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Stochastic differential equations
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Notice that two continuous random variables y and w in the same probability space are independent if the events {ω ∈ Ω: y(ω) ≤ z1} and {ω ∈ Ω: w(ω) ≤ z2} are independent for all z1,z2∈R. These two random variables form a random vector, (y,w):Ω→R2. We say that these random variables are identically distributed if they have the same CDF, and we write i.i.d. for “independent and identically distributed.” Correspondingly, we have the joint CDF given by F(z1, z2) = F(z1) F(z2).
Image Statistics
Published in Morton John Canty, Image Analysis, Classification, and Change Detection in Remote Sensing, 2019
More formally, let Z1, Z2 … Zm be independent random variables which all have the same distribution function P(z) with mean 〈Z〉 and variance var(Z). These random variables are referred to as a sample of the distribution and are said to be independent and identically distributed (i.i.d.). Any function of them is called a sample function and is itself a random variable. The pixel intensities contributing to Figure 2.4 are a particular realization of some sample of the distribution corresponding to the land cover category mixed forest. For our present purposes, the sample functions of interest are those which can be used to estimate the mean and variance of the distribution P(z). These are the sample mean () Z¯=1m∑i=1mZi
I
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
incremental encoder incremental encoder similar to an absolute encoder, except there are several radial lines drawn around the disc, so that pulses of light are produced on the light detector as the disc rotates. Thus incremental information is obtained relative to the starting position. A larger number of lines allows higher resolution, with one light detector. A once per rev counter may be added with a second light detector. See also encoder. incremental gain a system H : Xe Xe is said to be Lipschitz continuous, or simply continuous, if there exists a constant (H ) < , called incremental gain, such that (H ) = sup (H x1 )T - (H x2 )T (x1 )T - (x2 )T incrementally linear system a system that has a linear response to changes in the input, i.e., the difference in the outputs is a linear (additive and homogenous) function of the difference in the inputs. See linear system. independence a complete absence of any dependence between statistical quantities. In terms of probability density functions (PDFs), a set of random quantities are independent if their joint PDF equals the product of their marginal PDFs: px1 ,x2 ,... (x1 , x2 , . . .) = px1 (x1 ) · px2 (x2 ). . . . Independence implies uncorrelatedness. See correlation, probability density function. independent and identically distributed (IID) a term to describe a number of random variables, each of which exhibits identical statistical characteristics, but acts completely independently, such that the state or output of one random variable has no influence upon the state or output of any other. independent event event with the property that it gives no information about the occurrence of the other events. independent identically distributed process a random process x[i], where x[i] and x[ j] are independent for i = j, and where the probability distribution p(x[i]) for each element of the process is not a function of i. See independence, probability density function, random process. independent increments process a random process x(t), where the process increments over non-overlapping periods are independent. That is, for ti < si , si ti+1 , then (x(s1 ) - x(t1 )), (x(s2 ) - x(t2 )) , . . . are all independent. See independence, random process.
Uncertainty quantification for characterization of rock elastic modulus based on P-velocity
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Jian Liu, Quan Jiang, Dingping Xu, Hong Zheng, Fengqiang Gong, Jie Xin
Generally, Equation (1) can only be fully workable under ideal conditions due to the existence of various errors, i.e. model error and measurement error. If these errors are taken into consideration, Equation (1) is rewritten as: where is the deviation. Therefore, there are two parts in the regression model: the first part of (i.e. ) is determined or explained by the predictor variable, , and the second part of is left unexplained by the predictor variable, . The deviation, , is often taken as the independent and identically distributed (i.i.d) random variable that satisfies a normal distribution with zero mean: where is the standard deviation of a population parameter that is estimated by the samples. It is the average distance between individual values of and the mean of as described by the regression line. Bozorgzadeh, Escobar, and Harrison (2018) roughly treated as the standard deviation of .
Purchase decision process and information acquisition of zero-energy houses in Japan
Published in Journal of Asian Architecture and Building Engineering, 2023
Hitomu Kotani, Kazuyoshi Nakano
where a vector of intercepts , an unknown vector of factor loadings , and a random vector of measurement . is independent and identically distributed (i.i.d.) according to normal distribution with a mean of 0 and variance of 1, i.e., . Similarly, , , , , and are given by
A spatiotemporal prediction approach for a 3D thermal field from sensor networks
Published in Journal of Quality Technology, 2022
The local thermal distribution for fields at time is decomposed into two independent parts, namely, the latent function of the local thermal distribution for fields and the noise term Thus, we have where and are the sets in terms of and which denote the local thermal distribution, its corresponding latent function, and the noise at location and time of field respectively. is assumed to be an independent and identically distributed Gaussian white noise with the variance that is, Thus, we have