Inverse Problems With Errors in the Independent Variables Errors-in-Variables, Total Least Squares, and Bayesian Inference

Objectives To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping, that is, the coefficient matrix and the observation vector contain fixed quantities (or even fixed columns) and random quantities, and the random quantities at different positions are linearly related. Methods Starting from the errors-in-variables (EIV) function model, the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form, and the function model is reconstructed by the variable projection method. Then, a structured total least squares (STLS) estimation algorithm is derived by the Lagrange method. Results The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector. Compared with the weighted total least squares (WTLS) estimation algorithms, the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates. Compared with other STLS estimation algorithms, the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector. Conclusions This shows the effectiveness of the proposed algorithm. The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model, respectively.

BackgroundBayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.ResultsWith empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.ConclusionsOur results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.

This reference is for an abstract only. A full paper was not submitted for this conference. Abstract AVO (amplitude-varying-with-offset) is a seismic prestack inversion technique for estimating elastic parameters of the subsurface. The seismic AVO inversion problem can be formulated as a linearized or nonlinear inverse problem. It is a multidimensional and highly ill-posed inverse problem. It is affected by strong noise and measurement uncertainty. Therefore, in seismic AVO analysis, the goal is not only to find the model that best-fits the data, but also to characterize the uncertainty of the analysis. Uncertainty characterization of seismic analysis makes the geophysical interpretation more reliable. The Bayesian formalism generally constitutes a powerful approach for solving many seismic inverse problems. It allows combining available prior knowledge with the information contained in the seismic data. The solution of a Bayesian inverse problem is given by the joint posterior distribution of model parameters. Stochastic approximations of the true posterior are exemplified by the MCMC method and more recently by the particle filtering (PF) method. For most seismic inverse problems, stochastic approximation techniques are computationally unappealing. The maximum a posteriori (MAP) is the simplest Bayesian method for estimating model parameters. It provides the explanation that maximizes the posterior distribution. The maximum likelihood (ML) is a popular non-Bayesian method for estimating model parameters. It provides the explanation that maximizes the likelihood of the data. It assumes that the model parameters are deterministic and omits the pertinent prior knowledge on them. In practice, both MAP and ML point estimators have severe problems with over-fitting the data and model order estimation. In this paper, the recently-developed variational Bayesian learning is proposed as a new method for solving the linearized seismic AVO inversion problem. Variational Bayesian learning allows the true joint posterior distribution of model parameters to be approximated by a simpler approximating ensemble for which the required inferences are tractable. The main advantage in resorting to variational Bayesian learning is its robustness to the over-fitting problem, which is major in practice. This probabilistic machine learning method allows finding an optimum balance between the representation power and the model complexity of the data.

The paper addresses the identification of time-invariant bilinear system (BS) models in the errors-in-variables (EIV) framework. The proposed scheme is based on the structured total least norm (STLN) technique extended here to handle BS. The performance of the presented approach, i.e. the bilinear STLN (BSTLN) with its further extension incorporating the Tikhonov regularization is compared to several other EIV identification techniques via an extensive Monte-Carlo simulation study. The results obtained demonstrate a considerable noise robustness and therefore the applicability of the BSTLN algorithm in the EIV framework.

Fostered by advances in computing, simulation and data science have emerged as the third and fourth paradigms enabling scientific discovery. Many mathematical challenges arise when integrating these paradigms to produce new knowledge. One of the most important challenges is the inverse or calibration problem: find the inputs of a simulation such that its outputs agree with a given set of measurements. As both the cost of the complex simulations and the volume of measurement data sets increase, intuition-driven trial-and-error methods for calibration quickly become untenable; modern scientists need more computationally efficient and mathematically rigorous methods. When measurements and simulated predictions contain uncertainty, the Bayesian perspective on statistical inference provides a popular mathematical framework for developing computational methods to solve inverse problems. Markov chain Monte Carlo (MCMC) methods are a mainstay of this framework; they characterize a probability distribution on the space of model inputs that is conditioned on the given measurements. However, despite recent advances, MCMC methods remain impractical for expensive simulations with more than a handful of inputs. At its heart, MCMC searches the space of inputs for corresponding outputs that match measurements, and so it is afflicted with the curse of dimensionality. Additionally, MCMC methods become less efficient as the volume of the measurement data increases, particularly if the measurements are redundant. To make MCMC—and Bayesian methods, in general—amenable to today’s computer models with millions of inputs and terabytes of measurement data, one must devise methods to reduce the dimension of both the simulation inputs and the measurements. For many models used in practice, the outputs vary primarily on a low-dimensional manifold of the nominally high-dimensional input space. Recently developed active subspace methods discover this low-dimensional manifold and exploit it to enable analyses typically suited for low-dimensional settings (e.g., statistical inversion of complex systems). We hypothesize that active subspace methods can provide the necessary dimension reduction, enabling existing Bayesian methods to find solutions for a large class of otherwise intractable statistical inverse problems. We will test the hypothesis with a series of mathematical analyses and computational experiments that will proceed along three research veins: (i) integrating active subspace methods into methods for statistical inversion, (ii) addressing current challenges in active subspace methods, and (iii) incorporating methods for nonlinear dimensionality reduction into active subspace methods. We will apply the methods we develop to real inverse problems in chemical kinetics and turbulent flame modeling. Our team combines expertise in methods for inverse problems, dimension reduction, and reduced order modeling. With a history of active collaboration and thoughtful engagement, we bring enthusiasm and fresh perspectives to a well studied subject. We are excited by the potential opportunity to work together to extend and apply active subspace methods to data-intensive inverse problems.

Molecular phylogeny of Pompilinae (Hymenoptera: Pompilidae): Evidence for rapid diversification and host shifts in spider wasps

Statistical mediation analysis is used to investigate intermediate variables in the relation between independent and dependent variables. Causal interpretation of mediation analyses is challenging because randomization of subjects to levels of the independent variable does not rule out the possibility of unmeasured confounders of the mediator to outcome relation. Furthermore, commonly used frequentist methods for mediation analysis compute the probability of the data given the null hypothesis, which is not the probability of a hypothesis given the data as in Bayesian analysis. Under certain assumptions, applying the potential outcomes framework to mediation analysis allows for the computation of causal effects, and statistical mediation in the Bayesian framework gives indirect effects probabilistic interpretations. This tutorial combines causal inference and Bayesian methods for mediation analysis so the indirect and direct effects have both causal and probabilistic interpretations. Steps in Bayesian causal mediation analysis are shown in the application to an empirical example.

The molecular phylogeny of the Cimicoidea was reconstructed from an expanded sampling based on mitochondrial (16S, COI) and nuclear (18S, 28SD3) genes. The data were analyzed using maximum likelihood (ML), maximum parsimony (MP), and Bayesian inference (BI) phylogenetic frameworks. The phylogenetic relationships inferred by the model-based analyses (ML and BI) were largely congruent with those inferred by the MP analysis in terms of the monophyly of most of the higher taxonomic groups and the species-level relationships. The following clades were recovered in all analyses: Cimiciformes; Nabidae: Prostemmatinae; Nabidae: Nabinae; Plokiophilidae; Microphysidae; Lasiochilidae; Cimicidae: Cacodminae; Cimicidae; Lyctocoridae; Anthocoridae s. str.; Cardiastethini excluding Amphiareus; Almeidini; Scolopini; Anthocorini; Oriini; Curaliidae + Lasiochilidae; Almeidini + Xylocorini; Oriini + Cardiastethini; and Anthocorini + Amphiareus. Reconstructions of ancestral copulation states based on Bayesian and parsimony inference indicated that at least one shift from standard insemination (SI) to traumatic insemination (TI) occurred within Cimicoidea, and an investigation of the evolutionary correlation between TI and paragenitalia (PG) revealed that the acquisition of PG in cimicoid females was correlated with the TI habit. Additionally, our morphological examination of various types of PG suggested that even the same PG type may not constitute a homologous feature at various taxonomic levels, indicating the convergent evolution of female morphology to adapt to TI.

Estimation of the multivariate logistic risk function: A comparison of the discriminant function and maximum likelihood approaches

This paper deals with the use of Bayesian inference to regularize an inverse problem with a non linear transfer function. Bayesian inference is an useful tool to combine a statistical approach coming from a set of sensors and prior physical models of the system. Bayesian inference is applied to the estimation of combustion indicators. This is an inverse problem where only an indirect measurement from engine block vibrations is available and combustion models are necessary to extract relevant combustion parameters. Moreover the engine block is not a linear system and Bayesian inference can take into account this non linearity.

Coordinate transformation is the process of converting spatial data from a source coordinate to a target coordinate system. A set of control points, measured in the two coordinate systems, is used to estimate the transformation parameters. In general, more control points are measured, and the over-determined system is adjusted using the least squares method. However, the standard least squares method assumes that errors exist only in the measurements made at one coordinate system, or at the observation vector (y). This is not the case in many physical systems where errors exist in all the measurements made in both the source coordinate and the target coordinate systems. The Structured Total Least Norm (STLN) method is a relatively new mathematical concept developed to solve estimation problems of so-called Error-In-Variables (EIV) models. The method is specifically suitable for dealing with transformation problems, since it can handle the special structure of the data matrix (A). The STLN method is uniquely used to compute the parameters of common linear coordinate transformations (affine and similarity). A numerical example is presented to demonstrate the superiority of this technique in terms of accuracy and to compare the standard LS method, the generalized LS algorithm, and the STLN approach.

This paper deals with the estimation of reliability R = P[Y < X] when X and Y are two independent random variables with atwo-parameter bathtub shaped failure rate distribution with the samesecond shape parameter. Likelihood and Bayesian methods are proposedto make inferences about R. We obtain the likelihood interval andasymptotic confidence interval for R, and we consider Bayesianpoint estimates of R under both absolute and squared error loss,using either gamma or uniform priors for the three unknown modelparameters. An equal tail Bayesian credible interval for R isinvestigated. Analysis of a real data set is presented forillustrative purposes, and Monte Carlo simulations are performed tocompare: (1) the performance of Bayes estimates under two differentloss functions; and (2) the maximum likelihood and Bayesian methods.

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of dierent observation or measurement operators on the update.

The purpose of this study was to determine the level of success of the political marketing of the victory of the national party with its affiliated local party in Aceh. The political marketing concept of product, promotion, price and place is used in this study. Primary data was obtained by using a questionnaire. The research locations are two districts in the province of Aceh, namely Bireuen Regency and Pidie Jaya Regency, while sampling using Maximum Likelihood (ML) in the estimation model requires the use of a minimum sample size of 100. By setting the number of samples for each voter national parties and local party voters. The sample will be taken at convenience from each research location. The result of the study of the coefficient of determination (R2) is 0.269. This means that 26.9% changes in the dependent variable, namely the Decision to Choose a National Party, can be explained in the independent variables, namely Product, Promotion, Price, and Place. While the remaining 73.1% is explained by other variables outside the variables studied. This shows that the contribution of the independent variable has little impact on the dependent variable. While the results of the coefficient of determination of the Decision to Choose a Local Party is 0.672. This means that 67.2% changes in the dependent variable, namely the Decision to Choose a Local Party, can be explained in the independent variables, namely Product, Promotion, Price, and Place. While the remaining 32.8% is explained by other variables outside the variables studied. This shows that the contribution of the independent variable has an impact on the dependent variable.

The phylogeny of Orthoptera inferred from mtDNA and description of Elimaea cheni (Tettigoniidae: Phaneropterinae) mitogenome