# Gaussian Mixture Model

## Lecture Note

[GMM-EM class note](https://drive.google.com/file/d/1LCR-0fa8fYE8r24BKNTDADTinivBpTOe/view?usp=sharing)

## Gaussian Mixture Model

[Click here for PPT slides](https://people.engr.tamu.edu/rgutier/web_courses/csce666_s20/l14.pdf): TAMU

[Click here for PPT slides](https://drive.google.com/file/d/1v2M0Sk8eeyFZoIB_SJ7pqcjdnoULVW9h/view?usp=sharing): VTech

Gaussian Distribution

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-b021b1028fed283843dbf859b8652f20bf067d27%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-9209987c5bd3d53cc84c2cc5b8185d95de51cf12%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-239131290aa4eac76b8c180825c520521a64c934%2Fimage.png?alt=media)

The density can be estimated by multiples of Gaussian Kernels

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-752cafcd70542fe750210f98a42cbe8337619dfd%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-0a2be6b96d32c72e4a040a0f2413e3070068e81e%2Fimage.png?alt=media)

> $$\pi$$is prior, not likelihood.
>
> The mixing coefficients are themselves probabilities and must meet this condition: sum(pi)=1

**How to find the optimal parameter** $$\Theta$$**?**

Lets use the Maximum Likelihood Estimation to find the optimal parameter. This can be solved by E-M algorithm/

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-256e7c70df10da9b20033c9297f5d431f3e5d0f5%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-1bc943a23dc2eeabad85f876bf0375f95423ee97%2Fimage.png?alt=media)

### Example: Object segmentation

see [PPT slides](https://drive.google.com/file/d/1v2M0Sk8eeyFZoIB_SJ7pqcjdnoULVW9h/view?usp=sharing): VTech

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-219781abc3f3955df66fa66508f4beee82b23b4a%2Fimage.png?alt=media)

## E-M algorithm for Gaussian Mixture Kernel

[Read here for detail](https://towardsdatascience.com/gaussian-mixture-models-explained-6986aaf5a95)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-7b4f5c7e1fe57fc0aff563451138e9a4a9c2d471%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-d15ea19de4c0b48b1d8c0cc9385caae1eab78960%2Fimage.png?alt=media)

## Reference

The Hundred-Page Machine Learning Book [http://themlbook.com/wiki/doku.php](http://themlbook.com/wiki/doku.php)

머신러닝/패턴인식, 오일석

{% embed url="https://towardsdatascience.com/gaussian-mixture-models-explained-6986aaf5a95" %}
