By Kamil Feridun Turkman, Manuel González Scotto, Patrícia de Zea Bermudez (auth.)
This booklet deals an invaluable mixture of probabilistic and statistical instruments for examining nonlinear time sequence. Key beneficial properties of the e-book comprise a research of the extremal habit of nonlinear time sequence and a complete checklist of nonlinear types that tackle various facets of nonlinearity. a number of inferential tools, together with quasi chance equipment, sequential Markov Chain Monte Carlo tools and particle filters, also are incorporated so that it will offer an total view of the to be had instruments for parameter estimation for nonlinear types. A bankruptcy on integer time sequence versions according to numerous thinning operations, which brings jointly all contemporary advances made during this zone, is usually included.
Readers must have attended a previous direction on linear time sequence, and an excellent take hold of of simulation-based inferential equipment is usually recommended. This e-book deals a priceless source for second-year graduate scholars and researchers in facts and different medical components who want a simple realizing of nonlinear time series.
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Extra info for Non-Linear Time Series: Extreme Events and Integer Value Problems
We now look at certain aspects of nonlinear processes which may indicate how we should construct useful nonlinear models. 4 Sensitive Dependence on Initial Conditions, Lyapunov Exponents The most striking feature of nonlinear processes is the strong dependence on initial conditions and the noise amplification. Let us start with deterministic difference equations, representing some dynamic system in discrete time. xn 1 / defines a deterministic difference equation, for some function f . f . x0 //// be the value of the system after n iterations.
1 Parametric Models for the Conditional Mean These models represent the conditional mean function of the process Xt as a nonlinear function of the past observations, keeping the conditional variance constant. d. sequence and Â is an unknown parameter vector to be estimated. In some cases, the function f may also depend on other external processes. Several different forms of f give different classes of nonlinear models. One important subclass is the regime models or regime switching models. Models in this class are typically made up of several piecewise linear processes and the generating process switches from one linear model to another, depending on the value of an indicator.
Pij / for i; j D 1; 2. d. sequences. pj ; qj /. Hence the process will alternate among various linear models in accordance with the transient behavior of the unobserved process St . The estimation, identification and diagnostics for these models are complicated although not impossible, due to the fact that the process St is not observed. Note that if the residual process is made to depend on the unobserved Markov chain then the variance of the process also changes from one regime to another. In Fig.