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→Covariance Matrices: Added section
== Covariance Matrices ==
With 5 (or <code>numPOIs</code>) POIs picked out, we can build our multivariate distributions at each point for each Hamming weight. We need to write down two matrices for each weight:
* A mean matrix (<code>1 x numPOIs</code>) which stores the mean at each POI
* A covariance matrix (<code>numPOIs x numPOIs</code>) which stores the variances and covariances between each of the POIs
The mean matrix is easy to set up because we've already found the mean at every point. All we need to do is grab the right points:
<pre>
meanMatrix = np.zeros((9, numPOIs))
for HW in range(9):
for i in range(numPOIs):
meanMatrix[HW][i] = tempMeans[HW][POIs[i]]
</pre>
The covariance matrix is a bit more complex. We need a way to find the covariance between two 1D arrays. Helpfully, NumPy has the <code>cov(a, b)</code> function, which returns the matrix
<pre>
np.cov(a, b) = [[cov(a, a), cov(a, b)],
[cov(b, a), cov(b, b)]]
</pre>
We can use this to define our own covariance function:
<pre>
def cov(x, y):
# Find the covariance between two 1D lists (x and y).
# Note that var(x) = cov(x, x)
return np.cov(x, y)[0][1]
</pre>
As mentioned in the comments, this function can also calculate the variance of an array by passing the same array in for both <code>x</code> and <code>y</code>.
We'll use our function to build the covariance matrices:
<pre>
covMatrix = np.zeros((9, numPOIs, numPOIs))
for HW in range(9):
for i in range(numPOIs):
for j in range(numPOIs):
x = tempTracesHW[HW][:,POIs[i]]
y = tempTracesHW[HW][:,POIs[j]]
covMatrix[HW,i,j] = cov(x, y)
</pre>
Make sure these matrices look okay before continuing:
<pre>
print meanMatrix
print covMatrix[0]
</pre>
Note that you may not have enough data to complete the covariance matrix - check out the Gotchas at the end of this tutorial if this step blew up on you.
= Performing the Attack =
