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Bayesian prediction of the next glacial inception (Rev #3, changes)

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This paper tries to predict the next glacial cycle with a stochastic model, using a stochastic differential equation derived from a deterministic model from Saltzman and Maasch (1991).


The authors investigate a system with three dimensions, the three variables are

  • ice volume I

  • atmospheric CO 2CO_2 concentration μ\mu

  • deep-ocean temperature θ\theta

The dynamics of the system is described by three stochastic differential equations with uncorrelated additive noise terms:

dI t=(a 1(k μμ+k θθ+k RR(t))K II)dt+Σ 1dW 1,t dI_t = (-a_1 (k_{\mu} \mu + k_{\theta} \theta + k_R R(t)) - K_I I ) \; dt + \sqrt{\Sigma_1} \; dW_{1, t}

The term R(t)R(t) describes external forcing, caused, for example, by Milankovitch cycles.

dμ t=(b 1μb 2μ 2b 3μ 3b θθ)dt+Σ 2dW 2,t d\mu_t = (b_1 \mu - b_2 \mu^2 - b_3 \mu^3 - b_{\theta} \theta) \; dt + \sqrt{\Sigma_2} \; dW_{2, t}
dθ t=(c 1IK θθ)dt+Σ 3dW 3,t d\theta_t = (- c_1 I - K_{\theta} \theta) \; dt + \sqrt{\Sigma_3} \; dW_{3, t}

These equations correspond to the deterministic equations 5a, 5b and 5c of the paper, augmented by noise according to equation 8.

The authors treat some of the parameters of the model as fixed, and some as random variables.


  • Do the authors draw the parameters on every time step from the according distribution, or do they draw them once for each trajectory aka “particle”?

  • On page 25 there is the following statement:

At each time step, the algorithm considers n (here : n = 50000) samples of model parameters and states called ‘particles’. Particles are propagated forward in time according to a three-step process: (1) forward integration of the stochastic equations; (2) particle weight estimation as the product of prior weight and a likelihood taking into account the observation and its uncertainty; and (3) particle resampling in order keep a set of particles that all have approximately the same weight (importance resampling). We emphasise that this algorithm is a filter and not a smoother. Consequently, posterior estimates at any time t are only informed by data prior to time t.

Tim van Beek: I don’t understand anything after (2)…


This book covers many algorithms of the type used here: