Hasil untuk "Probabilities. Mathematical statistics"

Peter Filzmoser

In lieu of an abstract, here is the entry's first paragraph: Robust statistics are procedures that maintain nominal Type I error rates and statistical power in the presence of violations of the assumptions that underpin parametric inferential statistics. Since George Box coined the term in 1953, research on robust statistics has centered on the assumption of normality, although the violation of other parametric assumptions (e.g., homogeneity of variance) has their own implications for the accuracy of parametric procedures. This entry looks at the importance of robust statistics in educational and social science research and explains the robustness argument. It then describes robust descriptive statistics, their inferential extensions, and two common resampling procedures that are robust alternatives to classic parametric methods. Disciplines Statistics and Probability Comments This is an entry in: Frey, B. (2018). The SAGE encyclopedia of educational research, measurement, and evaluation (Vols. 1-4). Thousand Oaks,, CA: SAGE Publications, Inc.: https://dx.doi.org/10.4135/ 9781506326139 © 2018 by SAGE Publications, republished with permission. Content may not be distributed, resold, repurposed, used for commercial MOOCs, nor any other commercial purposes without permission. Please contact SAGE for any further usage or questions. This article is available at Fisher Digital Publications: https://fisherpub.sjfc.edu/statistics_facpub/8 1434 Robust Statistics Couch, A., & Keniston, K. (1960). Yeasayers and naysayers: Agreeing response set as a personality variable. Journal of Abnormal and Social Psychology, 60, 151-174. Gosling, S. D., Rentfrow, P. ]., & Swann, W. B., Jr. (2003). A very brief measure of the Big-Five personality domains. Journal of Research in Personality, 37, 504-528. Llorente, E., Warren, C. S., de Eulate, L. P., & Gleaves, D. H. (2013 ). A Spanish version of the sociocultural attitudes towards appearance questionnaire-3 (SATAQ-3 ): Translation and psychometric evaluation. Journal of Clinical Psychology, 69(3 ), 240-251. doi: 10.1002/jclp.21944 Rodebaugh, T. L, Woods, C. M, Heimberg, R. G., Liebowitz, M. R., & Schneier, F. R. (2006). The factor structure and screening utility of the Social Interaction Anxiety Scale. Psychological Assessment, 18(2), 231-237. Tay, L., & Drasgow, F. (2012). Theoretical, statistical, and substantive issues in the assessment of construct dimensionality: Accounting for the item response process. Organizational Research Methods, 15(3), 363-384. VonSonderen, E., Sanderman, R., & Coyne, J.C. (2013). Ineffectiveness of reverse wording of questionnaire items: Let's learn from cows in the rain. PloSONE, 8(7), e68967. doi:l0.1371/journal.pone.0068967 Weijters, B., Baumgartner, H., & Schillewaet, N. (2013). Reversed item bias: An integrative model. Psychological Method, 18(3), 320-334. doi:l0.1037/

H. Cramér

H. L. L. Roy, L. Lecam, J. Neyman

P. Rousseeuw, A. Leroy

R. Hogg, A. Craig

F. Dekking, C. Kraaikamp, H. P. Lopuhaä et al.

B. Ayyub, R. McCuen

Introduction Introduction Knowledge, Information, and Opinions Ignorance and Uncertainty Aleatory and Epistemic Uncertainties in System Abstraction Characterizing and Modeling Uncertainty Simulation for Uncertainty Analysis and Propagation Simulation Projects Data Description and Treatment Introduction Classification of Data Graphical Description of Data Histograms and Frequency Diagrams Descriptive Measures Applications Analysis of Simulated Data Simulation Projects Fundamentals of Probability Introduction Sets, Sample Spaces, and Events Mathematics of Probability Random Variables and Their Probability Distributions Moments Application: Water Supply and Quality Simulation and Probability Distributions Simulation Projects Probability Distributions for Discrete Random Variables Introduction Bernoulli Distribution Binomial Distribution Geometric Distribution Poisson Distribution Negative Binomial and Pascal Probability Distributions Hypergeometric Probability Distribution Applications Simulation of Discrete Random Variables A Summary of Distributions Simulation Projects Probability Distributions for Continuous Random Variables Introduction Uniform Distribution Normal Distribution Lognormal Distribution Exponential Distribution Triangular Distribution Gamma Distribution Rayleigh Distribution Beta Distribution Statistical Probability Distributions Extreme Value Distributions Applications Simulation and Probability Distributions A Summary of Distributions Simulation Projects Multiple Random Variables Introduction Joint Random Variables and Their Probability Distributions Functions of Random Variables Modeling Aleatory and Epistemic Uncertainty Applications Multivariable Simulation Simulation Projects Simulation Introduction Monte Carlo Simulation Random Numbers Generation of Random Variables Generation of Selected Discrete Random Variables Generation of Selected Continuous Random Variables Applications Simulation Projects Fundamentals of Statistical Analysis Introduction Properties of Estimators Method-of-Moments Estimation Maximum Likelihood Estimation Sampling Distributions Univariate Frequency Analysis Applications Simulation Projects Hypothesis Testing Introduction General Procedure Hypothesis Tests of Means Hypothesis Tests of Variances Tests of Distributions Applications Simulation of Hypothesis Test Assumptions Simulation Projects Analysis of Variance Introduction Test of Population Means Multiple Comparisons in the ANOVA Test Test of Population Variances Randomized Block Design Two-Way ANOVA Experimental Design Applications Simulation Projects Confidence Intervals and Sample-Size Determination Introduction General Procedure Confidence Intervals on Sample Statistics Sample Size Determination Relationship between Decision Parameters and Types I and II Errors Quality Control Applications Simulation Projects Regression Analysis Introduction Correlation Analysis Introduction to Regression Principle of Least Squares Reliability of the Regression Equation Reliability of Point Estimates of the Regression Coefficients Confidence Intervals of the Regression Equation Correlation versus Regression Applications of Bivariate Regression Analysis Simulation and Prediction Models Simulation Projects Multiple and Nonlinear Regression Analysis Introduction Correlation Analysis Multiple Regression Analysis Polynomial Regression Analysis Regression Analysis of Power Models Applications Simulation in Curvilinear Modeling Simulation Projects Reliability Analysis of Components Introduction Time to Failure Reliability of Components First-Order Reliability Method Advanced Second-Moment Method Simulation Methods Reliability-Based Design Application: Structural reliability of a Pressure Vessel Simulation Projects Reliability and Risk Analysis of Systems Introduction Reliability of Systems Risk Analysis Risk-Based Decision Analysis Application: System Reliability of a Post-Tensioned Truss Simulation Projects Bayesian Methods Introduction Bayesian Probabilities Bayesian Estimation of Parameters Bayesian Statistics Applications Appendix A: Probability and Statistics Tables Appendix B: Taylor Series Expansion Appendix C: Data for Simulation Projects Appendix D: Semester Simulation Project Index Problems appear at the end of each chapter.

E. Goligher, Anna Heath, M. Harhay

Frequentist and Bayesian statistics represent two differing paradigms for the analysis of data. Frequentism became the dominant mode of statistical thinking in medical practice during the 20th century. The advent of modern computing has made Bayesian analysis increasingly accessible, enabling growing use of Bayesian methods in a range of disciplines, including medical research. Rather than conceiving of probability as the expected frequency of an event (purported to be measurable and objective), Bayesian thinking conceives of probability as a measure of strength of belief (an explicitly subjective concept). Bayesian analysis combines previous information (represented by a mathematical probability distribution, the prior) with information from the study (the likelihood function) to generate an updated probability distribution (the posterior) representing the information available for clinical decision making. Owing to its fundamentally different conception of probability, Bayesian statistics offers an intuitive, flexible, and informative approach that facilitates the design, analysis, and interpretation of clinical trials. In this Review, we provide a brief account of the philosophical and methodological differences between Bayesian and frequentist approaches and survey the use of Bayesian methods for the design and analysis of clinical research.

Arif Fatahillah, Umi Mubarokah, Rafiantika Megahnia Prihandini et al.

Atherosclerosis is a leading cause of coronary heart disease. This study analyses how elliptically shaped stenoses alter blood-flow velocity in coronary arteries. The governing equations are discretised with the finite-volume method, coupling pressure and velocity through the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm and accelerating convergence with the Successive Over-Relaxation (SOR) technique. A weighted Gauss–Seidel iteration whose over-relaxation factor (  in this work) damps low-frequency error modes, cutting the number of iterations needed for residuals to fall below 10⁻⁴ by roughly 40 % compared with the standard Gauss–Seidel scheme. Simulations of 30 %, 50 %, and 70 % constrictions were carried out in MATLAB R2013a and ANSYS Fluent. Quantitative and qualitative cross-validation of the two software packages confirmed consistent velocity and pressure fields, though minor discrepancies arose from differing numerical schemes and model assumptions, underscoring the need for experimental verification. The highest centre-line velocity occurred at 70 % stenosis—0.72075 m/s in MATLAB versus 0.90 m/s in Fluent—while the lowest was recorded at 30 %. Velocity–pressure profiles showed that increasing inlet velocity or degree of narrowing elevates velocity but decreases pressure, with the largest drop (11492.4 Pa in MATLAB; 11747.32 Pa in Fluent) again at 70% stenosis. Study limitations include modelling blood as a Newtonian fluid and idealising arterial geometry; future work should incorporate non-Newtonian rheology and patient-specific shapes to enhance physiological accuracy.

Julian Granna, Wolfgang Brunauer, Stefan Lang et al.

Modeling real estate prices in the context of hedonic models often involves fitting a Generalized Additive Model, where only the mean of a (lognormal) distribution is regressed on a set of variables without taking other parameters of the distribution into account. Thus far, the application of regression models that model the full conditional distribution of the prices, has been infeasible for large data sets, even on powerful machines. Moreover, accounting for heterogeneity of effects regarding time and locale, is often achieved by naive stratification of the data rather than on a model basis.  A novel batchwise backfitting algorithm is applied in the context of a structured additive distributional regression model, which enables us to efficiently model all distributional parameters of the price distribution. Using a large German dataset of apartment asking prices with over one million observations, we employ a model-based clustering algorithm to capture the heterogeneity of covariate effects on the parameters with respect to dwelling locale. We thus identify clusters that are homogeneous with respect to the influence of dwelling locale on price. A boosting type algorithm of the batchwise backfitting algorithm is then used to automatically determine the variables relevant for modelling the location and scale parameters in each regional cluster. This allows for a different influence of variables on the distribution of prices depending on the locale and price segment of the dwelling.

G. Das, N. Goswami, B. Patir

In this paper, we derive some new fixed point results in C∗-algebra valued fuzzy metric space with the help of subadditive altering distance function with respect to a t-norm. Our results generalize some existing fixed point results in the literature. A common fixed point result is also derived for a pair of mappings on complete C∗-algebra valued fuzzy metric space. The results are supported by suitable examples along with the graphical demonstration of the used conditions. As application, we establish the solvability of a second order boundary value problem. Moreover, the results are also applied in control theory to study the possibility of optimally controlling the solution of an ordinary differential equation in dynamic programming.

Tingqing Ye, Haoran Zheng

Uncertain statistics is a set of mathematical techniques to collect, analyze and interpret data based on uncertainty theory. In addition, probability statistics is another set of mathematical techniques based on probability theory. In practice, when to use uncertain statistics and when to use probability statistics to model some quality depends on whether the distribution function of the quality is close enough to the actual frequency. If it is close enough, then probability statistics may be used. Otherwise, uncertain statistics is recommended. In order to illustrate it, this paper employs uncertain statistics, including uncertain time series analysis, uncertain regression analysis and uncertain differential equation, to model the birth rate in China, and explains the reason why uncertain statistics is used instead of probability statistics by analyzing the characteristics of the residual plot. In addition, uncertain hypothesis test is used to determine whether the estimated uncertain statistical models are appropriate.

Hafizah Ilma, Khairil Anwar Notodiputro, Bagus Sartono

Random forest is one of the most popular ensemble methods and has many advantages. However, random forest is a "black-box" model, so the model is difficult to interpret. This study discusses the interpretation of random forest with association rules technique using rules extracted from each decision tree in the random forest model. This analysis involves simulation and empirical data, to determine the factors that affect the poverty status of households in Tasikmalaya. The empirical data was sourced from Badan Pusat Statistik (BPS), the National Socio-Economic Survey (SUSENAS) data for West Java Province in 2019.  The results obtained are based on simulation data, the association rules technique can extract the set of rules that characterize the target variable. The application of interpretable random forest to empirical data shows that the rules that most distinguish the poverty status of households in Tasikmalaya are house wall materials and the main source of drinking water, house wall materials and cooking fuel, as well as house wall materials and motorcycle ownership.

V.I. Senashov, I.A. Paraschuk

The article discusses the possibility of recognizing a group by the bottom layer, that is, by the set of its elements of prime orders. The paper gives examples of groups recognizable by the bottom layer, almost recognizable by the bottom layer, and unrecognizable by the bottom layer. Results are obtained for recognizing a group by the bottom layer in the class of infinite groups under some additional restrictions. The notion of recognizability of a group by the bottom layer was introduced by analogy with the recognizability of a group by its spectrum (the set of orders of its elements). It is proved that all finite simple nonAbelian groups are recognizable by spectrum and bottom layer simultaneously in the class of finite simple non-Abelian groups.

Mekonnen Y. Legesse, K. Luneta, Tadele Ejigu

Abstract This study utilized discourse-based instruction as an alternative method of instruction that emphasizes the teaching of mathematics by actively engaging students in mathematical discourse practices. A quasi-experimental study was employed to determine the effectiveness of mathematical discourse-based instruction in enhancing eleventh-grade students’ conceptual and procedural understanding of probability and statistics. A researcher-constructed test instrument was used for data collection from the experimental and control groups. The data analysis performed using the Kruskal-Wallis test showed that the experimental group outperformed the control groups in terms of conceptual and procedural knowledge. Furthermore, the results suggest that discourse-based instruction when appropriately designed and implemented can increase students’ understanding of mathematical topics.

E Shine Lal, P Ramya

In this paper, we introduce a Tikhonov type regularization method for an ill-posed operator equation T x = y, where T is a closed densely defined unbounded operator on a Hilbert space H.

This paper discusses the development of an aided augmentative alternative communication (AAC) web-based system to assist speech delayed children between the age of four to nine years old that incorporates the gamifications techniques. The aim is to provide a set of strategies and techniques that can be used to support communications for children with communication needs. The research methodology used to develop the AAC web-based system with gamifications start with the preliminary investigation, proposing the framework of the prototype system that consists of two main modules, which are the speech recognition and spell session modules, design and development of the system and usability evaluation. The results of the usability testing that has been conducted to thirty speech delayed children showed that the ACC webbased system with gamifications have been proven satisfactory to support and assist speech delayed children in improving their speech and language learning.

Hendrik Bernd Petersen, Bubacarr Bah, Bubacarr Bah et al.

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

Axel Hutt, Axel Hutt

Ensemble Kalman filters are powerful tools to merge model dynamics and observation data. For large system models, they are known to diverge due to subsampling errors at small ensemble size and thus possible spurious correlations in forecast error covariances. The Local Ensemble Transform Kalman filter (LETKF) remedies these disadvantages by localization in observation space. However, its application to nonlocal observations is still under debate since it is still not clear how to optimally localize nonlocal observations. The present work studies intermittent divergence of filter innovations and shows that it increases forecast errors. Nonlocal observations enhance such innovation divergence under certain conditions, whereas similar localization radius and sensitivity function width of nonlocal observations minimizes the divergence rate. The analysis of the LETKF reveals inconsistencies in the assimilation of observed and unobserved model grid points which may yield detrimental effects. These inconsistencies inter alia indicate that the localization radius should be larger than the sensitivity function width if spatially synchronized system activity is expected. Moreover, the shift of observation power from observed to unobserved grid points hypothesized in the context of catastrophic filter divergence is supported for intermittent innovation divergence. Further possible mechanisms yielding such innovation divergence are ensemble member alignment and a novel covariation between background perturbations in location and observation space.

Setiyo Daru Cahyono, Tomi Tristono, Seno Aji et al.

Mathematical modelling assumes that the vehicle’s volume has a uniform pattern. Due to traffic lights settings, the number of vehicles queue grows linearly. The reality, the stochastic arrivals of the vehicles could be (1) in the randomized arrivals, (2) in the form of groups/ platoon, or (3) in the mixed arrivals. It is observed that the arrival of the vehicles in the queue tends to have a normal pattern. The objective of this research is to study the implications of the arrival categories to the travel delay. For simulation, it uses the numerical method referring to the real state. The result indicates, determining travel delay become precise for all vehicles. It is due to the travel delay formula is represented as a discrete function. The arrival time, departure time, and stop time for each vehicle at the signalized intersection are recorded in the device