China
Africa
Explore chapters and articles related to this topic
Multivariate Process Capability Indices for Asymmetric Specification Region
Published in Ashis Kumar Chakraborty, Moutushi Chatterjee, Handbook of Multivariate Process Capability Indices, 2021
Ashis Kumar Chakraborty, Moutushi Chatterjee
where χn−p−1 denotes the inverse chi distribution with n−p degrees of freedom. In this context, the inverse chi-square distribution is a continuous probability distribution of a positive valued random variable whose reciprocal (multiplicative inverse) follows chi-square distribution. The values of the density functions and the random number generators of the inverse chi-square distribution are available in geoR package of the R software [Ribeiro and Diggle [15]].
Concepts of Probability
Published in Richard Holland, Richard St. John, Statistical Electromagnetics, 2020
Richard Holland, Richard St. John
Thisis known as the chi distribution with n degrees of freedom. For n = 2, it becomes,. g(v)≡gχ(n)(v)=ve−v2/2v≥0which is also known as the Rayleigh distribution. For n = 3, we find g(v)≡gχ(3)(v)=2v22πe−v2/2v≥0
Polarimetric SAR Speckle Statistics
Published in Jong-Sen Lee, Eric Pottier, Polarimetric Radar Imaging, 2017
There are two ways of obtaining N-look amplitude images: (1) averaging the N amplitude images and (2) averaging the N intensity images and then taking the square root. In the first case, the PDF may be obtained by N convolutions of the Rayleigh distribution, but the result cannot be expressed in a closed form. However, the mean is the same as that of the 1-look amplitude mean, and the variance is 1/N of the same case. In the second case, NI from Equation 4.10 has a Chi distribution with 2N degrees of freedom. Therefore, the PDF of the N-look amplitude is given by NI
On guaranteed in-control performance for the Shewhart X and
Published in Journal of Quality Technology, 2018
Rob Goedhart, Marit Schoonhoven, Ronald J. M. M. Does
As an example, consider and . This means that we have , a = 1, and b = m(n − 1). We then find the desired control charting constant to be Note that the required quantiles, and consequently this formula, can easily be calculated in common statistical software programs such as R or Matlab. Note also that similar expressions for k can easily be obtained for different estimators. The only restrictions are that the estimator for location (approximately) follows a normal distribution and that the estimator for dispersion (approximately) follows a scaled chi distribution. This holds true for many other commonly used estimators, such as the grand sample median for location and the average sample standard deviation, the average sample range, and the grand sample standard deviation for dispersion. For more information on this, we refer to Goedhart, Schoonhoven, and Does (2017).
Estimation of aircraft distances using transponder signal strength information
Published in Cogent Engineering, 2018
For large N, an application of the central limit theorem shows that the resulting sum of the in-phase and quadrature components and Q(t) is a Gaussian random process. The in-phase and quadrature components have mean and variance respectively. The received signal amplitude, the quantity in which we are interested, is the square root of the sum of the squares of these components. Because the in-phase and quadrature components are both Gaussian, the square of their sum follows a chi-square distribution, and the square root follows a chi distribution. The chi distribution with two degrees of freedom is the Rayleigh distribution. In cases where the line-of-sight component is very weak, as one might expect from signals broadcast from low altitudes and received at low reception angles, the mean , and the received signal amplitude is approximately Rayleigh distributed. As the line-of-sight component strengthens, the signal amplitude more closely approximates a Rician distribution. However, the scattering resulting in the multipath interference assumed to be present in this particular case is expected to be predominately Rayleigh-distributed. Rayleigh scattering is associated with atmospheric phenomena such as precipitation, haze, and clouds, and with multipath interference; both result in a relatively weak line-of-sight signal component (Sang-Hoon, Jeong-Hun, & Young-Mok, 2016; Van Der Pryt & Vincent, 2015).
Generalized gradients for probabilistic/robust (probust) constraints
Published in Optimization, 2020
Wim van Ackooij, René Henrion, Pedro Pérez-Aros
In order to figure out those additional conditions finally guaranteeing differentiability, it turned out to be useful to investigate probability functions by means of tools from variational analysis and generalized differentiation [5,6,10,11]. A powerful tool to carry out this investigation in the context of Gaussian or Gaussian like distributions of ξ is the so-called spheric-radial decomposition of an m- dimensional Gaussian random vector , which allows to represent the Gaussian probability of a Borel measurable set M as where is the one-dimensional Chi-distribution with m degrees of freedom, is the uniform distribution on the unit sphere and L originates from a factorization of the covariance matrix of ξ. This decomposition has been used in the computation of Gaussian probabilities in early papers by Deák [12,13]. However, in the context of optimization problems, the set M would depend on the decision vector x and so, in addition to the computation of probabilities, one would also be interested in the sensitivity of this probability with respect to the decision vector. In [14] it was shown that the gradient with respect to x of the probability (2) (with M replaced by some explicitly described moving set ) can be represented – just by differentiation under the integral sign – as a spheric integral much like (2) but with a modified integrand. This allows for a simultaneous sampling scheme of the uniform distribution on the sphere in order to determine probabilities and sensitivities at a time in the framework of some nonliear optimization solver.