.  .                   Clustering by mixture model                            Pham The Thong                             A...
Outline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC . 2   Richardson&Green(1997): On B...
RJMCMC in clustering     Clustering overviewOutline . 1   RJMCMC in clustering       Clustering overview       Reversible ...
RJMCMC in clustering     Clustering overviewClustering overview          Divide the observations into groups.          Pre...
RJMCMC in clustering     Clustering overview  Clustering via mixture model  X = (x1 , · · · , xn ) be independent p-dimens...
RJMCMC in clustering     Clustering overviewSome approaches          Model Selection: Compare some model selection        ...
RJMCMC in clustering     Reversible Jump MCMCOutline . 1   RJMCMC in clustering       Clustering overview       Reversible...
RJMCMC in clustering     Reversible Jump MCMCOverview          First developed in Green(1995)          Has applications ra...
RJMCMC in clustering     Reversible Jump MCMCSome advantages of clustering byRJMCMC          Avoid the task of model selec...
RJMCMC in clustering     Reversible Jump MCMCGeneral ideas of RJMCMC I          Simulating a Markov Chain that converges t...
RJMCMC in clustering     Reversible Jump MCMCGeneral ideas of RJMCMC II          Split move: split one component into two ...
RJMCMC in clustering     Reversible Jump MCMCGeneral ideas of RJMCMC III          Calculate the acceptance probability A, ...
Richardson&Green(1997)    OverviewOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC ...
Richardson&Green(1997)    OverviewOverview          1-dimensional data.          Goal:                   Clustering data. ...
Richardson&Green(1997)    Split/Merge and Birth/Death MechanismOutline . 1   RJMCMC in clustering       Clustering overvie...
Richardson&Green(1997)    Split/Merge and Birth/Death MechanismSplit/Merge Mechanism          In Split move, select one co...
Richardson&Green(1997)    Split/Merge and Birth/Death MechanismBirth/Death Mechanism          Birth move                  ...
Richardson&Green(1997)    AlgorithmOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC...
Richardson&Green(1997)    Algorithm  One iteration contains      Gibbs Sampler:                   Updating the weights w  ...
Richardson&Green(1997)    ResultOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC . ...
Richardson&Green(1997)    ResultPost simulation  By processing the raw data come from the simulation,  one can      cluste...
Richardson&Green(1997)    ResultThe three dataset          Enzym data: enzymatic activity of one enzyme in          the bl...
Richardson&Green(1997)    ResultPham The Thong (   )          Clustering by mixture model   April 22, 2011   23 / 44
Richardson&Green(1997)    ResultPham The Thong (   )          Clustering by mixture model   April 22, 2011   24 / 44
Tadesse et.al.(2005)   OverviewOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC . 2...
Tadesse et.al.(2005)   OverviewOverview          High dimensional data          Goal:                   Variable selecting...
Tadesse et.al.(2005)   Variable SelectionOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jum...
Tadesse et.al.(2005)   Variable SelectionConcept          Perhaps not all variables are useful for clustering.          By...
Tadesse et.al.(2005)   Variable SelectionThe model of Tadesse et.al. I  Introduce γ = (γ1 , · · · , γp ): γj = 1 if the jt...
Tadesse et.al.(2005)   Variable SelectionThe model of Tadesse et.al. II  (η (γ c ) , Ω(γ c ) ): mean and covariance for th...
Tadesse et.al.(2005)   Variable SelectionSearching for γ          The problem of variable selection is re-casted as a     ...
Tadesse et.al.(2005)   RJMCMC MechanismOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump ...
Tadesse et.al.(2005)   RJMCMC MechanismDifficulties in high dimension          Unlike 1-dimensional case, there is no obviou...
Tadesse et.al.(2005)   RJMCMC MechanismApproach of Tadesse et.al.          Integrating out the mean vector and the covaria...
Tadesse et.al.(2005)   RJMCMC MechanismAlgorithm  One iteration contains      Metropolis search for γ      Gibbs sampler: ...
Tadesse et.al.(2005)   ResultOutline . 1   RJMCMC in clustering       Clustering overview       Reversible Jump MCMC . 2  ...
Tadesse et.al.(2005)   ResultPost simulation          Since the mean and covariance are integrated out,          there is ...
Tadesse et.al.(2005)   ResultMicroarray data          14 samples (samples are come from tissues).          Variables are g...
Tadesse et.al.(2005)   ResultPham The Thong (   )          Clustering by mixture model   April 22, 2011   39 / 44
Tadesse et.al.(2005)   ResultPham The Thong (   )          Clustering by mixture model   April 22, 2011   40 / 44
Tadesse et.al.(2005)   Weakness of the modelOutline . 1   RJMCMC in clustering       Clustering overview       Reversible ...
Tadesse et.al.(2005)   Weakness of the modelWeakness of the model [5]          The independence assumption would often lea...
Tadesse et.al.(2005)   Weakness of the modelReferences  [1]P.J.Green(1995), Reversible jump Markov chain Monte Carlo  comp...
Tadesse et.al.(2005)   Weakness of the model                   Thank you for your attentionPham The Thong (        )      ...
Upcoming SlideShare
Loading in …5
×

RJMCMC in clustering

1,865
-1

Published on

A 30-minute presentation about RJMCMC in clustering

Published in: Technology, Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
1,865
On Slideshare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
Downloads
12
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

RJMCMC in clustering

  1. 1. . . Clustering by mixture model Pham The Thong April 22, 2011Pham The Thong ( ) Clustering by mixture model April 22, 2011 1 / 44
  2. 2. Outline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 2 / 44
  3. 3. RJMCMC in clustering Clustering overviewOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 3 / 44
  4. 4. RJMCMC in clustering Clustering overviewClustering overview Divide the observations into groups. Predict group of a new observation. Model-based clustering: select a probabilistic model that underlying the observations and make statistical inferences based on that model. One popular model is the mixture model.Pham The Thong ( ) Clustering by mixture model April 22, 2011 4 / 44
  5. 5. RJMCMC in clustering Clustering overview Clustering via mixture model X = (x1 , · · · , xn ) be independent p-dimensional observations from G populations. ∑ G f (xi |w, θ) = wk f (xi |θk ) k=1 f (xi |θk ) is the density of an observation xi from the kth component. w = (w1 , · · · , wG )T are component weights. θ = (θ1 , · · · , θG )T are component parameters. Clustering is done via allocation vector y = (y1 , · · · , yn )T : yi = k if the ith observation xi comes from component k.Pham The Thong ( ) Clustering by mixture model April 22, 2011 5 / 44
  6. 6. RJMCMC in clustering Clustering overviewSome approaches Model Selection: Compare some model selection criteria of fixed-G models for various values of G to choose the best G . Inference on fixed-G model is often done via EM algorithm or Gibbs sampler. Nonparametric method: Use Dirichlet Process. Trans-dimensional Markov Chain Monte Carlo (MCMC): Allow G to be changed during the inference process by combining Gibbs sampler with MCMC moves that can change dimension of the model. Reversible jump MCMC (RJMCMC) is one possible scheme.Pham The Thong ( ) Clustering by mixture model April 22, 2011 6 / 44
  7. 7. RJMCMC in clustering Reversible Jump MCMCOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 7 / 44
  8. 8. RJMCMC in clustering Reversible Jump MCMCOverview First developed in Green(1995) Has applications ranged well beyond mixture model analysis. Mixture model analysis power first demonstrated in Richardson&Green(1997). They considered only the 1-dimensional case. Applied to multidimensional setting in Tadesse et.al. (2005).Pham The Thong ( ) Clustering by mixture model April 22, 2011 8 / 44
  9. 9. RJMCMC in clustering Reversible Jump MCMCSome advantages of clustering byRJMCMC Avoid the task of model selection. Provide a coherent Bayesian framework. The cluster number G is not treated as a special parameter. Can provide useful summary of data which is difficult to obtain by other methods.Pham The Thong ( ) Clustering by mixture model April 22, 2011 9 / 44
  10. 10. RJMCMC in clustering Reversible Jump MCMCGeneral ideas of RJMCMC I Simulating a Markov Chain that converges to the full posterior distribution p(G , y, w, θ|X). Hybrid sampler consist of Gibbs Sampler(the base) and jump moves (the extension). Gibbs sampler will sample (y, w, θ). Jump moves will sample the cluster number G . The jump moves come in pair: Split/Merge and Birth/DeathPham The Thong ( ) Clustering by mixture model April 22, 2011 10 / 44
  11. 11. RJMCMC in clustering Reversible Jump MCMCGeneral ideas of RJMCMC II Split move: split one component into two components. Merge move: combine two components into one component. Birth move: create an empty component. Death move: delete an empty component. At each iteration, propose to perform Split(Birth) move with some fixed probability bk and with probability 1 − bk propose to perform Merge(Death) move. In one proposal, calculate all the changes to the model as if the move was made.Pham The Thong ( ) Clustering by mixture model April 22, 2011 11 / 44
  12. 12. RJMCMC in clustering Reversible Jump MCMCGeneral ideas of RJMCMC III Calculate the acceptance probability A, which is the product of three terms: the ratio of the posterior of the new model to that of the old model the ratio of the probability of the way to go from the new model back to the old model to that of the way to go from old model to new model the Jacobian arises from the change of dimension To ensure convergence to the desired distribution, only actually carry out the move with probability min(1, A).Pham The Thong ( ) Clustering by mixture model April 22, 2011 12 / 44
  13. 13. Richardson&Green(1997) OverviewOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 13 / 44
  14. 14. Richardson&Green(1997) OverviewOverview 1-dimensional data. Goal: Clustering data. Estimating component parameters. Estimating the distribution of data. Predicting group of new data. Demonstrated in three real dataset: Enzym, Acid, and Galaxy.Pham The Thong ( ) Clustering by mixture model April 22, 2011 14 / 44
  15. 15. Richardson&Green(1997) Split/Merge and Birth/Death MechanismOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 15 / 44
  16. 16. Richardson&Green(1997) Split/Merge and Birth/Death MechanismSplit/Merge Mechanism In Split move, select one component (wj ∗ , µj ∗ , σj ∗ ) to split to 2 components (wj1 , µj1 , σj1 ) and (wj2 , µj2 , σj2 ). In Merge move, select two components (wj1 , µj1 , σj1 ) and (wj2 , µj2 , σj2 ) to merge into one new component (wj ∗ , µj ∗ , σj ∗ ). Equalizing the zeroth, first, second moment of the new component to those of a combination of the two old components.Pham The Thong ( ) Clustering by mixture model April 22, 2011 16 / 44
  17. 17. Richardson&Green(1997) Split/Merge and Birth/Death MechanismBirth/Death Mechanism Birth move Generate wj ∗ , µj ∗ , σj ∗ from some distributions. Rescale the weights. Death move Delete a randomly chosen empty component. Rescale the weights.Pham The Thong ( ) Clustering by mixture model April 22, 2011 17 / 44
  18. 18. Richardson&Green(1997) AlgorithmOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 18 / 44
  19. 19. Richardson&Green(1997) Algorithm One iteration contains Gibbs Sampler: Updating the weights w Updating the parameters µ, σ Updating the allocation y Split/Merge move Birth/Death movePham The Thong ( ) Clustering by mixture model April 22, 2011 19 / 44
  20. 20. Richardson&Green(1997) ResultOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 20 / 44
  21. 21. Richardson&Green(1997) ResultPost simulation By processing the raw data come from the simulation, one can clustering data by selecting the allocation vector y that has the highest frequency. estimating component parameters by their posterior mean. estimating the distribution of data. predicting group of new data.Pham The Thong ( ) Clustering by mixture model April 22, 2011 21 / 44
  22. 22. Richardson&Green(1997) ResultThe three dataset Enzym data: enzymatic activity of one enzyme in the blood of 245 unrelated people. The interest is identifying subgroups of slow or fast activity as a marker of genetic polymorphism in the general population(i.e. to some extent, people of the same subgroup may have similar genetic structure although they are unrelated). Acid data: acidity level of 155 lakes in Wisconsin. Galaxy data: velocities of 82 galaxies diverging from our galaxy.Pham The Thong ( ) Clustering by mixture model April 22, 2011 22 / 44
  23. 23. Richardson&Green(1997) ResultPham The Thong ( ) Clustering by mixture model April 22, 2011 23 / 44
  24. 24. Richardson&Green(1997) ResultPham The Thong ( ) Clustering by mixture model April 22, 2011 24 / 44
  25. 25. Tadesse et.al.(2005) OverviewOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 25 / 44
  26. 26. Tadesse et.al.(2005) OverviewOverview High dimensional data Goal: Variable selecting. Clustering data. Predicting group of new data. Applied to microarray data.Pham The Thong ( ) Clustering by mixture model April 22, 2011 26 / 44
  27. 27. Tadesse et.al.(2005) Variable SelectionOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 27 / 44
  28. 28. Tadesse et.al.(2005) Variable SelectionConcept Perhaps not all variables are useful for clustering. By throwing away non-discriminating variables (irrelevant variables) and clustering only on discriminating variables (relevant variables) we may improve clustering accuracy. We can think of variable selection as one way to generalize the basic approach “clustering by the full set of variables” to “clustering by a subset of variables”.Pham The Thong ( ) Clustering by mixture model April 22, 2011 28 / 44
  29. 29. Tadesse et.al.(2005) Variable SelectionThe model of Tadesse et.al. I Introduce γ = (γ1 , · · · , γp ): γj = 1 if the jth variable is a discriminating variable and 0 if it is not. Use (γ) and (γ c ) to index discriminating variables and non-discriminating variables. Three assumptions: The set of discriminating variables and the set of non-discriminating variables are independent. If we look only at (γ c ), the data X(γ c ) have a normal distribution(hence unsuitable for clustering). If we look only at (γ), the data X(γ) have a mixture distribution of G normal components (hence suitable for clustering).Pham The Thong ( ) Clustering by mixture model April 22, 2011 29 / 44
  30. 30. Tadesse et.al.(2005) Variable SelectionThe model of Tadesse et.al. II (η (γ c ) , Ω(γ c ) ): mean and covariance for the non-discriminating variables. (µk(γ) , Σk(γ) ): mean and covariance for the kth components Ck . The three assumptions can be written as ∏ n ( ) p(X|G , γ, w, y, µ, Σ, η, Ω) = N xi(γ c ) , η (γ c ) , Ω(γ c ) i=1 ∏G ∏ ( ) N xi(γ) , µk(γ) , Σk(γ) k=1 xi ∈CkPham The Thong ( ) Clustering by mixture model April 22, 2011 30 / 44
  31. 31. Tadesse et.al.(2005) Variable SelectionSearching for γ The problem of variable selection is re-casted as a problem of searching for the most probable binary vector γ. Use a Metropolis search(of which Simulated Annealing is one type) At each step randomly choosing one of the following two transitional moves: flip one bit or swap two bit of γ(and accept the ) move with probability new |X,y,w,G min 1, p(γ old |X,y,w,G )) . p(γPham The Thong ( ) Clustering by mixture model April 22, 2011 31 / 44
  32. 32. Tadesse et.al.(2005) RJMCMC MechanismOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 32 / 44
  33. 33. Tadesse et.al.(2005) RJMCMC MechanismDifficulties in high dimension Unlike 1-dimensional case, there is no obvious way to split a covariance matrix into two covariance matrix. Even if this could be done[4], the Jacobian may not have closed-form. The number of model parameters increases rapidly with order p 2 . The chain may converge very slowly.Pham The Thong ( ) Clustering by mixture model April 22, 2011 33 / 44
  34. 34. Tadesse et.al.(2005) RJMCMC MechanismApproach of Tadesse et.al. Integrating out the mean vector and the covariance matrix to obtain a marginalized posterior in which only G , w, γ,and y are involved. Despite being quite tedious, the math follows a standard framework: define conjugate priors for mean and covariance matrix and then take the integration. Only need to split or merge the weights of components in Split/Merge move. Birth/Death move are the same as in 1-dimensional case.Pham The Thong ( ) Clustering by mixture model April 22, 2011 34 / 44
  35. 35. Tadesse et.al.(2005) RJMCMC MechanismAlgorithm One iteration contains Metropolis search for γ Gibbs sampler: Updating the weights w Updating the allocation y Split/Merge move Birth/Death movePham The Thong ( ) Clustering by mixture model April 22, 2011 35 / 44
  36. 36. Tadesse et.al.(2005) ResultOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 36 / 44
  37. 37. Tadesse et.al.(2005) ResultPost simulation Since the mean and covariance are integrated out, there is no estimation for component parameters. Variable selection: Method 1: select the vector γ that have the highest frequency. Method 2: select all variables j that have p(γj |X, G ) greater than some threshold: p(γj |X, G ) ≥ a. Clustering and group prediction can be done in the same way as in the univariate case.Pham The Thong ( ) Clustering by mixture model April 22, 2011 37 / 44
  38. 38. Tadesse et.al.(2005) ResultMicroarray data 14 samples (samples are come from tissues). Variables are genes. There are 762 variables. By clustering the samples into subgroups, one may find out which genes are relevant to each subgroup.Pham The Thong ( ) Clustering by mixture model April 22, 2011 38 / 44
  39. 39. Tadesse et.al.(2005) ResultPham The Thong ( ) Clustering by mixture model April 22, 2011 39 / 44
  40. 40. Tadesse et.al.(2005) ResultPham The Thong ( ) Clustering by mixture model April 22, 2011 40 / 44
  41. 41. Tadesse et.al.(2005) Weakness of the modelOutline . 1 RJMCMC in clustering Clustering overview Reversible Jump MCMC . 2 Richardson&Green(1997): On Bayesian Analysis of Mixtures with an Unknown Number of Components Overview Split/Merge and Birth/Death Mechanism Algorithm Result . 3 Tadesse et.al.(2005): Bayesian Variable Selection in Clustering High-Dimensional Data Overview Variable Selection RJMCMC Mechanism Result Weakness of the modelPham The Thong ( ) Clustering by mixture model April 22, 2011 41 / 44
  42. 42. Tadesse et.al.(2005) Weakness of the modelWeakness of the model [5] The independence assumption would often lead to the wrongly case in which one irrelevant variable be identified as a discriminating one because it is related to some discriminating variables. It is not known whether one can relax this assumption while still being able to perform RJMCMC-based full Bayesian analysis.Pham The Thong ( ) Clustering by mixture model April 22, 2011 42 / 44
  43. 43. Tadesse et.al.(2005) Weakness of the modelReferences [1]P.J.Green(1995), Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrica 82,4,711-732. [2]S.Richardson and P.J.Green(1997), On Bayesian Analysis of Mixtures with an Unknown Number of Components, J.R.Statist. Soc.B 59, 4,731-792. [3]M.G.Tadesse, N.Sha, and M. Vannucci(2005), Bayesian Variable Selection in Clustering High-Dimensional Data,Journal of the American Statistical Association 100,470,602-617. [4]Petros Dellaportas and Ioulia Papageorgiou(2006), Multivariate mixtures of normals with unknown number of components,Statistics and Computing 16,1,57 - 68. [5]Maugis et.al.(2009), Variable Selection for Clustering with Gaussian Mixture Models, Biometrics 65, 701-709.Pham The Thong ( ) Clustering by mixture model April 22, 2011 43 / 44
  44. 44. Tadesse et.al.(2005) Weakness of the model Thank you for your attentionPham The Thong ( ) Clustering by mixture model April 22, 2011 44 / 44
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×