In Part I, the following analysis possibilities are described and illustrated using various synthetic datasets: (1) qualitative inspection of the estimated terms presented as fields, (2) reconstruction of the deterministic and stochastic evolution of the process and (3) approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained.
An estimator function was developed to calculate the parameter of the stochastic term, while the parameter of the deterministic term was estimated by the least squares method.
We approximated continuous-time model in Equation 4 by taking the first-order Taylor approximation of the deterministic term, which we write here as g (L ) ≡ − 2 λ sinh (L ).
So for noisy limit cycles, the deterministic term oscillates and the stochastic term perturbs these oscillations, whereas for quasi-cycles the deterministic term goes to a stable fixed point while oscillatory fluctuations persist in the stochastic term.
The deterministic term and the deterministic part of the stochastic term are treated like ODEs, as suggested by Bentele et al. [ 75]: (9) Δ x ^ i (t ) = ∑ j = 1 M n i j a j (x t ) Δ t (10) Δ x ˜ i (t ) = ∑ j = 1 M n i j a j (x (t ) ) Δ t.
The system noise term of an SDE is commonly referred to as the diffusion term, whereas the deterministic term is referred to as the drift term.
It is often assumed that the deterministic term has a linear structure, that is, η x, β) = f⊤(x) β, and that the random field ε(x) is Gaussian, allowing estimation of β and θ= by Maximum Likelihood.
We made first-order Taylor approximations of the deterministic terms in each model.
Figure 2 shows the term structure of economic capital and the term structure of the KVA obtained by a deterministic term structure approximation ES(1) as in (33) for economic capital and by the linear KVA formula (7) with ES(1) instead of EC there.
A regression model consists of two main elements; a deterministic term, linking the exposure to the outcome, and an error term which incorporates the residual variance of the observations, which is not expressed by the link function.
