Alternatives to the EM algorithm for ML estimation of location, scatter matrix, and degree of freedom of the Student t distribution
In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν , the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν that determines the tails of...
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| Abstract | In this paper, we consider maximum likelihood estimations of the degree of freedom parameter
ν
, the location parameter
μ
and the scatter matrix
Σ
of the multivariate Student
t
distribution. In particular, we are interested in estimating the degree of freedom parameter
ν
that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case
ν
→
∞
, for which the Student
t
distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed
ν
, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix
ν
, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for
ν
. We show how the objective function behaves for the different updates of
ν
and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student
t
noise. |
|---|---|
| AbstractList | In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν, the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case ν→∞, for which the Student t distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed ν, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix ν, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for ν. We show how the objective function behaves for the different updates of ν and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student t noise. In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν , the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case ν → ∞ , for which the Student t distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed ν , the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix ν , we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for ν . We show how the objective function behaves for the different updates of ν and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student t noise. In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν , the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case $\nu \rightarrow \infty $ ν → ∞ , for which the Student t distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed ν , the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix ν , we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for ν . We show how the objective function behaves for the different updates of ν and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student t noise. |
| Author | Laus, Friederike Hasannasab, Marzieh Steidl, Gabriele Hertrich, Johannes |
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| Cites_doi | 10.1145/321296.321305 10.1093/biomet/81.4.633 10.1007/s10851-018-0845-6 10.1371/journal.pone.0091381 10.1007/BFb0033290 10.1115/1.3625776 10.3390/s18041175 10.1137/19M1242203 10.1093/biomet/33.3.239 10.1080/03610919408813180 10.1007/s10851-017-0759-8 10.1007/s10851-018-0816-y 10.1080/10618600.2019.1594835 10.1137/15M1012682 10.1137/120874989 10.1137/140997816 10.1109/ICIP.2007.4378944 10.1016/j.imavis.2008.11.013 10.1016/j.amc.2018.08.014 10.1007/s10915-017-0460-5 10.1111/1467-9574.00211 10.2307/2332226 10.1111/1467-9868.00082 10.1023/A:1008981510081 10.1002/nla.617 10.1109/TMI.2011.2165342 |
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| References | LiuCRubinDBML estimation of the t distribution using EM and its extensions, ECM and ECMEStat. Sin.199551193913292870824.62047 KendallMGA new measure of rank correlationBiometrika1938301/2819310.2307/2332226 MengX-LVan DykDThe EM algorithm - an old folk-song sung to a fast new tuneJ. Royal Statis. Soc. :, Series B (Statis. Methodol.)1997593511567145202510.1111/1467-9868.00082 VaradhanRRolandCSimple and globally convergent methods for accelerating the convergence of any EM algorithm. ScandinavianJ. Statis. Theory Appli20083523353531164.65006 McLachlan, G., Krishnan, T.: The EM algorithm and extensions. John wiley and sons inc (1997) Kendall, M.G.: The treatment of ties in ranking problems. Biometrika 239–251 (1945) PetersenKBPedersenMSThe Matrix Cookbook2008Lecture NotesTechnical University of Denmark LangeKLLittleRJTaylorJMRobust statistical modeling using the t distributionJ. Am. Stat. Assoc.1989844088818961134486 PeelDMcLachlanGJRobust mixture modelling using the t distributionStat. Comput.200010433934810.1023/A:1008981510081 GerogiannisDNikouCLikasAThe mixtures of Student’s t-distributions as a robust framework for rigid registrationImage Vis. Comput.20092791285129410.1016/j.imavis.2008.11.013 DingMHuangTWangSMeiJZhaoXTotal variation with overlapping group sparsity for deblurring images under Cauchy noiseAppl. Math. Comput.201934112814738617441428.94021 LebrunMBuadesAMorelJ-MA nonlocal Bayesian image denoising algorithmSIAM J. Imag. Sci.20136316651688309703110.1137/120874989 NguyenTMWuQJRobust Student’s-t mixture model with spatial constraints and its application in medical image segmentationIEEE Trans. Med. Imaging201231110311610.1109/TMI.2011.2165342 LausFSteidlGMultivariate myriad filters based on parameter estimation of student-t distributionsSIAM J Imaging Sci201912418641904402981110.1137/19M1242203 Van DykDAConstruction, Implementation, and Theory of Algorithms Based on Data Augmentation and Model Reduction1995PhD ThesisThe University of Chicago SutourCDeledalleC-AAujolJ-FEstimation of the noise level function based on a nonparametric detection of homogeneous image regionsSIAM J. Imag. Sci.20158426222661342406710.1137/15M1012682 LausFStatistical Analysis and Optimal Transport for Euclidean and Manifold-Valued Data2020TU KaiserslauternPhD Thesis1433.62008 Byrne, C.L.: The EM algorithm: theory, applications and related methods. Lecture notes university of massachusetts (2017) LiuCRubinDBThe ECME algorithm: a simple extension of EM and ECM with faster monotone convergenceBiometrika1994814633648132641410.1093/biomet/81.4.633 SciacchitanoFDongYZengTVariational approach for restoring blurred images with Cauchy noiseSIAM J. Imag. Sci.20158318941922339742710.1137/140997816 YangZYangZGuiGA convex constraint variational method for restoring blurred images in the presence of alpha-stable noisesSensors2018184117510.3390/s18041175 MeiJ-JDongYHuangT-ZYinWCauchy noise removal by nonconvex ADMM with convergence guaranteesJ. Sci. Comput.2018742743766376127510.1007/s10915-017-0460-5 LanzaAMorigiSSciacchitanoFSgallariFWhiteness constraints in a unified variational framework for image restorationJ. Mathe. Imag. Vision201860915031526386474510.1007/s10851-018-0845-6 McLachlan, G., Peel, D.: Robust cluster analysis via mixtures of multivariate t-distributions. volume 1451 of Lecture Notes in Computer Science. Springer, New York (1998) AntoniadisALeporiniDPesquetJ-CWavelet thresholding for some classes of non-Gaussian noiseStatis. Neerlandica2002564434453202753510.1111/1467-9574.00211 Van Den OordASchrauwenBThe Student-t mixture as a natural image patch prior with application to image compressionJ. Mach. Learn. Res.20141512061208632316011319.62135 Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables, volume 55 Courier Corporation (1965) HendersonNCVaradhanRDamped Anderson acceleration with restarts and monotonicity control for accelerating EM and EM-like algorithmsJ. Comput. Graph. Stat.2019284834846404585210.1080/10618600.2019.1594835 AndersonDGIterative procedures for nonlinear integral equationsJ. Assoc. Comput. Mach.19651254756018444710.1145/321296.321305 BanerjeeAMajiPSpatially constrained Student’s t-distribution based mixture model for robust image segmentationJ. Mathe. Imag. Vision2018603355381376931510.1007/s10851-017-0759-8 ZhouZZhengJDaiYZhouZChenSRobust non-rigid point set registration using Student’s-t mixture modelPloS one201493e9138110.1371/journal.pone.0091381 Sfikas, G., Nikou, C., Galatsanos, N.: Robust image segmentation with mixtures of Student’s t-distributions. In: 2007 IEEE International Conference on Image Processing, volume 1, pages I – 273–I –276 (2007) LausFPierreFSteidlGNonlocal myriad filters for Cauchy noise removalJ. Math. Imag. Vision201860813241354385004210.1007/s10851-018-0816-y FangH-RSaadYTwo classes of multisecant methods for nonlinear accelerationNumer. Linear Algebra Appli.2009163197221248920310.1002/nla.617 KentJTTylerDEVardYA curious likelihood identity for the multivariate t-distributionCommunications in Statistics-Simulation and Computation1994232441453127967510.1080/03610919408813180 KL Lange (959_CR13) 1989; 84 DA Van Dyk (959_CR32) 1995 D Gerogiannis (959_CR8) 2009; 27 F Laus (959_CR17) 2019; 12 959_CR1 F Sciacchitano (959_CR28) 2015; 8 KB Petersen (959_CR27) 2008 Z Yang (959_CR34) 2018; 18 959_CR5 DG Anderson (959_CR2) 1965; 12 TM Nguyen (959_CR25) 2012; 31 C Sutour (959_CR30) 2015; 8 R Varadhan (959_CR33) 2008; 35 A Antoniadis (959_CR3) 2002; 56 959_CR11 A Van Den Oord (959_CR31) 2014; 15 X-L Meng (959_CR24) 1997; 59 JT Kent (959_CR12) 1994; 23 C Liu (959_CR19) 1994; 81 MG Kendall (959_CR10) 1938; 30 A Lanza (959_CR14) 2018; 60 H-R Fang (959_CR7) 2009; 16 F Laus (959_CR15) 2020 959_CR29 M Ding (959_CR6) 2019; 341 M Lebrun (959_CR18) 2013; 6 C Liu (959_CR20) 1995; 5 A Banerjee (959_CR4) 2018; 60 J-J Mei (959_CR23) 2018; 74 D Peel (959_CR26) 2000; 10 NC Henderson (959_CR9) 2019; 28 959_CR22 959_CR21 F Laus (959_CR16) 2018; 60 Z Zhou (959_CR35) 2014; 9 |
| References_xml | – reference: McLachlan, G., Krishnan, T.: The EM algorithm and extensions. John wiley and sons inc (1997) – reference: LanzaAMorigiSSciacchitanoFSgallariFWhiteness constraints in a unified variational framework for image restorationJ. Mathe. Imag. Vision201860915031526386474510.1007/s10851-018-0845-6 – reference: LangeKLLittleRJTaylorJMRobust statistical modeling using the t distributionJ. Am. Stat. Assoc.1989844088818961134486 – reference: HendersonNCVaradhanRDamped Anderson acceleration with restarts and monotonicity control for accelerating EM and EM-like algorithmsJ. Comput. Graph. Stat.2019284834846404585210.1080/10618600.2019.1594835 – reference: AndersonDGIterative procedures for nonlinear integral equationsJ. Assoc. Comput. Mach.19651254756018444710.1145/321296.321305 – reference: MeiJ-JDongYHuangT-ZYinWCauchy noise removal by nonconvex ADMM with convergence guaranteesJ. Sci. Comput.2018742743766376127510.1007/s10915-017-0460-5 – reference: MengX-LVan DykDThe EM algorithm - an old folk-song sung to a fast new tuneJ. Royal Statis. Soc. :, Series B (Statis. Methodol.)1997593511567145202510.1111/1467-9868.00082 – reference: PeelDMcLachlanGJRobust mixture modelling using the t distributionStat. Comput.200010433934810.1023/A:1008981510081 – reference: Van DykDAConstruction, Implementation, and Theory of Algorithms Based on Data Augmentation and Model Reduction1995PhD ThesisThe University of Chicago – reference: VaradhanRRolandCSimple and globally convergent methods for accelerating the convergence of any EM algorithm. ScandinavianJ. Statis. Theory Appli20083523353531164.65006 – reference: LausFStatistical Analysis and Optimal Transport for Euclidean and Manifold-Valued Data2020TU KaiserslauternPhD Thesis1433.62008 – reference: Byrne, C.L.: The EM algorithm: theory, applications and related methods. 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ν
, the location parameter
μ
and the scatter matrix
Σ
of the... In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν, the location parameter μ and the scatter matrix Σ of the... |
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| SubjectTerms | Algebra Algorithms Computer Science Degrees of freedom Iterative methods Maximum likelihood estimation Normal distribution Numeric Computing Numerical Analysis Original Paper Parameters Probability density functions Scattering Statistical analysis Theory of Computation |
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| Title | Alternatives to the EM algorithm for ML estimation of location, scatter matrix, and degree of freedom of the Student t distribution |
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