Parameters Of Normal Distribution Research Articles

Statistical Goodness Factor ‘ᴦ’ for Image Fusion Algorithm Based on UGGD Parameters

In this paper we propose a novel pyramid decomposition based Image fusion metric, Gamma Factor or Goodness of Fit ‘ᴦ’ which describes the statistically amount of information fused by the image fusion algorithm. We first apply steerable pyramid decomposition and then a fitting model for Univariate Generalised Gaussian Distribution (UGGD) parameter estimation. From the UGGD; P and S fitting model coefficients are computed. To estimate the optimum weights for computation a huge data set of complimentary images are used. Using these weights, amount of information contributed by each image to form a fused image can be estimated. Experimental results show the tremendous matching with the quantise information

Comparison of parameter estimation methods for normal inverse Gaussian distribution

This paper compares several methods for estimating parameters of normal inverse Gaussian distribution. Ordinary maximum likelihood estimation and the method of moment estimation often do not work properly due to restrictions on parameters. We examine the performance of adjusted estimation methods along with the ordinary maximum likelihood estimation and the method of moment estimation by simulation and real data application. We also see the effect of the initial value in estimation methods. The simulation results show that the ordinary maximum likelihood estimator is significantly affected by the initial value; in addition, the adjusted estimators have smaller root mean square error than ordinary estimators as well as less impact on the initial value. With real datasets, we obtain similar results to what we see in simulation studies. Based on the results of simulation and real data application, we suggest using adjusted maximum likelihood estimates with adjusted method of moment estimates as initial values to estimate the parameters of normal inverse Gaussian distribution.

Asymptotically Unbiased Estimation of Mean and Standard Deviation in the Presence of Outlying Errors

In this work, we develop a new method for estimating the mean and standard deviation of normally distributed populations when errors exist in the sample. The proposed estimators are asymptotically unbiased if all errors are outlying errors (i.e. all errors lie outside a region defined by the parameters of normal distribution). However, the proposed method requires having an upper bound for the percentage of errors in the sample. The proposed method is compared with the already existing and commonly used estimators in terms of bias, mean square error, and Pitman closeness criterion. The proposed method is found to be superior to the others when the sample size is not too small. A simulation study is designed to test the performance of proposed estimators when most but not all errors are outlying errors. The findings of the simulation study also indicate the superiority of the proposed method when the sample size is moderately large. As an application, we use our method in Phase I of designing a control chart to improve its performance. We apply the method on a dataset where the robustness of our proposed method is tested (and compared with the other estimators) against the presence of outlying errors in Phase I data. In the findings of the application, we notice that proposed estimators were the only ones that identified out of control data points in Phase II when Phase I samples are contaminated with the errors.

Description of the statistical variations of the measured strength for brittle ceramics: A comparison between two-parameter Weibull distribution and normal distribution

It is generally assumed that the measured strength of brittle ceramics follows a Weibull distribution. However, there seems to be few sound and direct evidences to support this assumption. Several previous studies have shown that other distributions, such as normal distribution and log-normal distribution may describe more appropriately the strength data than Weibull distribution. In this paper, the efficiency of using a normal distribution to describe the strength which follows a Weibull distribution is examined based on Monte-Carlo simulations. It was shown that there exist strong correlations between the parameters of normal distribution and those of Weibull distribution. For the designed fracture probability not lower than 0.01, analyses based on both normal distribution and Weibull distribution may give nearly identical predictions for the applicable stress levels. For lower fracture probabilities, the differences between the predictions of both distributions are not significant. It was suggested that, if there is no evidence to confirm that the measured strength follows a certain distribution, normal distribution and Weibull distribution seem to have the same efficiency in analysing the statistical variations in the measured strength of ceramics.

Multiple Chains Hidden Markov Models for Bivariate Dynamical Systems

We present a new modelling framework for the bi-variate hidden Markov model. The proposed specification is composed by five latent Markovian chains which drive the evolution of the parameters of a bi-variate Gaussian distribution. The maximum likelihood estimator is computed via an expectation conditional maximization algorithm with closed form conditional maximization steps, specifically developed for our model. Identification of model parameters, as well as consistency and asymptotic Normality of the maximum likelihood estimator are discussed. Finite sample properties of the estimator are investigated in an extensive simulation study. An empirical application with the bi-variate series of US stocks and bond returns illustrates the benefits of the new specification with respect to the standard hidden Markov model.

Statistics investigation of mechanical parameter of rock based on Bayesian reliability theory

In order to improve forecast accuracy of bearing capacity of rock, the forecast method of bearing capacity of rock was constructed on Jeffreys prior using MCMC method of Bayesian theory. The proposed approach was used to estimate the parameters of Normal distribution. Numerical simulation study was used to produce the pseudo samples. The maximum likelihood parameter estimation method and Bayesian statistical theory were used to estimate the forecast value of the Normal distribution, which has been done by comparing with the theoretical value of the pseudo sample of Normal distribution. The result indicates that the forecast model of Normal distribution is optimal than the maximum likelihood estimation, and a significant improvement were noticed with the number increasing of pseudo sample. At last, the proposed method was applied to estimate the Normal distribution of bearing capacity of rock, which shows that the proposed method in this paper has higher precision and good applicability.

Robust estimation and its application to a classification problem

In the article, a classification problem with two distributed classes is considered. The problem is solving using empirical discriminant functions for Gaussian classifier and estimators for unknown parameters of multivariate normal distribution. The three etimators, maximum likelihood estimator, Kulawik-Zontek estimator and minimum covariance determinant estimator, are compared in two different empirical examples (small size sample and large size sample).

Numerical modelling of anomalous microseismicity influenced by lithological heterogeneity in longwall top coal caving mining

Mining-induced microseismicity has been extensively used to evaluate the potential for rock bursts and coal and gas outbursts in underground coal mines. In a research project completed a few years ago, it was observed that characteristics of microseismicity around a working longwall panel were fairly consistent over the monitoring period until a lithological heterogeneity zone with a relatively high coal strength was reached. The current research presented in this paper aims at achieving a better understanding of the effect of lithological heterogeneity on microseismic activity in longwall coal mining. The heterogeneous zone inferred from the tomography measurements was first digitalised and implemented into a 3D geomechanical model. A microseismicity modelling approach which combines deterministic stress and failure analysis together with a stochastic fracture slip evaluation was used to simulate the evolution of microseismicity induced by the progressive face advance passing through the heterogeneous zone. The heterogeneity was taken into account by varying the material strength and the fracture attributes of the elements within the high strength zone. Results have shown that both the high rock strength of coal lithotype and low power law scaling exponent of fractures within this zone contribute to the reduction in fitted b values from frequency-magnitude distribution of microseismicity and the increase in fitted Gaussian distribution parameters to the logarithmic event energy. These deviations are believed to result from the combined effects of increased stress drops and slipped fracture sizes when the heterogeneous zone is approached.

Sensitivity analysis of the critical speed in a railway bogie system with uncertain parameters

An extended Dichotomy method is proposed for an accurate determination of the critical speed of railway vehicles. The high-dimensional Hermite orthogonal polynomial (HOP) based on the independent and normal distribution parameters is derived. Both the Monte Carlo (MC) method based on a Latin hypercube sampling (LHS) and the HOP expansion method using different collocation methods are compared to investigate the statistical characteristics of the critical speed of a Chinese railway bogie. After that, the relation between the critical speed and the random input parameters is approximately constructed. The relation is then used to conduct a relative local sensitivity analysis (LSA), a global sensitivity analysis (GSA) and a regional sensitivity analysis (RSA) of the bogie. It is indicated that RSA is a good method for sensitivity analysis of the critical speed due to its good efficiency and trusty precision. The results also show that the secondary lateral damper is the most important parameter to the critical speed for this bogie. By increasing its values and reducing its variation range, the critical speed can be increased and the variation is decreased. It may offer some instructions and practical values for vehicle design, operation, and comprehensive evaluation of the stability of railway vehicles.

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI.

Characterizing the System Impulse Response Function From Photon-Counting LiDAR Data

NASA’s Multiple Altimeter Beam Experimental LiDAR (MABEL) is an aircraft-based photon-counting laser altimeter designed as a simulator to test measurement techniques and algorithms for Advanced Topographic Laser Altimeter System (ATLAS), the sole instrument on NASA’s Ice, Cloud, and land Elevation Satellite-2 (ICESat-2) mission. By measuring the time of flight, pointing angle, and absolute position for individual photons, ICESat-2 provides detailed elevation measurements of earth’s surface. Calculating accurate and precise elevations requires an understanding of how photons interact with surfaces, and characterization of the photon distribution after returning from surfaces. Neither MABEL nor ATLAS records the transmitted laser pulse shape, relying instead on aggregating several pulses worth of photons, often using histograms, to characterize the pulse shape. In this paper, we assess the limitations of using histograms and propose a more robust method to describe MABEL’s system impulse-response function using an exponentially modified Gaussian distribution. We also provide standard error estimates for the arithmetic mean and standard deviation calculations, and for exponentially modified Gaussian parameters using a Monte Carlo sensitivity analysis. We apply this method to photon returns from a sea ice lead and from a dry salt lake bed as case studies for estimating the standard error associated with sample size for the arithmetic mean and standard deviation, and for the exponentially modified Gaussian parameters. We use these standard errors to calculate the minimum number of photons required to find both Gaussian and exponentially modified Gaussian distribution parameters within 3 cm of their parent population values.

Statistical Analysis of Location Parameter of Inverse Gaussian Distribution Under Noninformative Priors

Bayesian estimation for location parameter of the inverse Gaussian distribution is presented in this paper. Noninformative priors (Uniform and Jeffreys) are assumed to be the prior distributions for the location parameter as the shape parameter of the distribution is considered to be known. Four loss functions: Squared error, Trigonometric, Squared logarithmic and Linex are used for estimation. Bayes risks are obtained to find the best Bayes estimator through simulation study and real life data

An improved robust image-adaptive watermarking with two watermarks using statistical decoder

This paper presents an improved image-adaptive watermarking technique. Two image watermarks are embedded in the high entropy 8 × 8 blocks of the host image. DWT is applied on these blocks using the principle of sub band coding. This decomposes the high entropy blocks into four sub band coefficients, wherein the approximation and vertical frequency coefficients are modeled using Gaussian (or Normal) distribution. The two watermarks are inserted in the host image using Adjustable Strength Factor (ASF). It is calculated adaptively using the fourth statistical moment known as kurtosis. A limited side information is also transmitted along with the watermarked image. This side information consists of high entropy block positions and Gaussian distribution parameters. To extract both watermarks from the received watermarked image, the high entropy block positions sent in the side information help in applying DWT to calculate the approximation and vertical frequency coefficients. Gaussian (or Normal) distribution is similarly used for modeling and calculating the distribution parameters. This helps the Maximum Likelihood (ML) decoder to recover the watermarks successfully using a statistical approach. Two important contributions are presented in this paper. Firstly, adjustable kurtosis values are used which improves the capacity and robustness of the proposed technique. Secondly, the proposed work is implemented on medical applications and gives better performance as compared to the existing methods. Further, the efficiency of the proposed work is evaluated by better simulation results using PSNR, NCC, SSIM and GMSD under different attacks. The technique is highly robust as watermarks survive under different attacks. This increases security and ensures copyright protection.

Modeling of probability in obtaining intensity-duration-frequency relationships of rainfall occurrence for Pelotas, RS, Brazil

ABSTRACT Based on historical series, for each locality, equations can characterize the relationship between intensity, duration and frequency of rainfall occurrence. The objective of this study was to present two equations that can describe the occurrence of intense rainfall in Pelotas, RS state, over the period 1982-2015. The two equations were denominated conventional and hybrid, depending on the probabilistic model used. Following the conventional methodology, the parameters of Normal, Log-Normal, Gumbel and Gamma probability distributions were adjusted by the maximum likelihood method for return periods of 2, 5, 10, 20, 25, 50 and 100 years. The maximum intensity values for the hybrid equation were obtained using the empirical model of Weibull, considering return periods of 2, 5, 10, 20 and 25 years. On the other hand, the same theoretical distributions used in the conventional equation were applied to return periods of 50 and 100 years. The Kolmogorov-Smirnov test was used to select the best fitting distribution for the data. In order to verify the information acquired through the Weibull empirical model in comparison to the theoretical distributions, the t-test was applied to the angular coefficients. Significant differences were not verified between the values of maximum rainfall intensities obtained using the two methodologies, for the pre-established durations and return periods. Thus, considering the maximum rainfall intensities values (durations of 5-1440 min) and return periods of 2-100 years in the municipality of Pelotas, RS, Brazil, both the hybrid and the conventional intense rainfall equations can be used.

Including edge preserving smoothing filter within blocky- constrained, target-oriented AVA inversions

Resolving thin layers and achieve focused layer boundaries is one of the major challenges in seismic inversion. This translates into recovering a blocky solution with sparse spatial derivatives of model parameters. Here, we present two iterative focusing regularization techniques for target-oriented Amplitude Versus Angle (AVA) inversion. Target-oriented means that only the AVA responses of the reflections of interest are inverted for the simultaneous estimation of P-wave, S-wave and density reflectivities. The first approach imposes Cauchy constraints on the spatial model derivatives whereas, the second is inspired by the minimum gradient support regularization. Both the implemented algorithms enhance their focusing and edge-preserving abilities by exploiting an Edge Preserving Smoothing (EPS) filter that is used to compute both the model constraints and the model update. We include a-priori model information into the inversion kernel to guide the convergence of the algorithms toward physically plausible solutions. The two approaches are compared against the standard Bayesian inversion that simply considers Gaussian distributed model parameters, and with the well-known edge-preserving method that assumes Cauchy-distributed derivative of model parameters. For the lack of available field seismic data, we limit the attention to synthetic inversion experiments in which we simulate different signal-to noise (S/N) ratios. The inversion tests prove the suitability of the two proposed approaches for target-oriented AVA inversion and demonstrate their focusing and anti-noise abilities. In particular, the two implemented algorithms outperform the standard Bayesian inversion and the Cauchy approach in cases of low S/N ratios. The two implemented methods are also extremely flexible and can be applied to other linear or non-linear geophysical inverse problems.

Effects of modified posteromedial approach combined raft technique for posterior Pilon fractures with collapsed articular surface

Objective: To evaluate the effects of modified posteromedial approach combined raft technique in the treatment of posterior Pilon fractures with collapsed articular surface. Methods: A retrospective analysis was conducted on the clinical data of 51 patients with posterior Pilon fractures combined with collapsed articular surface treated in West China Hospital between January 2014 and June 2017. There were 30 males and 21 females with an average age of 49 years (range, 19-66 years).Subjective assessment of the results was performed according to the pain visual analog scale (VAS).The general function recovery was assessed with American Orthopaedic Foot & Ankle Society (AOFAS) ankle hindfoot scale at the latest follow-up. The fracture reduction quality was determined according to Burwell-Charnley imaging standard by the anteroposterior(AP)/mortise X-ray view and three-dimensional CT of the ankle on the 3rd day after operation. The t-test was used to compare the normal distribution parameters. Results: Primary healing of incision and bone were obtained in all patients without loosening, breakage or infection of internal fixation in (3.7±0.7) months (range, 3-6 months). According to the Burwell-Charnley's standards, 50 cases achieved anatomical reduction, 1 case achieved fair reduction. According to AOFAS scales, the results were excellent in 41 cases, good in 9, and fair in 1; the excellent rate was 98.0%. According to the AOFAS scores, the average score was 93.6±2.2. The preoperative VAS score was 7.5±1.1, and the postoperative score was 0.3±0.7, in which the difference was statistically significant (t=31.231, P=0.000). At the last follow-up, the angle of injured side were 13.3°±3.4° in dorsal extension, 33.5°±4.7° in plantar flexion, and 46.9°±6.1° in the range, while angel of uninjured side were 19.8°±2.3°, 36.0°±5.7° and 55.6°±2.7°, respectively; there were significant differences in the up-mentioned indexes between the both sides (t=78.932, 121.231, 113.432, all P<0.05). Conclusion: Excellent short-term effectiveness can be achieved through the modified medial approach in patients with posterior Pilon fractures, which can be used to restore the ankle joint surface under direct vision; and with the raft technique, a small plate is used to fix small fragments firmly with less soft tissue complications.

Methodology to Evaluate Fatigue Damage of High-Speed Train Welded Bogie Frames Based on On-Track Dynamic Stress Test Data

The current method of estimating the fatigue life of railway structures is to calculating the equivalent stress amplitude based on the measured stress data. However, the random of the measured data is not considered. In this paper, a new method was established to compute the equivalent stress amplitude to evaluate the fatigue damage based on the measurable randomness, since the equivalent stress is the key parameter for assessment of structure fatigue life and load derivation. The equivalent stress amplitude of a high-speed train welded bogie frame was found to obey normal distribution under uniform operation route that verified by on-track dynamic stress data, and the proposed model is, in effect, an improved version of the mathematical model used to calculate the equivalent stress amplitude. The data of a long-term, on-track dynamic stress test program was analyzed to find that the normal distribution parameters of equivalent stress amplitude values differ across different operation route. Thus, the fatigue damage of the high-speed train welded bogie frame can be evaluated by the proposed method if the running schedule of the train is known a priori. The results also showed that the equivalent stress amplitude of the region connected to the power system is more random than in other regions of the bogie frame.

Dynamic Multi-Factor Bid-Offer Adjustment Model: A Feedback Mechanism for Dealers (Market Makers) to Deal (Grapple) with the Uncertainty Principle of the Social Sciences

The objective is to come up with a model that alters the Bid-Offer, currently quoted by market makers, that varies with the market and trading conditions. The dynamic nature of financial markets and trading, as the rest of social sciences, where changes can be observed and decisions can be taken by participants to influence the system, means that our model has to be adaptive and include a feedback loop that alters the bid offer adjustment based on the modifications we are seeing in the market and trading conditions, without a significant time delay. We will build a sample model that incorporates such a feedback mechanism and also makes it possible to check the efficacy of the changes to the quotes being made, by gauging the impact on the Profits.The market conditions here refer to factors that are beyond the direct control of the market maker and this information is usually available publicly to other participants. Trading conditions refer to factors that can be influenced by the market maker and are dependent on the trading book being managed and will be privy only to the market maker and will be mostly confidential to others. The factors we use to adjust the spread are the price volatility, which is publicly observable; and trade count and volume, which are generally only known to the market maker, in various instruments over different historical durations in time. The contributions of each of the factors to the bid-offer adjustment are computed separately and then consolidated to produce a very adaptive bid-offer quotation. The ensuing discussion considers the calculations for each factor separately and the consolidation in detail. Any model that automatically updates the quotes is more suited for instruments that have a high number of transactions within short intervals, making it hard for traders to manually monitor and adjust the spread; though this is by no means a stringent requirement. We can use similar models for illiquid instruments as well and use the quotations provided by the model as a baseline for further human refinement. We have chosen the currency markets to build the sample model since they are extremely liquid, Over the Counter (OTC), and hence trading in them is not as transparent as other financial instruments like equities. The nature of currency trading implies that we do not have any idea on the actual volumes traded and the number of trades. We simulate the number of trades and the average size of trades from a log normal distribution. The parameters of the log normal distributions are chosen such that the total volume in a certain interval matches the volume publicly mentioned by currency trading firms. This methodology can be easily extended to other financial instruments and possibly to any product with an ability to make electronic price quotations or even be used to periodically perform manual price updates on products that are traded non-electronically.Thankfully, we are not at a stage where Starbucks will sell coffee using such an algorithm, since it can possibly lead to certain times of the day when it can be cheaper to have a cup of coffee and as people become wary of this, there can be changes to their buying habits, with the outcome that the time for getting a bargain can be constantly changing; making the joys of sipping coffee, a serious decision making affair.

Distribution-free estimation of f(E) in the distributed activation energy model based on matrix singular value decomposition method

The distribution activation energy model (DAEM) is commonly used for characterizing pyrolysis kinetics of different types of thermally active materials such as coal, biomass, charcoal, polymer, and solid waste. In this work, a new distribution-free method for estimating the activation energy distribution function f(E) in DAEM is developed by establishing a physical and mathematical analogy between DAEM and the integral equation in adsorption. A parameter sensibility analysis of the present method and algorithms for calculating f(E) at different Gaussian distribution parameter, heating rates and pre-exponential factors shows that the present method gives f(E) that leads to predictions of conversion in good agreement with the test data, and applicable to pyrolysis analysis of thermally active solid materials.

Spreadsheet for evaluation of global risks in conformity assessment of a multicomponent material or object

A user-friendly MS-Excel spreadsheet is developed for evaluation of global consumer's and producer's risks in conformity assessment of chemical composition of a multicomponent material or object, when up to four component concentrations are under control. These risks are probabilities of incorrect conformity decisions related to a material batch (lot or similar) randomly drawn from a statistical population of such batches. The probabilities characterize the material quality globally, allowing the prediction of false decisions on conformity of a future batch, based on the future measurement results. The spreadsheet program evaluates risks using Monte Carlo simulations. As input data, the following need to be provided to the software: parameters of normal or lognormal distribution of actual (‘true’) values of the component concentrations (prior distribution); parameters of the distribution of measurements results at the actual value of the component concentration (likelihood function); and correlation matrices for couples of the actual components' concentrations under control and also for corresponding measurement results. The spreadsheet is validated by comparison of the risk estimates with those calculated in R programing environment by numerical integration of the relevant analytical formulae. The developed Excel file and a demonstration videos of its use are available as electronic supplementary material.