods = all_periods( : ) ;

amplitude_changes = cell( max_period , 1 ) ;

for ii = min_period : max_period
  ix = find( all_periods == ii ) ;
  amplitude_changes{ ii , 1 } = all_amplitudes( ix , 1 ) ;
endfor

save all_amplitude_changes amplitude_changes ;

is used to get information about cycle amplitudes from real data and store it in a cell array.

This next box shows code that uses this cell array, plus earlier hidden markov modelling code for sine wave modelling, to produce synthetic price data

clear all ;
pkg load signal ;
pkg load statistics ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load data

raw_data = dlmread( 'raw_data_for_indices_and_strengths' ) ;
delete_hkd_crosses = [ 3 9 12 18 26 31 34 39 43 51 61 ] .+ 1 ; % +1 to account for date column
raw_data( : , delete_hkd_crosses ) = [] ;
raw_data( : , 1 ) = [] ; % delete date vector
all_g_c = dlmread( "all_g_mults_c" ) ; % the currency g mults
all_g_c( : , 7 ) = [] ; % delete hkd index
all_g_s = dlmread( "all_g_sv" ) ;      % the gold and silver g mults   
all_g_c = [ all_g_c all_g_s(:,2:3) ] ; % a combination of the above 2
all_g_c( : , 2 : end ) = cumprod( all_g_c( : , 2 : end) , 1 ) ;
all_g_c( : , 1 ) = [] ; % delete date vector
all_raw_data = [ raw_data all_g_c ] ; clear -x all_raw_data ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

load all_hmm_periods_daily ;
load all_amplitude_changes ;

% choose an ix for all_raw_data
ix = input( "ix? " ) ;
price = all_raw_data( : , ix ) ;
period = autocorrelation_periodogram( price ) ;
[ hp , trend ] = highpass_filter_basic( price ) ;

% estimate variance of cycle noise
hp_filt = sgolayfilt( hp , 2 , 7 ) ;
noise = hp( 101 : end ) .- hp_filt( 101 : end ) ; noise_var = var( noise ) ;
% zero pad beginning of noise vector
noise = [ zeros( 100 , 1 ) ; noise ] ;

% set for plotting purposes
hp( 1 : 50 ) = 0 ; trend( 1 : 50 ) = price( 1 : 50 ) ;

% create sine wave with known phase and period/angular frequency 
[ gen_period , ~ ] = hmmgenerate( length( price ) , transprobest , outprobest ) ;
gen_period = gen_period .+ hmm_min_period_add ;
phase = cumsum( ( 2 * pi ) ./ gen_period ) ; phase = mod( phase , 2 * pi ) ;
gen_sine = sin( phase ) ;

% amplitude vector
amp = ones( size( hp ) ) ;

% RMS initial amplitude 
amp( 100 ) = sqrt( 2 ) * sqrt( mean( hp( 100 - period( 100 ) : 100 ).^2 ) ) ;

% alter sine wave with known amplitude
for ii = 101 : length( hp )
  % get multiple for this period
  random_amp_mults = amplitude_changes{ gen_period( ii ) } ;
  rand_mult = random_amp_mults( randi( size( random_amp_mults , 1 ) , 1 ) ) ;
  amp( ii ) = rand_mult * sqrt( 2 ) * sqrt( mean( hp( ii - 1 - period( ii - 1 ) : ii - 1 ).^2 ) ) ; ;
  gen_sine( ii ) = amp( ii ) * gen_sine( ii ) ; 
endfor

% estimate variance of generated cycle noise
gen_filt = sgolayfilt( gen_sine , 2 , 7 ) ;
gen_noise = gen_sine( 101 : end ) .- gen_filt( 101 : end ) ; gen_noise_var = var( gen_noise ) ;

% a crude adjustment of generated cycle noise to match "real" noise by adding
% noise
gen_sine = gen_sine' .+ noise ;

% set beginning of gen_sine to real cycle for plotting purposes
gen_sine( 1 : 100 ) = hp( 1 : 100 ) ;
new_price = trend .+ gen_sine ;

% a simple correlation check
correlation_cycle_gen_sine = corr( hp , gen_sine ) ;
correlation_price_gen_price = corr( price , new_price ) ;

figure(1) ; plot( gen_sine , 'k' , 'linewidth' , 2 ) ; grid minor on ; title( 'The Generated Cycle' ) ; 
figure(2) ; plot( price , 'b' , 'linewidth' , 2 , trend , 'k' , 'linewidth' , 1 , new_price , 'r' , 'linewidth', 2 ) ; 
title( 'Comparison of Real and Generated Prices' ) ; legend( 'Real Price' , 'Real Trend' , 'Generated Price' ) ; grid minor on ;
figure(3) ; plot( hp , 'k' , 'linewidth' , 2 , gen_sine , 'r' , 'linewidth' , 2 ) ; title( 'Generated Cycle compared with Real Cycle' ) ; 
legend( 'Real Cycle' , 'Generated Cycle' ) ; grid minor on ;

Some typical chart output of this code is the synthetic price series in red compared with the real blue price series

and the underlying cycles.

For this particular run the correlation between the synthetic price series and the real price series is 0.99331, whilst the correlation between the relevant cycles is only 0.19988, over 1500 data points ( not all are shown in the above charts. )

Because the synthetic cycle is definitely a sine wave ( by construction ) with a known phase, angular frequency and amplitude I can run the above sinewave_indicator and RMS measurements on the synthetic prices, compare with the known synthetic cycle, and estimate the measurement noise covariance matrix R.

More in due course.