... | ... | @@ -261,7 +261,7 @@ cos(θ) & sin(θ) \\ |
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\end{bmatrix}
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```
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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:
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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:
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```math
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\sigma_1(θ) = \sum_{n}^{N} {[x_1(n) x_2(n)]\begin{bmatrix} cos(θ) \\ sin(θ) \end{bmatrix}}^2
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... | ... | @@ -272,10 +272,10 @@ The other principle component will be orthogonal: |
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\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
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```
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giving $`\sigma^{-1}`$ as:
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giving $`\Sigma^{-1}`$ as:
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```math
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\sigma^{-1} =
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\Sigma^{-1} =
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\begin{bmatrix}
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cos(θ) & 0 \\
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0 & cos(θ)
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... | ... | |