kalman.py: column vector or row vector?

[oyamad]

Hi,
in the study group session today, we found some unexpected behavior in the script given in the solution to Exercise 3 in "A First Look at the Kalman Filter".
I think I identified the problem: kn.current_x_hat has to be "flattened":

    # Wrong
    e1[t] = np.sum((x - kn.current_x_hat)**2)

    # Correct
    e1[t] = np.sum((x - kn.current_x_hat.flatten())**2)

kn.current_x_hat is a column vector (2x1), while x is a row vector (1x2), so x - kn.current_x_hat becomes 2x2, and hence np.sum((x - kn.current_x_hat)**2) gets doubled.

Here's how we got the "unexpected behavior":
http://nbviewer.ipython.org/gist/oyamad/1f9b66fda3679f85f794
(Please see the second half part starting with "Exercise 3".)

One may say that the problem is caused by the implementation design of Kalman, where current_xhat is implemented as a column vector. Mathematically it is legitimate, but in python we do not have to care in many cases as numpy guesses correctly to compute e.g. np.dot(A, x) even if x is a row vector. It might be better to consistently stick to row vectors.

[albop]

Could you define current_x_hat as a 1d vector instead ? Then it would be broadcasted to a 2x1 array or 1x2 array depending on the orientation of x. Would that work ?

To me, one good feature of numpy is that it has 1d vectors, while in matlab everything is a matrix by default. For the dot operator, it works likewise: provided that the dimensions match you can write dot(A,x) or dot(x,A) without caring whether x is a vector or a linear form.

[oyamad]

@albop As you suggest, I dropped conversion to 2d array in the Kalman class, then the solution script worked as expected without flatten(). (But I only tried one particular case with R = np.zeros((2, 2)) and x_hat = np.array([0, 0]).)
http://nbviewer.ipython.org/gist/oyamad/1f9b66fda3679f85f794/kalman_ex3_2.ipynb

[jstac]

Thanks for letting me know about this error. In the end I just corrected the exercise by flattening the vector ex post. I didn't flatten everything inside Kalman because I want the inner workings to be transparent --- this module is more instructive rather than for heavy duty use.