Normal Error Distribution Research Articles

C.F. Gauss and the method of least squares

Gauss introduced the MLSq and Helmert completed its development whereas Bessel made important discoveries in astronomy and geodesy but was often extremely inattentive. Gauss’ final condition of least variance led to effective estimators of the unknowns sought, jointly effective in case of the normal distribution of the observational errors. Gauss’ memoire of 1823 leads to the principle of least squares much easier than generally thought (abstrakt oryginalny)

Multivariate measurement error models using finite mixtures of skew-Student [formula omitted] distributions

In regression models, the classical assumption of normal distribution of the random observational errors is often violated, masking some important features of the variability present in the data. Some practical actions to solve the problem, like transformation of variables to achieve normality, are often of doubtful utility. In this work we present a proposal to deal with this issue in the context of the simple linear regression model when both the response and the explanatory variable are observed with error. In such models, the experimenter observes a surrogate variable instead of the covariate of interest. We extend the classical normal model by jointly modeling the unobserved covariate and the random errors by a finite mixture of a skewed version of the Student t distribution. This approach allows us to model data with great flexibility, accommodating skewness, heavy tails and multi-modality. We develop a simple EM-type algorithm to perform maximum likelihood inference of the parameters of the proposed model, and compare the efficiency of our method with some competitors through the analysis of some artificial and real data.

The Bootstrap Estimation of the Accuracy of Preliminary Orbit Determination

From the point of view of the non-parametric statistics, a general estimation method of the accuracy and confidence interval of preliminary orbit determination is proposed for the occasion without any other information but observational data. Based on the bootstrap method, the estimation relies only on the observational data and does not require the precise orbit determination as a reference, or the assumption of normal distribution of observational errors. Numerical experiments show that this method is very simple in implementa- tion, and may serve as an easy accuracy evaluation for the preliminary orbit determination and for the follow-up employments.

Probabilistic Model of Payment Cost Minimization Considering Wind Power and Its Uncertainty

The penetration of wind energy sources to power systems has significantly increased in recent years. With variable and uncertain wind power output, the payment and market-clearing price (MCP) may vary in different cases. In this paper, a methodology to quantitatively model the payment cost minimization (PCM) considering the effects of wind power from a probabilistic viewpoint is presented. The autoregressive moving average (ARMA) method with normal distribution of wind forecast error is used to model a time series of wind speed. Based on the wind turbine power curve, the probability distribution of wind power output can be obtained. Then, Monte Carlo simulation (MCS) is used to produce random samples of wind speed, and the genetic algorithm is applied to solve PCM for each sample. The proposed methodology and its solution are verified with simulation studies of two sample systems. The probabilistic distribution results can give consumers an overview of how much they should pay in a probabilistic sense. Further, the simulation results can serve as a lookup table to provide useful input for more refined unit commitment, and also provide a benchmark for future research works on PCM considering wind power.

Energy savings from using mobile smart technologies

This paper presents the most recent results of energy saving benefits from the convergence of consumer products into a multi-function smart device, such as a smartphone or tablet, compared to single-function products (e.g., an electronic clock). Although individual users are predominantly driven only by the marked trends and the individual energy savings are moderate, the sheer number of users makes this global trend promising and a model for sustainable energy. The energy consumption of selected smart devices was tested using 42 frequently used applications and utilities (e.g., portable gaming, scanning, music players, etc.). A range of Operating Systems (OS) were selected for testing: Android OS (Google), Blackberry OS (RIM), iOS (Apple), and Windows 7 OS (Microsoft). Testing was conducted using the following smartphones: Blackberry Curve 9300, iPhone 4, Samsung Focus, Samsung Galaxy S; and tablets: iPad 2 and Samsung Galaxy Tablet 7″. In order to investigate the battery consumption, two programs were developed in-house for iOS and Android platforms. For the Windows phone, the built-in program was utilized, while a third-party application was used for the Blackberry phone. To further test the battery, each device was also decoupled from its battery and hooked up through a measuring device, and experiments were then repeated. This study shows smart devices save consumers up to $150 annually compared to single-function devices in a range of applications. The results from this study demonstrate how smart devices are transforming our society by providing superior functionality at a lesser cost in energy, paving the way for other technologies to be integrated into smart devices, and energy reduction to continue. The error in measurement was estimated using standard techniques utilizing 95% confidence scoring for each of the measurements. According to these results that assume normal distribution of measurement error, we show that our measurements were done correctly and the results are significant to within margin of error of ±0.0249 and ±0.0775 for intrusive measurements for smart phones and tablets and for non-intrusive measurements for smart phones and tablets, respectively, of ±0.497 and ±0.325.

Association of elevation error with surface type, vegetation class and data origin in discrete-returns airborne LiDAR

Airborne LiDAR (light detection and ranging) data are now commonly regarded as the most accurate source of elevation data for medium-scale topographical modelling applications. However, quoted LiDAR elevation error may not necessarily represent the actual errors occurring across all surfaces, potentially impacting the reliability of derived predictions in Geographical Information Systems (GIS). The extent to which LiDAR elevation error varies in association with land cover, vegetation class and LiDAR data source is quantified relative to dual-frequency global positioning system survey data captured in a 400-ha area in Ireland, where four separate classes of LiDAR point data overlap. Quoted elevation errors are found to correspond closely with the minimum requirement recommended by the American Society of Photogrammetry and Remote Sensing for the definition of 95% error in urban areas only. Global elevation errors are found to be up to 5 times the quoted error, and errors within vegetation areas are found to be even larger, with errors in individual vegetation classes reaching up to 15 times the quoted error. Furthermore, a strong skew is noted in vegetated areas within all the LiDAR data sets tested, pushing errors in some cases to more than 25 times the quoted error. The skew observed suggests that an assumption of a normal error distribution is inappropriate in vegetated areas. The physical parameters that were found to affect elevation error most fundamentally were canopy depth, canopy density and granularity. Other factors observed to affect the degree to which actual errors deviate from quoted error included the primary use for which the data were acquired and the processing applied by data suppliers to meet these requirements.

A Simulation Study to Evaluate Accuracy and Precision of Blinded Independent Central Reviews of Progression-free Survival in Cancer Clinical Trials

Purpose: To model the effect of accuracy and precision on hazard ratios, sample size and overall trial cost in Progression-Free Survival (PFS) trials using different Blinded Independent Review (BICR) paradigms. Method: PFS times were simulated based on a tumor growth model using a median 180 days in the treatment arm; the control arm varied according to Hazard Ratios (HR) from 0.7-0.85. False positives were added for varying accuracy (1-False positive rate) and a log normal distribution of measurement error was used for varying precision. Local Evaluation (LE) accuracy of 70% and measurement error’s standard deviation of 0.30 were compared to different BICR paradigms with varying accuracy and precision (i.e., accuracy =70,90% & precision .30,.25,.20). Results: Compared to LE, all BICR paradigms decreased overall trial costs by $ 0.0037 - 26.6.3×106 and sample sizes by 12-435 with effect magnitude being greatest at higher HRs, higher accuracy and lower measurement error. Conclusion: Our study indicates that for trials with PFS based on a radiological assessment, BICR can be a cost-effective strategy by decreasing sample sizes and trial costs. More importantly, the study provides a quantitative indication of how changing accuracy and precision can alter sample size projections and trial cost.

Time-varying betas in Central and Eastern European markets: a bivariate BEKK GARCH approach

In this paper, we analyse country risk of eight Central and Eastern European (CEE) countries by calculating time-varying betas. We have used daily closing prices of indices from 3 June 2002 through 2 December 2011, resulting in 2021 observations. The time-varying betas where calculated by applying multivariate GARCH BEKK models under multivariate normal distribution of errors. Results indicate that BEKK models are appropriate on estimating time-varying beta for Czech Republic, Hungary, Estonia, Lithuania, Poland, Romania, and Russia, except for Latvia. Moreover, the behaviour of time-varying betas during the crisis differs between CEE markets. Countries with flat exchange rates seem to eliminate some of the country risk. For Romania and Hungary, time-varying beta increase was eliminated due to assistance of the International Monetary Fund during the crisis.

Estimating reaction model parameter uncertainty with Markov Chain Monte Carlo

Predicting the performance of chemical reactions with a mechanistic model is desired during the development of pharmaceutical and other high value chemical syntheses. Model parameters usually must be regressed to experimental observations. However, experimental error may not follow conventional distributions and the validity of common statistical assumptions used for regression should be examined when fitting mechanistic models.This paper compares different techniques to estimate parameter confidence for reaction models encountered in pharmaceutical manufacturing, simulated with either normally distributed or experimentally measured noise. Confidence intervals were calculated following standard linear approaches and two Markov Chain Monte Carlo algorithms utilizing a Bayesian approach to parameter estimation: one assuming a normal error distribution, and a new non-parametric likelihood function. While standard frequentist approaches work well for simpler nonlinear models and normal distributions, only MCMC accurately estimates uncertainty when the system is highly nonlinear, and can account for any measurement bias via customized likelihood functions.

Relative efficiency of ols and lav estimators

A new approach is proposed to comparing ordinary least squares and least absolute value methods by estimating the relative efficiency of sample mean and median. The results confirm the previous conclusions concerning the relative efficiency of these methods for estimating the location parameters, which have been obtained by other authors using more time-consuming techniques. The proposed approach is applied to the common cases of normal and Laplace error distributions, but it can be extended also to other distributions.

Factors affecting hospital admission and recovery stay duration of in-patient motor victims in Spain

Hospital expenses are a major cost driver of healthcare systems in Europe, with motor injuries being the leading mechanism of hospitalizations. This paper investigates the injury characteristics which explain the hospitalization of victims of traffic accidents that took place in Spain. Using a motor insurance database with 16,081 observations a generalized Tobit regression model is applied to analyse the factors that influence both the likelihood of being admitted to hospital after a motor collision and the length of hospital stay in the event of admission. The consistency of Tobit estimates relies on the normality of perturbation terms. Here a semi-parametric regression model was fitted to test the consistency of estimates, concluding that a normal distribution of errors cannot be rejected. Among other results, it was found that older men with fractures and injuries located in the head and lower torso are more likely to be hospitalized after the collision, and that they also have a longer expected length of hospital recovery stay.

江戸共通語の可能性の模索

This paper presents a vehicle motion predictor for an autonomous racing system to assure safety when driving in limit handling situation. The proposed algorithm consists of two sequential parts; (1) Tire model identifier (2) Prediction of vehicle state. In the tire model identifier, parameters of a nonlinear tire model are identified through nonlinear optimization. In the prediction of vehicle sate, future vehicle states are calculated through numerical integration of a 2- DOF bicycle model that includes previously identified tire model. Moreover, assuming normal distribution of prediction error of each state, the probability of lateral instability and track escape of the future can be calculated. The proposed algorithm has been validated through computer simulations. The results show that the algorithm well predicts future vehicle states.

How to optimize intracardiac blood flow tracking by echocardiographic particle image velocimetry? Exploring the influence of data acquisition using computer-generated data sets

Echocardiographic particle image velocimetry (EPIV) has been used for tracking contrast-enhanced intracavitary blood flow. Little is known, however, how basic imaging parameters (line density, frame rate, contrast bubble density) affect the quality of such tracking results. Our study aimed at investigating this by using simulated echo data sets. A computational three-dimensional (3D) blood flow field of the left ventricle (LV) was built using Fluent 12.1 (ANSYS Inc., USA). Then, the 3D motion of contrast microbubbles was simulated and 2D B-mode image loops were obtained (f = 4.5 MHz; 50 sector angle) and analysed using flow tracking software (Omega Flow, Siemens, USA). Vorticity and the resulting in-plane velocity vector field was calculated at different frame rates (227, 113, 76, and 57 fps) and bubble densities (100, 63, 36, 19, 10, and 3 bubbles/mL) and compared with the ground truth known from the computational LV flow model. The normal distribution of the amplitude error and angle error histograms confirmed the overall good performance of the tracking method. In the standard deviation analysis of error histograms, tracked velocity amplitudes correlated best with the ground truth at 10 bubbles/mL and 227 fps (45.81 ± 3.43%, P < 0.05), while the best performance of flow direction estimates was at 10 bubbles/mL and 76 fps (25.41 ± 1.22°, P < 0.05). The correlation of estimated and true vorticity tended to grow with increasing frame rate and was optimal at 19 bubbles/mL and 113 fps (r = 0.79 ± 0.02). To achieve accurate vorticity measurements, frame rate acquisitions as 113 fps and contrast bubble density of 19 bubbles/mL are needed.

Maximum sustainable yield estimates of Ladypees, Sillago sihama (Forsskål), fishery in Pakistan using the ASPIC and CEDA packages

Using surplus production model packages of ASPIC (a stock-production model incorporating covariates) and CEDA (Catch effort data analysis), we analyzed the catch and effort data of Sillago sihama fishery in Pakistan. ASPIC estimates the parameters of MSY (maximum sustainable yield), F msy (fishing mortality), q (catchability coefficient), K (carrying capacity or unexploited biomass) and B1/K (maximum sustainable yield over initial biomass). The estimated non-bootstrapped value of MSY based on logistic was 598 t and that based on the Fox model was 415 t, which showed that the Fox model estimation was more conservative than that with the logistic model. The R 2 with the logistic model (0.702) is larger than that with the Fox model (0.541), which indicates a better fit. The coefficient of variation (cv) of the estimated MSY was about 0.3, except for a larger value 88.87 and a smaller value of 0.173. In contrast to the ASPIC results, the R 2 with the Fox model (0.651–0.692) was larger than that with the Schaefer model (0.435–0.567), indicating a better fit. The key parameters of CEDA are: MSY, K, q, and r (intrinsic growth), and the three error assumptions in using the models are normal, log normal and gamma. Parameter estimates from the Schaefer and Pella-Tomlinson models were similar. The MSY estimations from the above two models were 398 t, 549 t and 398 t for normal, log-normal and gamma error distributions, respectively. The MSY estimates from the Fox model were 381 t, 366 t and 366 t for the above three error assumptions, respectively. The Fox model estimates were smaller than those for the Schaefer and the Pella-Tomlinson models. In the light of the MSY estimations of 415 t from ASPIC for the Fox model and 381 t from CEDA for the Fox model, MSY for S. sihama is about 400 t. As the catch in 2003 was 401 t, we would suggest the fishery should be kept at the current level. Production models used here depend on the assumption that CPUE (catch per unit effort) data used in the study can reliably quantify temporal variability in population abundance, hence the modeling results would be wrong if such an assumption is not met. Because the reliability of this CPUE data in indexing fish population abundance is unknown, we should be cautious with the interpretation and use of the derived population and management parameters.

Robust M- and L-Estimators of Scale Parameter

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.

Natural error patterns enable transfer of motor learning to novel contexts

Successful behavior demands motor learning to be transferable in some cases (e.g., adjusting walking patterns as we develop and age) and context specific in others (e.g., learning to walk in high heels). Here we investigated differences in motor learning transfer in people learning a new walking pattern on a split-belt treadmill, where the legs move at different speeds. We hypothesized that transfer of the newly acquired walking pattern on the treadmill to natural over ground walking might depend on the pattern of errors experienced during learning. Error patterns within a person's natural range might be experienced as endogenous (i.e., produced by the body), encouraging general adjustments that transfer across contexts. On the other hand, larger errors might be experienced as exogenous (i.e., produced by the environment), indicating unusual conditions requiring context-specific learning. To test this, we manipulated the distribution of errors experienced during learning to lie either within or outside the normal distribution of walking errors. We found that restriction of errors to the natural range produced transfer of the new walking pattern from the treadmill to natural walking off the treadmill, while larger errors prevented transfer. This result helps explain how transfer of motor learning is controlled, and it offers an important strategy for clinical rehabilitation, where transfer of motor learning to other contexts is essential.

Uncertainties in modeling hazardous gas releases for emergency response

In case of an accidental release of toxic gases the emergency responders need fast information about the affected area and the maximum impact. Hazard distances calculated with the models MET, ALOHA, BREEZE, TRACE and SAMS for scenarios with chlorine, ammoniac and butane releases are compared in this study. The variations of the model results are measures for uncertainties in source estimation and dispersion calculation. Model runs for different wind speeds, atmospheric stability and roughness lengths indicate the model sensitivity to these input parameters. In-situ measurements at two urban near-traffic sites are compared to results of the Integrated Nowcasting through Comprehensive Analysis (INCA) in order to quantify uncertainties in the meteorological input. The hazard zone estimates from the models vary up to a factor of 4 due to different input requirements as well as due to different internal model assumptions. None of the models is found to be 'more conservative' than the others in all scenarios. INCA wind-speeds are correlated to in-situ observations at two urban sites in Vienna with a factor of 0.89. The standard deviations of the normal error distribution are 0.8 ms ―1 in wind speed, on the scale of 50 degrees in wind direction, up to 4°C in air temperature and up to 10 % in relative humidity. The observed air temperature and humidity are well reproduced by INCA with correlation coefficients of 0.96 to 0.99. INCA is therefore found to give a good representation of the local meteorological conditions. Besides of real-time data, the INCA-short range forecast for the following hours may support the action planning of the first responders.

A bootstrap algorithm for deriving the archeomagnetic field intensity variation curve in the Middle East over the past 4 millennia BC

Recent compilations of archeomagnetic intensity data have shown the existence of outliers. Acknowledging this problem, we propose a new way of generating and studying the uniqueness of regional archeomagnetic master curves bypassing the need to assume a normal data error distribution. Our approach lessens the weight of outliers by applying an iteratively re‐weighted least‐squares method combined with a bootstrap algorithm. Given a particular set of archeomagnetic data associated with experimental and dating uncertainties, we produce an ensemble of curves that are designed to sample the conditional probability distribution of the ‘true’ master curve. Using this technique, we propose an archeointensity master curve with its probability density for the Middle East region that covers the past four millennia BC.

New Calibration Methods with Asymmetric Data

In this paper, two new inverse regression methods are introduced. One is a distance based method, and the other is a likelihood based method. While a model is fitted by minimizing the sum of squared prediction errors of y's and x's in the classical and inverse methods, respectively. In the new distance based method, we simultaneously minimize the sum of both squared prediction errors. In the likelihood based method, we propose an inverse regression with Arnold-Beaver Skew Normal(ABSN) error distribution. Using the cross validation method with an asymmetric real data set, two new and two existing methods are studied based on the relative prediction bias(RBP) criteria.

CITT and inverse analyses applied to the study of the mushroom drying process

A study and modeling of the mass transfer phenomenon in the drying process of agricultural products, using Fick’s second law of diffusion adapted to infinite flat plate geometry, are described. An analytical solution methodology, the classical integral transform technique (CITT), is employed. Predicted results are compared with experimental data on the drying of sliced mushrooms in different operating conditions. A phenomenological analysis is made using graphics and tables. The data obtained by CITT and experimentally show negligible differences. The mass diffusion coefficient during hot air drying of mushrooms is estimated. Experimental drying kinetics are applied to the product at different air temperatures and air flow rates aiming to find a solution for the mass transfer equation, using an inverse problem with the Levenberg–Marquardt optimization technique in successive trials. Experimental data are also modified by a normal distribution of random errors. Statistical analyses show good agreement between experimental and predicted curves.