A Gaussian-mixture ensemble transform filter

We generalize the popular ensemble Kalman filter to an ensemble transform filter, in which the prior distribution can take the form of a Gaussian mixture or a Gaussian kernel density estimator. The design of the filter is based on a continuous formulation of the Bayesian filter analysis step. We call the new filter algorithm the ensemble Gaussian-mixture filter (EGMF). The EGMF is implemented for three simple test problems (Brownian dynamics in one dimension, Langevin dynamics in two dimensions and the three-dimensional Lorenz-63 model). It is demonstrated that the EGMF is capable of tracking systems with non-Gaussian uni- and multimodal ensemble distributions. Copyright © 2011 Royal Meteorological Society

On-line Gaussian mixture density estimator for adaptive minimum bit-error-rate beamforming receivers

We develop an on-line Gaussian mixture density estimator (OGMDE) in the complex-valued domain to facilitate adaptive minimum bit-error-rate (MBER) beamforming receiver for multiple antenna based space-division multiple access systems. Specifically, the novel OGMDE is proposed to adaptively model the probability density function of the beamformer’s output by tracking the incoming data sample by sample. With the aid of the proposed OGMDE, our adaptive beamformer is capable of updating the beamformer’s weights sample by sample to directly minimize the achievable bit error rate (BER). We show that this OGMDE based MBER beamformer outperforms the existing on-line MBER beamformer, known as the least BER beamformer, in terms of both the convergence speed and the achievable BER.

Experimentation with different thermodynamic cycles used for pKa calculations on carboxylic acids using Complete Basis Set and Gaussian-n Models combined with CPCM Continuum Solvation Methods

Complete basis set and Gaussian-n methods were combined with Barone and Cossi's implementation of the polarizable conductor model (CPCM) continuum solvation methods to calculate pKa values for six carboxylic acids. Four different thermodynamic cycles were considered in this work. An experimental value of −264.61 kcal/mol for the free energy of solvation of H+, ΔGs(H+), was combined with a value for Ggas(H+) of −6.28 kcal/mol, to calculate pKa values with cycle 1. The complete basis set gas-phase methods used to calculate gas-phase free energies are very accurate, with mean unsigned errors of 0.3 kcal/mol and standard deviations of 0.4 kcal/mol. The CPCM solvation calculations used to calculate condensed-phase free energies are slightly less accurate than the gas-phase models, and the best method has a mean unsigned error and standard deviation of 0.4 and 0.5 kcal/mol, respectively. Thermodynamic cycles that include an explicit water in the cycle are not accurate when the free energy of solvation of a water molecule is used, but appear to become accurate when the experimental free energy of vaporization of water is used. This apparent improvement is an artifact of the standard state used in the calculation. Geometry relaxation in solution does not improve the results when using these later cycles. The use of cycle 1 and the complete basis set models combined with the CPCM solvation methods yielded pKa values accurate to less than half a pKa unit. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

Accurate pKa Calculations for Carboxylic Acids Using Complete Basis Set and Gaussian-n Models Combined with CPCM Continuum Solvation Methods”

Complete Basis Set and Gaussian-n methods were combined with CPCM continuum solvation methods to calculate pKa values for six carboxylic acids. An experimental value of −264.61 kcal/mol for the free energy of solvation of H+, ΔGs(H+), was combined with a value for Ggas(H+) of −6.28 kcal/mol to calculate pKa values with Cycle 1. The Complete Basis Set gas-phase methods used to calculate gas-phase free energies are very accurate, with mean unsigned errors of 0.3 kcal/mol and standard deviations of 0.4 kcal/mol. The CPCM solvation calculations used to calculate condensed-phase free energies are slightly less accurate than the gas-phase models, and the best method has a mean unsigned error and standard deviation of 0.4 and 0.5 kcal/mol, respectively. The use of Cycle 1 and the Complete Basis Set models combined with the CPCM solvation methods yielded pKa values accurate to less than half a pKa unit.

Accurate relative pKa calculations for carboxylic acids using complete basis set and Gaussian-n models combined with continuum solvation methods

The complete basis set methods CBS-4, CBS-QB3, and CBS-APNO, and the Gaussian methods G2 and G3 were used to calculate the gas phase energy differences between six different carboxylic acids and their respective anions. Two different continuum methods, SM5.42R and CPCM, were used to calculate the free energy differences of solvation for the acids and their anions. Relative pKa values were calculated for each acid using one of the acids as a reference point. The CBS-QB3 and CBS-APNO gas phase calculations, combined with the CPCM/HF/6-31+G(d)//HF/6-31G(d) or CPCM/HF/6-31+G(d)//HF/6-31+G(d) continuum solvation calculations on the lowest energy gas phase conformer, and with the conformationally averaged values, give results accurate to ½ pKa unit. © 2001 American Institute of Physics.

Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.

Global Optimization with Sparse and Local Gaussian Process Models

We present a novel surrogate model-based global optimization framework allowing a large number of function evaluations. The method, called SpLEGO, is based on a multi-scale expected improvement (EI) framework relying on both sparse and local Gaussian process (GP) models. First, a bi-objective approach relying on a global sparse GP model is used to determine potential next sampling regions. Local GP models are then constructed within each selected region. The method subsequently employs the standard expected improvement criterion to deal with the exploration-exploitation trade-off within selected local models, leading to a decision on where to perform the next function evaluation(s). The potential of our approach is demonstrated using the so-called Sparse Pseudo-input GP as a global model. The algorithm is tested on four benchmark problems, whose number of starting points ranges from 102 to 104. Our results show that SpLEGO is effective and capable of solving problems with large number of starting points, and it even provides significant advantages when compared with state-of-the-art EI algorithms.

Fully adaptive gaussian mixture metropolis-hastings algorithm

Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multimodal and multidimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.

Application of mixture models to detect differentially expressed genes

An important and common problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As this problem concerns the selection of significant genes from a large pool of candidate genes, it needs to be carried out within the framework of multiple hypothesis testing. In this paper, we focus on the use of mixture models to handle the multiplicity issue. With this approach, a measure of the local FDR (false discovery rate) is provided for each gene. An attractive feature of the mixture model approach is that it provides a framework for the estimation of the prior probability that a gene is not differentially expressed, and this probability can subsequently be used in forming a decision rule. The rule can also be formed to take the false negative rate into account. We apply this approach to a well-known publicly available data set on breast cancer, and discuss our findings with reference to other approaches.

Using mixture models to detect differentially expressed genes

An important and common problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As this problem concerns the selection of significant genes from a large pool of candidate genes, it needs to be carried out within the framework of multiple hypothesis testing. In this paper, we focus on the use of mixture models to handle the multiplicity issue. With this approach, a measure of the local false discovery rate is provided for each gene, and it can be implemented so that the implied global false discovery rate is bounded as with the Benjamini-Hochberg methodology based on tail areas. The latter procedure is too conservative, unless it is modified according to the prior probability that a gene is not differentially expressed. An attractive feature of the mixture model approach is that it provides a framework for the estimation of this probability and its subsequent use in forming a decision rule. The rule can also be formed to take the false negative rate into account.

Mixture models for detecting differentially expressed genes in microarrays

An important and common problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As this problem concerns the selection of significant genes from a large pool of candidate genes, it needs to be carried out within the framework of multiple hypothesis testing. In this paper, we focus on the use of mixture models to handle the multiplicity issue. With this approach, a measure of the local FDR (false discovery rate) is provided for each gene. An attractive feature of the mixture model approach is that it provides a framework for the estimation of the prior probability that a gene is not differentially expressed, and this probability can subsequently be used in forming a decision rule. The rule can also be formed to take the false negative rate into account. We apply this approach to a well-known publicly available data set on breast cancer, and discuss our findings with reference to other approaches.

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.