scholarly journals Maximum Likelihood for Matrices with Rank Constraints

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Jonathan Hauenstein
Keyword(s):  
Algebraic Geometry ◽  
Maximum Likelihood ◽  
Optimization Problem ◽  
Likelihood Estimation ◽  
Numerical Algebraic Geometry ◽  
Likelihood Functions ◽  
Determinantal Varieties ◽  
Discrete Random Variables ◽  
Rank Constraints ◽  
Bounded Rank

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.

scholarly journals Word Embedding as Maximum A Posteriori Estimation

2019 ◽  
Vol 33 ◽  
pp. 6562-6569 ◽  
Author(s):  
Shoaib Jameel ◽  
Zihao Fu ◽  
Bei Shi ◽  
Wai Lam ◽  
Steven Schockaert
Keyword(s):  
Global Optimization ◽  
Maximum Likelihood ◽  
Optimization Problem ◽  
Likelihood Estimation ◽  
Word Embedding ◽  
Maximum A Posteriori ◽  
Context Word ◽  
Global Optimization Problem ◽  
A Posteriori ◽  
Posteriori Estimation

The GloVe word embedding model relies on solving a global optimization problem, which can be reformulated as a maximum likelihood estimation problem. In this paper, we propose to generalize this approach to word embedding by considering parametrized variants of the GloVe model and incorporating priors on these parameters. To demonstrate the usefulness of this approach, we consider a word embedding model in which each context word is associated with a corresponding variance, intuitively encoding how informative it is. Using our framework, we can then learn these variances together with the resulting word vectors in a unified way. We experimentally show that the resulting word embedding models outperform GloVe, as well as many popular alternatives.

scholarly journals Algebraic boundaries of Hilbert’s SOS cones

2012 ◽  
Vol 148 (6) ◽  
pp. 1717-1735 ◽  
Author(s):  
Grigoriy Blekherman ◽  
Jonathan Hauenstein ◽  
John Christian Ottem ◽  
Kristian Ranestad ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
K3 Surfaces ◽  
Sums Of Squares ◽  
Numerical Algebraic Geometry ◽  
Hankel Matrices ◽  
Severi Variety ◽  
The Difference ◽  
Rank Constraints ◽  
Extreme Rays

AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

scholarly journals Some Notes on Concordance between Optimization and Statistics

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Weiyan Mu ◽  
Qiuyue Wei ◽  
Shifeng Xiong
Keyword(s):  
Maximum Likelihood ◽  
General Framework ◽  
Optimization Problem ◽  
Theoretical Study ◽  
Optimization Problems ◽  
Likelihood Estimation ◽  
Statistical Optimization ◽  
Comparison Theorems ◽  
Engineering Problems ◽  
Statistical Sense

Many engineering problems require solutions to statistical optimization problems. When the global solution is hard to attain, engineers or statisticians always use the better solution because we intuitively believe a principle, called better solution principle (BSP) in this paper, that a better solution to a statistical optimization problem also has better statistical properties of interest. This principle displays some concordance between optimization and statistics and is expected to widely hold. Since theoretical study on BSP seems to be neglected by statisticians, this paper presents a primary discussion on BSP within a relatively general framework. We demonstrate two comparison theorems as the key results of this paper. Their applications to maximum likelihood estimation are presented. It can be seen that BSP for this problem holds under reasonable conditions; i.e., an estimator with greater likelihood is better in some statistical sense.

A Cautionary Note on Incremental Fit Indices Reported by LISREL

Methodology ◽  
2005 ◽  
Vol 1 (2) ◽  
pp. 81-85 ◽  
Author(s):  
Stefan C. Schmukle ◽  
Jochen Hardt
Keyword(s):  
Factor Analysis ◽  
Maximum Likelihood ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Null Model ◽  
Likelihood Estimation ◽  
Parameter Estimates ◽  
Fit Indices ◽  
Cautionary Note ◽  
Confirmatory Factor

Abstract. Incremental fit indices (IFIs) are regularly used when assessing the fit of structural equation models. IFIs are based on the comparison of the fit of a target model with that of a null model. For maximum-likelihood estimation, IFIs are usually computed by using the χ2 statistics of the maximum-likelihood fitting function (ML-χ2). However, LISREL recently changed the computation of IFIs. Since version 8.52, IFIs reported by LISREL are based on the χ2 statistics of the reweighted least squares fitting function (RLS-χ2). Although both functions lead to the same maximum-likelihood parameter estimates, the two χ2 statistics reach different values. Because these differences are especially large for null models, IFIs are affected in particular. Consequently, RLS-χ2 based IFIs in combination with conventional cut-off values explored for ML-χ2 based IFIs may lead to a wrong acceptance of models. We demonstrate this point by a confirmatory factor analysis in a sample of 2449 subjects.

Estimating the Cost of Car Warranty in Indonesia using the Gertsbakh-Kordonsky Method

2020 ◽  
Vol 2 (1) ◽  
pp. 33-40
Author(s):  
Anggis Sagitarisman ◽  
Aceng Komarudin Mutaqin
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Type A ◽  
Claim Data ◽  
Likelihood Method ◽  
Selling Price ◽  
One Dimensional ◽  
Warranty Period ◽  
Kolmogorov Smirnov ◽  
Warranty Costs

AbstractCar manufacturers in Indonesia need to determine reasonable warranty costs that do not burden companies or consumers. Several statistical approaches have been developed to analyze warranty costs. One of them is the Gertsbakh-Kordonsky method which reduces the two-dimensional warranty problem to one dimensional. In this research, we apply the Gertsbakh-Kordonsky method to estimate the warranty cost for car type A in XYZ company. The one-dimensional data will be tested using the Kolmogorov-Smirnov to determine its distribution and the parameter of distribution will be estimated using the maximum likelihood method. There are three approaches to estimate the parameter of the distribution. The difference between these three approaches is in the calculation of mileage for units that do not claim within the warranty period. In the application, we use claim data for the car type A. The data exploration indicates the failure of car type A is mostly due to the age of the vehicle. The Kolmogorov-Smirnov shows that the most appropriate distribution for the claim data is the three-parameter Weibull. Meanwhile, the estimated using the Gertsbakh-Kordonsky method shows that the warranty costs for car type A are around 3.54% from the selling price of this car unit without warranty i.e. around Rp. 4,248,000 per unit.Keywords: warranty costs; the Gertsbakh-Kordonsky method; maximum likelihood estimation; Kolmogorov-Smirnov test.                                   AbstrakPerusahaan produsen mobil di Indonesia perlu menentukan biaya garansi yang bersifat wajar tidak memberatkan perusahaan maupun konsumen. Beberapa pendekatan statistik telah dikembangkan untuk menganalisis biaya garansi. Salah satunya adalah metode Gertsbakh-Kordonsky yang mereduksi masalah garansi dua dimensi menjadi satu dimensi. Pada penelitian ini, metode Gertsbakh-Kordonsky akan digunakan untuk mengestimasi biaya garansi untuk mobil tipe A pada perusahaan XYZ. Data satu dimensi hasil reduksi diuji kecocokan distribusinya menggunakan uji kecocokan Kolmogorov-Smirnov dan taksiran parameter distribusinya menggunakan metode penaksir kemungkinan maksimum. Ada tiga pendekatan yang digunakan untuk menaksir parameter distribusi. Perbedaan dari ketiga pendekatan tersebut terletak pada perhitungan jarak tempuh untuk unit yang tidak melakukan klaim dalam periode garansi. Sebagai bahan aplikasi, kami menggunakan data klaim unit mobil tipe A. Hasil eksplorasi data menunjukkan bahwa kegagalan mobil tipe A lebih banyak disebabkan karena faktor usia kendaraan. Hasil uji kecocokan distribusi untuk data hasil reduksi menunjukkan bahwa distribusi yang cocok adalah distribusi Weibull 3-parameter. Sementara itu, hasil perhitungan taksiran biaya garansi menunjukan bahwa taksiran biaya garansi untuk unit mobil tipe A sekitar 3,54% dari harga jual unit mobil tipe A tanpa garansi, atau sekitar Rp. 4.248.000,- per unit.Kata Kunci: biaya garansi; metode Gertsbakh-Kordonsky; penaksiran kemungkinan maksimum; uji Kolmogorov-Smirnov.

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