Question

I’m trying to code this expression in python but I’m having some difficulty.

This is the code I have so far and wanted some advice.

 x = 1x2 vector
 mu = 1x2 vector
 Sigma = 2x2 matrix 

 xT = (x-mu).transpose()
 sig = Sigma**(-1)
 dotP = dot(xT ,sig )
 dotdot = dot(dotP, (x-mu))
  E = exp( (-1/2) dotdot  )

Am I on the right track? Any suggestions?

Answer 1

• Sigma ** (-1) isn’t what you want. That would raise each element of Sigma to the -1 power, i.e. 1 / Sigma, whereas in the mathematical expression it means the inverse, which is written in Python as np.linalg.inv(Sigma).

• (-1/2) dotdot is a syntax error; in Python, you need to always include * for multiplication, or just do - dotdot / 2. Since you’re probably using python 2, division is a little wonky; unless you’ve done from __future__ import division (highly recommended), 1/2 will actually be 0, because it’s integer division. You can use .5 to get around that, though like I said I do highly recommend doing the division import.

• This is pretty trivial, but you’re doing the x-mu subtraction twice where it’s only necessary to do once. Could save a little speed if your vectors are big by doing it only once. (Of course, here you’re doing it in two dimensions, so this doesn’t matter at all.)

• Rather than calling the_array.transpose() (which is fine), it’s often nicer to use the_array.T, which is the same thing.

• I also wouldn’t use the name xT; it implies to me that it’s the transpose of x, which is false.

I would probably combine it like this:

# near the top of the file
# you probably did some kind of `from somewhere import *`.
# most people like to only import specific names and/or do imports like this,
# to make it clear where your functions are coming from.
import numpy as np

centered = x - mu
prec = np.linalg.inv(Sigma)
E = np.exp(-.5 * np.dot(centered.T, np.dot(prec, centered)))