Parameter estimation in finite mixture of α -Stable Distributions

1. Parameter estimation in finite mixture of α-Stable Distributions Presented by Yiding Han D. Salas-Gonzalez, E.E. Kuruoglu, D.P. Ruiz, Finite mixture of alpha Stable distributions, Digital Signal Processing (2007), doi: 10.1016/j.dsp.2007.11.004

2. Gaussian Distribution

3. Still Gaussian?

4. α-Stable Distributions

5. α-Stable Noises

6. Characteristic Function • Definition: A univariate distribution function F(x) is stable if and only if its characteristic function has the form where and

7. Parameters in α-Stable distribution • α, called the characteristic exponent, measures the “thickness” of the tails of the distribution. • β is the symmetry (skewness) parameter. When β=0, the distribution is symmetric about a. In this case the distribution is called symmetric α-stable (SαS) • γ is the scale parameter, also known as the dispersion, which is similar to the variance of the Gaussian distribution. • ais a location parameter. For SαS distribution it’s the mean when 1<α≤2 and the median when 0< α<1.

8. α-Stable density function • The α-Stable density function is the inverse Fourier transform of the characteristic function • Unfortunately, the above expression doesn’t have analytical solution other than some special cases, such as when α=2. • For the standard α-stable distribution (γ=1,a=0), the stable density function is given by:

9. mixture model for α-Stable density The mixture of α-Stable density is given by: pY(y) is the pdf (probability density function) of vector y. wj is the weight or mixture proportion for component j Also an allocation variable zi ∈ [1,2,…k] is introduced, we have: thus we have:

10. Bayesian stable mixture model • Bayes’ rule • Bayesian mixture model Directed Acyclic Graph(DAG) (*Snapshot from the paper)

11. MCMC and RJMCMC implementation • Bayesian methods provides an adequate framework, but we need to solve multidimensional integrals that normally don’t have closed form: • The solution can be calculated using numerical Monte Carlo methods based on Markov Chains.

12. Scheme of obtaining Samples for every parameter at every iteration

13. Update weights w Vector w is sampled from a Dirichlet distribution with parameters ζ+nj where which is the number of samples assigned to the component j

14. Update α-Stable parameters θ={α,β,γ,a} • Parameter α It is estimated using the Metropolis-Hasting algorithm Suppose the most recent value sampled is . To follow the Metropolis-Hastings algorithm, we next draw a new proposal sample from a proposal distribution , and calculate a value where is the likelihood ratio between the proposed sample and the previous sample , and is the ratio of the proposal density in two directions. Then the new state is chosen according to the following rules. (next page)

15. Update α-Stable parameter α … Where in our case is equal to the following

16. Update α-Stable parameter α … The equation can be simplified to: if we choose our proposal distribution q as a symmetrical one. In particular, a Normal distribution centered in the previous variable and with variance σθis chosen: The other α-Stable parameters, such as skewness, location and dispersion, can be obtained using the same strategy.

17. Update the allocation z In order to find which subpopulation the data yi is more likely to belong to, we need to calculate the allocation at each iteration.

18. Reversible jump move for the number of components k • At every iteration, two new components are proposed by splitting from one former component with probability bk , otherwise one new component is proposed by combining two former components with probability dk = 1 – bk . • It is accepted with probability min{1,A}

19. Reversible jump move for the number of components k… where A is equal to:

20. simulation • A mixture of α-Stable distributed samples are generated in the pdf of: