Statistical Error Model Research Articles

Reliability of buildings in service limit state for maximum horizontal displacements

Brazilian design code ABNT NBR6118:2003 - Design of Concrete Structures - Procedures - [1] proposes the use of simplified models for the consideration of non-linear material behavior in the evaluation of horizontal displacements in buildings. These models penalize stiffness of columns and beams, representing the effects of concrete cracking and avoiding costly physical non-linear analyses. The objectives of the present paper are to investigate the accuracy and uncertainty of these simplified models, as well as to evaluate the reliabilities of structures designed following ABNT NBR6118:2003[1&] in the service limit state for horizontal displacements. Model error statistics are obtained from 42 representative plane frames. The reliabilities of three typical (4, 8 and 12 floor) buildings are evaluated, using the simplified models and a rigorous, physical and geometrical non-linear analysis. Results show that the 70/70 (column/beam stiffness reduction) model is more accurate and less conservative than the 80/40 model. Results also show that ABNT NBR6118:2003 [1] design criteria for horizontal displacement limit states (masonry damage according to ACI 435.3R-68(1984) [10]) are conservative, and result in reliability indexes which are larger than those recommended in EUROCODE [2] for irreversible service limit states.

Performance of the Weather Research and Forecasting Model for Month-Long Pan-Arctic Simulations

Abstract The performance of the Weather Research and Forecasting (WRF) model was evaluated for month-long simulations over a large pan-Arctic model domain. The evaluation of seven different WRF (version 3.1) configurations for four months (January, April, July, and October 2007) indicated that WRF produces reasonable simulations of the Arctic atmosphere. Ranking of the model error statistics, calculated relative to the NCEP/Department of Energy Global Reanalysis 2 (NCEP-2), for sea level pressure, 500- and 300-hPa geopotential height, 2-m air temperature, and precipitation identified the model configurations that consistently produced the best pan-Arctic simulations. For all WRF configurations considered, large errors in circulation are evident in the North Pacific. The errors in the North Pacific are manifested as an overly weak and westward-shifted Aleutian low and overly strong subtropical Pacific high simulated by WRF. These circulation errors are nearly barotropic, with a slight increase in magnitude with height, and they vary slightly in magnitude and position as the WRF physics options and domain size are changed. It is concluded that the circulation errors are likely due to errors in the treatment of the model-top boundary. The use of a higher model top (10 hPa rather than 50 hPa) or spectral nudging of wavenumbers 1–3 in the top half of the model domain results in significantly reduced circulation biases. Simulations with WRF version 3.2 also show reduced errors.

Optimized Preprocessing of Ultra-Performance Liquid Chromatography/Mass Spectrometry Urinary Metabolic Profiles for Improved Information Recovery

Ultra-performance liquid chromatography coupled to mass spectrometry (UPLC/MS) has been used increasingly for measuring changes of low molecular weight metabolites in biofluids/tissues in response to biological challenges such as drug toxicity and disease processes. Typically samples show high variability in concentration, and the derived metabolic profiles have a heteroscedastic noise structure characterized by increasing variance as a function of increased signal intensity. These sources of experimental and instrumental noise substantially complicate information recovery when statistical tools are used. We apply and compare several preprocessing procedures and introduce a statistical error model to account for these bioanalytical complexities. In particular, the use of total intensity, median fold change, locally weighted scatter plot smoothing, and quantile normalizations to reduce extraneous variance induced by sample dilution were compared. We demonstrate that the UPLC/MS peak intensities of urine samples should respond linearly to variable sample dilution across the intensity range. While all four studied normalization methods performed reasonably well in reducing dilution-induced variation of urine samples in the absence of biological variation, the median fold change normalization is least compromised by the biologically relevant changes in mixture components and is thus preferable. Additionally, the application of a subsequent log-based transformation was successful in stabilizing the variance with respect to peak intensity, confirming the predominant influence of multiplicative noise in peak intensities from UPLC/MS-derived metabolic profile data sets. We demonstrate that variance-stabilizing transformation and normalization are critical preprocessing steps that can benefit greatly metabolic information recovery from such data sets when widely applied chemometric methods are used.

Imposed and Neglected Uncertainty in the Global Average Surface Air Temperature Index

The statistical error model commonly applied to monthly surface station temperatures assumes a physically incomplete climatology that forces deterministic temperature trends to be interpreted as measurement errors. Large artefactual uncertainties are thereby imposed onto the global average surface air temperature record. To illustrate this problem, representative monthly and annual uncertainties were calculated using air temperature data sets from globally distributed surface climate stations, yielding ±2.7 C and ±6.3 C, respectively. Further, the magnitude uncertainty in the 1961–1990 global air temperature annual anomaly normal, entirely neglected until now, is found to be ±0.17 C. After combining magnitude uncertainty with the previously reported ±0.46 C lower limit of measurement error, the 1856–2004 global surface air temperature anomaly with its 95% confidence interval is 0.8±0.98 C, Thus, the global average surface air temperature trend is statistically indistinguishable from 0 C. Regulatory policies aimed at influencing global surface air temperature are not empirically justifiable.

Co-registration and bias corrections of satellite elevation data sets for quantifying glacier thickness change

Abstract. There are an increasing number of digital elevation models (DEMs) available worldwide for deriving elevation differences over time, including vertical changes on glaciers. Most of these DEMs are heavily post-processed or merged, so that physical error modelling becomes difficult and statistical error modelling is required instead. We propose a three-step methodological framework for assessing and correcting DEMs to quantify glacier elevation changes: (i) remove DEM shifts, (ii) check for elevation-dependent biases, and (iii) check for higher-order, sensor-specific biases. A simple, analytic and robust method to co-register elevation data is presented in regions where stable terrain is either plentiful (case study New Zealand) or limited (case study Svalbard). The method is demonstrated using the three global elevation data sets available to date, SRTM, ICESat and the ASTER GDEM, and with automatically generated DEMs from satellite stereo instruments of ASTER and SPOT5-HRS. After 3-D co-registration, significant biases related to elevation were found in some of the stereoscopic DEMs. Biases related to the satellite acquisition geometry (along/cross track) were detected at two frequencies in the automatically generated ASTER DEMs. The higher frequency bias seems to be related to satellite jitter, most apparent in the back-looking pass of the satellite. The origins of the more significant lower frequency bias is uncertain. ICESat-derived elevations are found to be the most consistent globally available elevation data set available so far. Before performing regional-scale glacier elevation change studies or mosaicking DEMs from multiple individual tiles (e.g. ASTER GDEM), we recommend to co-register all elevation data to ICESat as a global vertical reference system.

Performance of the Weather Research and Forecasting (WRF) Model for Month-long pan-Arctic Simulations

The performance of the Weather Research and Forecasting (WRF) model was evaluated for month-long simulations over a large pan-Arctic model domain. The evaluation of seven different WRF (version 3.1) configurations for four months (January, April, July, and October 2007) indicated that WRF produces reasonable simulations of the Arctic atmosphere. Ranking of the model error statistics, calculated relative to the NCEP/Department of Energy Global Reanalysis 2 (NCEP-2), for sea level pressure, 500- and 300-hPa geopotential height, 2-m air temperature, and precipitation identified the model configurations that consistently produced the best pan-Arctic simulations. For all WRF configurations considered, large errors in circulation are evident in the North Pacific. The errors in the North Pacific are manifested as an overly weak and westward-shifted Aleutian low and overly strong subtropical Pacific high simulated by WRF. These circulation errors are nearly barotropic, with a slight increase in magnitude with height, and they vary slightly in magnitude and position as the WRF physics options and domain size are changed. It is concluded that the circulation errors are likely due to errors in the treatment of the model-top boundary. The use of a higher model top (10 hPa rather than 50 hPa) or spectral nudging of wavenumbers 1–3 in the top half of the model domain results in significantly reduced circulation biases. Simulations with WRF version 3.2 also show reduced errors.

Rainfall uncertainty in hydrological modelling: An evaluation of multiplicative error models

Summary This paper presents an investigation of rainfall error models used in hydrological model calibration and prediction. Traditional calibration methods assume input error to be negligible: an assumption which can lead to bias in parameter estimation and compromise model predictions. In response, a growing number of studies now specify an error model for rainfall input, usually simple in form due to both difficulties in understanding sampling errors in rainfall, and to computational constraints during parameter estimation. Such rainfall error models have not typically been validated against experimental evidence. In this study we use data from a dense gauge/radar network in the Mahurangi catchment (New Zealand) to directly evaluate the form of basic statistical rainfall error models. For this catchment, our results confirm the suitability of a multiplicative error formulation for correcting mean catchment rainfall values during high-rainfall periods (e.g., intensities over 1 mm/h); or for longer timesteps at any rainfall intensity (timestep 1 day or greater). We show that the popular lognormal multiplier distribution provides a relatively close approximation to the true error characteristics but does not capture the distribution tails, especially during heavy rainfall where input errors would have important consequences for runoff prediction. Our research highlights the dependency of rainfall error structure on the data timestep.

Sensitivity-Based Pole and Input-Output Errors of Linear Filters as Indicators of the Implementation Deterioration in Fixed-Point Context

Input-output or poles sensitivity is widely used to evaluate the resilience of a filter realization to coefficients quantization in an FWL implementation process. However, these measures do not exactly consider the various implementation schemes and are not accurate in general case. This paper generalizes the classical transfer function sensitivity and pole sensitivity measure, by taking into consideration the exact fixed-point representation of the coefficients. Working in the general framework of the specialized implicit descriptor representation, it shows how a statistical quantization error model may be used in order to define stochastic sensitivity measures that are definitely pertinent and normalized. The general framework of MIMO filters and controllers is considered. All the results are illustrated through an example.

Development of the ElarmS methodology for earthquake early warning: Realtime application in California and offline testing in Japan

In July 2009, the California Integrated Seismic Network concluded a three-year study of earthquake early warning systems in California. Three algorithms were expanded and examined during the study. Here we discuss the history, methodology, and performance of one of the algorithms, ElarmS. Earthquake Alarm Systems, or ElarmS, uses peak displacement and maximum predominant frequency of the P-wave to detect earthquakes and quantify their hazard in the seconds after rupture begins. ElarmS was developed for Northern and Southern California, and now processes waveforms in realtime from 603 seismic sensors across the state. We outline the methodology as currently implemented, present several example events from different regions of California, and summarize the performance in terms of false and missed alarms. ElarmS was also tested offline with a dataset of 84 large magnitude earthquakes from Japan. The results from the Japan dataset were used to create a statistical error model for the algorithm. The model can be used to provide realtime uncertainty estimates at any stage in processing. In August 2009 the CISN embarked on a second three-year study of earthquake early warning. As part of this ongoing research, we identify the technological and methodological challenges facing ElarmS. Telemetry latencies and false alarm rates are two key opportunities for improvement.

Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals

Abstract In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a scaling factor for the observation error correlation length scale. As an application, the resulting adaptive filter is used to estimate the time evolution of ocean mesoscale signals using observations of the ocean dynamic topography. To check the behavior of the adaptive mechanism, this is done in the context of idealized experiments, in which model error and observation error statistics are known. This ideal framework is particularly appropriate to explore the ill-conditioned situations (inadequate prior assumptions or uncontrollability of the parameters) in which adaptivity can be misleading. Overall, the experiments show that, if used correctly, the efficient optimal adaptive algorithm proposed in this paper introduces useful supplementary degrees of freedom in the estimation problem, and that the direct control of these statistical parameters by the observations increases the robustness of the error estimates and thus the optimality of the resulting Kalman filter.

Assimilation of Argo temperature and salinity profiles using a bias-aware localized EnKF system for the Pacific Ocean

In this study, Argo profiles of temperature and salinity for the period from January 2005 to December 2007 are assimilated into a primitive equation model of the Pacific Ocean using a bias-aware localized ensemble Kalman filter (EnKF) with a sequence of 5-day assimilation cycles. Some other in situ observations, including XBT, TAO/TRITON and CTD profiles, used to supplement, are also assimilated into the model. To improve the assimilation performance, several strategies addressing the computational expense and model error statistics are incorporated into the assimilation scheme. Validation is performed by comparing the analyzed ocean states with independent data, including withheld Argo profiles, satellite remote sensing sea level height anomalies (SLA) and the NCEP ocean state re-analysis products. The results show that the assimilation system is capable of significantly reducing the bias and RMSE of ocean temperature and salinity compared with the control run. It can also improve the simulation of zonal currents and SLAs along the equator, especially during strong ENSO events. In addition, a hybrid coupled ENSO prediction model initialized by the assimilation analysis improves the ENSO prediction skill compared against that initialized by the control run without data assimilation.

Model Representation of Local Air Quality Characteristics

Abstract Daily (24 h) and hourly air quality data at several sites are used to examine the performance of the fifth-generation Pennsylvania State University–NCAR Mesoscale Model (MM5)–Community Multiscale Air Quality Model (CMAQ) system over a 3-month period in 2003. A coarse (36 km) model grid was expected to provide relatively poor performance for ozone and comparatively better performance for fine particles, especially the more regional sulfate and carbonaceous aerosols. However, results were different from this expectation. Modeling showed significant skill for ozone at several locations but very little skill for particulate species. Modeling did poorly identifying surface wind directions associated with the highest and lowest pollutant exposures at most sites, although results varied widely by location. Model skill appeared to be better for ozone when spatial–temporal (S–T) patterns were examined, due in part to the ability of the model to reproduce much of the temporal variance associated with the diurnal photochemical cycle. At some sites the modeling even performed well in replicating the directional variability of hourly ozone despite relatively low spatial resolution. MM5–CMAQ spatial (directional) representation of 24-h-average particulate data was not good in most cases, but model skill improved somewhat when hourly data were examined. Modeling exhibited skill for sulfate at only one of nine sites using 24-h data averaged by daily resultant wind direction, at two of six sites when hourly data were averaged by direction, and at four of six sites when the combined spatial and temporal variance of sulfate was examined. Results were generally poorer for total carbon aerosol mass and total mass of particulate matter with diameter of less than 2.5 μm (PM2.5). The primary result of this study is that an S–T analysis of pollutant patterns reveals model performance insights that cannot be realized by only examining model error statistics as is typically done for regulatory applications. Use of this S–T analysis technique is recommended for better understanding model performance during longer simulation periods, especially when using grids of finer spatial resolution for applications supporting local air quality management studies. Of course, using this approach will require measuring semicontinuous fine particle data at more sites and for longer periods.

A Stable Bayesian Vector Network Analyzer Calibration Algorithm

A new overdetermined vector network analyzer (VNA) calibration algorithm is presented. The new algorithm shows significant advantages in the measurement of very high-impedance devices such as carbon nanotube transistors and can be applied to all types of VNA calibration. It was found that, for high-impedance devices, the new algorithm yields up to four times more accurate results. The focus of this study is on the accuracy and robustness of the algorithm. A statistical error model of calibration, which includes errors in the calibration standards and errors in the VNA, is converted into a formula for calibration by Bayes' theorem. The numerical implementation of this formula makes use of nonlinear optimization techniques and Monte Carlo integration. The resulting new algorithm is compared against various other algorithms. Benchmarking shows that the presented calibration algorithm is robust and more accurate than all other tested algorithms in all tested calibration scenarios.

An adaptive estimation of forecast error covariance parameters for Kalman filtering data assimilation

An adaptive estimation of forecast error covariance matrices is proposed for Kalman filtering data assimilation. A forecast error covariance matrix is initially estimated using an ensemble of perturbation forecasts. This initially estimated matrix is then adjusted with scale parameters that are adaptively estimated by minimizing −2log-likelihood of observed-minus-forecast residuals. The proposed approach could be applied to Kalman filtering data assimilation with imperfect models when the model error statistics are not known. A simple nonlinear model (Burgers’ equation model) is used to demonstrate the efficacy of the proposed approach.

Surface salinity response to changes in the model parameters and forcings in a climatological simulation of the eastern North-Atlantic Ocean

The surface salinity response to changes of various external forcings and model parameters is investigated in a regional 1/3° configuration of the NEMO-OPA model implemented over the eastern North-Atlantic subtropical Ocean. Fourteen realistic climatological simulations are run. By default, neither relaxation to climatological surface salinity nor temperature is included. Forcing fields and parameters expected to impact the surface salinity are modified. These include: the wind stress, the surface temperature, wind speed and relative humidity entering the bulk calculation for sensible and latent heat fluxes, the precipitation, the data specified at open boundaries, the lateral viscosity operator, the salt lateral diffusivity, the vertical mixing scheme, the on/off switch of the double diffusion parameterization, the river runoffs and the SSS/SST restoring terms. The SSS standard deviation over this ensemble of model simulations is of the order of 0.1 psu, which is also the order of the annual cycle of the surface salinity field in this area. In this experimental framework, the wind stress is found to have the largest impact on the model SSS, both in terms of mean field and annual variability. The sensitivity to the precipitation, atmospheric temperature, open boundary external data, and to the relaxation to climatological SSS and SST, is also significant. Of all the model parameterizations, the lateral salt diffusivity is the one associated to the strongest surface salinity model response. This work allows the identification of the main sources of error for model surface salinity. It paves the way for a more comprehensive investigation of SSS model error statistics, which is needed for future data assimilation experiments motivated by the close launch of the new SSS-observing SMOS and Aquarius satellite missions.

Cloud and Precipitation Parameterizations in Modeling and Variational Data Assimilation: A Review

Abstract This paper first reviews the current status, issues, and limitations of the parameterizations of atmospheric large-scale and convective moist processes that are used in numerical weather prediction and climate general circulation models. Both large-scale (resolved) and convective (subgrid scale) moist processes are dealt with. Then, the general question of the inclusion of diabatic processes in variational data assimilation systems is addressed. The focus is put on linearity and resolution issues, the specification of model and observation error statistics, the formulation of the control vector, and the problems specific to the assimilation of observations directly affected by clouds and precipitation.

Hybrid Wordlength Optimization Methods of Pipelined FFT Processors

Quickly and accurately predicting the performance based on the requirements for IP-based system implementations optimizes the design and reduces the design time and overall cost. This study describes a novel hybrid method for the word-length optimization of pipelined FFT processors that is the arithmetic kernel of OFDM-based systems. This methodology utilizes the rapid computing of statistical analysis and the accurate evaluation of simulation-based analysis to investigate a speedy optimization flow. A statistical error model for varying word-lengths of PE stages of an FFT processor was developed to support this optimization flow. Experimental results designate that the word-length optimization employing the speedy flow reduces the percentage of the total area of the FFT processor that increases with an increasing FFT length. Finally, the proposed hybrid method requires a shorter prediction time than the absolute simulation-based method does and achieves more accurate outcomes than a statistical calculation does.

A statistical error model for image sensors and its testing

AbstractSince images obtained from image sensor devices are commonly tested by performing a manual inspection or by means of procedures that involve large amounts of image processing, the growth in testing costs as the size of such devices increases poses a significant problem. Several approaches to automating the testing process have been proposed with a view to reducing testing costs but trial and error is needed to set the parameters required by such test procedures. In this paper we propose an error model for images taken by image sensor devices that can perform a quantitative evaluation with a small number of free parameters. Image quality testing can be performed relatively easily using this model since it replaces a previously ambiguous error model by one that can be determined uniquely. In addition, since the proposed error model is defined statistically in terms of the brightness levels of pixels in each of the individual images, even if there are individual differences between the image sensor devices, these have no influence on the testing procedure. Experimental results demonstrate that an image quality testing based on the proposed error model has the same level of accuracy as manual inspections performed by an experienced technician. © 2007 Wiley Periodicals, Inc. Syst Comp Jpn, 38(11): 1– 11, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20804

Topics in data assimilation: Stochastic processes

Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability. The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified. The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly.

Kernel Density Estimation-Based Data Correlation

Calibration is the process of identifying and correcting the most likely error in sensor measurements. The basis for the authors' calibration procedure is to construct a statistical error model that captures the characteristics of the measurement errors. Such an error model can be constructed either offline or online and is derived using the nonparametric kernel-density-estimation techniques. Models constructed using various forms of the kernel smoothing functions are compared using statistical evaluation methods. Based on the selected error model, they propose four alternatives to make the transition from the error model to the calibration model, which is represented by piecewise polynomials. In addition, statistical validation and evaluation methods such as resubstitution, is used in order to establish the interval of confidence for both the error model and the calibration model. Traces of the acoustic signal-based distance measurements recorded by infield deployed sensors are used as their demonstrative example. Finally, they discuss the broad range of applications of the error models and provide an example on how adopting statistical error model as the optimization objective impacts the accuracy of the location discovery problem in wireless sensor networks