Identification method of stochastic nonlinear dynamics using dynamical phase analysis—application to Forsmark data

Maximum likelihood estimators for generalized Cauchy processes

“…39,40 The parameters ␤ and ␥ are natural ones. In this case, the MLE relations become ͗R 2 ͘ = ͑␥ +1͒ /2␤ and ͗ln R͘ = ͑͑␥ +1͒ /2͒ − ͑1/2͒ln ␤.…”

“…In this case, one can also estimate successfully values of these parameters by combining ͗R 2 ͘ and ͗R 2 ln R͘ c with simple algebraic calculation and without bypassing the digamma function ͑z͒. 39,40 The method of parameter estimation from time series data in the GCP should be recursive even in the case of independent noise since there is the non-natural parameter a in the pdf for the GCP. One can take the MLEs ͗ln͑x 2 + a 2 ͒͘ = ͑b͒ − ͑b −1/2͒ +2 ln a in Eq.…”

“…Consider the stochastic Ginzburg-Landau model under the influence of multiplicative noise F͑t͒͑=F x ͑t͒ + iF y ͑t͒͒ in the complex number, 39,40 …”

“…7,39,40 However, the detailed description of parameter estimation for the pdf of the phase-velocity has not been given in detail.…”

A Combined Model-Based and Data-Driven Prognostic Approach for Aircraft System Life Management

“…Although no general method exists for inferring the parameters of stochastic nonlinear dynamical models from measurements, various schemes have been proposed [33,20,22,35,19,42,48,45]. An important numerical technique, suggested in [43,20], is based on estimating drift and diffusion coefficients at a number of points in the phase space of the dynamical system.…”