DDE-1/Dimensionality-reduction.md

@@ -251,11 +251,36 @@ where $`x`$ is the measured signal. Note that n is the elements in X. In the nex
 ```math
 θ = 0.5 tan^{-1}(-2\sum{x_1x_2}/\sum(x_2^2-x_1^2))
 ```
-The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and  $`x_1`$. Next, we express $`U^*`$ with θ:
+The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and  $`x_1`$. Once θ is found, we express $`U^*`$ as:
 
 ```math
-U^* = \begin{bmatrix} cos(θ) & sin(θ) \\ -sin(θ) & cos(θ) \end{bmatrix}
+U^* = 
+\begin{bmatrix} 
+cos(θ) & sin(θ) \\ 
+-sin(θ) & cos(θ) 
+\end{bmatrix}
 ```
+
+The second action was the stretching ($`\sigma^{-1}`$, see the figure above). This step is very easy to calculate as we already found the first principle direction and its variance. Let's call it $`\sigma_1`$ this time:
+
+```math
+\sigma_1(θ) = \sum_{n}^{N} {[x_1(n) x_2(n)]\begin{bmatrix} cos(θ) \\ sin(θ) \end{bmatrix}}^2
+```
+The other principle component will be orthogonal:
+
+```math
+\sigma_2(θ) = \sum_{n}^{N} {[x_1(n) x_2(n)]\begin{bmatrix} cos(θ-\pi/2) \\ sin(θ-\pi/2) \end{bmatrix}}^2
+```
+
+giving $`\sigma^{-1}`$ as:
+
+```math
+\sigma^{-1} =
+\begin{bmatrix} 
+cos(θ) & 0 \\ 
+0 & cos(θ) 
+\end{bmatrix}
+``` 
    
 
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