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

Fast identification algorithms for Gaussian process model

A class identification algorithms is introduced for Gaussian process(GP)models.The fundamental approach is to propose a new kernel function which leads to a covariance matrix with low rank,a property that is consequently exploited for computational efficiency for both model parameter estimation and model predictions.The objective of either maximizing the marginal likelihood or the Kullback–Leibler (K–L) divergence between the estimated output probability density function(pdf)and the true pdf has been used as respective cost functions.For each cost function,an efficient coordinate descent algorithm is proposed to estimate the kernel parameters using a one dimensional derivative free search, and noise variance using a fast gradient descent algorithm. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

Observation impact in data assimilation: the effect of non-Gaussian observation error.

Data assimilation methods which avoid the assumption of Gaussian error statistics are being developed for geoscience applications. We investigate how the relaxation of the Gaussian assumption affects the impact observations have within the assimilation process. The effect of non-Gaussian observation error (described by the likelihood) is compared to previously published work studying the effect of a non-Gaussian prior. The observation impact is measured in three ways: the sensitivity of the analysis to the observations, the mutual information, and the relative entropy. These three measures have all been studied in the case of Gaussian data assimilation and, in this case, have a known analytical form. It is shown that the analysis sensitivity can also be derived analytically when at least one of the prior or likelihood is Gaussian. This derivation shows an interesting asymmetry in the relationship between analysis sensitivity and analysis error covariance when the two different sources of non-Gaussian structure are considered (likelihood vs. prior). This is illustrated for a simple scalar case and used to infer the effect of the non-Gaussian structure on mutual information and relative entropy, which are more natural choices of metric in non-Gaussian data assimilation. It is concluded that approximating non-Gaussian error distributions as Gaussian can give significantly erroneous estimates of observation impact. The degree of the error depends not only on the nature of the non-Gaussian structure, but also on the metric used to measure the observation impact and the source of the non-Gaussian structure.

Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach

The analysis step of the (ensemble) Kalman filter is optimal when (1) the distribution of the background is Gaussian, (2) state variables and observations are related via a linear operator, and (3) the observational error is of additive nature and has Gaussian distribution. When these conditions are largely violated, a pre-processing step known as Gaussian anamorphosis (GA) can be applied. The objective of this procedure is to obtain state variables and observations that better fulfil the Gaussianity conditions in some sense. In this work we analyse GA from a joint perspective, paying attention to the effects of transformations in the joint state variable/observation space. First, we study transformations for state variables and observations that are independent from each other. Then, we introduce a targeted joint transformation with the objective to obtain joint Gaussianity in the transformed space. We focus primarily in the univariate case, and briefly comment on the multivariate one. A key point of this paper is that, when (1)-(3) are violated, using the analysis step of the EnKF will not recover the exact posterior density in spite of any transformations one may perform. These transformations, however, provide approximations of different quality to the Bayesian solution of the problem. Using an example in which the Bayesian posterior can be analytically computed, we assess the quality of the analysis distributions generated after applying the EnKF analysis step in conjunction with different GA options. The value of the targeted joint transformation is particularly clear for the case when the prior is Gaussian, the marginal density for the observations is close to Gaussian, and the likelihood is a Gaussian mixture.

The effect of rapid changes in ionospheric flow on velocity vectors deduced from radar beam-swinging experiments

The effects on the horizontal ionospheric velocity vectors deduced from radar beam-swinging experiments, which occur when changes in the flow take place on short time scales compared with the experiment cycle time, are analysed in detail. The further complications which arise in the interpretation of beam-swinging data, due to longitudinal gradients in the flow and to field-aligned flows, are also considered. It is concluded that these effects are unlikely to seriously compromise statistical determinations of the response time of the flow, e.g. to changes in the north-south component of the IMF, such as have been recently reported by Etemadiet al. (1988, Planet. Space Sci.36, 471), using EISCAT ‘Polar’ data.

Estimation of Gaussian process regression model using probability distance measures

A new class of parameter estimation algorithms is introduced for Gaussian process regression (GPR) models. It is shown that the integration of the GPR model with probability distance measures of (i) the integrated square error and (ii) Kullback–Leibler (K–L) divergence are analytically tractable. An efficient coordinate descent algorithm is proposed to iteratively estimate the kernel width using golden section search which includes a fast gradient descent algorithm as an inner loop to estimate the noise variance. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

Gaussian processes autoencoder for dimensionality reduction

Learning low dimensional manifold from highly nonlinear data of high dimensionality has become increasingly important for discovering intrinsic representation that can be utilized for data visualization and preprocessing. The autoencoder is a powerful dimensionality reduction technique based on minimizing reconstruction error, and it has regained popularity because it has been efficiently used for greedy pretraining of deep neural networks. Compared to Neural Network (NN), the superiority of Gaussian Process (GP) has been shown in model inference, optimization and performance. GP has been successfully applied in nonlinear Dimensionality Reduction (DR) algorithms, such as Gaussian Process Latent Variable Model (GPLVM). In this paper we propose the Gaussian Processes Autoencoder Model (GPAM) for dimensionality reduction by extending the classic NN based autoencoder to GP based autoencoder. More interestingly, the novel model can also be viewed as back constrained GPLVM (BC-GPLVM) where the back constraint smooth function is represented by a GP. Experiments verify the performance of the newly proposed model.

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.

Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

High-precision measurements of the co-polar correlation coefficient: non-Gaussian errors and retrieval of the dispersion parameter µ in rainfall

The co-polar correlation coefficient (ρhv) has many applications, including hydrometeor classification, ground clutter and melting layer identification, interpretation of ice microphysics and the retrieval of rain drop size distributions (DSDs). However, we currently lack the quantitative error estimates that are necessary if these applications are to be fully exploited. Previous error estimates of ρhv rely on knowledge of the unknown "true" ρhv and implicitly assume a Gaussian probability distribution function of ρhv samples. We show that frequency distributions of ρhv estimates are in fact highly negatively skewed. A new variable: L = -log10(1 - ρhv) is defined, which does have Gaussian error statistics, and a standard deviation depending only on the number of independent radar pulses. This is verified using observations of spherical drizzle drops, allowing, for the first time, the construction of rigorous confidence intervals in estimates of ρhv. In addition, we demonstrate how the imperfect co-location of the horizontal and vertical polarisation sample volumes may be accounted for. The possibility of using L to estimate the dispersion parameter (µ) in the gamma drop size distribution is investigated. We find that including drop oscillations is essential for this application, otherwise there could be biases in retrieved µ of up to ~8. Preliminary results in rainfall are presented. In a convective rain case study, our estimates show µ to be substantially larger than 0 (an exponential DSD). In this particular rain event, rain rate would be overestimated by up to 50% if a simple exponential DSD is assumed.

Cone-beam computerized tomographic, radiographic, and histologic evaluation of periapical repair in dogs` post-endodontic treatment

Objective. To evaluate the periapical repair after root canal treatment in the teeth of dogs using CT and conventional radiography and to compare these findings with the gold standard microscopic evaluation. Study design. The animals were divided into three groups according to endodontic treatment performed: Group 1, single-visit endodontic treatment in teeth without apical periodontitis; Group 2, single-visit endodontic treatment in teeth with apical periodontitis; and Group 3, endodontic treatment in teeth with apical periodontitis using calcium hydroxide as a root canal dressing. Group 4 consisted of teeth with apical periodontitis not submitted to root canal treatment and Group 5 consisted of healthy teeth without periapical disease. Radiographic, tomographic, and microscopic evaluations were performed by blind examiners. At 180 days experimental time, CT and radiographic measurements of periapical disease were compared with the gold standard microscopic measurement using intraclass correlation coefficient. Intergroup comparisons considering different methods of periapical lesions measurement or different clinical protocols of root canal treatment were performed by Kruskal Wallis test followed by Dunn. Integrity of lamina dura, presence of radiolucent areas, and presence of root resorption were analyzed by Fisher`s exact test. Results. There was discontinuity of the lamina dura and CPD in all teeth from Groups 2, 3, and 4 evaluated by tomography and radiography 45 days after CPD induction. Radiographically, 180 days after root canal treatment, there was no periapical lesion in teeth from Groups 1 and 3, different from groups 2 and 4 (p < .05). The highest reduction in the CPD size was observed on Group 3 (p < .05). According to the tomographic results, there was decrease of the size of the CPD on Group 3 but not on Groups 2 or 4. However, in all groups the periapical lesions presented larger mesio-distal extension if compared with radiography, both 45 days after CPD induction and 180 days after root canal treatment. At 180 days, CT measurements were closely related to microscopic results (ICC = 0.95) differently from radiographic evaluation (ICC = 0.86). Conclusion. CT Scan evaluation of periapical repair following root canal treatment provided similar information than that obtained by microscopic analysis, whereas radiographic evaluation underestimated the size do periapical lesion. (Oral Surg Oral Med Oral Pathol Oral Radiol Endod 2009; 108:796-805)

Accuracy of Periapical Radiography and Cone-Beam Computed Tomography Scans in Diagnosing Apical Periodontitis Using Histopathological Findings as a Gold Standard

Introduction: The aim of this study was to evaluate the accuracy of two imaging methods in diagnosing apical periodontitis (AP) using histopathological findings as a gold standard. Methods: The periapex of 83 treated or untreated roots of dogs` teeth was examined using periapical radiography (PR), cone-beam computed tomography (CBCT) scans, and histology. Sensitivity, specificity, predictive values, and accuracy of PR and CBCT diagnosis were calculated. Results: PR detected AP in 71% of roots, a CBCT scan detected AP in 84%, and AP was histologically diagnosed in 93% (p = 0.001). Overall, sensitivity was 0.77 and 0.91 for PR and CBCT, respectively. Specificity was 1 for both. Negative predictive value was 0.25 and 0.46 for PR and CBCT, respectively. Positive predictive value was 1 for both. Diagnostic accuracy (true positives + true negatives) was 0.78 and 0.92 for PR and CBCT (p = 0.028), respectively. Conclusion: A CBCT scan was more sensitive in detecting AP compared with PR, which was more likely to miss AP when it was still present. (J Endod 2009;35:1009-1012)

Outcome of Root Canal Treatment in Dogs Determined by Periapical Radiography and Cone-Beam Computed Tomography Scans

The purpose of this study was to compare the favorable outcome of root canal treatment determined by periapical radiographs (PRs) and cone beam computed tomography (CBCT) scans. Ninety-six roots of dogs` teeth were used to form four groups (n = 24). In group 1, root canal treatments were performed in healthy teeth. Root canals in groups 2 through 4 were infected until apical periodontitis (AP) was radiographically confirmed. Roots with AP were treated by one-visit therapy in group 2, by two-visit therapy in group 3, and left untreated in group 4. The radiolucent area in the PRs and the volume of CBCT-scanned periapical lesions were measured before and 6 months after the treatment. In groups 1, 2, and 3, a favorable outcome (lesions absent or reduced) was shown in 57 (79%) roots using PRs but only in 25 (35%) roots using CBCT scans (p = 0.0001). Unfavorable outcomes occurred more frequently after one-visit therapy than two-visit therapy when determined by CBCT scans (p = 0.023). (J Endod 2009; 35:723-726)

Conservative force fields in non-Gaussian statistics

In this Letter, we determine the kappa-distribution function for a gas in the presence of an external field of force described by a potential U(r). In the case of a dilute gas, we show that the kappa-power law distribution including the potential energy factor term can rigorously be deduced in the framework of kinetic theory with basis on the Vlasov equation. Such a result is significant as a preliminary to the discussion on the role of long range interactions in the Kaniadakis thermostatistics and the underlying kinetic theory. (C) 2008 Elsevier B.V. All rights reserved.