scholarly journals Polynomial Representations of High-Dimensional Observations of Random Processes

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 123
Author(s):  
Pavel Loskot
Keyword(s):  
Random Variables ◽  
Mean Value ◽  
Weighted Sum ◽  
Computationally Efficient ◽  
First Order ◽  
Statistical Measures ◽  
Polynomial Representations ◽  
Univariate Polynomial ◽  
The Mean ◽  
Numerical Complexity

The paper investigates the problem of performing a correlation analysis when the number of observations is large. In such a case, it is often necessary to combine random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is obtained by approximating the Taylor expansion of a general multivariate scalar symmetric function by a univariate polynomial in the variable given as a simple sum of the original random variables. The mean value of the polynomial is then a weighted sum of statistical central sum-moments with the weights being application dependent. Computing the sum-moments is computationally efficient and amenable to mathematical analysis, provided that the distribution of the sum of random variables can be obtained. Among several auxiliary results also obtained, the first order sum-moments corresponding to sample means are used to reduce the numerical complexity of linear regression by partitioning the data into disjoint subsets. Illustrative examples provided assume the first and the second order Markov processes.

Unified uncertainty analysis by the mean value first order saddlepoint approximation

2012 ◽  
Vol 46 (6) ◽  
pp. 803-812 ◽  
Author(s):  
Ning-Cong Xiao ◽  
Hong-Zhong Huang ◽  
Zhonglai Wang ◽  
Yu Liu ◽  
Xiao-Ling Zhang
Keyword(s):  
Uncertainty Analysis ◽  
Saddlepoint Approximation ◽  
Mean Value ◽  
First Order ◽  
The Mean

scholarly journals A New Conformable Fractional Derivative and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ahmed Kajouni ◽  
Ahmed Chafiki ◽  
Khalid Hilal ◽  
Mohamed Oukessou
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Conformable Fractional Derivative ◽  
First Order ◽  
Rolle’S Theorem ◽  
Power Rule ◽  
The Mean ◽  
Definition Of

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

scholarly journals On the Accuracy of Error Propagation Calculations by Analytic Formulas Obtained for the Inverse Transformation

2019 ◽  
Vol 64 (3) ◽  
pp. 217
Author(s):  
V. I. Romanenko ◽  
N. V. Kornilovska
Keyword(s):  
Error Propagation ◽  
Random Variable ◽  
Mean Value ◽  
Inverse Transformation ◽  
First Order ◽  
Order Of Magnitude ◽  
The Mean ◽  
Normally Distributed

The accuracy of error propagation calculations is estimated for the transformation x → y = f(x) of the normally distributed random variable x. The estimation is based on the formulas for the error propagation obtained for the inverse transformation y → x of the normally distributed random variable y. In the general case, the calculation accuracy for the mean value and the variance of the random variable y is shown to be of the first order of magnitude in the variance of the random variable x.

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Uniqueness Theorem ◽  
Differential Operators ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Initial Value ◽  
First Order ◽  
Fractional Differential ◽  
The Mean ◽  
Fractional Differential Operators ◽  
First Order Differential Equations

AbstractWe generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

On Two First Order Reliability Methods for Computing the Non-Probabilistic Reliability Index

2014 ◽  
Vol 551 ◽  
pp. 648-652
Author(s):  
Xin Zhou Qiao
Keyword(s):  
Reliability Index ◽  
Limit State ◽  
Mean Value ◽  
Point Method ◽  
State Function ◽  
Design Point ◽  
First Order ◽  
Reliability Methods ◽  
The Mean ◽  
Mean Value Method

The two first order reliability methods (FORM) for computing the non-probabilistic reliability index, namely the mean-value method and the design-point method, are investigated. A performance comparison is presented between these two methods. The results show that: (1) the value of the reliability index of the mean-value method depends on the specific form of the limit state function, whereas the value of the reliability index of the design-point one does not;(2) the design-point method should be preferentially used in structural reliability assessment. The conclusions are verified by a numerical example.

scholarly journals EVALUATION OF THE CONVERGENCE SPEED IN THE LAW OF LARGE NUMBERS FOR GAMMA-DISTRIBUTED SEQUENCES

2018 ◽  
pp. 28-54 ◽  
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Mean Value ◽  
Minimum Volume ◽  
Average Value ◽  
Large Numbers ◽  
The Mean ◽  
The Law ◽  
The Difference

The paper considers the problem of estimating the rate of convergence in the law of large numbers for the case when the initial set of random variables is distributed according to the law of the gamma distribution. The problem is urgent due to the fact that with a small number of initial random variables, accurate and close to the true values are the values obtained on the basis of averaging, in particular, if the receipt of each additional value is associated with significant resource costs. The main result of the paper contains estimates for the modulus of difference in distribution function of the mean value for the set of N random variables in the original population, where N is arbitrary, and distribution function of their limiting value, which is a constant (mean value). The result includes three cases: when the argument of distribution function is greater than the average value; when it is equal to it and when it is less than the average value. Estimates are obtained for the modulus of difference of distributions, which depend not only on the number of random variables N, but also on the argument of distribution function. The dependence of the obtained estimate on the argument of distribution function has an exponential character, and on the volume of the set N this dependence makes about the root of N. For convenience of practical application, and also for solving the inverse problem on the basis of the obtained result, estimating the modulus of the difference of distributions is simplified. On the basis of the simplified estimates obtained, the solution of the following inverse problem is given: to find the minimum volume of the string N at which the modulus of the difference of distributions (the accuracy of estimating the mean value on the basis of the mean value) does not exceed a given (small) value. The paper presents a formula for finding the specified minimum volume N, and an algorithm for finding the exact value of N for the estimate under consideration.

Unscented MPSP for Optimal Control of a Class of Uncertain Nonlinear Dynamic Systems

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
S. Mathavaraj ◽  
Radhakant Padhi
Keyword(s):  
Optimal Control ◽  
Initial Conditions ◽  
Mean Value ◽  
Control Synthesis ◽  
Benchmark Problems ◽  
Computationally Efficient ◽  
Efficient Manner ◽  
The Mean ◽  
Low Dimensional ◽  
Time Invariant

A new computationally efficient nonlinear optimal control synthesis technique, named as unscented model predictive static programming (U-MPSP), is presented in this paper that is applicable to a class of problems with uncertainties in time-invariant system parameters and/or initial conditions. This new technique is a fusion of two recent ideas, namely MPSP and Riemann–Stieltjes optimal control problems. First, unscented transform is utilized to construct a low-dimensional finite number of deterministic problems. The philosophy of MPSP is utilized next so that the solution can be obtained in a computational efficient manner. The control solution not only ensures that the terminal constraint is met accurately with respect to the mean value, but it also ensures that the associated covariance matrix (i.e., the error ball) is minimized. Significance of U-MPSP has been demonstrated by successfully solving two benchmark problems, namely the Zermelo problem and inverted pendulum problem, which contain parametric and initial condition uncertainties.

A Mean Value Reliability Method for Bimodal Distributions

2017 ◽  
Author(s):  
Zhangli Hu ◽  
Xiaoping Du
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Random Variable ◽  
Mean Value ◽  
Traffic Load ◽  
First Order ◽  
Reliability Method ◽  
Basic Random Variable ◽  
Two Peaks

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. For example, the random load of a bridge may have two peaks, with a distribution consisting of a weighted sum of two normal distributions, suggested by traffic load data. When binomial variables are involved, traditional reliability methods, such as the First Order Second Moment (FOSM) method and the First Order Reliability Method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation for bimodal variables and then employs a mean value reliability method to accurately predict the reliability. A limit-state function is at first approximated with the first order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The saddlepoint approximation is then applied to estimate the reliability. Examples show that the new method is more accurate than FOSM and FORM.

RESPONSE STATISTICS OF LINEAR STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS UNDER GAUSSIAN STOCHASTIC INPUT

2011 ◽  
Vol 11 (04) ◽  
pp. 775-804 ◽  
Author(s):  
GIUSEPPE MUSCOLINO ◽  
ALBA SOFI
Keyword(s):  
Structural Engineering ◽  
Structural Parameters ◽  
Taylor Series Expansion ◽  
Mean Value ◽  
Linear Structures ◽  
Gaussian Random Process ◽  
Second Order Statistics ◽  
First Order ◽  
Stiffness Properties ◽  
The Mean

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas. To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.

Histamine Levels in Commercially Processed Fish in Morocco

1986 ◽  
Vol 49 (11) ◽  
pp. 904-908 ◽  
Author(s):  
L. ABABOUCH ◽  
M.M. ALAOUI ◽  
F.F. BUSTA
Keyword(s):  
Shelf Life ◽  
Regional Differences ◽  
Reaction Rates ◽  
Mean Value ◽  
Tuna Fish ◽  
Fresh Fish ◽  
First Order ◽  
First Order Kinetics ◽  
Fish Samples ◽  
The Mean

Histamine levels were determined in 248 samples of fish commercially processed in Morocco. Concentrations ranging from <0.01 to 694 mg/100 g of fish (mg%) were observed. The mean value was 12.33 mg% (sardines, 9.75; mackerel, 13.74; tuna 9.86) and the standard deviation was 55.28 mg% (sardines, 43.21; mackerel, 71.99; tuna, 25.05). The bulk of the samples (85.5%) had low histamine levels (<10 mg%); 26 samples (10.5%) had levels within the range 10–50 mg% and should be classified as not from fresh fish or of low quality; 10 samples (4%) had toxicologically significant levels, above 50 mg%. Tuna fish was more susceptible to histamine development than were sardines or mackerel; 7% of tuna fish samples contained levels above 50 mg% as compared to 3.7% and 3.2% for sardines and mackerel, respectively. The percentage of samples containing levels above 50 mg% was somewhat higher for fish processed in the central region (7.1%) than the southern (4.3%) or northern (1.3%) regions; however, statistically the regional differences were not significantly different. Histamine development in sardines demonstrated first-order kinetics. Reaction rates ranged from 0.00200 to 0.000421 mn−1. Refrigeration controlled histamine development. Fish held at 8°C showed a shelf life 12 h longer than fish held at 17°C. A combination of salting and refrigeration was more effective. Fish held at 8°C and salted at a level of 5 or 8% showed a shelf life 35 h longer than fish held at 17°C with no salt.

Sign in / Sign up

Export Citation Format

Share Document