The Kalman filter states in this variable acceleration model are position, velocity, acceleration and three constants that are used to calculate the acceleration; b, c and d. The state transition matrix for this model is

```
F = [ 1 1 0 1/2 1/6 1/12 ; % position
0 1 0 1 1/2 1/3 ; % velocity
0 0 0 1 1 1 ; % acceleration
0 0 0 1 0 0 ; % b
0 0 0 0 1 0 ; % c
0 0 0 0 0 1 ] ; % d
```

with a measurement/observation matrix of```
H = [ 1 0 0 0 0 0 ;
0 1 0 0 0 0 ;
0 0 1 0 0 0 ;
0 0 0 1 0 0 ;
0 0 0 0 1 0 ;
0 0 0 0 0 1 ] ; % measurement matrix
```

Of course there are no "measurements" of b, c and d that can be directly taken from a price series, so my solution to this is to use Octave's fminunc function to minimise this```
function [ J ] = kalman_bcd_constant_minimisation( bcd , position_measurements , velocity_measurements , accel_measurements )
pos_proj = position_measurements( 1 ) + velocity_measurements( 1 ) + [ 1/2 1/6 1/12 ] * bcd ;
vel_proj = velocity_measurements( 1 ) + [ 1 1/2 1/3 ] * bcd ;
accel_proj = sum( bcd ) ;
pos_proj_cost = ( pos_proj - position_measurements( 2 ) )^2 ;
vel_proj_cost = ( vel_proj - velocity_measurements( 2 ) )^2 ;
accel_proj_cost = ( accel_proj - accel_measurements( 2 ) )^2 ;
J = pos_proj_cost + vel_proj_cost + accel_proj_cost ;
endfunction
```

where position, velocity and acceleration measurements are calculated as described in my previous post. These cost minimised b, c and d constants are used as the noisy measurements for filter input purposes.One implementation bug I experienced is that I was unable to use the ALS package to model the process noise as the dlqe function it calls kept crashing my session - so as a result I went with the crude, hand constructed covariance matrix as per the code in my previous post.

I wrote above that the tests on this model were the most successful to date. What I mean by this is that the Innovation Magnitude Bound test, the Innovation Zero-centred tests and Innovation auto correlation plot zero-centred tests on the position, velocity and acceleration states were passed 100% and about half of the Innovation auto correlation plot bound tests passed. The Innovation Whiteness (Auto correlation) tests varied from low single figure to 60+% percentage pass rates. However, I'm not 100% certain that I have correctly implemented this last set of tests so these results are suspect. Despite this reservation I think that I can say the Variable Acceleration Model expressed as a polynomial in time is the simplest kinematic model that is suitable for use on financial time series.

More in due course.

## 1 comment:

Have a look at Ehlers article, "Optimal Tracking Filters."

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