This document provides an overview of mixture models and related machine learning concepts such as the Gaussian mixture model, EM algorithm, and their application to speaker identification and diarization. It first reviews Gaussian distributions and introduces multivariate Gaussian distributions. It then discusses latent variable models and mixture models for representing multiple sub-populations within an overall population. The EM algorithm is presented as a method for solving mixture models like the Gaussian mixture model. Applications to speaker diarization using Mel frequency cepstrum coefficients as features are also mentioned.

1. Mixture Model
Hong Chulju

2. Index
• Gaussian Distribution
• Multivariate Gaussian Distribution
• Latent Variable Model
• Mixture Model
• EM Algorithm
• Gaussian Mixture Model Example
• Gaussian Mixture Model vs k-Means Clustering
• Mel Frequency Cepstrum Coefficient
• Speaker identification
• Speaker diarization

3. Review : Gaussian Distribution
a.k.a. Normal Distribution

4. Gaussian Distribution Likelihood
N(165, 5)
N(165, 10)
N(165, 15)
Which one is the fittest?

5. Gaussian Distribution Likelihood

6. Multivariate Gaussian Distribution

7. Latent Variable Model a.k.a. Hidden Variable Model
관찰되지 않고 숨겨져 있는 변수
관찰된 변수로부터 추정
Latent Variable
Latent Variable Model
숨겨진 변수로부터 관찰된 변수가 도출되었다고 가정
관찰된 변수 + 숨겨진 변수를 이용하여 모델링

8. Latent Variable Model a.k.a. Hidden Variable Model
Observable: Height
Hidden: Gender

9. Mixture Model
전체 모집단에 있는 여러 개의 부분 모집단을 표현하기 위한 확률 모델

10. Mixture Model Example

11. Gaussian Mixture Model
Guassian distribution pdf
Mixture Model Revisited

12. Gaussian Mixture Model
How to solve it?

13. Expectation-Maximization Algorithm

14. EM Algorithm : GMM example

15. EM Algorithm : GMM example

16. EM Algorithm : GMM example

17. GMM vs k-Means Clustering
GMM
k-Means Clustering
Latent Variable based clustering
Observable Variable based clustering

18. More
Generalization of Mixture Model
(http://en.wikipedia.org/wiki/Hidden_Markov_model)
Hidden Markov Model
Algorithm for decoding HMM
(http://en.wikipedia.org/wiki/Viterbi_algorithm)
Viterbi Algorithm

19. MFCCs
Mel Frequency Cepstrum Coefficients
1. Mel Frequency?
2. Cepstrum?

20. Cepstrum
Spectrum > cepStrum
http://www.speech.cs.cmu.edu/15-492/slides/03_mfcc.pdf
1. Spectrum?
2. Why Cepstral analysis?

21. Mel Frequency
Mel-Frequency analysis of speech is based on human perception experiments

22. MFCCs revisited
Mel Frequency Cepstrum Coefficients
1. 주어진 신호(time-amplitude)를 푸리에 변환한다.
2. 푸리에 변환된 신호(frequency-amplitude)를 Mel-Scale로 변환한다.
3. Mel-Scale로 변환된 신호에 log를 취한다.
4. 여기에 이산 코사인 변환을 취한다.
구하는 방법

23. Speaker Diarization
Features (Observable): MFCCs 12+a elements, etc
Latent Variable : Speaker
#Components : # Speakers
http://cslu.ohsu.edu/~bedricks/courses/cs655/hw/hw4/hw4.html

24. Practice 1: GMM

25. Practice 2: Speaker Diarization
Hmm

26. Questions?