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  • Data Driven EngineeringData Driven Engineering
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  • Dde 1
  • Dimensionality reduction

Dimensionality reduction · Changes

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Update Dimensionality reduction authored Nov 30, 2021 by cihan.ates's avatar cihan.ates
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DDE-1/Dimensionality-reduction.md
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......@@ -249,9 +249,13 @@ We are ready to explore the math behind the curtains. The first step was to figu
where $`x`$ is the measured signal. Note that n is the elements in X. In the next step, we take the derivative with respect to θ and make it equal to zero. The maximum will give us the first principle component, where the minimum will give the second principle component, which are orthogonal. In practice, it does not matter whether we find the min or max with the derivative; they differ by 90 degree here. After couple of calculation steps, we can get the angle θ:
```math
θ = 1/2 tan^{-1}(-2\sum{x_1x_2}/\sum(x_2^2-x_1^2))
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:
```math
U^* = \begin{bmatrix} cos(θ) & sin(θ) \\ -sin(θ) & cos(θ) \end{bmatrix}
```
The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and $`x_1`$.
...
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