Commit d8c67d11 authored by Ying-Qiu Zheng's avatar Ying-Qiu Zheng
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Update 2021JUL21.md

parent c61680c6
......@@ -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
```math
\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
```math
\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} )
```
- $`\mu_{k}^{\text{new}} = \frac{1}{N_{k}}((\Sigma^{H}_{k})^{-1} + (\Sigma^{L}_{k})^{-1} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})((\Sigma_{k}^{H})^{-1}\mathbf{Ux}^{H}_{n} + (\Sigma_{k}^{L})^{-1}\mathbf{x}_{n}^{L} )`$
- $`\Sigma_{k}^{L} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})^{T}`$
- $`\Sigma_{k}^{H} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})^{T}`$
- $`\pi_k = \frac{N_{k}}{N}`$
- $`\mathbf{U}=`$
### Simulation results
#### We considered three scenarios
......
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