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  • Data Driven EngineeringData Driven Engineering
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  • 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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......@@ -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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