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Hyperbolastic regressions are statistical models that utilize standard hyperbolastic functions to model a dichotomous or multinomial outcome variable. The purpose of hyperbolastic regression is to predict an outcome using a set of explanatory (independent) variables.
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
Growth equations. Like exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in important respects.These functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic behavior (behavior as input gets large) differs dramatically:
Hyperbolic discounting is an alternative mathematical model that agrees more closely with these findings. [ 5 ] According to hyperbolic discounting, valuations fall relatively rapidly for earlier delay periods (as in, from now to one week), but then fall more slowly for longer delay periods (for instance, more than a few days).
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
The distance function for the Beltrami–Klein model is a Cayley–Klein metric.Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b, so that | aq | > | ap | and | pb | > | qb |.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
Through circular trigonometric functions {= = , = , <, , . With the tangent slope as parameter: A parametric representation, which uses the slope m {\displaystyle m} of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case b 2 {\displaystyle b^{2}} by − b 2 ...