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Ying-Qiu Zheng
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edafd11f
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edafd11f
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Jul 25, 2021
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Ying-Qiu Zheng
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Update 2021JUL21.md
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@@ -26,13 +26,13 @@ Algorithm 1. EM for the Fusion of GMMs
3.
Initialise the transformation matrix $
`\mathbf{U}`
$ using Algorithm 3.
4.
For iteration = $
`1, 2, ...`
$, do
-
**E-step.**
Evaluate the responsibilities using the current parameter values
-
$
`\gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}`
$
-
$
`\gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}`
$
-
**M-step.**
Re-estimate the parameters using the current responsibilities by setting the derivatives to zero
-
$
`\mu_{k}^{\text{new}} = \frac{1}{N_{k}}((\Sigma^{H}_{k})^{-1} + (\Sigma^{L}_{k})^{-1} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})((\Sigma_{k}^{H})^{-1}\mathbf{Ux}^{H}_{n} + (\Sigma_{k}^{L})^{-1}\mathbf{x}_{n}^{L} )`
$
-
$
`\Sigma_{k}^{L} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})^{T}`
$
-
$
`\Sigma_{k}^{H} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})^{T}`
$
-
$
`\pi_k = \frac{N_{k}}{N}`
$
-
$
`\mathbf{U}=`
$
-
$
`\mu_{k}^{\text{new}} = \frac{1}{N_{k}}((\Sigma^{H}_{k})^{-1} + (\Sigma^{L}_{k})^{-1} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})((\Sigma_{k}^{H})^{-1}\mathbf{Ux}^{H}_{n} + (\Sigma_{k}^{L})^{-1}\mathbf{x}_{n}^{L} )`
$
-
$
`\Sigma_{k}^{L} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})^{T}`
$
-
$
`\Sigma_{k}^{H} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})^{T}`
$
-
$
`\pi_k = \frac{N_{k}}{N}`
$
-
$
`\mathbf{U}=`
$
### Simulation results
#### We considered three scenarios
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