Estimation Of Linear Models Research Articles

Data-Driven Identification of Dissipative Linear Models for Nonlinear Systems

We consider the problem of identifying a dissipative linear model of an unknown nonlinear system that is known to be dissipative, from time-domain input–output data. We first learn an approximate linear model of the nonlinear system using standard system identification techniques and then perturb the system matrices of the linear model to enforce dissipativity, while closely approximating the dynamical behavior of the nonlinear system. Further, we provide an analytical relationship between the size of the perturbation and the radius in which the dissipativity of the linear model guarantees local dissipativity of the unknown nonlinear system. We demonstrate the application of this identification technique through two examples.

Assessment of approaches to estimate scapular orientation in children with brachial plexus birth injury

BackgroundChallenges in measuring dynamic scapular orientation limit assessment of scapulothoracic and glenohumeral contributions to shoulder function in children with brachial plexus birth injury (BPBI). Double calibration acromion marker cluster (D-AMC) and linear model approaches have been validated to estimate scapular motion in healthy adults, but neither has been evaluated in BPBI. Research questionAre the linear model and D-AMC approaches able to accurately estimate scapular orientation in children with BPBI at functional arm postures? MethodsSeventeen children with BPBI positioned their affected limbs in 11 static positions while their segment orientations were measured with motion capture. Linear model and D-AMC estimates of scapular orientation were compared against palpation at six of the static positions with functional relevance to BPBI using a three-way repeat measures ANOVA and a comparison of root mean square errors (RMSE) against literature AMC values for healthy adults. ResultsThe D-AMC was similar to palpation across all positions and scapular axes while the linear model differed from palpation in a few instances. RMSEs of the D-AMC (3.7–14.8°) and particularly the linear model (4.6–24.8°) were generally at or beyond the upper range of past AMC analyses on healthy adults (1.6–14.2°), especially for more complex, multiplanar arm postures. Despite the D-AMC outperforming the linear model, this approach still produced clinically meaningful (>10°) errors for roughly (12.7–22.5%) of subjects. SignificanceCurrent methods for estimating dynamic scapular orientation remain less than ideal for BPBI. Use of the D-AMC may be appropriate to gain broad insights into general dynamic scapulothoracic and glenohumeral function; however, given their potential for producing clinically meaningful errors, the D-AMC and linear model are not recommended for diagnostic purposes or outcomes assessment on an individual patient basis unless their patient-specific accuracy has been evaluated and confirmed prior to use.

A Generalized Linear Joint Trained Framework for Semi-Supervised Learning of Sparse Features

The elastic net is among the most widely used types of regularization algorithms, commonly associated with the problem of supervised generalized linear model estimation via penalized maximum likelihood. Its attractive properties, originated from a combination of ℓ1 and ℓ2 norms, endow this method with the ability to select variables, taking into account the correlations between them. In the last few years, semi-supervised approaches that use both labeled and unlabeled data have become an important component in statistical research. Despite this interest, few researchers have investigated semi-supervised elastic net extensions. This paper introduces a novel solution for semi-supervised learning of sparse features in the context of generalized linear model estimation: the generalized semi-supervised elastic net (s2net), which extends the supervised elastic net method, with a general mathematical formulation that covers, but is not limited to, both regression and classification problems. In addition, a flexible and fast implementation for s2net is provided. Its advantages are illustrated in different experiments using real and synthetic data sets. They show how s2net improves the performance of other techniques that have been proposed for both supervised and semi-supervised learning.

Hyper-differential sensitivity analysis for inverse problems governed by ODEs with application to COVID-19 modeling

We consider inverse problems governed by systems of ordinary differential equations (ODEs) that contain uncertain parameters in addition to the parameters being estimated. In such problems, which are common in applications, it is important to understand the sensitivity of the solution of the inverse problem to the uncertain model parameters. It is also of interest to understand the sensitivity of the inverse problem solution to different types of measurements or parameters describing the experimental setup. Hyper-differential sensitivity analysis (HDSA) is a sensitivity analysis approach that provides tools for such tasks. We extend existing HDSA methods by developing methods for quantifying the uncertainty in the estimated parameters. Specifically, we propose a linear approximation to the solution of the inverse problem that allows efficiently approximating the statistical properties of the estimated parameters. We also explore the use of this linear model for approximate global sensitivity analysis. As a driving application, we consider an inverse problem governed by a COVID–19 model. We present comprehensive computational studies that examine the sensitivity of this inverse problem to several uncertain model parameters and different types of measurement data. Our results also demonstrate the effectiveness of the linear approximation model for uncertainty quantification in inverse problems and for parameter screening.

Using ground-based thermal imagery to estimate debris thickness over glacial ice: fieldwork considerations to improve the effectiveness

AbstractDebris-covered glaciers are an important component of the mountain cryosphere and influence the hydrological contribution of glacierized basins to downstream rivers. This study examines the potential to make estimates of debris thickness, a critical variable to calculate the sub-debris melt, using ground-based thermal infrared radiometry (TIR) images. Over four days in August 2019, a ground-based, time-lapse TIR digital imaging radiometer recorded sequential thermal imagery of a debris-covered region of Peyto Glacier, Canadian Rockies, in conjunction with 44 manual excavations of debris thickness ranging from 10 to 110 cm, and concurrent meteorological observations. Inferring the correlation between measured debris thickness and TIR surface temperature as a base, the effectiveness of linear and exponential regression models for debris thickness estimation from surface temperature was explored. Optimal model performance (R2 of 0.7, RMSE of 10.3 cm) was obtained with a linear model applied to measurements taken on clear nights just before sunrise, but strong model performances were also obtained under complete cloud cover during daytime or nighttime with an exponential model. This work presents insights into the use of surface temperature and TIR observations to estimate debris thickness and gain knowledge of the state of debris-covered glacial ice and its potential hydrological contribution.

Incremental adaptive optimal control for nonlinear systems with disturbance and input time-delay

In this paper, a model-free incremental adaptive optimal control scheme is proposed for nonlinear systems in presence of disturbance and input time-delay. The linear time-varying approximate model of the nonlinear system is obtained by incremental method, in which the relevant matrix parameters are identified by recursive least squares (RLS) estimation. A time-delay matrix function is constructed by neural network (NN) to eliminate the input time-delay, and the external disturbance of nonlinear system is dealt with by [Formula: see text] optimal control; incremental adaptive dynamic programming (IADP) algorithm is used to get the control law by solving Hamilton–Jacobi–Isaacs (HJI) equation. Convergence analysis of the proposed control scheme is provided. Simulations are given to verify the effectiveness of the proposed control scheme.

Direct constraint control for EM-based miniaturization of microwave passives

Handling constraints imposed on physical dimensions of microwave circuits has become an important design consideration over the recent years. It is primarily fostered by the needs of emerging application areas such as 5G mobile communications, internet of things, or wearable/implantable devices. The size of conventional passive components is determined by the guided wavelength, and its reduction requires topological modifications, e.g., transmission line folding, or utilization of compact cells capitalizing on the slow-wave phenomenon. The resulting miniaturized structures are geometrically complex and typically exhibit strong cross coupling effects, which cannot be adequately accounted for by analytical or equivalent network models. Consequently, electromagnetic (EM)-driven parameter tuning is necessary, which is computationally expensive. When the primary objective is size reduction, the optimization task becomes far more challenging due to the presence of constraints related to electrical performance figures (bandwidth, power split ratio, etc.), which are all costly to evaluate. A popular solution approach is to utilize penalty functions. Therein, possible violations of constraints degrade the primary objective, thereby enforcing their satisfaction. Yet, the appropriate setup of penalty coefficients is a non-trivial problem by itself, and is often associated to extra computational expenses. In this work, we propose an explicit approach to constraint handling, which is combined with the trust-region gradient-search procedure. In our technique, the decision about the adjustment of the search radius is determined based on the reliability of rendering the feasible region boundary by linear approximation models of the constraints. Comprehensive numerical experiments conducted using three miniaturized coupler structures demonstrate superiority of the presented method over the penalty function paradigm. Apart from the efficacy, its appealing features include algorithmic simplicity, and no need for tailoring the procedure for a particular circuit to be optimized.

Two-way ANOVA by using Cholesky decomposition and graphical representation

In general, the coefficient estimates of linear models are carried out using the ordinary least squares (OLS) method. Since the analysis of variance is also a linear model, the coefficients can be estimated using the least-squares method. In this study, the coefficient estimates in the two-way analysis of variance were performed by using the Cholesky decomposition. The purpose of using the Cholesky decomposition in finding coefficient estimates make variables used in model being orthogonal such that important variables can be easily identified. The sum of squares in two-way analysis of variance (row, column, interaction) were also found by using the coefficient estimates obtained as a result of the Cholesky decomposition. Thus, important variables that affect the sum of squares can be determined more easily because the Cholesky decomposition makes the variables in the model orthogonal. By representing the sum of squares with vectors, how the prediction vector in two-way ANOVA model was created was shown. It was mentioned how the Cholesky decomposition affected the sum of squares. This method was explained in detail on a sample data and shown geometrically.

Self-Coupling and Mutual Pulling in Phase-Locked Loops

The influence of coupling of oscillators in phase-locked loops (PLLs) was studied here. Particularly, numerical analyses based on nonlinear differential equations of the self- and mutual coupling systems were performed and compared with the results of the linear approximation model under synchronous conditions. The analysis revealed that coupling using the reference clock caused greater PLL instability than that caused by self-coupling for a given coupling strength. This shows that the phase error of mutual coupling exhibits the same behavior as that of reference-clock coupling in the single PLL under synchronous conditions, and the noise sensitivity of mutual coupling is twice that of the reference-clock coupling. In addition, the phase error of mutual coupling in the quasi-synchronous condition strongly depends on the relationship between the frequency difference of the two PLLs and the PLL bandwidth. We designed and fabricated the test circuits for theoretical verification using the 0.18- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> standard CMOS process. From the measurements, adjustment of the timing difference between the two PLLs is effective in mitigating mutual coupling under synchronous conditions. Moreover, we implemented mutual injections between two oscillators in the test circuit and confirmed their effectiveness. The proposed methods will be useful for achieving the compact and effective cancellation of the mutual coupling between PLLs.

A robust omnichannel pricing and ordering optimization approach with return policies based on data-driven support vector clustering

This work considers an omnichannel retailer selling a product with market demand uncertainties to customers in different regions through an e-store and multiple brick-and-mortar stores. The retailer also manages online product returns by selecting appropriate return policies. A robust omnichannel pricing and ordering optimization model is proposed with two, i.e., a full-refund and a no-refund, return policies. The online demand and the offline demand depend on the prices of the e-store and the brick-and-mortar stores, where the online demand also depends on the refund of the e-store. A linearization technique is adopted to deal with nonlinearity of the model. A data-driven robust optimization approach is used to construct uncertainty sets based on available historical data using support vector clustering to handle demand uncertainties. Furthermore, the proposed model is transformed into an approximate mixed integer linear programming model which can be solved by using commercial software. An electronics retailer in China is used as a case study to illustrate the effectiveness and practicality of the proposed model and the solution method. A comparison with the box uncertainty set reveals that the data-driven uncertainty set is less conservative and performs better by obtaining higher profit for the retailer. Furthermore, sensitive analysis results indicate that return policies and the return rate of the product affect the optimal pricing and ordering decisions and the total profit.

A data-driven linearization approach to analyze the three-phase unbalance in active distribution systems,

Due to unevenly distributed energy resources between phases, it is expected that active distribution networks (ADNs) may experience a high degree of three-phase unbalance. Accurate analysis tools play an essential role in addressing the issues caused by the three-phase unbalance. To address existing computational challenges in the assessment and mitigation of three-phase unbalance in ADNs, we propose a direct and an indirect linear approximation models of three-phase voltage unbalance metrics (VUF, LVUR and PVUR), which are obtained using a support vector regression (SVR)-based three-phase voltage unbalance metric linearization approach. The linearized three-phase voltage unbalance metrics are incorporated in an improved optimal dispatch problem with voltage unbalance constraints. The resulting optimization problem is a convex quadratic program with low computational complexity and high accuracy. Numerical analyses demonstrate that the proposed approach achieves satisfactory accuracy, which indicates its promise for inclusion in three-phase unbalance analysis in ADNs.

Genetic parameters for dairy calf and replacement heifer wellness traits and their association with cow longevity and health indicators in Holstein cattle

High mortality and involuntary culling rates cause great economic losses to the worldwide dairy cattle industry. However, there is low emphasis on wellness traits in replacement animals (dairy calves and replacement heifers) during their development stages in modern dairy cattle breeding programs. Therefore, the main objectives of this study were to estimate genetic parameters of wellness traits in replacement cattle (replacement wellness traits) and obtain their genetic correlations with 12 cow health and longevity traits in the Chinese Holstein population. Seven replacement wellness traits were analyzed, including birth weight, survival from 3 to 60 d (Sur1), survival from 61 to 365 d (Sur2), survival from 366 d to the first calving (Sur3), calf diarrhea, calf pneumonia, and calf serum total protein (STP). Single and bivariate animal models were employed to estimate (co)variance components using the data from 189,980 Holstein cattle. The genetic correlations between replacement wellness traits and cow longevity, health traits were calculated by employing bivariate models, including 6 longevity traits and 6 health traits (clinical mastitis, metritis, ketosis, displaced abomasum, milk fever, and hoof health or hoof disease). The estimated heritabilities (± SE) were 0.335 (± 0.008), 0.088 (± 0.005), 0.166 (± 0.006), 0.102 (±0 .006), 0.048 (± 0.003), 0.063 (± 0.004), and 0.170 (± 0.019) for birth weight, Sur1, Sur2, Sur3, pneumonia, diarrhea, and STP, respectively. The majority of the genetic correlations among the 7 replacement wellness traits were negligible. The genetic correlations among Sur1, Sur2, and Sur3 ranged from 0.112 (Sur1 and Sur3) to 0.445 (Sur1 and Sur2) when fitting a linear model (estimates in the observed scale), and from 0.560 (Sur1 and Sur3) to 0.773 (Sur1 and Sur2) when fitting a threshold model (estimates in the liability scale). The genetic correlations between replacement wellness and cow longevity were low (absolute value lower than 0.30), but some of them were significantly different from zero. Compared with other replacement wellness traits, Sur3 and STP had relatively high genetic correlations with cow longevity. Replacement wellness traits are heritable and can be improved through direct genetic and genomic selection. The results from the current study will contribute for better balancing dairy cattle breeding goals to genetically improve dairy cattle wellness in the period from birth to first calving.

Estimation of Hand Grip Strength Using Foot motion Measured by In-shoe Motion Sensor.

There is a strong need to assess frailty in daily living. Hand grip strength (HGS) has been proven to be a very important factor for identifying frailty, however it is always assessed under the guidance of facility clinicians. Our purpose is to demonstrate the possibility of providing HGS estimation by using foot-motion signals measured by an in-shoe motion sensor (IMS) embedded in an insole to achieve high precision HGS assessment in daily living. The foot-motion signals were collected from 62 elder participants (27 men and 35 women). Their HGSs were assessed by a hand dynamometer. Gait parameters, individual properties, and predictors derived from foot-motion signal features in one gait cycle were selected as candidates. Statistical parametric mapping analyses were used to generate predictors from the foot-motion signals. Prior to estimation construction, least absolute shrinkage and selection operator was applied to reduce redundant predictors from candidates. Linear regression models for HGS estimation of men and women were constructed. As the results, we discovered new effective predictors for HGS estimation from foot motions and successfully constructed HGS estimation models that achieved "excellent" agreement with the reference according to intra-class coefficients, and mean absolute errors of 2.96 and 2.57 kg for men and women in leave-one-subject-out cross-validation, respectively. These results suggest that HGS can be estimated with high precision by IMS-measured foot motion and more effective frailty identification in daily living is possible through wearing an IMS.

Disentangling the varying associations between systolic blood pressure and health outcomes in the very old: an individual patient data meta-analysis.

Objectives:While randomized controlled trials have proven the benefits of blood pressure (BP) lowering in participating octogenarians, population-based observational studies suggest an association between low systolic blood pressure (SBP) and faster overall decline. This study investigates the effects of BP-lowering treatment, a history of cardiovascular diseases (CVD), and cognitive and physical fitness on the associations between SBP and health outcomes in the very old.Methods:Five cohorts from the Towards Understanding Longitudinal International older People Studies (TULIPS) consortium were included in a two-step individual participant data meta-analysis (IPDMA). We pooled hazard ratios (HR) from Cox proportional-hazards models for 5-year mortality and estimates of linear mixed models for change in cognitive and functional decline. Models were stratified by BP-lowering treatment, history of CVD, Mini-Mental State Examination scores, grip strength (GS) and body mass index (BMI).Results:Of all 2480 participants (59.9% females, median 85 years), median baseline SBP was 149 mmHg, 64.3% used BP-lowering drugs and 47.3% had a history of CVD. Overall, higher SBP was associated with lower all-cause mortality (pooled HR 0.91 [95% confidence interval 0.88–0.95] per 10 mmHg). Associations remained irrespective of BP-lowering treatment, history of CVD and BMI, but were absent in octogenarians with above-median MMSE and GS. In pooled cohorts, SBP was not associated with cognitive and functional decline.Conclusion:While in the very old with low cognitive or physical fitness a higher SBP was associated with a lower all-cause mortality, this association was not evident in fit octogenarians. SBP was not consistently associated with cognitive and functional decline.

Age group prediction with panoramic radiomorphometric parameters using machine learning algorithms

The aim of this study is to investigate the relationship of 18 radiomorphometric parameters of panoramic radiographs based on age, and to estimate the age group of people with permanent dentition in a non-invasive, comprehensive, and accurate manner using five machine learning algorithms. For the study population (209 men and 262 women; mean age, 32.12 ± 18.71 years), 471 digital panoramic radiographs of Korean individuals were applied. The participants were divided into three groups (with a 20-year age gap) and six groups (with a 10-year age gap), and each age group was estimated using the following five machine learning models: a linear discriminant analysis, logistic regression, kernelized support vector machines, multilayer perceptron, and extreme gradient boosting. Finally, a Fisher discriminant analysis was used to visualize the data configuration. In the prediction of the three age-group classification, the areas under the curve (AUCs) obtained for classifying young ages (10–19 years) ranged from 0.85 to 0.88 for five different machine learning models. The AUC values of the older age group (50–69 years) ranged from 0.82 to 0.88, and those of adults (20–49 years) were approximately 0.73. In the six age-group classification, the best scores were also found in age groups 1 (10–19 years) and 6 (60–69 years), with mean AUCs ranging from 0.85 to 0.87 and 80 to 0.90, respectively. A feature analysis based on LDA weights showed that the L-Pulp Area was important for discriminating young ages (10–49 years), and L-Crown, U-Crown, L-Implant, U-Implant, and Periodontitis were used as predictors for discriminating older ages (50–69 years). We established acceptable linear and nonlinear machine learning models for a dental age group estimation using multiple maxillary and mandibular radiomorphometric parameters. Since certain radiomorphological characteristics of young and the elderly were linearly related to age, young and old groups could be easily distinguished from other age groups with automated machine learning models.

A two-parameter estimator in linear measurement error model

ABSTRACT This article is concerned with the parameter estimation in linear measurement error model when there is ill-conditioned data. To deal with the multicollinearity problem, a new two-parameter estimator is proposed. The asymptotic properties of the new estimator are considered using the mean squared error matrix. Finally, a Monte Carlo simulation is presented to show the performances of the estimators in terms of simulated mean squared error criteria. According to the results, the new estimator can be suggested as an alternative to the other existing estimators in the presence of ill-conditioned data.

Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation à la Fenichel.

The linear noise approximation models the random fluctuations from the mean-field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order [Formula: see text], where [Formula: see text] is the size of the chemical system (usually the volume). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the slow scale linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis -Menten reaction mechanism.

Monitoring Spatiotemporal Distribution of the GDP of Major Cities in China during the COVID-19 Pandemic.

Monitoring the fine spatiotemporal distribution of urban GDP is a critical research topic for assessing the impact of the COVID-19 outbreak on economic and social growth. Based on nighttime light (NTL) images and urban land use data, this study constructs a GDP machine learning and linear estimation model. Based on the linear model with better effect, the monthly GDP of 34 cities in China is estimated and the GDP spatialization is realized, and finally the GDP spatiotemporal correction is processed. This study analyzes the fine spatiotemporal distribution of GDP, reveals the spatiotemporal change trend of GDP in China’s major cities during the current COVID-19 pandemic, and explores the differences in the economic impact of the COVID-19 pandemic on China’s major cities. The result shows: (1) There is a significant linear association between the total value of NTL and the GDP of subindustries, with R2 models generated by the total value of NTL and the GDP of secondary and tertiary industries being 0.83 and 0.93. (2) The impact of the COVID-19 pandemic on the GDP of cities with varied degrees of development and industrial structures obviously varies across time and space. The GDP of economically developed cities such as Beijing and Shanghai are more affected by COVID-19, while the GDP of less developed cities such as Xining and Lanzhou are less affected by COVID-19. The GDP of China’s major cities fell significantly in February. As the COVID-19 outbreak was gradually brought under control in March, different cities achieved different levels of GDP recovery. This study establishes a fine spatial and temporal distribution estimation model of urban GDP by industry; it accurately monitors and assesses the spatial and temporal distribution characteristics of urban GDP during the COVID-19 pandemic, reveals the impact mechanism of the COVID-19 pandemic on the economic development of major Chinese cities. Moreover, economically developed cities should pay more attention to the spread of the COVID-19 pandemic. It should do well in pandemic prevention and control in airports and stations with large traffic flow. At the same time, after the COVID-19 pandemic is brought under control, they should speed up the resumption of work and production to achieve economic recovery. This study provides scientific references for COVID-19 pandemic prevention and control measures, as well as for the formulation of urban economic development policies.

Abortion Legalization in Uruguay: Effects on Adolescent Fertility.

The Latin American and Caribbean adolescent fertility rates are among the highest in the world: about 1.7 million children are born to teen mothers every year, and most of them are declared unintended pregnancies. The region also has the highest rate of unintended pregnancy of any world region, and nearly half of such pregnancies end in abortion. However, fewer than 18 percent of the region's women live in countries where abortion is broadly legal. This paper estimates the causal effect of abortion legalization on adolescent fertility in Uruguay using official data on legal abortions provided after the 2012 reform. We employed a difference-in-differences strategy, classifying states by whether they are responsive or unresponsive to the reform. The results suggest that abortion reform had a negative impact on the adolescent birth rate by 2.5-2.8 births per thousand adolescents aged 15-19 (a 4 percent decrease from the preintervention average). Additionally, we exploited variation in reform implementation intensity through the estimation of fixed-effect linear regression models and found consistent results. Our findings are robust to controlling for a concurrent large-scale program of contraceptive implants. We conclude that legislation aimed at enhancing rights and reducing avoidable deaths and complications from unsafe abortions may also have spillover effects that help reduce adolescent fertility.

Bagging, optimized dynamic mode decomposition for robust, stable forecasting with spatial and temporal uncertainty quantification.

Dynamic mode decomposition (DMD) provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of temporal, or spatio-temporal, data. A variety of regression techniques have been developed for producing the linear model approximation whose solutions are exponentials in time. For spatio-temporal data, DMD provides low-rank and interpretable models in the form of dominant modal structures along with their exponential/oscillatory behaviour in time. The majority of DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. The optimized DMD algorithm minimizes the model bias with a variable projection optimization, thus leading to stabilized forecasting capabilities. Here, the optimized DMD algorithm is improved by using statistical bagging methods whereby a single set of snapshots is used to produce an ensemble of optimized DMD models. The outputs of these models are averaged to produce a bagging, optimized dynamic mode decomposition (BOP-DMD). BOP-DMD improves performance by stabilizing and cross-validating the DMD model by ensembling; it also robustifies the model and provides both spatial and temporal uncertainty quantification (UQ). Thus, unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic, or Bayesian, forecasting with comprehensive UQ metrics. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.