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A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

Thermally induced errors significantly affect the accuracy of the CNC machining process as they account for 40–70% of overall machining errors. The amount of thermal error is a function of the temperatures of the machine tool mixed with that of the environment, posing difficulty to accurately predict the thermal errors. To address this, this work compares prediction models under variations of cutting parameters and environmental temperatures and summarizes the strategy of manipulating the input data to attain a robust estimation of the thermal offset error. In particular, the accuracy of a prediction model can vary under varying ambient temperatures, and therefore, the robustness and generalizability are rather important in evaluating model’s efficacy. Here, four models, namely linear regression (LR), eXtreme gradient boosting (XGBoost), back propagation neural network (BPNN), and gated recurrent unit (GRU), are compared using the variance and average of the root mean square error (RMSE) for three spindle speed profiles. The averaged RMSEs fall into the range between 3.5 and 4.2 μm across all models and among them, the SFS–LR model yields the lowest variance. Finally, the estimated error is then being compensated using the zero-point shift function on the FANUC controller of the machine tool through the VMX platform developed by the Industrial Technology Research Institute (ITRI) in Taiwan. The resultant thermal error based on the SFS–LR model reduces 21.6 to 8.8 μm after compensation.

Forecasting monsoon rainfall using dynamical climate models has met with little success, partly due to models’ inability to represent the monsoon climatological state accurately. In this article the nature and dynamical causes of their biases are investigated. The approach is to analyze errors in multimodel-mean climatological fields determined from CMIP5, and to carry out sensitivity experiments using a coupled model [the Coupled Model for the Earth Simulator (CFES)] that does represent the monsoon realistically. Precipitation errors in the CMIP5 models persist throughout the annual cycle, with positive (negative) errors occurring over the near-equatorial western Indian Ocean (South Asia). Model errors indicate that an easterly wind stress bias Δτ along the equator begins during April–May and peaks during November; the severity of the Δτ is that the Wyrtki jets, eastward-flowing equatorial currents during the intermonsoon seasons (April–May and October–November), are almost eliminated. An erroneous east–west SST gradient (warm west and cold east) develops in June. The structure of the model errors indicates that they arise from Bjerknes feedback in the equatorial Indian Ocean (EIO). Vertically integrated moisture and moist static energy budgets confirm that warm SST bias in the western EIO anchors moist processes that cause the positive precipitation bias there. In CFES sensitivity experiments in which Δτ or warm SST bias over the western EIO is artificially introduced, errors in the EIO are similar to those in the CMIP5 models; moreover, precipitation over South Asia is reduced. An overall implication of these results is that South Asian rainfall errors in CMIP5 models are linked to errors of coupled processes in the western EIO, and in coupled models correct representation of EIO coupled processes (Bjerknes feedback) is a necessary condition for realistic monsoon simulation.

The NUTS classification, established by Eurostat, divides the European territories into three levels: NUTS 1 (major regions), NUTS 2 (basic regions), and NUTS 3 (small regions). Our study investigated regional disparities in mortality across 232 NUTS 2 regions in Europe by analysing the function of their spatial health services. Using a spatial error model, we assessed the influence of healthcare expenditures and the number of hospital beds and medical doctors on death rates across eight major disease categories. We employed global and local spatial statistics to capture spatial disparities in resource allocation and death rates. Spatial clustering techniques revealed distinctive but differing patterns regarding mortality and resource allocation, with central and East Europe experiencing higher mortality from circulatory and digestive diseases, with mental and neurological conditions being more prevalent in the more affluent West. Our findings demonstrated decreasing returns at scale across all resources, with varied elasticities depending on disease type. Improved financial resources significantly reduced mortality for most illnesses except for mental or neurological disorders, while outcomes with respect to neoplasms depended on systemic factors beyond spending levels. The number of hospital beds often correlated positively with mortality, indicating system strain and reactive action rather than with preventive healthcare factors. Access to doctors reduced mortality only for mental and neurological conditions, highlighting the importance of specialised, continuous care. Regional affluence was found to consistently reduce mortality for several disease categories, underscoring the role of socioeconomic context in public health. These insights offer crucial guidance for more equitable and disease-specific resource allocation in health policy.

Accuracy design constitutes an important role in machine tool designing. It is used to determine the permissible level of each error parameter of a machine tool, so that any criterion can be optimized. Geometric, thermal-induced, and cutting force-induced errors are responsible for a large number of comprehensive errors of a machine tool. These errors not only influence the machining accuracy but are also of great importance for accuracy design to be performed. The aim of this paper is the proposal of a general approach that simultaneously considered geometric, thermal-induced, and cutting force-induced errors, in order for machine tool errors to be allocated. By homogeneous transformation matrix (HTM) application, a comprehensive error model was developed for the machining accuracy of a machine tool to be acquired. In addition, a generalized radial basis function (RBF) neural network modeling method was used in order for a thermal and cutting force-induced error model to be established. Based on the comprehensive error model, the importance sampling method was applied for the reliability and sensitivity analysis of the machine tool to be conducted, and two mathematical models were presented. The first model predicted the reliability of the machine tool, whereas the second was used to identify and optimize the error parameters with larger effect on the reliability. The permissible level of each geometric error parameter can therefore be determined, whereas the reliability met the design requirement and the cost of this machining was optimized. An experiment was conducted on a five-axis machine tool, and the results confirmed the proposed approach being able to display the accuracy design of the machine tool.

In survival analysis, covariate measurement error has been studied extensively for the Cox model. In this article, we propose a corrected profile likelihood approach, and show that many existing methods can be unified by our approach. Furthermore, we extend our discussion to general measurement error and Berkson models, as opposed to the classical additive error model that has been widely used in the literature. We investigate the impact of model misspecification of the measurement error process and uncover interesting findings. Empirical studies are carried out to illustrate the usage of the proposed methods and to assess their performance. En analyse de survie, les conséquences des erreurs de mesure sur les covariables ont été largement étudiées pour le modèle de Cox. Les auteures proposent une solution basée sur une correction du profil de vraisemblance et montrent que leur approche permet d'unifier plusieurs méthodes existantes pour traiter les erreurs de mesure. Elles étendent leur discussion au modèle général d'erreurs de mesure et au modèle de Berkson, plutôt qu'au modèle d'erreur additive omniprésent dans la littérature. Elles étudient l'impact d'une mauvaise spécification du processus produisant les erreurs de mesure et font des découvertes intéressantes. Elles procèdent finalement à des études empiriques afin d'illustrer l'usage de la méthode proposée et d'en évaluer la performance.

This paper evaluates the performance of different small area estimation methods using model and design-based simulation experiments. Design-based simulation experiments are carried out using the Mexican Intra Censal survey as a census of roughly 3.9 million households from which 500 samples are drawn using a two-stage selection procedure similar to that of Living Standards Measurement Study (LSMS) surveys. The estimation methods considered are that of Elbers, Lanjouw and Lanjouw (2003), the empirical best predictor of Molina and Rao (2010), the twofold nested error extension presented by Marhuenda et al. (2017), and finally an adaptation, presented by Nguyen (2012), that combines unit and area level information, and which has been proposed as an alternative when the available census data is outdated. The findings show the importance of selecting a proper model and data transformation so that model assumptions hold. A proper data transformation can lead to a considerable improvement in mean squared error (MSE). Results from design-based validation show that all small area estimation methods represent an improvement, in terms of MSE, over direct estimates. However, methods that model unit level welfare using only area level information suffer from considerable bias. Because the magnitude and direction of the bias is unknown ex ante, methods relying only on aggregated covariates should be used with caution, but may be an alternative to traditional area level models when these are not applicable.

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error-prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.

Differences in the predicted carbon and water fluxes by different global land models have been quite large and have not decreased over the last two decades. Quantification and attribution of the uncertainties of global land surface models are important for improving the performance of global land surface models, and are the foci of this study. Here we quantified the model errors by comparing the simulated monthly global gross primary productivity (GPP) and latent heat flux (LE) by two global land surface models with the model‐data products of global GPP and LE from 1982 to 2005. By analyzing model parameter sensitivities within their ranges, we identified about 2–11 most sensitive model parameters that have strong influences on the simulated GPP or LE by two global land models, and found that the sensitivities of the same parameters are different among the plant functional types (PFT). Using parameter ensemble simulations, we found that 15%–60% of the model errors were reduced by tuning only a few (<4) most sensitive parameters for most PFTs, and that the reduction in model errors varied spatially within a PFT or among different PFTs. Our study shows that future model improvement should optimize key model parameters, particularly those parameters relating to leaf area index, maximum carboxylation rate, and stomatal conductance. Key Points: There are 2–11 most sensitive model parameters in two land surface models Fifteen percent to 60% of the model errors can be reduced by tuning the few (<4) most sensitive parameters Future model improvement should optimize model parameters, particularly those parameters relating to leaf area index, maximum carboxylation rate, and stomatal conductance.

This paper is concerned with the estimation of forecast error, particularly in relation to insurance loss reserving. Forecast error is generally regarded as consisting of three components, namely parameter, process and model errors. The first two of these components, and their estimation, are well understood, but less so model error. Model error itself is considered in two parts: one part that is capable of estimation from past data (internal model error), and another part that is not (external model error). Attention is focused here on internal model error. Estimation of this error component is approached by means of Bayesian model averaging, using the Bayesian interpretation of the LASSO. This is used to generate a set of admissible models, each with its prior probability and likelihood of observed data. A posterior on the model set, conditional on the data, may then be calculated. An estimate of model error (for a loss reserve estimate) is obtained as the variance of the loss reserve according to this posterior. The population of models entering materially into the support of the posterior may turn out to be “thinner” than desired, and bootstrapping of the LASSO is used to increase this population. This also provides the bonus of an estimate of parameter error. It turns out that the estimates of parameter and model errors are entangled, and dissociation of them is at least difficult, and possibly not even meaningful. These matters are discussed. The majority of the discussion applies to forecasting generally, but numerical illustration of the concepts is given in relation to insurance data and the problem of insurance loss reserving.