Pearson System Research Articles

개선된 피어슨 시스템을 이용한 신뢰성기반 최적설계

Since conventional optimization that is classified as a deterministic method does not consider the uncertainty involved in a modeling or manufacturing process, an optimum design is often determined to be on the boundaries of the feasible region of constraints. Reliability-based design optimization is a method for obtaining a solution by minimizing the objective function while satisfying the reliability constraints. This method includes an optimization process and a reliability analysis that facilitates the quantization of the uncertainties related to design variables. Moment-based reliability analysis is a method for calculating the reliability of a system on the basis of statistical moments. In general, on the basis of these statistical moments, the Pearson system estimates seven types of distributions and determines the reliability of the system. However, it is technically difficult to practically consider the Pearson Type IV distribution. In this study, we propose an enhanced Pearson Type IV distribution based on a kriging model and validate the accuracy of the enhanced Pearson Type IV distribution by comparing it with a Monte Carlo simulation. Finally, reliability-based design optimization is performed for a system with type IV distribution by using the proposed method.

A simple derivation and classification of common probability distributions based on information symmetry and measurement scale

Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His approach was phenomenological: a differential equation that generated common distributions without any underlying conceptual basis for why common distributions have particular forms and what explains the familial relations. Pearson's system and its descendants remain the most popular systematic classification of probability distributions. Here, we unify the disparate forms of common distributions into a single system based on two meaningful and justifiable propositions. First, distributions follow maximum entropy subject to constraints, where maximum entropy is equivalent to minimum information. Second, different problems associate magnitude to information in different ways, an association we describe in terms of the relation between information invariance and measurement scale. Our framework relates the different continuous probability distributions through the variations in measurement scale that change each family of maximum entropy distributions into a distinct family. From our framework, future work in biology can consider the genesis of common patterns in a new and more general way. Particular biological processes set the relation between the information in observations and magnitude, the basis for information invariance, symmetry and measurement scale. The measurement scale, in turn, determines the most likely probability distributions and observed patterns associated with particular processes. This view presents a fundamentally derived alternative to the largely unproductive debates about neutrality in ecology and evolution.

Application of Elastic-Plastic Static Friction Models to Rough Surfaces With Asymmetric Asperity Distribution

Asymmetric height distribution in surface roughness is important in many engineering surfaces, such as in constant velocity (CV) joints, where specific manufacturing processes could result in such surfaces. Even if the initial surfaces exhibit symmetric roughness, the running-in and sliding processes could result in asymmetric roughness distributions. In this paper, the effect of asymmetric asperity height distribution on the static friction coefficient is investigated theoretically and experimentally. The asymmetry of the surface roughness is modeled using the Pearson system of frequency curves. Two elastic-plastic static friction models, the Kogut–Etsion (KE) and Cohen–Kligerman–Etsion (CKE) models are adapted to account for asymmetric roughness and employed to obtain the tangential and normal contact forces. Static friction experiments using CV joint roller and housing surfaces, which exhibit different levels of surface roughness, were performed and directly compared with the KE and CKE static friction models using both a symmetric Gaussian as well as Pearson distributions of asperity heights. It is found that the KE model with the Pearson distribution compares favorably with the experimental measurements.

Evaluation of Fatigue Life Reliability of Steering Knuckle Using Pearson Parametric Distribution Model

Steering module is a part of automotive suspension system which provides a means for an accurate vehicle placement and stability control. Components such as steering knuckle are subjected to fatigue failures due to cyclic loads arising from various driving conditions. This paper intends to give a description of a method used in the fatigue life reliability evaluation of the knuckle used in a passenger car steering system. An accurate representation of Belgian pave service loads in terms of response-time history signal was obtained from accredited test track using road load data acquisition. The acquired service load data was replicated on durability test rig and the SN method was used to estimate the fatigue life. A Pearson system was developed to evaluate the predicted fatigue life reliability by considering the variations in material properties. Considering random loads experiences by the steering knuckle, it is found that shortest life appears to be in the vertical load direction with the lowest fatigue life reliability between 14000–16000 cycles. Taking into account the inconsistency of the material properties, the proposed method is capable of providing the probability of failure of mass-produced parts.

59325 RELIABILITY ANALYSIS OF THE TRUCK FRAME FE MODEL CONSIDERING THE STATISTICALLY DISTRIBUTED DESIGN VARIABLES(Flexible Multibody Dynamics)

Reliability analysis using the Pearson System considering the variance of tolerance for the design variables of the truck frame structure is performed in this paper. First, the fatigue life of the structure is estimated by three steps. Multibody dynamic model of 9-DOF vehicle to calculate constraint forces applied on the frame is simulated. Then the quasi-static analysis of the truck frame to calculate the stress coefficient of structure is performed. Finally using the stress histories based on the linear combination of constraint forces and stress coefficients, fatigue analysis to calculate the fatigue life is done. It is assumed that the design variables related to the multibody model such as mass, spring and damping coefficients and the dimension of the section in FE model are statistically distributed. By employing the Pearson System, the reliability analysis of the structure based on the statistical moment is performed. Then the variance of the probability density function considering the tolerance size of the design variables is estimated to analyze the effect of the design variable on the variance of fatigue failure probability. From these procedures, most sensitive design variables are known and the proper tolerance size for each design variables could be determined compared to the required reliability of structure.

A Single Form for t-Distributions and Symmetric Beta Distributions

A single parametric form is given for the symmetric distributions in the Pearson system with finite variance. In effect, these are Student's t-distributions with ν > 2 and all centered symmetric beta distributions. A different parametrization allows the inclusion of the t-distributions with ν ≤2 at the expense of symmetric beta distributions with a low shape parameter.

On defining [formula omitted]-values

The Fisherian prescription of reporting P -values as a summary of a result, as compared to the Neyman–Pearson system of acceptance or rejection of a null hypothesis, is more common in applied science. This popularity is largely due to the fact that the P -value provides a more complete, meaningful and useful evidence regarding the null hypothesis. Conventionally, P -values are defined in the context of one-sided alternatives, although there exist some ideas in the literature concerning two-sided alternatives; see e.g. [Gibbons, J.D., Pratt, J.W., 1975. P -values: Interpretation and methodology. American Statistician 24, 20–25; George, E.O., Mudholkar, G.S., 1990. P -values for two-sided tests. Biometrical Journal 32, 747–751]. This note takes an axiomatic approach for defining P -values which involves at most ordering of the alternatives but is not restricted by their nature. It also involves a correspondence between a P -value and the associated level α test for each α . A P -value turns out to be valid if and only if the associated level α test is unbiased in the traditional sense for each α . Furthermore, it is shown that the resulting optimal tests agree with those given by the Neyman-Person framework when the ordering is stochastic. Thus, a theory based on optimal P -values parallels to the Neyman–Pearson theory and bridges the two approaches to testing of hypotheses.

INTERPRETATION OF THE GENERALIZED ZIPF-MANDELBROT LAW PARAMETERS

This article provides an interpretation of the parameters of the generalized Zipf-Mandelbrot law, which the authors have called the generalized range-frequency law which has been developed in 2009. Said law models the distribution of frequencies of words in school textbooks obtained using a Pearson system. The significance of the parameters gathered was obtained by analyzing certain variables. The fact that the statistical law can be applied to texts written using natural language means that it may be applicable to ecological models, these being considered as a particular case of mathematical models written as texts in a formal language, as developed in 1999. After establishing certain hypotheses regarding the significance of the parameters of the generalized law obtained for natural language, this study was designed to analyze whether this law is applicable to models considered as texts written in formal language. If this were so, the law would be a very useful tool for comparing models.

Further results on the approximation of log-normal power sum via pearson type IV distribution: a general formula for log-moments computation

In, we have recently proposed a general approach for approximating the power sum of Log-Normal Random Variables (RVs) by using the Pearson system of distributions. Therein, we have also highlighted the main advantages of using Pearson approximation instead of the usual Log-Normal one, and compared the proposed method with other approaches available in the open technical literature. However, despite being very accurate, the proposed method may be, in some circumstances, computational demanding since a non-linear least-squares problem needs to be solved numerically to get an accurate approximation. Motivated by the above consideration, the aim of this Letter is to provide an alternative approach for computing the parameters of the approximating Pearson distribution. The proposed solution is based on the Method of Moments (MoMs) in the logarithmic domain. In particular, by using some known properties of the Laplace transform, we will show that the MGF of the Log-Normal power sum in the logarithmic domain (denoted as log-MGF) can be obtained from the Mellin transform of the MGF of the Log-Normal power sum in the linear domain. From the estimated log-MGF, we will then compute the desired log-moments required for Pearson approximation. Numerical results will be also shown in order to substantiate the accuracy of the proposed method.

Karl Pearson in Russian Contexts

Summary The confluence of statistics and probability into mathematical statistics in the Russian Empire through the interaction, 1910–1917, of A.A. Chuprov and A.A. Markov was influenced by the writings of the English Biometric School, especially those of Karl Pearson. The appearance of the Russian‐language exposition of Pearsonian ideas by E. E. Slutsky in 1912 was instrumental in this confluence. Slutsky's predecessors in such writings (Lakhtin, Orzhentskii, and Leontovich) were variously of mathematical, political economy, and biological backgrounds. Work emanating from the interpolational nature of Pearson's system of frequency curves was continued subsequently through the work of Markov, Bernstein, Romanovsky, and Kravchuk (Krawtchouk), who laid a solid probabilistic foundation. The correlational nature in the interpolational early work of Chebyshev, and work of the English Biometric School in the guise of linear least‐squares fitting exposited as the main component of Slutsky's book, was developed in population as well as sample context by Chuprov. He also championed the expectation operation in providing exact relations between sample and population moments, in direct interaction with Karl Pearson. Romanovsky emerges as the most adaptive and modern mathematical statistician.

Karl Pearson and the Origin of Kurtosis

Summary Although the kurtosis index proposed by Karl Pearson in 1905 is introduced in statistical textbooks at all levels, the measure is not easily interpreted and has been a subject of considerable debate. In this study, the theoretical development of kurtosis is surveyed from a historical perspective of Pearson's work on evolution. It surprisingly emerges that there was no emphasis in Pearson's papers on kurtosis as measuring (in part) tail heaviness. However, it is found that Pearson used to frequently adjust the formalisation of kurtosis depending on his changing needs. This complex development partly explains the confusion that would surround kurtosis in subsequent literature. Our conclusion is that most misunderstandings arise from improper use of the kurtosis coefficient outside the Pearson system of frequency curves.

Approximating the linear combination of log-normal RVs via pearson type IV distribution for UWB performance analysis

Approximating the sum of log-normal random variables (RVs) is a long-standing open issue and many studies in the old and recent literature resort to assume that the "power sum" can be well modeled via another log-normal RV. Such a log-normal approximation can be effectively used for accurate outage probability (Pout) analysis and modeling of signal-to-(noise+interference) ratio (SNIR) behaviors in the context of radio resource management in wireless systems; nevertheless, it is recognized that i) an adequate accuracy can only be obtained for a sub-range of values taken by the distribution; ii) the achievable accuracy depends on the parameters of the set of RVs; iii) log-normal approximation may not provide a satisfactory accuracy for estimating bit error probabilities. Moreover, all contributions in the open technical literature are referred to model the "power sum" distribution, i.e., the weighted linear combination of log-normal RVs with weights that may take only positive values. While this setup is consistent with typical scenarios where summands add incoherently, it has recently been shown that in some high frequency-selective multipath channels with log-normal distributed shadow- and fast-fading, e.g., in impulse radio ultra wide band (IR-UWB) systems, the signal- to-noise ratio (SNR) is more accurately modeled by the weighted linear combination of log-normal RVs with weights that may assume both positive and negative values. Motivated by the above considerations, and moving from recent results, which show that approximating the "power sum" of log-normal RVs can boil down to a model selection problem within the pearson system of distributions, the specific contribution of this paper can be summarized as follows, i) We show that the pearson type IV distribution can not only be used to accurately approximate the probability density function (PDF) of the "power sum" of log-normal RVs with positive weights, but it can also accurately model the scenario with both positive and negative weights, as well as the scenario with generically correlated RVs. ii) For the scenario with independent RVs, we propose a numerical technique, which is based on the moment generating function (MGF) matching principle and uses a least- squares criterion, to accurately estimate the four parameters of the pearson type IV distribution, iii) A byproduct of this analysis.

Proton-Transfer Reactions of Nitroalkanes: The Role of aci-Nitro Species

Proton-transfer reactions of two systems, ionization of a series of small carbon acids in water (the Pearson system) and reactions of substituted phenylnitromethanes, were examined in detail computationally. Comparison of experimental reactivity and pK(a) with calculated relative activation barrier and reaction energy for the Pearson system suggested that the origin of the well-know nitroalkane anomaly does not reside in the reactivity but in the equilibrium. For the reactions of substituted phenylnitromethanes, proton transfers among three species, PhCH(2)NO(2), PhCHNO(2)(-), and PhCH=NO(2)H, were examined, and the role of the aci-nitro species (PhCH=NO(2)H) was evaluated on the basis of its stability and reactivity. Protonation of PhCHNO(2)(-) by H(2)O was suggested to occur kinetically on the oxygen site, but due to its instability PhCH=NO(2)H does not contribute to the overall reaction energetics. The protonation of PhCHNO(2)(-) under acidic conditions occurs on the oxygen site to give PhCH=NO(2)H both kinetically and thermodynamically. The aci-nitro species thus formed appears to give PhCH(2)NO(2) via intramolecular H(2)O-mediated proton transfer, but a possibility of the route through PhCHNO(2)(-)-C-protonation would not be fully eliminated.

MODELLING OF STATISTICAL LAWS FOR TEXTS USING A PEARSON SYSTEM

In this article, we have obtained a generalization of statistical laws for texts for application to natural languages, in order to determine whether this general law is applicable to model texts in accordance with the definition given in Villacampa et al. (1999a). The studies were carried out in the context of a system whose components generated model texts, through which we confirmed that the laws of Zipf and Mandelbrot are appropriate for explaining the texts. On the basis of a Pearson System, we also obtained a unique function for the distribution of frequencies, with said distribution being a generalization of the above-mentioned laws. For the problem analyzed in this article, the text or text system constitutes a model of the cognitive development of individuals. We can thus infer the cognitive characteristics of different individuals by studying text systems.

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

Abstract. The Pearson diffusions form a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well‐known instances are the Ornstein–Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy‐tailed and skew marginals are included. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. Explicit optimal martingale estimating functions are found. The discussion covers GMM, quasi‐likelihood, non‐linear weighted least squares estimation and likelihood inference too. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions and Pearson stochastic volatility models. For the non‐Markov models, explicit optimal prediction‐based estimating functions are found. The estimators are shown to be consistent and asymptotically normal.

Application of maximum entropy principle for reliability-based design optimization

The maximum entropy principle (MEP) is used to generate a natural probability distribution among the many possible that have the same moment conditions. The MEP can accommodate higher order moment information and therefore facilitate a higher quality PDF model. The performance of the MEP for PDF estimation is studied by using more than four moments. For the case with four moments, the results are compared with those by the Pearson system. It is observed that as accommodating higher order moment, the estimated PDF converges to the original one. A sensitivity analysis formulation of the failure probability based on the MEP is derived for reliability-based design optimization (RBDO) and the accuracy is compared with that by finite difference method (FDM). Two RBDO examples including a realistic three-dimensional wing design are solved by using the derived sensitivity formula and the MEP-based moment method. The results are compared with other methods such as TR-SQP, FAMM + Pearson system, FFMM + Pearson system in terms of accuracy and efficiency. It is also shown that an improvement in the accuracy by including more moment terms can increase numerical efficiency of optimization for the three-dimensional wing design. The moment method equipped with the MEP is found flexible and well adoptable for reliability analysis and design.

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating functions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and non-linear weighted least squares estimation too, and it is discussed how explicit likelihood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formula for the conditional moments and the polynomial eigenfunctions are derived. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators.

A CLOSED‐FORM EXPRESSION FOR THE PEARSON TYPE IV DISTRIBUTION FUNCTION

SummaryA closed‐form expression is presented for the probability integral of the Pearson Type IV distribution, and a corresponding method of evaluation is given. This analysis addresses a long‐standing gap in the theory of the Pearson system of distributions. In addition, a simple derivation is given of an expression for the normalizing constant in the Type IV integral.

Reliability analysis using an enhanced response surface moment method

Moment methods, which are powerful and simple techniques for analyzing the reliability of a system, evaluate the statistical moments of a system response function and use information from the probability distribution in the analysis. The full factorial moment method (FFMM) performs reliability analysis by using a 3n full factorial design of experiments (DOE) and the Pearson system for random variables. To overcome the inefficiency of FFMM, the response surface moment method (RSMM) has been proposed, which is based on a response surface model (RSM) that is updated by adding cross product terms into the simple quadratic model. In this paper, we propose the enhanced RSMM (RSMM+) that modifies the procedure of selecting a cross product term in the RSMM and adds a process of judging whether the response surface model can be established before performing an additional experiment. We apply the proposed method to several examples and show that it gives better results in efficiency.

Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis

This paper presents the eigenvector dimension reduction (EDR) method for probability analysis that makes a significant improvement based on univariate dimension reduction (DR) method. It has been acknowledged that the DR method is accurate and efficient for assessing statistical moments of mildly nonlinear system responses in engineering applications. However, the recent investigation on the DR method has found difficulties of instability and inaccuracy for highly nonlinear system responses while maintaining reasonable efficiency. The EDR method integrates the DR method with three new technical components: (1) eigenvector sampling, (2) one-dimensional response approximation, and (3) a stabilized Pearson system. First, 2N+1 and 4N+1 eigenvector sampling schemes are proposed to resolve correlated and asymmetric random input variables. The eigenvector samples are chosen along the eigenvectors of the covariance matrix of random parameters. Second, the stepwise moving least squares (SMLS) method is proposed to accurately construct approximate system responses along the eigenvectors with the response values at the eigenvector samples. Then, statistical moments of the responses are estimated through recursive numerical integrations. Third, the stabilized Pearson system is proposed to predict probability density functions (PDFs) of the responses while eliminating singular behavior of the original Pearson system. Results for some numerical and engineering examples indicate that the EDR method is a very accurate, efficient, and stable probability analysis method in estimating PDFs, component reliabilities, and qualities of system responses.