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Taylor expansions for the moments of functions of random variables. In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
Less than or equal to the sum of individual entropies. The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example of subadditivity. This inequality is an equality if and only if and are statistically independent. [3]: 30.
In probability theory, the first-order second-moment (FOSM) method, also referenced as mean value first-order second-moment (MVFOSM) method, is a probabilistic method to determine the stochastic moments of a function with random input variables. The name is based on the derivation, which uses a first-order Taylor series and the first and second ...
v. t. e. Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations.
The sample Taylor diagram shown in Figure 1 [15] provides a summary of the relative skill with which several global climate models simulate the spatial pattern of annual mean precipitation. Eight models, each represented by a different letter on the diagram, are compared, and the distance between each model and the point labeled “observed ...