scholarly journals Computing tail areas for a high-dimensional Gaussian mixture

2019 ◽  
Vol 13 (3) ◽  
pp. 871-882
Author(s):  
Burcin Simsek ◽  
Satish Iyengar
Keyword(s):  
Importance Sampling ◽  
Exponential Family ◽  
Random Variable ◽  
Gaussian Mixture ◽  
High Dimensional ◽  
Low Density ◽  
Gaussian Mixtures ◽  
Tail Probabilities ◽  
Exponential Tilting ◽  
The Third

We consider the problem of computing tail probabilities - that is, probabilities of regions with low density - for high-dimensional Gaussian mixtures. We consider three approaches: the first is a bound based on the central and non-central ?2 distributions; the second uses Pearson curves with the first three moments of the criterion random variable U; the third embeds the distribution of U in an exponential family, and uses exponential tilting, which in turn suggests an importance sampling distribution. We illustrate each method with examples and assess their relative merits.

The Bender Equation of State and Virial Coefficients

2000 ◽  
Vol 65 (9) ◽  
pp. 1464-1470 ◽  
Author(s):  
Anatol Malijevský ◽  
Tomáš Hujo
Keyword(s):  
Equation Of State ◽  
Virial Coefficient ◽  
Virial Coefficients ◽  
Low Density ◽  
Second Virial Coefficients ◽  
The Third ◽  
Temperature Dependences ◽  
Experimental Values

The second and third virial coefficients calculated from the Bender equation of state (BEOS) are tested against experimental virial coefficient data. It is shown that the temperature dependences of the second and third virial coefficients as predicted by the BEOS are sufficiently accurate. We conclude that experimental second virial coefficients should be used to determine independently five of twenty constants of the Bender equation. This would improve the performance of the equation in a region of low-density gas, and also suppress correlations among the BEOS constants, which is even more important. The third virial coefficients cannot be used for the same purpose because of large uncertainties in their experimental values.

scholarly journals The Generalized Bayes Method for High-Dimensional Data Recognition with Applications to Audio Signal Recognition

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 19
Author(s):  
Hsiuying Wang
Keyword(s):  
Gaussian Mixture Model ◽  
Mixture Model ◽  
Conventional Method ◽  
High Dimensional Data ◽  
Audio Signal ◽  
Gaussian Mixture ◽  
High Dimensional ◽  
Signal Recognition ◽  
Bayes Method ◽  
Generalized Bayes

High-dimensional data recognition problem based on the Gaussian Mixture model has useful applications in many area, such as audio signal recognition, image analysis, and biological evolution. The expectation-maximization algorithm is a popular approach to the derivation of the maximum likelihood estimators of the Gaussian mixture model (GMM). An alternative solution is to adopt a generalized Bayes estimator for parameter estimation. In this study, an estimator based on the generalized Bayes approach is established. A simulation study shows that the proposed approach has a performance competitive to that of the conventional method in high-dimensional Gaussian mixture model recognition. We use a musical data example to illustrate this recognition problem. Suppose that we have audio data of a piece of music and know that the music is from one of four compositions, but we do not know exactly which composition it comes from. The generalized Bayes method shows a higher average recognition rate than the conventional method. This result shows that the generalized Bayes method is a competitor to the conventional method in this real application.

scholarly journals Beyond the Third Dimension: Visualizing High-Dimensional Data with Projections

2016 ◽  
Vol 18 (5) ◽  
pp. 98-107 ◽  
Author(s):  
Renato R.O. da Silva ◽  
Paulo E. Rauber ◽  
Alexandru C. Telea
Keyword(s):  
High Dimensional Data ◽  
High Dimensional ◽  
The Third ◽  
Third Dimension

scholarly journals Optimal Coordination of Last Trains for Maximum Transfer Accessibility with Heterogeneous Walking Time

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Yao Chen ◽  
Baohua Mao ◽  
Yun Bai ◽  
Tin Kin Ho ◽  
Zhujun Li
Keyword(s):  
Dwell Time ◽  
Random Variable ◽  
Minimal Extension ◽  
The Third ◽  
Train Timetable ◽  
Urban Rail ◽  
Optimal Coordination ◽  
Transfer Stations ◽  
The Given ◽  
Walking Time

Last train coordination aims to synchronize the arrival and departure times of the last feeder trains and the last connecting trains at transfer stations to improve the transfer accessibility of urban rail networks. This study focuses on the transfer accessibility between last trains with considering heterogeneous transfer walking time. Three mathematical models are developed on the last train timetable optimization. The first model fine-tunes the last train timetable under the given bound of the dwell time. The second one aims to allow the mutual transfers with the prolonged dwell time to maximize the transfer accessibility. A biobjective function is proposed to seek the trade-off between the maximal transfer accessibility and the minimal extension of dwell time. The third model considers the heterogeneity of transfer walking time that is represented as a random variable following a probability distribution. A discrete approximation method is proposed to reformulate the nonlinear model. The embedded Branch & Cut algorithm of CPLEX is applied to solve the models. A real case on the Shenzhen metro network is conducted to demonstrate the performance of the models. The three models all provide better last train timetable than the current timetable in practice. The sensitivity analysis manifests that the third model are always advantageous in the optimization of successful transfer passengers.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

scholarly journals Regularized Parameter Estimation in High-Dimensional Gaussian Mixture Models

2011 ◽  
Vol 23 (6) ◽  
pp. 1605-1622 ◽  
Author(s):  
Lingyan Ruan ◽  
Ming Yuan ◽  
Hui Zou
Keyword(s):  
Parameter Estimation ◽  
Mixture Models ◽  
Gaussian Mixture Models ◽  
Expectation Maximization Algorithm ◽  
Real Data ◽  
Gaussian Mixture ◽  
High Dimensional ◽  
Model Based Clustering ◽  
Text Type ◽  
Effective Dimensionality

Finite gaussian mixture models are widely used in statistics thanks to their great flexibility. However, parameter estimation for gaussian mixture models with high dimensionality can be challenging because of the large number of parameters that need to be estimated. In this letter, we propose a penalized likelihood estimator to address this difficulty. The [Formula: see text]-type penalty we impose on the inverse covariance matrices encourages sparsity on its entries and therefore helps to reduce the effective dimensionality of the problem. We show that the proposed estimate can be efficiently computed using an expectation-maximization algorithm. To illustrate the practical merits of the proposed method, we consider its applications in model-based clustering and mixture discriminant analysis. Numerical experiments with both simulated and real data show that the new method is a valuable tool for high-dimensional data analysis.

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