1. What is the first procedure to use?
Their first procedure is to use a fixed number of iterations of Metropolis-Hastings227 (M-H) proposals/acceptance steps comprised of a fixed number of “burn-in” steps228 followed by a fixed number of iteration steps.
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2. What is the largest global Lyapunov exponent?
Their knowledge of this dynamical system (Kostuk361 2012) indicates that the largest global Lyapunov exponent is approximately 1.2 in362 the time units indicated by ∆t.
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3. What is the NI initial path for the Monte Carlo search?
To complete their choice of initial paths, the authors now split the state variables xa(n)209 into those observed a = 1, 2, ..., L and those unobserved a > L. The latter we210 call the ‘rest’ and write them as xR(n); R = L + 1, L + 2, ..., D. The dynamical211 equations (in discrete time) can now be written212xl(n+ 1) = fl(xl(n), xR(n)) xR(n+ 1) = fR(xl(n), xR(n)).
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4. What is the CC BY 4.0 license for the MCMC procedure?
CC BY 4.0 License.model is increased, if the model is consistent with the data and the number of271 observed measurements, L, at each τk is large enough, the action level plot values272 will become independent of Rf and one will stand out as lower than the rest.
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![Figure 3: The values of the actions Eq. (16), the measurement error, and the model error for the D = 20 dimensional Lorenz96 model when L = 12 of the dynamical variables x(t) are observed; the observed variables are [x1(t), x2(t), x4(t), x6(t), x7(t), x9(t), x11(t), x12(t), x14(t), x16(t), x17(t), x19(t)]. The actions, the measurement error, and the model error are evaluated as a function of β = logα[Rf/Rf0] where α = 1.4 and Rf0 = 1.0. We perform the Precision Annealing Monte Carlo (PAMC) calculation starting with NI initial paths at each Rf . We used NI = 50 in these calculations; on display here are NI action, measurement error, and model error values at each Rf (or β). These are evaluated along the expected path resulting from the accepted paths generated during the Metropolis-Hastings procedures from each of the NI initial paths. In this case, when L = 12, the model error becomes much smaller than the measurement error as β is increased. This leads the action to become effectively equal to the action itself and essentially independent of Rf . We have seen this before in the precision annealing variational principle calculations (Quinn 2010; Ye 2016; Ye, Rey, et al. 2015; Ye, Kadakia, et al. 2015).](/figures/figure-3-the-values-of-the-actions-eq-16-the-measurement-2hzouxty.webp)
![Figure 2: The values of the actions Eq. (16) for the D = 20 dimensional Lorenz96 model when L = 5 of the dynamical variables x(t) are observed. The actions are evaluated as a function of β = logα[Rf/Rf0] where α = 1.4 and Rf0 = 1.0. We perform the Precision Annealing Monte Carlo (PAMC) calculation starting with NI initial paths at each Rf . We used NI = 50 in these calculations. Displayed here are NI action values at each Rf (or β). These actions are evaluated along the expected path resulting from the accepted paths generated during the MetropolisHastings procedures from each of the NI initial paths.](/figures/figure-2-the-values-of-the-actions-eq-16-for-the-d-20-dz9tooo0.webp)
