Following the method described in Section 3.8 of Chapter 3, the ABC-SMC inputs are defined as follows. The other steps of the Sequential ABC-UKF, i.e.
the UKF initialization, the data generating process and the optimization methods settings, are the same of Sections 4.1 - 4.3. The number of particles is fixed to N = 1, 000. The prior distribution for the parameters is a Uniform and the extremes of the interval are the 400% deviation from the true values. To avoid instability, the parameter β ∈ R+, and, hence, the left extreme value of the Uniform for β is trun-cated at zero. The perturbation kernel is a Gaussian distribution centered at λ∗∗ and with variance τt2. The distance function is the Euclidean norm, while the thresholds
(a)
(b)
(c)
Figure 4.7: Deterministic Duffing system: log-likelihood of measurements for different offset. (a) High offset (400% of deviation from the true values). (b) Medium offset (250%
of deviation from the true values). (c) Low offset (100% of deviation from the true values).
The white spaces are due to numerical instability when inverting the Kalman gain matrix.
are generated as linearly spaced vector in the summary statistics space.
Let us consider three summary statistics: (i) the Ratio of the Peaks (RP), which is the ratio between the amplitude of the first peak and the amplitude of the last peak of the oscillation, (ii) the dominant frequency (DF) of the fast Fourier transform (FFT), and (iii) the number of zero crossings (ZC) of the signal. The choice of the statistics relies on the characteristics of the three Duffing parameters. The DF detects the predominant frequency of oscillation, which is captured by α, while the RP is more suitable to identify the damping term. The ZC statistics is evaluated to find
SEQUENTIAL ABC-UKF ESTIMATES 79
Figure 4.8: Computational costs of EI and UCB acquisition functions.
Figure 4.9: Objective function evaluation and number of acquisition function computa-tions. Comparison between EI and UCB.
β, which influences the stability of the trivial fixed points in the phace-space and, so, the convergence to zero of the oscillation in the time domain, as pointed out in Chapter 2.
The univariate and bivariate posterior distributions of the ABC-SMC are shown in Figures 4.10 and 4.11. The ABC-SMC posterior distributions for α and c are
(a) (b) (c)
Figure 4.10: Deterministic Duffing system: univariate first and posterior distributions of parameters in the ABC-SMC scheme initialized with 400% offset. (a) Estimate of parameter α. (b) Estimate of parameter β. (c) Estimate of parameter c.
(a) (b) (c)
Figure 4.11:Deterministic Duffing system: bivariate posterior distributions of parameters after ABC-SMC scheme with 400 % offset. (a) Distribution of (α, β). (b) Distribution of (c, β). (c) Distribution of (c, α).
peaked near their true values. The most difficult parameter to infer is β, the term associated to the chaotic behaviour. In order to obtain 1,000 particles per population in the SMC intermediate distributions, the overall generating process consists of about 500,000 particles and the final acceptance rate is 0.04. The number of effective particles at the end of the intermediate distributions is the 95% of the original number of particles. The median values of the posterior distributions of the ABC-SMC are the starting values for the UKF.
Table 4.3 summarizes the inference achievements with the Sequential ABC-UKF in the parameter space. The new method converges to the true values in the sense that the true parameters lie in the estimated confidence interval (Figure 4.12).
Com-SEQUENTIAL ABC-UKF ESTIMATES 81
(a) (b)
(c) (d)
Figure 4.12: Deterministic Duffing system: Sequential ABC-UKF estimates in the time domain. (a) Signal estimate. (b) Estimate of parameter α. (c) Estimate of parameter β. (d) Estimate of parameter c.
pared with the default UKF, the Sequential ABC-UKF shows a massive improve-ment: the first let increase the distance between the estimates and the true values of 239%, while the latter cuts the same distance of 6.65. The estimates of α and c at the end of the filtering phase always lie within the predicted standard error around the estimate. Instead, the value of β is more affected by uncertainty. Figure 4.13 shows the RSS in functional and parameter space. The Signal RSS is low but not equal to zero; indeed, from the top-left panel of Figure 4.12 it is noticeable that the signal reconstruction is smooth and it reproduces the observation path, except for some peaks in the measurements. The Solution RSS is higher than the Signal RSS, mean-ing that insertmean-ing the final parameter estimates of the filtermean-ing phase in the ordinary differential equation comes out in more jagged path.
As far as the comparison between optimizing methods is concerned, Table 4.3
Norm in parameter space Optimization method Before inference After inference
Default UKF 6.90 16.50
UKF with EI 6.90 3.60
UKF with UCB 6.90 4.43
UKF with Grid 6.90 4.09
Table 4.2: Deterministic Duffing system: standardized Euclidean norm in the parameter space before and after UKF filtering with 400% offset. Comparison between default sigma points location and optimizing methods.
Norm in parameter space
Method Before inference After inference
Default UKF 6.90 16.50
Sequential ABC-UKF with EI 6.90 0.25
Sequential ABC-UKF with UCB 6.90 0.27
Sequential ABC-UKF with Grid 6.90 0.26
Table 4.3: Deterministic Duffing system: standardized Euclidean norm in the parameter space before and after UKF filtering. Comparison between the default UKF and Sequential ABC-UKF method with different optimization schemes. Initialization is 400% offset.
points out that there is little difference among them. Tables 4.2 and 4.3 highlight that the discrepancies between optimization algorithm disappear as the UKF reaches a good initialization. The EI acquisition function comes out to have the smallest Euclidean distance in the parameter space but its predominance on the UCB and discrete grid shrinks if compared with the results in Table 4.2. The evaluation of the UKF filtering performance coupled with optimized sigma points location and with respect to a thoughtful research of starting values brings out the conclusion that the initialization is more determinant than sigma points placement to convergence to the true parameters.
DISCUSSION 83
Figure 4.13: Deterministic Duffing system: RSS and ARB of Sequential ABC-UKF esti-mates in functional and parameter space.