Confidence Bands Research Articles

State-of-the-art nonstationary hypersurface damage assessment approach for energy harvesters

Since EH (Energy Harvesters) constitute nowadays a vital part of renewable energy engineering, experimental research is required in addition to numerical modeling, serving reliable structural design and ensuring prolonged device service time. The performance of GPEH (GalloPing EH) has been examined in this case study, utilizing comprehensive laboratory wind tunnel tests, carried out under realistic windspeed conditions. Novel structural multivariate risks assessment methodology, presented here, being feasible for nonstationary nonlinear GPEH dynamic systems, that had been either physically measured over a representative period, providing jointly quasi-ergodic time-series, or directly numerically MCS (Monte Carlo Simulated). Based on laboratory-measured GPEH dynamics, the presented analysis demonstrates that the proposed multivariate hypersurface methodology offers robust predictions of the structural failure/damage risks. Furthermore, when dealing with raw measured timeseries, representing the high-dimensional dynamic system, existing risk assessment techniques struggle to handle nonlinear inter-correlations between GPEH critical components. This case study's main objective has been to validate and benchmark the novel multimodal risk assessment methodology, which utilizes multivariate nonstationary lab-recorded time histories to extract relevant design information from the underlying GPEH dynamics.The proposed state-of-the-art nonstationary hypersurface reliability approach being of a generic nature, offering additional capacity for damage/failure risks prognostics for a wide range of nonlinear multidimensional nonstationary systems. Forecasted damage and failure risks have been supplied with confidence bands, demonstrating the experimental setup's robustness, as well as the useful design features of the presented nonstationary hypersurface risks assessment methodology. It should be noted that the presented reliability methodology being mathematically exact, and it does not rely on simplifying assumptions.

Sulcardine, an investigational multichannel blocker for atrial fibrillation with uniquely intense JTp inhibition to enhance safety

Abstract Background Sulcardine is a multi-ion channel blocker for treating atrial fibrillation (AF) with similar inhibitory potency on peak/late sodium, L-type calcium, and rapidly activating delayed-rectifier potassium (IKr) currents. The therapeutic effect of inhibiting IKr includes QT prolongation (accompanied by a proportional J to T peak [JTp] interval prolongation), which, when excessive, may cause malignant arrhythmias. As previously reported, sulcardine strongly mitigates the JTp increase that may contribute to excessive QT prolongation by inhibiting late sodium and L-type calcium currents, an intrinsic mechanism to protect from excessive IKr block. Purpose We explored the JTp mechanism, pharmacokinetics (PK), electrocardiogram (ECG) changes, safety, and drug-drug interaction (DDI) of a single sulcardine (as HBI-3000) intravenous (IV) infusion in healthy subjects. Methods An open-label Phase 1 study evaluated the PK interaction between sulcardine, dosed as a 350 mg 30-minute infusion of the sulfate salt HBI-3000, and a cytochrome P450 CYP2D6 enzyme inhibitor (paroxetine). Triplicate ECGs were extracted from continuous 12-lead Holter ECG recordings at baseline and 10 time points thereafter. Mean baseline-subtracted and heart rate-corrected QTcF (dQTcF), JTpc (dJTpc), and other ECG changes were calculated at each time point. Left ventricular ejection fraction (LVEF) was assessed at baseline, 30-, and 120-minutes post-infusion start by transthoracic echocardiography (TTE). Results 39 subjects received HBI-3000, 33 in the paired PK population. Expected ECG parameter changes proportional to sulcardine plasma concentrations were observed in the HBI-3000-alone portion of the study, including modest QTcF prolongation (Fig 1, blue slope, confidence band) and JTpc inhibition (Fig 1, red). The usual JTpc increase associated with IKr blockers, even those with mitigating channel block (such as the late sodium current), was not only reduced but actually reversed. We calculated the dQTcF change that would have been observed in the absence of JTpc reduction by sulcardine (Fig 1, green). Sulcardine reduced the QTcF increase that might have occurred by about 21 msec at the highest concentrations, and this mitigating effect was observed at all concentrations. No clinically significant safety findings or arrhythmias were observed. The most frequent TEAEs were nausea and headache (n=2 each [5.1%]). No LVEF changes were clinically significant. Sulcardine had minimal metabolic interactions. Conclusion Acute IV HBI-3000 administration resulted in no clinically significant safety findings, LVEF depression, or DDI with CYP2D6. Sulcardine facilitates QT prolongation, which has a therapeutic benefit in terminating AF, with simultaneous uncoupling and reversing of the JTpc interval increase, a potential mechanism for reducing proarrhythmic risk due to excessive QT prolongation.Figure 1.ECG Intervals vs Sulcardine Conc

Analyzing and forecasting air pollution concentration in the capital and Southern Thailand using a lag-dependent Gaussian process model.

The air pollution problem has now amassed worldwide attention due to its multifaceted harm to human health. Exploring the concentration of air pollution and improving forecast have important consideration worldwide. In this research, we analyze the air pollution concentration of Southern Thailand and compare it with the central region. Also, we proposed a methodology based on the lag-dependent Gaussian process (LDGP), a Bayesian non-parametric machine learning model, with a stable optimization approach, which is a cluster-based multi-starter technique based on the Nelder-Mead optimizer. This model also provides the confidence band for forecasted values. We also used autoregressive deep neural network (AR-DNN), autoregressive random forest (AR-RF), gradient boosting (GB), and K-nearest neighbors (KNN) models. A comparison of the proposed methodology was performed on the daily air pollution data collected from the southern provinces and also from the capital of Thailand from 1 January 2018 to 31 December 2022. We used well-established performance evaluation measures to compare the performance of the models. To evaluate the bias due to overfit, we performed a tenfold cross-validation for all the pollutants in each region and compared the models to choose the best one. Moreover, we explored the concentration of air pollution in these regions. Results of descriptive analysis revealed that Bangkok had a much higher concentration of air pollution as compared to the southern region. However, the southern region had higher exposure to PM air pollutants as per WHO recommendations and also had higher exposure to O3 and CO levels. The proposed LDGP model outperformed the other machine learning models for forecasting all air pollutants. Hence, it is recommended to be used by experts for further research and studies with different kernel functions. This research is also expected to contribute to local government planning and prevention and worldwide use of the same methodology for the sustainability of public health.

Longitudinal benefit:risk analysis through the desirability of outcome ranking (DOOR) with application to ACTT-1 Trial

A clinical trial aims to uncover high-quality evidence for both efficacy and safety for an experimental treatment. Traditionally, statisticians evaluate the two aspects marginally, where, for example, efficacy considers the proportion of patients cured and safety considers the proportion of patients not showing adverse events. However, separate analyses could end up with misleading results, as the cumulative nature of clinical outcomes and the correlation between efficacy and safety endpoints are neglected. The desirability of outcome ranking (DOOR) method addresses such issues and provides a patient-centric approach to benefit:risk evaluation. A patient’s outcome is ranked based on pre-specified clinical criteria, where the most desirable rank represents a good outcome with no side effects and the least desirable rank is the worst possible clinical outcome. As the DOOR outcome is a temporal state that can have repeated measures, we propose a longitudinal approach that estimates and infers the temporal treatment effects. We develop a methodology for constructing simultaneous confidence bands by accounting for the correlations across time points. Additionally, we propose a weighted Mann-Whitney-U statistic to evaluate the treatment effect over the entire trial period. The performance of the proposed methodologies is examined through simulations and an application to a COVID-19 trial.

Bootstrap inference for linear time-varying coefficient models in locally stationary time series

Time-varying coefficient models can capture evolving relationships. However, constructing asymptotic confidence bands for coefficient curves in these models is challenging due to slow convergence rates and the presence of various nuisance parameters. A residual-based sieve bootstrap method has recently been proposed to address these issues. While it successfully produces confidence bands with accurate empirical coverage, its applicability is restricted to strictly stationary processes. We introduce a new bootstrap scheme, the local blockwise wild bootstrap (LBWB), that allows for locally stationary processes. The LBWB can replicate the distribution of the parameter estimates while automatically accounting for nuisance parameters. An extensive simulation study reveals the superior performance of the LBWB compared to various benchmark approaches. It also shows the potential applicability of the LBWB in broader scenarios, including time-varying cointegrating models. We then examine herding effects in the Chinese renewable energy market using the proposed methods. Our findings strongly support the presence of herding behaviors before 2016, aligning with earlier studies. However, contrary to previous research, we find no significant evidence of herding between around 2018 and 2021. Online supplementary materials are available for this article.

Limit hypersurface state-of-the-art damage assessment approach for a galloping energy harvester, accounting for memory effects

Energy harvesters (i.e., EH) constituting nowadays an essential part of renewable energy engineering; hence, apart from numerical modeling, experimental studies are necessary, constituting reliable input for durable design and structural safety assessments. In the current study, galloping EH’s performance was analyzed, utilizing extensive laboratory wind-tunnel tests, conducted under realistic windspeed conditions. The novel structural multivariate risks assessment methodology, presented in the current study is particularly feasible for multi-dimensional nonlinear EH dynamic systems, that have been either directly Monte Carlo (i.e., MC) numerically simulated or physically measured over a representative temporal lapse, providing piecewise ergodic time series. As shown in this analysis, the suggested multivariate methodology accurately enables accurate predictions of the EH dynamic system’s failure/hazard or damage risks, based on laboratory-measured EH system’s dynamics. Furthermore, nonlinear inter-correlations between various systems’ critical components are not always easily handled by classic risk assessment techniques, when dealing with a high-dimensional system’s raw time series. The primary goal of this study was validation and benchmarking the novel risk assessment methodology, which can extract pertinent information, contained within the EH system’s dynamics, based on lab-recorded time histories. In conclusion, the novel hypersurface methodology presented in this study is generic, providing additional capability to accurately, yet efficiently predict damage/failure risks for a variety of nonlinear EH multidimensional systems. Relatively narrow confidence bands have been reported for the forecasted damage and failure levels, indicating both the robustness of the experimental setup, as well as practical design virtues of the advocated Gaidai hypersurface risks assessment methodology. Note that the presented methodology being mathematically exact does not rely on pre-assumptions and yet it is of general purpose.

Recognition and Characterization of Nanoscale Phases: Modulus Mapping of Asphalt Film in Pavement Mixture Cores.

The objective of this study is to recognize and characterize the nanoscale phase modulus mapping of the asphalt film in pavement mixture cores using atomic force microscopy quantitative nanomechanical technology. The pavement core samples from the upper and middle layers of four highways and laboratory samples were taken as the research object. The phase modulus-macro property correlation of recovered asphalt was analyzed using mathematical statistics. The results showed that the pavement core samples had more significant multi-phase and diversified phase characteristics compared to lab samples. This indicated that the asphalt in the pavement core had an obvious phase separation phenomenon due to aging. The phase modulus of each sample was distributed across a relatively wide numerical range, and there were also many numerical points with large fluctuations. Especially for the mixture sample containing SBS (Styrene-Butadiene-Styrene)-modified asphalt, the phase modulus distribution mappings presented a multi-peak phenomenon. Hence, considering the distribution characteristics of the data, the box plot method was introduced. Compared with quantified results from laboratory samples, the phase modulus of SBS-modified asphalt increased by 0.96 times, 1.18 times and 1.15 times, and that of base asphalt increased by 0.59 times, 0.56 times, 0.42 times, 1.24 times and 0.39 times, respectively. This indicates that the aging degree of asphalt in the upper layer was generally greater than that of the asphalt in the middle layer and that there was an aging gradient in the direction of pavement depth. All points were within the 95% confidence band in terms of correlation fitting, indicating a better fitting effect between phase modulus and complex shear modulus, as well as between phase modulus and penetration. This research provides innovative ideas for future multi-scale numerical simulation and cross-scale performance model development of asphalt binders.

Concurrent Functional Linear Regression Via Plug-in Empirical Likelihood

AbstractFunctional data with non-smooth features (e.g., discontinuities in the functional mean and/or covariance) and monotonicity arise frequently in practice. This paper develops simultaneous inference for concurrent functional linear regression in this setting. We construct a simultaneous confidence band for a functional covariate effect of interest. Along with a Wald-type formulation, our approach is based on a powerful nonparametric likelihood ratio method. Our procedures are flexible enough to allow discontinuities in the coefficient functions and the covariance structure, while accounting for discretization of the observed trajectories under a fixed dense design. A simulation study shows that the proposed likelihood ratio-based procedure outperforms the Wald-type procedure in moderate sample sizes. We apply the proposed methods to studying the effect of age on the occupation time curve derived from wearable device data obtained in an NHANES study.

Tests and confidence bands for multiple one-sided comparisons

One-sided inference should be used in some applications, but Stata has limited support for one-sided tests and confidence intervals for a single compari­son and almost no support for one-sided tests and confidence bands for multiple comparisons. In this article, we provide an introduction to one-sided tests and confidence intervals for a single hypothesis and to one-sided tests and confidence intervals for multiple comparisons. We also discuss extensions of the sotable com­mand introduced in Drukker (2023, Stata Journal 23: 518-544) to cover one-sided tests and confidence bands for a single comparison and for multiple comparisons. We also provide examples of how to use sotable to perform multiple tests against values other than zero and how to perform multiple tests after commands like margins and nlcom that support the post option.

Impact of the LLM on the Google

In recent years, the emergence and development of large language models has occurred. The authors of this paper are intrigued by how this will affect the stock market of Google. Therefore, they have written this thesis paper to investigate the effect of emergence of LLMs on the stock prices of Google. Using the method of event study, this paper sets an event window to observe the stock market of Google when encountering the emergence of different LLMs. The authors have calculated the AR (abnormal return) and CAR (cumulative abnormal return) of Google. Next, they put these statistics into various models to investigate the impact LLMs have caused. They determine the significance of the impact by checking if AR or CAR have exceeded confidence band. After carrying out the investigation, the authors have concluded that LLMs apply limited influence on the stock market of Google.The purpose of this paper is to inform large old-school technology companies like Google to stick to its own plan. They can rest reassured as LLMs are unable to truly threaten them for a while, and should also accept the advancement of LLMs.

Modelling climate change-induced nonstationarity in rainfall extremes: A comprehensive approach for hydrological analysis

Climate change and global warming have induced a dynamic shift in extreme rainfall patterns, leading to nonstationary behaviour. This alteration in behaviour challenges conventional hydrologic design, which assumes stationarity and can yield misleading outcomes. This study aims to address nonstationarity by modelling distribution parameters with covariates. Utilizing a 70-year (1951–2020) high-resolution India Meteorological Department (IMD) gridded dataset, extreme annual rainfall across diverse Indian cities was extracted and modelled. Previous research and goodness-of-fit tests favour the Generalized Extreme Value (GEV) distribution for modelling extremes. This study incorporates indices like Nino3.4, dipole mode index, global and local temperature, CO2, and time to characterize nonstationarity in extreme annual rainfall, leveraging climate cycles and global warming. Performance assessment employs the Akaike information criterion, Bayesian information criterion and Likelihood ratio test, while quantile reliability is evaluated through confidence intervals (CIs). Findings reveal widespread nonstationary trends in most grid points, translating to wider CIs in estimated quantiles for chosen non-exceedance probability and covariates in fitted models. Generally, nonstationary conditions are associated with broader confidence bands in return levels, highlighting nonstationary model weaknesses compared to stationary models. However, the results showed that the rainfall extremes follow a nonstationary pattern. Hence, there is a strong need to develop nonstationary models of low uncertainty.

Assessing treatment effects with adjusted restricted mean time lost in observational competing risks data

BackgroundAccording to long-term follow-up data of malignant tumor patients, assessing treatment effects requires careful consideration of competing risks. The commonly used cause-specific hazard ratio (CHR) and sub-distribution hazard ratio (SHR) are relative indicators and may present challenges in terms of proportional hazards assumption and clinical interpretation. Recently, the restricted mean time lost (RMTL) has been recommended as a supplementary measure for better clinical interpretation. Moreover, for observational study data in epidemiological and clinical settings, due to the influence of confounding factors, covariate adjustment is crucial for determining the causal effect of treatment.MethodsWe construct an RMTL estimator after adjusting for covariates based on the inverse probability weighting method, and derive the variance to construct interval estimates based on the large sample properties. We use simulation studies to study the statistical performance of this estimator in various scenarios. In addition, we further consider the changes in treatment effects over time, constructing a dynamic RMTL difference curve and corresponding confidence bands for the curve.ResultsThe simulation results demonstrate that the adjusted RMTL estimator exhibits smaller biases compared with unadjusted RMTL and provides robust interval estimates in all scenarios. This method was applied to a real-world cervical cancer patient data, revealing improvements in the prognosis of patients with small cell carcinoma of the cervix. The results showed that the protective effect of surgery was significant only in the first 20 months, but the long-term effect was not obvious. Radiotherapy significantly improved patient outcomes during the follow-up period from 17 to 57 months, while radiotherapy combined with chemotherapy significantly improved patient outcomes throughout the entire period.ConclusionsWe propose the approach that is easy to interpret and implement for assessing treatment effects in observational competing risk data.

Impact of Measurement Uncertainty on Fault Diagnosis Systems: A Case Study on Electrical Faults in Induction Motors.

Classification systems based on machine learning (ML) models, critical in predictive maintenance and fault diagnosis, are subject to an error rate that can pose significant risks, such as unnecessary downtime due to false alarms. Propagating the uncertainty of input data through the model can define confidence bands to determine whether an input is classifiable, preferring to indicate a result of unclassifiability rather than misclassification. This study presents an electrical fault diagnosis system on asynchronous motors using an artificial neural network (ANN) model trained with vibration measurements. It is shown how vibration analysis can be effectively employed to detect and locate motor malfunctions, helping reduce downtime, improve process control and lower maintenance costs. In addition, measurement uncertainty information is introduced to increase the reliability of the diagnosis system, ensuring more accurate and preventive decisions.

Simultaneous predictive bands for functional time series using minimum entropy sets

Functional Time Series (FTS) are sequences of dependent random elements taking values on some functional space. Most of the research on this domain focuses on producing a predictor able to forecast the next function, having observed a part of the sequence. For this, the Autoregressive Hilbertian process is a suitable framework. Here, we address the problem of constructing simultaneous predictive confidence bands for a stationary FTS. The method is based on an entropy measure for stochastic processes. To construct predictive bands, we use a Reproducing Kernel Hilbert Space (RKHS) to represent the functions and a functional bootstrap procedure that allows us to estimate the prediction law and a Reproducing Kernel Hilbert Spaces (RKHS) to represent the functions, considering then the basis associated to the reproducing kernel. We then classify the points on the projected space according to those that belong to the minimum entropy set (MES) and those that do not. We map the minimum entropy set back to the functional space and construct a band using the regularity property of the RKHS. The proposed methodology is illustrated through artificial and real data sets.

Difference-in-differences Estimator of Quantile Treatment Effect on the Treated

We propose a new difference-in-differences (DID) estimator of the quantile treatment effect on the treated (QTT). The model assumes a common time effect on the cumulative distribution functions of untreated potential outcomes, allowing for covariates. This condition holds if and only if the net change in the untreated outcome densities is common across treated and control groups. Unlike the Changes-in-Changes model proposed by Athey and Imbens (2006), our model is compatible with the usual DID assumption for means, and it provides a computationally simple and straightforward way to control for covariates. We establish uniform consistency and weak convergence of the proposed estimator of QTT and the related functions. The estimators and the simultaneous confidence bands remain valid even for discrete outcome variables. As an empirical application, the distributional impact of the earned income tax credit on birth weight is investigated. We provide a STATA ado file package.

Functional index coefficient models for locally stationary time series

In the analysis of nonlinear time series, we propose a novel functional index coefficient model for the locally stationary data. The proposed model can effectively capture the dynamic interaction effects between variables and the nature of data evolution. Drawing the idea from the spline-backfitted method, we propose a three-step estimation procedure and establish the asymptotic properties of the resulting nonparametric estimators. We further construct simultaneous confidence bands for the time-varying functions to explore the global variation of the original data. We also develop a test statistic to check the time-varying properties based on a bootstrap procedure. Simulation studies have been conducted to investigate the finite sample performance of the proposed methods. Two real applications in the finance market and the Hong Kong respiratory and circulatory disease data are analysed for illustration.

Seasonal solar irradiance forecasting using artificial intelligence techniques with uncertainty analysis

Renewable integration in utility grid is crucial in the current energy scenario. Optimized utilization of renewable energy can minimize the energy consumption from the grid. This demands accurate forecasting of renewable contribution and planning. Most of the researches aim to find a suitable forecasting model in terms of accuracy and error metrics. However, the uncertainty and variability in these forecasts are also significant. This work combines point forecast with interval forecast to provide comprehensive information about the forecast uncertainty. In this work, solar irradiance forecasting is carried out using artificial intelligence (AI) techniques. Forecasting is done using seasonal auto-regressive moving average with exogenous factors (SARIMAX), support vector regression (SVR), long short term memory (LSTM) techniques and performance is evaluated. SVR model exhibited the best performance with R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} values of 0.97 and 0.96 for winter and summer respectively and 0.85 for monsoon and post-monsoon seasons. This is followed by forecast error distribution studies and uncertainty analysis. For this, SVR forecast error data is fitted using laplace distribution. Uncertainty study is carried out using confidence intervals and coverage rates. Excellent coverage rates are obtained for various confidence levels for all seasons, indicating the appropriate fitting of error distribution. For the narrow 85% confidence band, coverage rates of 89%, 95%, 90%, and 88% are obtained for winter, summer, monsoon and post-monsoon respectively. The work emphasizes the need for error-distribution studies, modeling of forecast errors and their application in providing reliable forecast intervals with the perspective of enhancing system reliability.

Statistical inference for innovation distribution in ARMA and multi-step-ahead prediction via empirical process

The kernel distribution estimator (KDE) is proposed based on residuals of the innovation distribution in the autoregressive moving-average (ARMA) time series. The deviation between KDE and the innovation distribution function is shown to converge to the Brownian bridge, leading to the construction of a Kolmogorov–Smirnov smooth simultaneous confidence band for the innovation distribution function. Additionally, an empirical cumulative distribution function (CDF) based on prediction residuals is introduced for the multi-step-ahead prediction error distribution function. This empirical process weakly converges to a Gaussian process with a specific covariance function. Furthermore, a quantile estimator is derived from the empirical CDF of prediction residuals, and multi-step-ahead prediction intervals (PIs) for future observations are established using these estimated quantiles. The PIs achieve the nominal prediction level asymptotically for the finite variance ARMA model. Simulation studies and analysis of crude oil price data confirm the validity of the asymptotic theory.

Confidence sets for intraclass correlation coefficients in test–retest curve measurements

Evaluating test–retest reliability is crucial in biomechanical research, as it validates experimental results. While methods for reliability of scalar outcome variables are well-established, methods to assess reliability of continuous curve data (such as joint angle trajectories during gait) remain less explored. This study investigates methods for constructing confidence sets for curve-level intraclass correlation coefficients (ICC), which can be expressed as either an ICC curve or an integrated ICC. Currently, no standardised guidelines exist in biomechanics for reporting curve-level ICC uncertainty. Nonparametric bootstrapping techniques are proposed for both the ICC curve’s confidence bands and the integrated ICC’s confidence intervals, and these methods are validated through Monte Carlo simulations, covering various effect sizes and curve characteristics. Additionally, these methods are applied to assess the test–retest reliability of knee kinematics in three different planes during landing of one-leg hops, where less uncertainty is observed for the ICC curve and integrated ICC in the frontal plane compared to other planes. When the entire time domain is of primary empirical interest, we recommend using a rank-based bootstrap confidence band to express ICC uncertainty, as it yields increasingly precise and valid results as the number of individuals increases, with the coverage rate approaching the correct level of 95%. When a single summary metric is of primary interest, we recommend using the integrated ICC along with a typical bootstrap confidence interval based on the normal distribution, as the coverage rate remains adequately accurate and stable at around the correct level of 95% across varying number of individuals.

Inference for ARMA time series with mildly varying trend

Statistical inference is studied for ARMA time series with a mildly varying smooth trend. Following properly calibrated B-spline estimation of the trend, approximates of the unobserved ARMA series are used instead to estimate ARMA parameters by maximum likelihood. The parameter estimates are shown to be oracally efficient in the sense that they are asymptotically as efficient as if the true trend function was known and removed to obtain the true ARMA series. The mildly varying trend function is estimated by Nadaraya–Watson method with asymptotically correct simultaneous confidence band (SCB). The SCB sprawls over an interval that eventually covers the half real line, instead of a bounded interval in the existing literature, with asymptotic width of the SCB dependent on ARMA coefficients. Simulation experiments corroborate the theoretical findings.