Linear Gaussian Model Research Articles

Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

In this paper, we deal with some anisotropic extensions of the multifractional Brownian fields that account for spatial phenomena whose properties of regularity and directionality may both vary in space. Our aim is to set statistical tests to decide whether an observed field of this kind is heterogeneous or not. The statistical methodology relies upon a field analysis by quadratic variations, which are averages of square field increments. Specific to our approach, these variations are computed locally in several directions. We establish an asymptotic result showing a linear Gaussian relationship between these variations and parameters related to regularity and directional properties of the model. Using this result, we then design a test procedure based on Fisher statistics of linear Gaussian models. Eventually we evaluate this procedure on simulated data.

Semi-supervised optimal recursive filtering and smoothing in non-Gaussian Markov switching models

Filtering and smoothing in switching state-space models are important in numerous applications. The classic family of conditionally Gaussian linear state space models (CGLSSMs) is a natural extension of the Gaussian linear system by introducing its dependence on switches. In spite of their simplicity, recursive filtering and smoothing are no longer feasible in CGLSSMs and approximate methods must be used. Conditionally Markov switching hidden linear models (CMSHLMs) are alternative models which allow recursive optimal exact filtering and smoothing. We introduce an original family of CMSHLMs defined with copulas and we address the problem of their identification. The proposed identification method chooses a model in a family of admissible parametric models and estimates the parameters. It is applied to a learning sample containing observations and states, while the switches are unknown. The interest of the proposed ”semi-unsupervised” filtering and smoothing is validated via experiments on simulated data.

Trend of Commodity Prices and Exchange Rate in Australian Economy: Time Varying Parameter Model Approach

Here we investigate the relationship between export commodity prices and AUD/USD exchange rate fluctuation using time varying parameter model. Using monthly data for over 30 years we found that exchange rate is determined by commodity prices and Australian base metal indices is highly correlated with country’s exchange rate. We have considered linear Gaussian state space model where common variance is treated as a stochastic time varying variable which gets considered for modeling economic time series.

Identifiability of Gaussian linear structural equation models with homogeneous and heterogeneous error variances

In this work, we consider the identifiability assumption of Gaussian linear structural equation models (SEMs) in which each variable is determined by a linear function of its parents plus normally distributed error. It has been shown that linear Gaussian structural equation models are fully identifiable if all error variances are the same or known. Hence, this work proves the identifiability of Gaussian SEMs with both homogeneous and heterogeneous unknown error variances. Our new identifiability assumption exploits not only error variances, but edge weights; hence, it is strictly milder than prior work on the identifiability result. We further provide a structure learning algorithm that is statistically consistent and computationally feasible, based on our new assumption. The proposed algorithm assumes that all relevant variables are observed, while it does not assume causal minimality and faithfulness. We verify our theoretical findings through simulations and real multivariate data, and compare our algorithm to state-of-the-art PC, GES and GDS algorithms.

Bayesian State Space Models in Macroeconometrics

State space models play an important role in macroeconometric analysis and the Bayesian approach has been shown to have many advantages. This paper outlines recent developments in state space modelling applied to macroeconomics using Bayesian methods. We outline the directions of recent research, specifically the problems being addressed and the solutions proposed. After presenting a general form for the linear Gaussian model, we discuss the interpretations and virtues of alternative estimation routines and their outputs. This discussion includes the Kalman filter and smoother, and precision based algorithms. As the advantages of using large models have become better understood, a focus has developed on dimension reduction and computational advances to cope with high-dimensional parameter spaces. We give an overview of a number of recent advances in these directions. Many models suggested by economic theory are either non-linear or non-Gaussian, or both. We discuss work on the particle filtering approach to such models as well as other techniques that use various approximations - to either the time t state and measurement equations or to the full posterior for the states - to obtain draws.

Bellman Filtering for State-Space Models

This article presents a filter for state-space models based on Bellman's dynamic programming principle applied to the mode estimator. The proposed Bellman filter generalises the Kalman filter including its extended and iterated versions, while remaining equally inexpensive computationally. The Bellman filter is also (unlike the Kalman filter) robust under heavy-tailed observation noise and applicable to a wider range of (nonlinear and non-Gaussian) models, involving e.g. count, intensity, duration, volatility and dependence. The Bellman-filtered states are shown to be convergent, in quadratic mean, towards a small region around the true state. (Hyper)parameters are estimated by numerically maximising a filter-implied log-likelihood decomposition, which is an alternative to the classic prediction-error decomposition for linear Gaussian models. Simulation studies reveal that the Bellman filter performs on par with (or even outperforms) state-of-the-art simulation-based techniques, e.g. particle filters and importance samplers, while requiring a fraction (e.g. 1%) of the computational cost, being straightforward to implement and offering full scalability to higher dimensional state spaces.

Safety and efficacy of topically administered netarsudil (Rhopressa™) in normal and glaucomatous dogs with ADAMTS10-open-angle glaucoma (ADAMTS10-OAG).

To evaluate safety and efficacy of topically administered 0.02% netarsudil ophthalmic solution (Rhopressa™; Aerie Pharmaceutical) in normal and glaucomatous dogs with ADAMTS10-open-angle glaucoma (ADAMTS10-OAG). Five normal and 5 glaucomatous Beagle dogs with ADAMTS10-OAG. In each dog, left or right eye was randomly selected for netarsudil treatment. Contralateral eyes were sham-treated with balanced salt solution (BSS). Following a 1-week baseline period, dogs were treated once daily (q24h) during week 2, and twice daily (q12h) during week 3; week 4 served as washout period. Efficacy was measured by diurnal intraocular pressure (IOP) and pupil diameter. Safety was assessed by routine ophthalmic examination, gonioscopy, and pachymetry. Differences in least square means of quantitative outcome measures were compared between netarsudil and BSS sham-treated eyes by linear Gaussian model. Baseline IOPs were 18.5±0.5mmHg (mean±SEM) in normal and 27.8±1.0mmHg in OAG dogs. Even though mean IOPs were lower in netarsudil- vs sham-treated eyes, the overall differences were neither significant nor clinically relevant, regardless of treatment frequency (q24h-normal: sham 16.4±1.1mmHg vs treatment 15.6±1.0mmHg; q24hr-OAG: sham 25.8±2.3mmHg vs. treatment 25.7±2.4mmHg; q12hr-normal: sham 15.4±0.8mmHg vs. treatment 14.4±0.8mmHg; q12hr-OAG: sham 26.3±1.7mmHg vs. treatment 25.4±1.8mmHg). Netarsudil administration was well tolerated but resulted in significant, moderate-to-severe conjunctival hyperemia (P<.001). Once or twice daily administration of netarsudil resulted in marginal and clinically irrelevant IOP decreases in normal and OAG-affected dogs. Except for conjunctival hyperemia, the drug was well tolerated.

Mobile Device Applications for Electronic Patient Portals in Oncology.

Mobile devices provide individuals with rapid and frequent access to electronic patient portals. We investigated how oncology patients use this technology to review test results and communicate with providers. We performed a retrospective study of patients enrolled in the MyChart electronic health portal associated with the Epic electronic medical record at the Harold C. Simmons Comprehensive Cancer Center from 2012 to 2017. We recorded type of portal access according to year and patient characteristics. Associations among patient characteristics and types of portal access were tested using Mann-Whitney U test, χ2 test, and linear Gaussian regression models. Since the availability of a mobile device application in 2012, 2,524 patients with cancer accessed MyChart from a mobile device at least once, which accounted for 291,526 mobile log-ins. The number of patients with MyChart mobile application log-ins increased from 4% in 2012 to 13% in 2017 ( P = .004). Among these patients, the median proportion of log-ins that occurred through mobile device use increased from 22% to 72% during this time period ( P < .001). Mobile access occurred more frequently among younger ( P < .001), black ( P = .002), and Hispanic ( P = .004) patients. Since 2012, total portal log-in frequency increased approximately 110% among patients who used the mobile application compared with 25% among those who did not use the mobile application ( P < .001). Mobile access to electronic health portals has increased patient portal use, particularly among traditionally underserved populations. How this widely and immediately available technology affects patient expectations and experiences warrants additional study.

Variance Prior Forms for High-Dimensional Bayesian Variable Selection

Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori. In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori. Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Rockova and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation. The unknown variance implementation of the Spike-and-Slab Lasso is provided in the publicly available R package SSLASSO (Rockova and Moran, 2017).

Bayesian Inference in Spatial Stochastic Volatility Models with an Application to House Price Returns in Chicago

In this study, we propose a spatial stochastic volatility model in which the latent log-volatility terms follow a spatial autoregressive process. Though there is no spatial correlation in the outcome equation (the mean equation), the spatial autoregressive process defined for the log-volatility terms introduces spatial dependence in the outcome equation. To introduce a Bayesian Markov chain Monte Carlo (MCMC) estimation algorithm, we transform the model so that the outcome equation takes the form of log-squared terms. We approximate the distribution of the log-squared error terms in the outcome equation with a finite mixture of normal distributions so that the transformed model turns into a linear Gaussian state-space model. Our simulation results indicate that the Bayesian estimator has satisfactory finite sample properties. We investigate the practical usefulness of our proposed model and estimation method by using the price returns of residential properties in the broader Chicago Metropolitan area.

Nonequilibrium Oscillations, Probability Angular Momentum, and the Climate System

Though the Boltzmann-Gibbs framework of equilibrium statistical mechanics has been successful in many arenas, it is clearly inadequate for describing many interesting natural phenomena driven far from equilibrium. The simplest step towards that goal is a better understanding of nonequilibrium steady-states (NESS). Here we focus on one of the distinctive features of NESS, persistent probability currents, and their manifestations in our climate system. We consider the natural variability of the steady-state climate system, which can be approximated as a NESS. These currents must form closed loops, which are odd under time reversal, providing the crucial difference between systems in thermal equilibrium and NESS. Seeking manifestations of such current loops leads us naturally to the notion of probability angular momentum and oscillations in the space of observables. Specifically, we will relate this concept to the asymmetric part of certain time-dependent correlation functions. Applying this approach, we propose that these current loops give rise to preferred spatio-temporal patterns of natural climate variability that take the form of climate oscillations such as the El-Ni\~{n}o Southern Oscillation (ENSO) and the Madden-Julien Oscillation (MJO). In the space of climate indices, we observe persistent currents and define a new diagnostic for these currents: the probability angular momentum. Using the observed climatic time series of ENSO and MJO, we compute both the averages and the distributions of the probability angular momentum. These results are in good agreement with the analysis from a linear Gaussian model. We propose that, in addition to being a new quantification of climate oscillations across models and observations, the probability angular momentum provides a meaningful characterization for all statistical systems in NESS.

Multirate Dynamic Process Monitoring Based on Multirate Linear Gaussian State-Space Model

Multivariate statistical process monitoring (MSPM) has been widely used in modern industries and most of traditional MSPM methods are developed using uniformly sampled measurements. However, process variables are often sampled with different rates in practical industries. On the other hand, most of the industries are dynamic processes in which the measurements are highly autocorrelated. Thus, it is difficult to build a dynamic process model with incomplete data sets in multirate processes. In this paper, a multirate linear Gaussian state-space model is exploited to deal with the above issues. Both the offline model training and online process monitoring schemes are developed in the present of incomplete multirate process data sets. The proposed method is validated through a numerical example and the Tennessee Eastman benchmark process. Note to Practitioners —Process monitoring is essential to industrial process maintenance and safety. Typical monitoring methods focus on uniformly sampled variables which are rarely satisfied in practical industries. On the other hand, most industries are dynamic processes which bring more difficulties to process monitoring with multirate variables. This paper proposed a new scheme for multirate dynamic process monitoring based on the multirate linear Gaussian state-space model. To fully implement this approach: 1) the multirate process data need to be well divided into multiblocks. Each of the block should contain at least one variable; 2) the amount of latent variable’s dimension needs to be properly determined; and 3) the sampling rates of the process cannot be extremely distinguished otherwise will lead to a poor model performance. Two cases studied on both numerical example and Tennessee Eastman benchmark are given to validate the efficiency of the proposed algorithm on multirate dynamic process monitoring.

Minimax Rate of Testing in Sparse Linear Regression

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the l2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form $$\sqrt {\left( {s/n} \right)\log \left( {\sqrt p /s} \right)}$$ . We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.

Variations of power-expected-posterior priors in normal regression models

The power-expected-posterior (PEP) prior is an objective prior for Gaussian linear models, which leads to consistent model selection inference, under the M-closed scenario, and tends to favour parsimonious models. Recently, two new forms of the PEP prior were proposed which generalize its applicability to a wider range of models. The properties of these two PEP variants within the context of the normal linear model are examined thoroughly, focusing on the prior dispersion and on the consistency of the induced model selection procedure. Results show that both PEP variants have larger variances than the unit-information g-prior and that they are M-closed consistent as the limiting behaviour of the corresponding marginal likelihoods matches that of the BIC. The consistency under the M-open case, using three different model misspecification scenarios is further investigated.

Selective inference via marginal screening for high dimensional classification

Post-selection inference is a statistical technique for determining salient variables after model or variable selection. Recently, selective inference, a kind of post-selection inference framework, has garnered the attention in the statistics and machine learning communities. By conditioning on a specific variable selection procedure, selective inference can properly control for so-called selective type I error, which is a type I error conditional on a variable selection procedure, without imposing excessive additional computational costs. While selective inference can provide a valid hypothesis testing procedure, the main focus has hitherto been on Gaussian linear regression models. In this paper, we develop a selective inference framework for binary classification problem. We consider a logistic regression model after variable selection based on marginal screening, and derive the high dimensional statistical behavior of the post-selection estimator. This enables us to asymptotically control for selective type I error for the purposes of hypothesis testing after variable selection. We conduct several simulation studies to confirm the statistical power of the test, and compare our proposed method with data splitting and other methods.

Simulation of Ultrasonic Beam Propagation From Phased Arrays in Anisotropic Media Using Linearly Phased Multi-Gaussian Beams.

Phased array ultrasonic testing is widely used to test structures for flaws due to its ability to produce steered and focused beams. The inherent anisotropic nature of some materials, however, leads to skewing and distortion of the phased array beam and consequently measurement errors. To overcome this, a quantitative model of phased array beam propagation in such materials is required, so as to accurately model the skew and the distortion. The existing phased array beam models which are based on exact methods or numerical methods are computationally expensive or time consuming. This article proposes a modeling approach based on developing the linear phased multi-Gaussian beam (MGB) approach to model beam steering in anisotropic media. MGBs have the advantages of being computationally inexpensive and remaining non-singular. This article provides a comparison of the beam propagation modeled by the developed ordinary Gaussian beam and linear phased Gaussian beam models through transversely isotropic austenitic steel for different steering angles. It is shown that the linear phased Gaussian beam model outperforms the ordinary one, especially at steering angles higher than 20° in anisotropic solids. The proposed model allows us to model the beam propagation from phased arrays in both isotropic and anisotropic media in a way that is computationally inexpensive. As a further step, the developed model has been validated against a finite element model (FEM) computed using COMSOL Multiphysics.

Measuring Parkinson's disease over time: The real-world within-subject reliability of the MDS-UPDRS.

BackgroundAn important challenge in Parkinson's disease research is how to measure disease progression, ideally at the individual patient level. The MDS‐UPDRS, a clinical assessment of motor and nonmotor impairments, is widely used in longitudinal studies. However, its ability to assess within‐subject changes is not well known. The objective of this study was to estimate the reliability of the MDS‐UPDRS when used to measure within‐subject changes in disease progression under real‐world conditions.MethodsData were obtained from the Parkinson's Progression Markers Initiative cohort and included repeated MDS‐UPDRS measurements from 423 de novo Parkinson's disease patients (median follow‐up: 54 months). Subtotals were calculated for parts I, II, and III (in on and off states). In addition, factor scores were extracted from each part. A linear Gaussian state space model was used to differentiate variance introduced by long‐lasting changes from variance introduced by measurement error and short‐term fluctuations. Based on this, we determined the within‐subject reliability of 1‐year change scores.ResultsOverall, the within‐subject reliability ranged from 0.13 to 0.62. Of the subscales, parts II and III (OFF) demonstrated the highest within‐subject reliability (both 0.50). Of the factor scores, the scores related to gait/posture (0.62), mobility (0.45), and rest tremor (0.43) showed the most consistent behavior.ConclusionsOur results highlight that MDS‐UPDRS change scores contain a substantial amount of error variance, underscoring the need for more reliable instruments to forward our understanding of the heterogeneity in PD progression. Focusing on gait and rest tremor may be a promising approach for an early Parkinson's disease population. © 2019 The Authors. Movement Disorders published by Wiley Periodicals, Inc. on behalf of International Parkinson and Movement Disorder Society.

A Novel Robust Gaussian–Student's t Mixture Distribution Based Kalman Filter

In this paper, a novel Gaussian-Student's t mixture (GSTM) distribution is proposed to model non-stationary heavy-tailed noises. The proposed GSTM distribution can be formulated as a hierarchical Gaussian form by introducing a Bernoulli random variable, based on which a new hierarchical linear Gaussian state-space model is constructed. A novel robust GSTM distribution based Kalman filter is proposed based on the constructed hierarchical linear Gaussian state-space model using the variational Bayesian approach. The Kalman filter and robust Student's t based Kalman filter (RSTKF) with fixed distribution parameters are two existing special cases of the proposed filter. The novel GSTM distributed Kalman filter has the important advantage over the RSTKF that the adaptation of the mixing parameter is much more straightforward than learning the degrees of freedom parameter. Simulation results illustrate that the proposed filter has better estimation accuracy than those of the Kalman filter and RSTKF for a linear state-space model with non-stationary heavy-tailed noises.

Dynamic Monitoring and Control of Irreversible Chronic Diseases with Application to Glaucoma.

To manage chronic disease patients effectively, clinicians must know (1) how to monitor each patient (i.e., when to schedule the next visit and which tests to take), and (2) how to control the disease (i.e., what levels of controllable risk factors will sufficiently slow progression). Our research addresses these questions simultaneously and provides the optimal solution to a novel linear quadratic Gaussian state space model. For the objective of minimizing the relative change in state over time (i.e., disease progression), which is necessary for managing irreversible chronic diseases while also considering the cost of tests and treatment, we show that the classical two-way separation of estimation and control holds. This makes a previously intractable problem solvable by decomposition into two separate, tractable problems while maintaining optimality. The resulting optimization is applied to the management of glaucoma. Based on data from two large randomized clinical trials, we validate our model and demonstrate how our decision support tool can provide actionable insights to the clinician caring for a patient with glaucoma. This methodology can be applied to a broad range of irreversible chronic diseases to devise patient-specific monitoring and treatment plans optimally.

Computing exact score vectors for linear Gaussian state space models

A recursive formula for computing the exact value of score vectors is proposed for a general form of the linear Gaussian state space model, which is more desirable than approximate values in some statistical analyses. Unlike most extant methods, our formula calculates all components of the score vector simultaneously. This approach significantly simplifies its programing, in particular, with some matrix-oriented programing languages, such as MATLAB. We also consider a way of handling initial conditions that depend on unknown parameters. This issue has not yet been explicitly addressed in the existing literature in the context of exact score computing for a general case, such as the one that we consider in this paper. It is also shown that our formula is especially useful for calculating score tests with an outer product of gradient asymptotic covariance matrix estimator.