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Random Variables, Distributions & Linear Regression
Published in Nailong Zhang, A Tour of Data Science, 2020
In the previous section, we saw the PDF for multivariate normal distribution in (4.3). A multivariate distribution is also called joint distribution, since the multivariate random variable can be viewed as a joint of multiple univariate random variables. Joint PDF gives the probability density of a set of random variables. Sometimes we may only be interested in the probability distribution of a single random variable from a set. And that distribution is called marginal distribution. The PDF of a marginal distribution can be obtained by integrating the joint PDF over all the other random variables. For example, the integral of (4.3) gives the PDF of a univariate normal distribution.
Estimation of the trajectory and attitude of railway vehicles using inertial sensors with application to track geometry measurement
Published in Vehicle System Dynamics, 2023
J. González-Carbajal, Pedro Urda, Sergio Muñoz, José L. Escalona
Maximum Likelihood Estimation (MLE) is a very common approach for tuning the parameters of statistical models — the meaning of the word constrained in the title of this subsection will be seen at the end of it. The basic idea is the following. Consider a generic multivariate random variable whose probability density function, , depends on a set of parameters . Given a specific observation , we estimate as the set of parameters that maximises . That is to say, we choose the parameters that are most likely to generate the observed data. For a more complete explanation of the MLE concept, see Ref. [36].
Distribution inference from early-stage stationary data streams by transfer learning
Published in IISE Transactions, 2022
Kai Wang, Jian Li, Fugee Tsung
We again represent by b basis functions, each of which is now defined in the d-dimensional space as where is the bandwidth matrix and can be determined using Silverman’s rule of thumb. Then the estimation of is equivalent to the minimization over as in Section 2.2. When the distribution of the multivariate random variable can be factorized as a product of several marginal distributions and conditional distributions according to some dependence structures (e.g., Bayesian networks), our CDF inference can be simplified into several independent tasks conducted in some spaces of dimension lower than d.
Testing hypotheses for multivariate normal distribution with fuzzy random variables
Published in International Journal of Systems Science, 2022
Gholamreza Hesamian, Mohamad Ghasem Akbari
Let be a probability space. A mapping is called a fuzzy multivariate random variable (FMRV) if each α-values of (Hesamian & Shams, 2016) that is is an ordinary multivariate random variable for any . Two FMRVs of and are also said to be independent and identically distributed if and are independent and identically distributed random variables for each . Moreover, is a fuzzy multivariate random sample (FMRS) if 's are independent and identically distributed fuzzy random variables.