The reasons for this failure are unclear to me, and I don't intend to spend any more time investigating, but some educated guesses are that the underlying model of sinusoid is mismatched and my estimation of the process and/or measurement noise variances is lacking; either way, the end result is that the EKF is diverging and my earlier leading signal 1, leading signal 2 and leading signal 3 posts outline what I think will be a more promising line of investigation in the future.

Nevertheless, below I provide the rough working code that I have been using in my EKF work, and maybe readers will find something of value in it. There is a lot of commenting and some blocked out code and I'm afraid readers will have to wade through this as best they can.

```
clear all ;
1 ;
pkg load signal ;
% function declarations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Y = ekf_sine_h( x )
% Measurement model function for the sine signal.
% Takes the state input vector x of sine, phase, angular frequency and amplitude and
% calculates the current value of the sine given the state vector values.
f = x( 2 , : ) ; % phase value in radians
a = x( 4 , : ) ; % amplitude
Y = a .* sin( f ) ; % the sine value
endfunction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ cost ] = sine_ekf_Q_optim( Q_mult , data , targets , Q , R )
% initial state from noisy data input measurements
s = data( : , 1 ) ;
n = size( data , 1 ) ;
% initial state covariance
P = eye( n ) ;
% allocate memory
xV = zeros( size( data ) ) ; % record estmates of states
% basic Jacobian of state transition matrix
A = [ 0 0 0 0 ; % sine value
0 1 1 0 ; % phase
0 0 1 0 ; % angular frequency
0 0 0 1 ] ; % amplitude
H = eye( n ) ; % measurement matrix
Q = Q_mult .* Q ;
for k = 2 : size( data , 2 )
% do ekf
% nonlinear update x and linearisation at current state s
x = s ;
x( 2 ) = x( 2 ) + x( 3 ) ; % advance phase value by angular frequency
x( 2 ) = mod( x( 2 ) , 2 * pi ) ; % limit phase values to range 0 --- 2 * pi
x( 1 ) = x( 4 ) * sin( x( 2 ) ) ; % sine value calculation
% update the 1st row of the jacobian matrix at state vector s values
A( 1 , : ) = [ 0 s( 4 ) * cos( s( 2 ) ) 0 sin( s( 2 ) ) ] ;
P = A * P * A' + Q ; % state transition model update of covariance matrix P
measurement_residual = [ data( 1 , k ) - x( 1 ) ; % sine residual
data( 2 , k ) - x( 2 ) ; % phase residual
data( 3 , k ) - x( 3 ) ; % angular frequency residual
data( 4 , k ) - x( 4 ) ] ; % amplitude residual
innovation_residual_covariance = H * P * H' + R ;
kalman_gain = P * H' / innovation_residual_covariance ;
% update the state vector s with kalman_gain
s = x + kalman_gain * measurement_residual ;
% some reality based post hoc adjustments
s( 2 ) = abs( s( 2 ) ) ; % no negative phase values
s( 3 ) = abs( s( 3 ) ) ; % no negative angular frequencies
s( 4 ) = abs( s( 4 ) ) ; % no negative amplitudes
% update the state covariance matrix P
% NOTE
% The Joseph formula is given by P+ = ( I − KH ) P− ( I − KH )' + KRK', where I is the identity matrix,
% K is the gain, H is the measurement mapping matrix, R is the measurement noise covariance matrix,
% and P−, P+ are the pre and post measurement update estimation error covariance matrices, respectively.
% The optimal linear unbiased estimator (equivalently the optimal linear minimum mean square error estimator)
% or Kalman filter often utilizes simplified covariance update equations such as P+ = (I−KH)P− and P+ = P− −K(HP−H'+R)K'.
% While these alternative formulations require fewer computations than the Joseph formula, they are only valid
% when K is chosen as the optimal Kalman gain. In engineering applications, situations arise where the optimal
% Kalman gain is not utilized and the Joseph formula must be employed to update the estimation error covariance.
% Two examples of such a scenario are underweighting measurements and considering states.
% Even when the optimal gain is used, the Joseph formulation is still preferable because it possesses
% greater numerical accuracy than the simplified equations.
P = ( eye( n ) - kalman_gain * H ) * P * ( eye( n ) - kalman_gain * H )' + kalman_gain * R * kalman_gain' ;
xV( : , k ) = s ; % save estimated updated states
endfor
Y = ekf_sine_h( xV ) ;
cost = mean( ( Y( 4 : end - 3 ) .- targets( 4 : end - 3 ) ).^2 ) ;
%cost = mean( ( Y .- targets ).^2 ) ;
%output = xV( 1 , : ) ;
endfunction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function output = run_sine_ekf_Q_optim( data , Q , R )
% initial state from noisy data input measurements
s = data( : , 1 ) ;
n = size( data , 1 ) ;
% initial state covariance
P = eye( n ) ;
% allocate memory
xV = zeros( size( data ) ) ; % record estmates of states
% basic Jacobian of state transition matrix
A = [ 0 0 0 0 ; % sine value
0 1 1 0 ; % phase
0 0 1 0 ; % angular frequency
0 0 0 1 ] ; % amplitude
H = eye( n ) ; % measurement matrix
for k = 2 : size( data , 2 )
% do ekf
% nonlinear update x and linearisation at current state s
x = s ;
x( 2 ) = x( 2 ) + x( 3 ) ; % advance phase value by angular frequency
x( 2 ) = mod( x( 2 ) , 2 * pi ) ; % limit phase values to range 0 --- 2 * pi
x( 1 ) = x( 4 ) * sin( x( 2 ) ) ; % sine value calculation
% update the 1st row of the jacobian matrix at state vector s values
A( 1 , : ) = [ 0 s( 4 ) * cos( s( 2 ) ) 0 sin( s( 2 ) ) ] ;
P = A * P * A' + Q ; % state transition model update of covariance matrix P
measurement_residual = [ data( 1 , k ) - x( 1 ) ; % sine residual
data( 2 , k ) - x( 2 ) ; % phase residual
data( 3 , k ) - x( 3 ) ; % angular frequency residual
data( 4 , k ) - x( 4 ) ] ; % amplitude residual
innovation_residual_covariance = H * P * H' + R ;
kalman_gain = P * H' / innovation_residual_covariance ;
% update the state vector s with kalman_gain
s = x + kalman_gain * measurement_residual ;
% some reality based post hoc adjustments
s( 2 ) = abs( s( 2 ) ) ; % no negative phase values
s( 3 ) = abs( s( 3 ) ) ; % no negative angular frequencies
s( 4 ) = abs( s( 4 ) ) ; % no negative amplitudes
% update the state covariance matrix P
% NOTE
% The Joseph formula is given by P+ = ( I − KH ) P− ( I − KH )' + KRK', where I is the identity matrix,
% K is the gain, H is the measurement mapping matrix, R is the measurement noise covariance matrix,
% and P−, P+ are the pre and post measurement update estimation error covariance matrices, respectively.
% The optimal linear unbiased estimator (equivalently the optimal linear minimum mean square error estimator)
% or Kalman filter often utilizes simplified covariance update equations such as P+ = (I−KH)P− and P+ = P− −K(HP−H'+R)K'.
% While these alternative formulations require fewer computations than the Joseph formula, they are only valid
% when K is chosen as the optimal Kalman gain. In engineering applications, situations arise where the optimal
% Kalman gain is not utilized and the Joseph formula must be employed to update the estimation error covariance.
% Two examples of such a scenario are underweighting measurements and considering states.
% Even when the optimal gain is used, the Joseph formulation is still preferable because it possesses
% greater numerical accuracy than the simplified equations.
P = ( eye( n ) - kalman_gain * H ) * P * ( eye( n ) - kalman_gain * H )' + kalman_gain * R * kalman_gain' ;
xV( : , k ) = s ; % save estimated updated states
endfor
output = xV ;
endfunction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load data
cd /home/dekalog/Documents/octave/indices ;
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
% so now
% 1 2 3 4 5 6 7 8 9 10 11 12 13 14
% aud_cad aud_chf aud_jpy aud_nzd aud_sgd aud_usd cad_chf cad_jpy cad_sgd chf_jpy eur_aud eur_cad eur_chf eur_gbp
%
% 15 16 17 18 19 20 21 22 23 24 25 26 27 28
% eur_jpy eur_nzd eur_sgd eur_usd gbp_aud gbp_cad gbp_chf gbp_jpy gbp_nzd gbp_sgd gbp_usd nzd_cad nzd_chf nzd_jpy
%
% 29 30 31 32 33 34 35 36 37 38 39 40 41 42
% nzd_sgd nzd_usd sgd_chf sgd_jpy usd_cad usd_chf usd_jpy usd_sgd xag_aud xag_cad xag_chf xag_eur xag_gbp xag_jpy
%
% 43 44 45 46 47 48 49 50 51 52 53 54 55
% xag_nzd xag_sgd xag_usd xau_aud xau_cad xau_chf xau_eur xau_gbp xau_jpy xau_nzd xau_sgd xau_usd xau_xag
% aud_x = x(1) ; cad_x = x(2) ; chf_x = x(3) ; eur_x = x(4) ; gbp_x = x(5) ; hkd_x = x(6) ;
% jpy_x = x(7) ; nzd_x = x(8) ; sgd_x = x(9) ; usd_x = x(10) ; gold_x = x(11) ; silver_x = x(12) ;
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
% so now index ix are
% aud_x = 56 ; cad_x = 57 ; chf_x = 58 ; eur_x = 59 ; gbp_x = 60 ; jpy_x = 61 ;
% nzd_x = 62 ; sgd_x = 63 ; usd_x = 64 ; gold_x = 65 ; silver_x = 66 ;
all_raw_data = [ raw_data all_g_c ] ; clear -x all_raw_data ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
cd /home/dekalog/Documents/octave/kalman/ekf ;
## >> bs_phase_error = mean( all_bootstrap_means(:,1)) + 2 * std( all_bootstrap_means(:,1))
## bs_phase_error = 0.50319
## >> bs_period_ang_frequency_error = mean( all_bootstrap_means(:,2)) + 2 * std( all_bootstrap_means(:,2))
## bs_period_ang_frequency_error = 0.17097
## >> bs_amp_error = mean( all_bootstrap_means(:,3)) + 2 * std( all_bootstrap_means(:,3))
## bs_amp_error = 0.18179
price_ix = 65 ;
price = all_raw_data( : , price_ix ) ; % plot(price) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% get measurements of price action
period = autocorrelation_periodogram( price ) ;
measured_angular_frequency = ( 2 * pi ) ./ period ; % plot( measured_angular_frequency ) ;
trend = sma( price , period ) ;
cycle = price .- trend ;
%smooth_cycle = sgolayfilt( cycle , 2 , 7 ) ;
smooth_cycle = smooth_2_5( cycle ) ;
% plot( price,'k',trend,'r') ;
[ ~ , ~ , ~ , ~ , ~ , ~ , measured_phase ] = sinewave_indicator( cycle ) ; % figure(1) ; plot( deg2rad( measured_phase ) ) ;
% figure(2) ; plot( sin( deg2rad(measured_phase)) ) ;
measured_phase = mod( unwrap( deg2rad( measured_phase ) ) , 2 * pi ) ; % figure(1) ; hold on ; plot( measured_phase , 'r' ) ; hold off ;
% figure(2) ; hold on ; plot( sin( measured_phase)) ; hold off ;
measured_amplitude = cycle ;
for ii = 50 : length( cycle ) ;
measured_amplitude( ii ) = sqrt( 2 ) * sqrt( mean( cycle( ii - period( ii ) : ii ).^2 ) ) ;
endfor % end ii loop
% plot(measured_amplitude) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% run the ekf optimisation
ekf_cycle = cycle ;
% complete values for R matrix for this data set
data = [ smooth_cycle( 101 : end ) measured_phase( 101 : end ) measured_angular_frequency( 101 : end ) measured_amplitude( 101 : end ) ]' ;
%measured_sine = sgolayfilt( data( 1 , : ) , 2 , 7 ) ; % figure(1) ; plot( data( 1 , : ) , 'k' , measured_sine , 'r' ) ;
measured_sine = smooth_cycle( 101 : end )' ;
R( 1 , 1 ) = mean( ( data( 1 , : ) .- measured_sine ).^2 ) ; % variance of sine wave value measurements
R( 3 , 3 ) = mean( ( data( 3 , : ) .* 0.17 ).^2 ) ; % variance of angular frequency measurements
R( 4 , 4 ) = mean( ( data( 4 , : ) .* 0.18 ).^2 ) ; % variance of amplitude measurements
% get initial process covariance matrix Q
Q = analytical_shrinkage( data' ) ;
Q = 0.1 .* Q ;
lookback = 200 ;
%for ii = 301 : length( price )
% complete values for R matrix for this data set
data = [ smooth_cycle(ii-lookback:ii) measured_phase(ii-lookback:ii) measured_angular_frequency(ii-lookback:ii) measured_amplitude(ii-lookback:ii) ]' ;
%measured_sine = sgolayfilt( data( 1 , : ) , 2 , 7 ) ; % figure(1) ; plot( data( 1 , : ) , 'k' , measured_sine , 'r' ) ;
measured_sine = cycle( ii - lookback : ii )' ;
R( 1 , 1 ) = mean( ( data( 1 , : ) .- measured_sine ).^2 ) ; % variance of sine wave value measurements
R( 3 , 3 ) = mean( ( data( 3 , : ) .* 0.17 ).^2 ) ; % variance of angular frequency measurements
R( 4 , 4 ) = mean( ( data( 4 , : ) .* 0.18 ).^2 ) ; % variance of amplitude measurements
% Q = analytical_shrinkage( data' ) ;
% optimise the Q matrix for this data set
% declare optimisation function
f = @( Q_mult ) sine_ekf_Q_optim( Q_mult , data , smooth_cycle( ii - lookback : ii )' , Q , R ) ;
% Set options for fminunc
options = optimset( "MaxIter" , 50 ) ;
% initial value
Q_mult = 1 ;
Q_mult = fminunc( f , Q_mult , options ) ;
% adjust Q by optimised Q_mult
Q = Q_mult .* Q ;
output = run_sine_ekf_Q_optim( data , Q , R ) ;
%ekf_cycle( ii ) = ekf_sine_h( output( : , end ) ) ;
ekf_cycle = ekf_sine_h( output ) ;
figure(1) ; subplot( 4 , 1 , 1 ) ; plot( data( 1 , : ) , 'k' , 'linewidth' , 2 , ekf_sine_h( output ) , 'r' , 'linewidth' , 2 ) ;
title( 'High Pass' ) ; legend( 'Measured' , 'EKF Estimated' ) ;
figure(1) ; subplot( 4 , 1 , 2 ) ; plot( data( 2 , : ) , 'k' , 'linewidth' , 2 , output( 2 , : ) , 'r' , 'linewidth' , 2 ) ;
title( 'High Pass Phase' ) ; legend( 'Measured' , 'EKF Estimated' ) ;
figure(1) ; subplot( 4 , 1 , 3 ) ; plot( data( 3 , : ) , 'k' , 'linewidth' , 2 , output( 3 , : ) , 'r' , 'linewidth' , 2 ) ;
title( 'High Pass Angular Frequency' ) ; legend( 'Measured' , 'EKF Estimated' ) ;
figure(1) ; subplot( 4 , 1 , 4 ) ; plot( data( 4 , : ) , 'k' , 'linewidth' , 2 , output( 4 , : ) , 'r' , 'linewidth' , 2 ) ;
title( 'High Pass Amplitude' ) ; legend( 'Measured' , 'EKF Estimated' ) ;
%endfor
figure(1) ; plot( cycle(101:end)' , 'k' , 'linewidth' , 2 , ekf_cycle , 'r' , 'linewidth' , 2 , output(1,:) , 'b' ) ;
figure(2) ; plot( price(101:end) , 'k' , 'linewidth' , 2 , trend(101:end).+ekf_cycle' , 'r' , 'linewidth' , 2 ) ;
```

One thing I will point out is the use of a function called *analytical_shrinkage*, which I have taken directly from a recent paper, Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices, the MATLAB code being provided as an appendix in the paper. Readers may find this to be particularly useful outside the use I have tried putting it to.

Next week I shall go on my customary, summer working holiday and be away from home until August: therefore there will be a hiatus in blog posts until my return.

## 2 comments:

I think the stock market data series are just not applicable to the function domain covered by the Kalman filter

Try to highpass the price data first to remove the longer frequencies. However, if your aim is to trend trade, then perhaps a sine wave is the wrong model to begin with. Look at fractional differencing the data first to get a stationary time series but one that retains trend information.

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