Smooth functional tempering for nonlinear differential equation models

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Stat ComputFig. 6 Bias in point estimatesfor the FitzHugh-Nagumoparameters α (top), β (middle)and γ (bottom) of Sect. 4.35 Nylon exampleIn this section, we model the production of nylon in a heatedreactor where its constituents, amine (A) and carboxyl (C),combine to produce the polymer, nylon (L), and water (W),which escapes as steam. At the same time, before escapingthe system as steam, W decomposes L into A and C in themolten nylon mixture, giving the symbolic competing reactionsA + C ⇋ L + W . In the experiment of Zheng et al.(2005), steam is bubbled through molten nylon to maintainan approximately constant concentration of W in the systemcausing A, C and L to move towards equilibrium concentrationswith W. Within each of the i = 1,...,6 experimentalruns, the pressure of input steam was held at a high level untiltime τ i1 , and then reduced until time τ i2 , at which point itreturned to its original level for the remainder of the experiment.Each experiment was performed at a constant temperatureT i which, along with the input water pressure, determinesthe equilibrium concentration of water in the moltennylon mixture, W eq . Using reaction rates k p and K a , the dynamicsof the model are described with the following systemof differential equations:− dL = dA = dC =−k p (CA − LW/K a )10 −3 , (14)dt dt dtdW= k p 10 −3 (CA − LW/K a ) − 24.3(W − W eq ). (15)dtThe reaction rate K a is allowed to change with T i andW eq through relationships depending on the reference temperatureT 0 = 549.15 giving four ODE parameters: θ =[k p ,γ,K a0 ,H] by the following expansion of K a :{K a = 1 + W eq γ 10 −3} ( HK T [K a0 ]l8.314), (16){ 1l(m) = exp(−m10 3 − 1 }), (17)T i T 0(K T = 20.97 exp −9.624 + 3613 ). (18)T iFigure 7 shows the data for each of the 6 experimentalruns. The plot shows the observed components A and C aswell as input W eq . Due to the mass balance of this system,given any three components the fourth can be computed exactly.Because only A and C are observed, we must estimatethe unobserved W(t) for each experimental run. Furthermore,since the components are chemical reactions, theyare constrained to take on non-negative values. In the nylonsystem, x 0 increases the dimension of the parameter spacefrom 6 parameters in [θ,σ 2 ], to 24 parameters; [θ, x 0 ,σ 2 ].We set our prior distribution to be uniform on the set offunctions taking values between 0 and 250, where 250 wasselected because it its about 10% larger than the largest observation.We chose this interval to be wider than is likely tobe necessary to place non-zero mass over realistic functionsfor this problem. It is likely that values of the unobservedW would remain close to the values of W eq (all take onvalue less than 100) but the more conservative value of 250was used throughout for the states A, C and W . Additionally,the prior distributions on 1/σA 2 and 1/σ C 2 were chosento be independent Gamma densities with mean 9 and variance27. These are pessimistic priors relative to the measurementerror variance estimates from additional experimentsby Zheng et al. (2005).We implemented SFT1 and SFT2 using evenly spacedknots placed at a rate of 3 per hour of experimental duration.In anticipation of sharp dynamics after the step change in inputW eq , an additional 9 knots were evenly spaced at timesτ +[0.1, 0.2,...,0.9] after the input change. The discontinuousfirst derivative induced by the step input change wasaccommodated by the addition of knots at the time of thestep change. SFT2 was implemented with values λ 1 = 100,and λ 2 = 10, 000. SFT1 required four times the number ofparameters of the SFT2 model and consequently M = 3chains were used, with tempering values λ 1 = 200,λ 2 =500.
Stat ComputFig. 7 The nylon observations along with the fit to the data. Temperatures of the experimental runs are given above component A in degreesKelvin. Vertical axes are in concentration units and horizontal axes are in hoursFig. 8 A comparison of theposterior density estimates forthe nylon parameters usingSFT1 (black line) and SFT2(dashed line)The small values of λ 1 in both methods produced considerablerobustness with respect to values used to initializethe Markov Chains. The kernel density estimate of 40,000posterior draws from the M th chain of SFT1 and SFT2 forθ,σA 2 and σ C 2 (after discarding burn-in) are shown in Fig. 8.Estimates for the marginal posterior densities of θ are similarbetween the methods, and the values for the integratedsquared difference between marginal posteriors comparingSFT1 with SFT2 are 0.057, 0.016, 0.018 and 0.0073 fork p0 , γ,K a0 and H respectively. The squared discrepancybetween the marginal posterior density estimates deviatesslightly more for σA 2 and σ C 2 giving values equal to 0.11 and0.15 respectively. The reason for this discrepancy may lie inthe marginal posterior density estimates of x 0 , estimated bySFT1 and shown in Fig. 9. The dynamics of the system arequite fast, so that the impact for some of the experimentalruns on moving W 0 from near 0 to near 250 only affects thefit to the first few data points, leading to some relatively flat(and uninformative) posterior distributions. SFT1 exploresthe distribution of X 0 and, in the process, finds more valuesthat allow a better fit to A in exchange for a decrease in fit toC giving the shifted densities for σC 2 and σ A 2 showninFig.8.SFT1 also allows for new insights into the vast uncertaintyin W 0 . The advantages of SFT2 are the reduced dimensionof the problem and, in this case, a five fold computationaltime reduction.5.1 Comparison of computational effortThe simulation studies of Sect. 4 were performed with highresolution, homoscedastic data from a fully observed systemin order to highlight the algorithmic performance inthe face of specific posterior topological difficulties. Datasuch as this are unlikely to be observed often in practice.In this section, we examine the variability in computationtime of the SFT1, SFT2 and PT algorithms using more realisticreplications of the same algorithmic process wherethe only difference between algorithmic trials is the randomseed initializations. This is in contrast to the simulation studiesof Sect. 4 where each trial used a different random dataset. Variability in the computation time required for a singleiteration of the sampler depends on the parameter valuesbeing proposed because they determine the stiffness (magnitudeof the derivative) of the differential equation system.When numerically solving an ODE, a stiff system is considerablyslower than a non-stiff system and the parametervalues determine the stiffness (Deuflhard and Bornemann2000). Consequently, we use repeated algorithmic simulationson a fixed real dataset to compare the algorithmic differencesin methods and examine the practical advantages ofthe proposed methods.The full nylon model in (14)–(18) requires multiple experimentalruns to estimate the temperature dependency ofk p and K a , however for this study we use a single experimentalrun of the nylon dataset, (shown in Fig. 7 with tem-
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Smooth functional tempering for nonlinear differential equation models

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