...  ...  @@ 240,10 +240,10 @@ We need to think about the last rotation: U. We need to rotate it back via $`U^* 


<img src="uploads/1b71bb1a15e8c1a2e1849e8ea14a9b55/ica_3.png" width="600">



</div>






We are ready to explore the math behind the curtains. The first step was to figure out the last rotation step with U. Herein, we are after the angle $`\teta`$. If you look at the 2D example above, it is easy to see that after these transformations, data is to be oriented with respect to variance (this was the objective in PCA, maximize the variance). In other words, we are looking for the angle that gives the maximum variance. So, we need to formulate how the variance changes as we look at different $`\teta`$ values. In the above 2D example:



We are ready to explore the math behind the curtains. The first step was to figure out the last rotation step with U. Herein, we are after the angle $`θ`$. If you look at the 2D example above, it is easy to see that after these transformations, data is to be oriented with respect to variance (this was the objective in PCA, maximize the variance). In other words, we are looking for the angle that gives the maximum variance. So, we need to formulate how the variance changes as we look at different $`θ`$ values. In the above 2D example:






```math



\sigma(\teta) = \sum_{n}^{N} [x_1(i) x_2(i)]



\sigma(θ) = \sum_{n}^{N} [x_1(i) x_2(i)]\begin{bmatrix} cos(θ) \\ sin(θ) \end{bmatrix}



```






where $`x_j`$ is the measured signal.

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