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In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments. [1]
The first few central moments have intuitive interpretations: The "zeroth" central moment μ 0 is 1. The first central moment μ 1 is 0 (not to be confused with the first raw moment or the expected value μ). The second central moment μ 2 is called the variance, and is usually denoted σ 2, where σ represents the standard deviation.
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph.If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia.
In the above moment-cumulant formula\ = (:) for joint cumulants, one sums over all partitions of the set { 1, ..., n}. If instead, one sums only over the noncrossing partitions , then, by solving these formulae for the κ {\textstyle \kappa } in terms of the moments, one gets free cumulants rather than conventional cumulants treated above.
In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability ...
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random ...
In this normalized space, the interest points' parameters (spatial location, integration scale and differentiation scale) are refined using methods similar to the Harris–Laplace detector. The second-moment matrix is computed in this normalized reference frame and should have an isotropic measure close to one at the final iteration.
The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed -norm. Jürgen Moser [ 4 ] firstly introduced the concept of normalized solutions in the study of regularity properties of solutions to elliptic partial differential equations (elliptic PDEs).