In my last post I said that this next post would report the results of tests on a Constant Acceleration model Kalman filter, and the results are: fail, just like the Constant Velocity model, so I won’t bore readers with reporting the details of the failed tests. However, tests of a Variable Acceleration model have been more successful, and so this post is about the results of tests on this model.

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.