Signal decompositions using trans-dimensional Bayesian methods: Alireza Roodaki's Ph.D. defense

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These are the slides I have used during the defense of my thesis (see https://sites.google.com/site/alireza4702/publications/phd-thesis for more information).

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Signal decompositions using trans-dimensional Bayesian methods: Alireza Roodaki's Ph.D. defense

  1. 1. Relabeling and summarizing posterior distributions Proposed approach Results Conclusion Signal decompositions using trans-dimensional Bayesian methods Alireza Roodaki Ph.D. Thesis Defense Department of Signal Processing and Electronic Systems 2012, May 14th Advisors: Julien Bect and Gilles Fleury Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
  2. 2. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project Ultra high energy particles coming from space (E ∼ 1019 eV) How and where? What is their composition (Proton, Iron)?Figure: A conceptual shower(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
  3. 3. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project Ultra high energy particles coming from space (E ∼ 1019 eV) How and where? What is their composition (Proton, Iron)?Figure: A conceptual shower(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
  4. 4. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project Ultra high energy particles coming from space (E ∼ 1019 eV) How and where? What is their composition (Proton, Iron)?Figure: A conceptual shower(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
  5. 5. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) muons are generated when particles cross the atmosphere the number k of muons and their arrival times t µ are indicators of the originFigure: A conceptual shower and and composition of particledetectors (water tanks)(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
  6. 6. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) muons are generated when particles cross the atmosphere the number k of muons and their arrival times t µ are indicators of the originFigure: A conceptual shower and and composition of particledetectors (water tanks)(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
  7. 7. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) muons are generated when particles cross the atmosphere the number k of muons and their arrival times t µ are indicators of the originFigure: A conceptual shower and and composition of particledetectors (water tanks)(http://auger.org). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
  8. 8. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) Figure: Water tank detector. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 3/ 52
  9. 9. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) #PE 20 10 0 1.5 intensity 1 0.5 0 100 200 300 400 500 600 t[ns] Figure: Observed signal (n) Prof. Balázs Kégl from LAL, University of Paris 11. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 4/ 52
  10. 10. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 1: Detection and estimation of muons in theAuger project (Contd.) #PE 20 10 0 1.5 intensity 1 0.5 0 100 200 300 400 500 600 t[ns] Figure: Observed signal (n) Prof. Balázs Kégl from LAL, University of Paris 11. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 4/ 52
  11. 11. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 2: Spectral analysis 10 0 −10 Applications 0 10 20 30 40 50 60 time RADAR / SONAR 150 100 Array signal processingPower 50 Vibration analysis 0 ... 0 0.5 1 1.5 2 2.5 3 radial frequencyFigure: Observed signal (top) and itsperiodogram (bottom). Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 5/ 52
  12. 12. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 2: Spectral analysis (Cont.) Detection and estimation of sinusoids in white noise model the observed signal y by sinusoidal components Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
  13. 13. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 2: Spectral analysis (Cont.) Detection and estimation of sinusoids in white noise model the observed signal y by sinusoidal components observed signal k Mk : y [i] = aj cos[ωj i] + bj sin[ωj i] + n[i]. j=1 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
  14. 14. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 2: Spectral analysis (Cont.) Detection and estimation of sinusoids in white noise model the observed signal y by sinusoidal components observed signal k Mk : y [i] = aj cos[ωj i] + bj sin[ωj i] + n[i]. j=1 Joint model selection and parameter estimation problem Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
  15. 15. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionTrans-dimensional problems Def. The problems in which the number of things that we don′ t know is one of the things that we don′ t know [Green, 2003.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
  16. 16. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionTrans-dimensional problems Def. The problems in which the number of things that we don′ t know is one of the things that we don′ t know [Green, 2003.] space X = k ∈K {k } × Θk with points x = (k , θk ) « k ∈ K denotes number of components « θk ∈ Θk is a vector of component-specific parameters Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
  17. 17. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionTrans-dimensional problems Def. The problems in which the number of things that we don′ t know is one of the things that we don′ t know [Green, 2003.] space X = k ∈K {k } × Θk with points x = (k , θk ) « k ∈ K denotes number of components « θk ∈ Θk is a vector of component-specific parameters Applications: « Spectral Analysis (Array signal processing) « (Gaussian) Mixture modeling & Clustering « Object detection and recognition Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
  18. 18. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  19. 19. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Prior distribution Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  20. 20. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Prior distribution Posterior distribution Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  21. 21. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Prior distribution Posterior distribution x = (k , θk ) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  22. 22. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Prior distribution Posterior distribution x = (k , θk ) « both detection and estimation problems Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  23. 23. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionBayesian inference Likelihood p(y | x) p(x) p(x | y) = X p(y | x ′ ) p(x ′ )dx ′ Prior distribution Posterior distribution x = (k , θk ) « both detection and estimation problems high-dimensional / intractable integrals Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
  24. 24. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionMarkov Chain Monte Carlo (MCMC) methods generate samples from the posterior distribution of interest (target distribution), say, π. construct a Markov chain (x (1) , . . . , x (M) ) that under some conditions converges to π. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
  25. 25. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionMarkov Chain Monte Carlo (MCMC) methods generate samples from the posterior distribution of interest (target distribution), say, π. construct a Markov chain (x (1) , . . . , x (M) ) that under some conditions converges to π. Famous algorithms: « Metropolis-Hastings (MH) sampler [Metropolis, et al. 1953, Hastings, 1970.] « Gibbs sampler [Geman and Geman, 1984.] « RJ-MCMC sampler [Green, 1995.] [Robert and Casella, 2004.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
  26. 26. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionMarkov Chain Monte Carlo (MCMC) methods generate samples from the posterior distribution of interest (target distribution), say, π. construct a Markov chain (x (1) , . . . , x (M) ) that under some conditions converges to π. Famous algorithms: « Metropolis-Hastings (MH) sampler [Metropolis, et al. 1953, Hastings, 1970.] « Gibbs sampler [Geman and Geman, 1984.] « RJ-MCMC sampler [Green, 1995.] [Robert and Casella, 2004.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
  27. 27. Relabeling and summarizing posterior distributions Proposed approach Results ConclusionExample 2: spectral analysis (Cont.) RJ-MCMC sampler ⇒ variable dimensional samples 0.8 0.6 ωk 0.4 4 3 k 2 160 170 180 190 200 Iteration number Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 10/ 52
  28. 28. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization
  29. 29. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies
  30. 30. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies 3 Results Detection and estimation of sinusoids in white noise Detection and estimation of muons in the Auger project
  31. 31. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies 3 Results Detection and estimation of sinusoids in white noise Detection and estimation of muons in the Auger project 4 Conclusion
  32. 32. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies 3 Results Detection and estimation of sinusoids in white noise Detection and estimation of muons in the Auger project 4 Conclusion
  33. 33. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  34. 34. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  35. 35. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) (RJ)-MCMC sampler Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  36. 36. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) (RJ)-MCMC sampler « What to do with the generated samples? Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  37. 37. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) (RJ)-MCMC sampler « What to do with the generated samples? Summarization human readable summaries Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  38. 38. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) (RJ)-MCMC sampler « What to do with the generated samples? Summarization human readable summaries interpretable figures (e.g., histograms) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  39. 39. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionsummarizing posterior distributions Posterior = all the information « It is a complex mathematical object (not easy to manipulate) (RJ)-MCMC sampler « What to do with the generated samples? Summarization human readable summaries interpretable figures (e.g., histograms) statistical measures (e.g., mean and variance) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
  40. 40. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionFixed-dimensional problems uni-modal uni-variate case 0.6 p 0.4 0.2 0 0 2 4 Samples report location (mean and median) and dispersion (variance and confidence intervals) parameters µ σ2 2.0 0.25 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 13/ 52
  41. 41. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionLabel-switching issue Additive mixture: lack of identifiability the likelihood is invariant under relabeling of components Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
  42. 42. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionLabel-switching issue Additive mixture: lack of identifiability the likelihood is invariant under relabeling of components the posterior distribution is invariant under permutation of components Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
  43. 43. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionLabel-switching issue Additive mixture: lack of identifiability the likelihood is invariant under relabeling of components the posterior distribution is invariant under permutation of components Marginal posteriors ofComp. #1 10 component-specific 0 parameters are identical!Comp. #2 10 0Comp. #3 10 0 0.5 0.75 1 ω Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
  44. 44. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionLabel-switching issue Additive mixture: lack of identifiability the likelihood is invariant under relabeling of components the posterior distribution is invariant under permutation of components Marginal posteriors ofComp. #1 10 component-specific 0 parameters are identical!Comp. #2 10 How to summarize the 0 posterior information?Comp. #3 10 0 0.5 0.75 1 ω Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
  45. 45. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionstrategies to deal with label-switching imposing artificial “identifiability constraints” Exp: sorting the components [Richardson and Green, 1997.] Comp. #1 10 0 Comp. #2 10 0 Comp. #3 10 0 0.5 0.75 1 Figure: components are sorted based on ω. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 15/ 52
  46. 46. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization Conclusionstrategies to deal with label-switching imposing artificial “identifiability constraints” Exp: sorting the components [Richardson and Green, 1997.] Comp. #1 10 0 Comp. #2 10 0 Comp. #3 10 0 0.5 0.75 1 Figure: components are sorted based on ω. relabeling algorithms [Celeux, et al. 1998, Stephens, 2000, Jasra, et al, 2005, Sperrin et al, 2010, Yao, 2011.]. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 15/ 52
  47. 47. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionExample 2: spectral analysis (Cont.) Variable-dimensional posterior distribution 4 3 k 2 0 0.3 0.6 0.5 0.75 1 p(k |y) ω Figure: Posteriors of k and sorted radial frequencies, ωk , given k. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 16/ 52
  48. 48. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionClassical Bayesian approaches Bayesian Model Selection (BMS) One model is selected (estimated) by looking at the MAP, ˆ i.e. k = argmax p(k |y). Component-specific parameters are summarized given ˆ k = k. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 17/ 52
  49. 49. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionBayesian Model Selection (BMS) 3 4 3 k 2 0 0.3 0.6 0.5 0.75 1 p(k |y) ω Figure: The model with k = 2 is selected. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 18/ 52
  50. 50. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionClassical Bayesian approaches Bayesian Model Selection (BMS) One model is selected (estimated) by looking at the MAP, ˆ i.e. k = argmax p(k |y). Component-specific parameters are summarized given ˆ k = k. Bayesian Model Averaging (BMA) Use the information from all possible models: p(∆|y) = k p(∆|k , y)p(k |y) However, ∆ cannot be ωk as its size changes from model to model. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 19/ 52
  51. 51. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionBayesian Model Averaging (BMA) Binned data representation (∆ = N(Bj )): kmax E(N(Bj ) | y) = E(N(Bj ) | k , y) · p(k | y) k =1 1 expected nbr comp where .75 j = 1, . . . , Nbin .5 and E(N(Bj )) is the .25 expected number of components in bin 0 0 0.5 1 1.5 2 2.5 3 Bj . ω Figure: Expected number of components using BMA. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 20/ 52
  52. 52. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionAre BMS and BMA approaches satisfactory? Bayesian Model Selection (BMS) « selects a model ⇒ component-specific parameters « losing information from the discarded models « ignoring the uncertainties about the presence of components. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 21/ 52
  53. 53. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionAre BMS and BMA approaches satisfactory? (Cont.) Bayesian Model Averaging (BMA) « appropriate for signal reconstruction and prediction « does not provide information about component-specific parameters Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 22/ 52
  54. 54. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionA novel approach is needed! A novel approach is needed! Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 23/ 52
  55. 55. Relabeling and summarizing posterior distributions Proposed approach Label-switching issue Results Variable-dimensional summarization ConclusionA novel approach is needed! A novel approach is needed! Properties of an “ideal” approach information from all (plausible) models « interpretable summaries for component-specific parameters uncertainties about the presence of components Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 23/ 52
  56. 56. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies 3 Results Detection and estimation of sinusoids in white noise Detection and estimation of muons in the Auger project 4 Conclusion
  57. 57. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionBig picture: relabeling and summarizing posteriordistributions True posterior f = p(· | y) Approximate posterior qη 2 2 1 1 0 0 0 1 2 3 0 1 2 3 Parametric family {qη , η ∈ N} Measure of “distance” [Stephens, 2000.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 24/ 52
  58. 58. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionBig picture: relabeling and summarizing posteriordistributions True posterior f = p(· | y) Samples Approximate posterior qη 2 2 2 1 1 1 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Parametric family {qη , η ∈ N} Measure of “distance” Samples [Stephens, 2000.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 24/ 52
  59. 59. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionFixed-dimensional problems uni-modal uni-variate case 0.6 p 0.4 0.2 0 0 2 4 Samples report location (mean and median) and dispersion (variance and confidence intervals) parameters µ σ2 2.0 0.25 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 25/ 52
  60. 60. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionvariable-dimensional approximate posterior An original (variable-dimensional) parametric model qη Four main requirements: 1 Must be defined on the same space X = k ∈K {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
  61. 61. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionvariable-dimensional approximate posterior An original (variable-dimensional) parametric model qη Four main requirements: 1 Must be defined on the same space X = k ∈K {k } × Θk 2 Must be permutation invariance Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
  62. 62. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionvariable-dimensional approximate posterior An original (variable-dimensional) parametric model qη Four main requirements: 1 Must be defined on the same space X = k ∈K {k } × Θk 2 Must be permutation invariance 3 Must be “simple” (small number of parameters) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
  63. 63. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionvariable-dimensional approximate posterior An original (variable-dimensional) parametric model qη Four main requirements: 1 Must be defined on the same space X = k ∈K {k } × Θk 2 Must be permutation invariance 3 Must be “simple” (small number of parameters) 4 Must be able to capture the main features of the posterior distributions typically met in practice. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
  64. 64. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view ··· 1 2 L x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  65. 65. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view ··· 1 2 L u1 u2 uL ul ∈ Θ x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  66. 66. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view ··· 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  67. 67. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view ··· 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} {u l | ξl = 1} x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  68. 68. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view ··· 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} {u l | ξl = 1} random arrangement x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  69. 69. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view π1 π2 πL ··· 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} {u l | ξl = 1} ξl ∼ B(πl ) random arrangement x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  70. 70. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view π1 π2 πL µ1 Σ1 µ2 Σ2 ··· µL ΣL 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} {u l | ξl = 1} ξl ∼ B(πl ) u l ∼ N (µl , Σl ) random arrangement x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  71. 71. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view π1 π2 πL µ1 Σ1 µ2 Σ2 ··· µL ΣL 1 2 L ξ1 u1 ξ2 u2 ξL uL ul ∈ Θ ξl ∈ {0, 1} {u l | ξl = 1} ξl ∼ B(πl ) u l ∼ N (µl , Σl ) random arrangement ηl = {πl , µl , Σl } x = (k , θk ) ∈ X = k {k } × Θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
  72. 72. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  73. 73. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  74. 74. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: 18 L = 3 12 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) s2 = (0.05, 0.02, 0.1) 6 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  75. 75. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (0, 1, 1) ⇒ k = 2 & θk = (0.52, 1.05) 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  76. 76. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (0, 1, 0) ⇒ k = 1 & θk = 0.49 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  77. 77. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (1, 0, 1) ⇒ k = 2 & θk = (0.27, 1.03) 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  78. 78. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (0, 0, 1) ⇒ k = 1 & θk = 1.05 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  79. 79. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (1, 1, 1) ⇒ k = 3 & θk = (0.27, 0.53, 1.15) 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  80. 80. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionA generative model point of view 1 for l = 1, . . . , L generate binary number ξl ∼ B (πl ) end L 2 set k = l=1 ξl ; 3 for each l such that ξl = 1 generate random sample u l ∼ N (µl , Σl ) end 4 Random arrangement ⇒ θk Example: L = 3 18 π = (0.4, 0.9, 0.7) µ = (0.2, 0.5, 1) 12 s2 = (0.05, 0.02, 0.1) 6 ξ = (0, 0, 0) ⇒ k = 0 & θk = () 0.4 0.8 1.2 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
  81. 81. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionBig picture: relabeling and summarizing posteriordistributions True posterior f = p(· | y) Samples Approximate posterior qη 2 2 2 1 1 1 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Parametric family {qη , η ∈ N} Measure of “distance” Samples [Stephens, 2000.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 29/ 52
  82. 82. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionfitting the parametric model qη to the posterior f minimizing the Kullback-Leibler divergence f (x) J (η) DKL (f (x) qη (x)) = f (x) log dx qη (x) Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
  83. 83. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionfitting the parametric model qη to the posterior f minimizing the Kullback-Leibler divergence f (x) J (η) DKL (f (x) qη (x)) = f (x) log dx qη (x) A key point: samples x (i) , i = 1, 2, · · · , M, are generated from f , so M 1 J (η) ≃ − log qη (x (i) ) + Const. M i=1 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
  84. 84. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies Conclusionfitting the parametric model qη to the posterior f minimizing the Kullback-Leibler divergence f (x) J (η) DKL (f (x) qη (x)) = f (x) log dx qη (x) A key point: samples x (i) , i = 1, 2, · · · , M, are generated from f , so M 1 J (η) ≃ − log qη (x (i) ) + Const. M i=1 M Objective: η = argmaxη ˆ i=1 log qη (x (i) ) . Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
  85. 85. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionExpectation Maximization (EM) latent (hidden) variable: « Binary indicator vector ξ « Random permutation Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
  86. 86. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionExpectation Maximization (EM) latent (hidden) variable: « Binary indicator vector ξ « Random permutation « Define z = (z1 , . . . , zk ) as an allocation vector for x « zj = l ⇒ x j comes from component l Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
  87. 87. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionExpectation Maximization (EM) latent (hidden) variable: « Binary indicator vector ξ « Random permutation « Define z = (z1 , . . . , zk ) as an allocation vector for x « zj = l ⇒ x j comes from component l idea: use EM to maximize the likelihood. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
  88. 88. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionExpectation Maximization (EM) latent (hidden) variable: « Binary indicator vector ξ « Random permutation « Define z = (z1 , . . . , zk ) as an allocation vector for x « zj = l ⇒ x j comes from component l idea: use EM to maximize the likelihood. The E-step is computationally expensive! For example, assuming L = 15 and k (i) = 10, then, it contains 1.1 × 1010 terms. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
  89. 89. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionStochastic EM (SEM) at iteration (r + 1) Stochastic (S)-step: for i = 1, . . . , M generate z(i) from p( · | x (i) , η (r ) ) end ˆ M M-step: η (r +1) = argmaxη ˆ i=1 log(p(x (i) , z(i) | η)) [Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt, 1993.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
  90. 90. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionStochastic EM (SEM) at iteration (r + 1) Stochastic (S)-step: for i = 1, . . . , M generate z(i) from p( · | x (i) , η (r ) ) end ˆ M M-step: η (r +1) = argmaxη ˆ i=1 log(p(x (i) , z(i) | η)) [Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt, 1993.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
  91. 91. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionStochastic EM (SEM) at iteration (r + 1) Stochastic (S)-step: for i = 1, . . . , M generate z(i) from p( · | x (i) , η (r ) ) end ˆ M M-step: η (r +1) = argmaxη ˆ i=1 log(p(x (i) , z(i) | η)) [Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt, 1993.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
  92. 92. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionStochastic EM (SEM) at iteration (r + 1) Stochastic (S)-step: for i = 1, . . . , M generate z(i) from p( · | x (i) , η (r ) ) end ˆ M M-step: η (r +1) = argmaxη ˆ i=1 log(p(x (i) , z(i) | η)) [Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt, 1993.] S-step To draw z(i) ∼ p( · | x (i) , η (r ) ) we developed an I-MH sampler. ˆ Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
  93. 93. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionExample 2: spectral analysis (Cont.) Variable-dimensional posterior distribution 4 3 k 2 0 0.3 0.6 0.5 0.75 1 p(k |y) ω Figure: Posteriors of k and sorted radial frequencies, ωk , given k. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 33/ 52
  94. 94. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionRobustifying solutions Add a Poisson point process component To capture the “outliers” λ is the mean parameter points are uniformly distributed on Θ Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
  95. 95. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionRobustifying solutions Add a Poisson point process component To capture the “outliers” λ is the mean parameter points are uniformly distributed on Θ Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
  96. 96. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionRobustifying solutions Add a Poisson point process component To capture the “outliers” λ is the mean parameter points are uniformly distributed on Θ Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
  97. 97. Relabeling and summarizing posterior distributions An original variable-dimensional parametric model Proposed approach Estimating the model parameters (SEM-type algorithms) Results Robustifying strategies ConclusionRobustifying solutions Add a Poisson point process component To capture the “outliers” λ is the mean parameter points are uniformly distributed on Θ Other possibilities Robust estimates in the M-step. « Median instead of mean « interquartile range instead of variance Using another divergence measure. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
  98. 98. Outline 1 Relabeling and summarizing posterior distributions Label-switching issue Variable-dimensional summarization 2 Proposed approach An original variable-dimensional parametric model Estimating the model parameters (SEM-type algorithms) Robustifying strategies 3 Results Detection and estimation of sinusoids in white noise Detection and estimation of muons in the Auger project 4 Conclusion
  99. 99. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionExample 2: Spectral analysis 10 0 −10 0 10 20 30 40 50 60 time 150 100 Power 50 0 0 0.5 1 1.5 2 2.5 3 radial frequency Figure: Observed signal (top) and its periodogram (bottom). k Mk : y [i] = aj cos[ωj i] + bj sin[ωj i] + n[i]. j=1 [Andrieu and Doucet, 1999.] Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 35/ 52
  100. 100. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionExample 2: spectral analysis (Cont.) Variable-dimensional posterior distribution 4 3 k 2 0 0.3 0.6 0.5 0.75 1 p(k |y) ω Figure: Posteriors of k and sorted radial frequencies, ωk , given k. Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 36/ 52
  101. 101. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionS-step: randomized allocation procedure 0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72 z1 z2 z1 z2 z3 z4 z1 z2 z3M-step norm. density 1 0.5 0 0.5 0.75 1 ω ˆ λ = 0.1 Iter = 0 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
  102. 102. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionS-step: randomized allocation procedure 0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72 3 1 4 2 3 1 1 4 2M-step norm. density 1 0.5 0 0.5 0.75 1 ω ˆ λ = 0.1 Iter = 0 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
  103. 103. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionS-step: randomized allocation procedure 0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72 3 1 4 2 3 1 1 4 2M-stepnorm. density 1 0.5 0 0.5 0.75 1 ω ˆ λ = 0.167 Iter = 1 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
  104. 104. Relabeling and summarizing posterior distributions Proposed approach Detection and estimation of sinusoids in white noise Results Detection and estimation of muons in the Auger project ConclusionS-step: randomized allocation procedure 0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72 3 1 4 2 3 1 1 3 2M-stepnorm. density 1 0.5 0 0.5 0.75 1 ω ˆ λ = 0.167 Iter = 1 Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52

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