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In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically, [22] the invariant measure is ().
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.
The metalog distribution is generalization of the logistic distribution, in which power series expansions in terms of are substituted for logistic parameters and . The resulting metalog quantile function is highly shape flexible, has a simple closed form, and can be fit to data with linear least squares.
An animated cobweb diagram of the logistic map = (), showing chaotic behaviour for most values of >. A cobweb plot , known also as Lémeray Diagram or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions , such as the logistic map .
When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point (see [1]). For the case where C = 1 {\displaystyle C=1} ,
If X has a log-logistic distribution with scale parameter and shape parameter then Y = log(X) has a logistic distribution with location parameter and scale parameter /. As the shape parameter β {\displaystyle \beta } of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) logistic distribution .
The logistic map is + = where is a function of the (discrete) time =,,, …. [3] The parameter is assumed to lie in the interval [,], in which case is bounded on [,].. For between 1 and 3, converges to the stable fixed point = /.
The unknown parameters in each vector β k are typically jointly estimated by maximum a posteriori (MAP) estimation, which is an extension of maximum likelihood using regularization of the weights to prevent pathological solutions (usually a squared regularizing function, which is equivalent to placing a zero-mean Gaussian prior distribution on ...