Imperative jump regularization: A W IENER exponential example

The question whether it is necessary to include the jump regularization term in the objective instead of omitting them completely shall be shortly addressed here. On first sight, it might seem reasonable to view the interval solutions as independent problems, which are only coupled by the common parameters. If there is plenty of measurement data containing enough and the right information (e.g. many full-state measurements with small measurement error), this, indeed, may succeed. However, the following example shows that this approach might also fail even in the case of error-free full-state measurements, and that the inclusion of the jump regularization term in the objective leads to good estimates.

We investigate a WIENER-driven exponential, given as

dXt= pXtdt + D dWt, X0= 1.0, p = 0.25, D = 2.0, t∈ [0, 10], (5.32)

whose deterministic counterpart has the solution x(t) = X0· ept.

Figure 5.4 shows simulation results for the above system. On the left, in (a), the deter- ministic interpretation without diffusion is shown. On the right, in (b), a certain SDE solution together with the underlying realization of the driving WIENER process is depicted.

Notably is the fact that this realization of Wtdrives the system into negative states, giving

the impression of a negative initial value.

From this SDE solution, we take measurements at points _{{k+0.5, k+0.55} k = 0, ..., 9}, from which the parameter p and initial state X0 shall be recovered. The time horizon is

equidistantly split in 10 shooting intervals; all jump regularization weights ω2_k are set to 1.0. The values sk at the shooting nodes and the parameter p, in this example 11 unknowns, are

estimated by solving the parameter estimation problem 5.18 using the 20 state measurements. Table 5.1 gives a description of the experimental set-up and shows the estimation results. The method we propose in this thesis recovers both, the parameter p and the initial state X0.

Removing the jump regularization gives a much better fit in terms of residual reduction (a residual norm of approx. 0.31 compared to 10.2 for exact measurements) but leads to improper estimates.

The visualization in figure 5.5 sheds light on the cause. Without jump regularization, the coupling between the interval solution is only by the parameter p, i.e. the state varibles at the shooting nodes are independent from the state values at the previous interval’s end, and can be freely chosen to minimize the residual, leading to a trajectory with large jumps at the grid points. Moreover, by comparing figure 5.5c and figure 5.5d, and the respective results in table 5.1, we see that without jump regularization, the results are not robust to measurement noise: In both cases the residuals are small, but the estimates differ immensely.

(a) deterministic interpretation (b) with driving WIENERprocess

Figure 5.4: Trajectories of a WIENER exponential. Deterministic and stochastic interpretation of the

exponential system described in eq. (5.32), with p = 0.25 and initial value X0= 1.0.

The impact of the driving WIENERprocess, displayed as second graph in the right figure, is clearly visible in the WIENERexponential and manifests in jitter and even switching signs, changing it into a negativ exponential despite a positiv initial value.

Also, given for comparison, fitting the deterministic exponential x(t) = X0 · ept is not

convincing. The residual norm is high, the estimate for p is poor, and the estimate of the initial state X0 has the wrong sign. Clearly, as this realization of the WIENER process drives

the system into the negative halfspace, the initial value for a continuous exponential has to be negative, i.e. the initial state X0 = 1.0 cannot be recovered.

As can be seen in figures 5.5a and 5.5b, the new method with jump regularization is able to deliver good estimates, even in situations where other methods fail, and further

robustly approximates the trajectory for exact and noisy measurements in this

WIENER exponential example. We refer to chapter 6 for further successful applications of the

proposed method.

We show in appendix S.3.2, how this problem may be set up and solved using the software package :sfit developed in this thesis.

Table 5.1: Estimation results for the WIENER exponential with and without jump regularization

for the system described by eq. (5.32). In the first column, exact state measurements, taken at time points_{{k+0.5, k+0.55} k = 0, ..., 9} have been have been used for the estimation. In the second col- umn, every datapoint was additively disturbed by a random value drawn from a normal distribution with zero mean and a variance of 0.25% of the respective measurement value. The shooting node variables at time points _{{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} were initialized by the temporally most proximate measurement} data; the initial guess for the parameter wasp0= 0.5. Measurement variances σi and jump regularization weights ω2

k were chosen as 1.0.

The letters (a)–(e) refer to the respective picture in figure 5.5; the values R and J refer to the 2-norm of the residual vector and of the weighted jumps, respectively. Values rounded. Also see appendix S.3.2.

p∗= 0.25 true exact measurements with measurement error

X∗₀= 1.0 values fig. estimate Res./Jmp. fig. estimate Res./Jmp.

with jump regularization (a) p = 0.2896 R = 10.2 (b) p = 0.2629 R = 20.4 X0= 1.1841 J = 10.5 X0 = 1.0429 J = 14.8 without jump regularization (c) p = 0.5133 R = 0.31 (d) p =−1.3901 R = 2.85 X0= 0.7317 J = 68.7 X0 = 1.5294 J = 1651.7 continuous trajectory (e) p = 0.3461 R = 61.9 (f) p = 0.3371 R = 82.6 X0 =−0.8410 J ≈ 10−10 X0=−0.9192 J ≈ 10−10

(a)with jump regularization, exact measurements (b) with jump regularization, with measurement error

(c) without jump regularization, exact measurements (d) without jump regularization, with measurement error

(e) continuous trajectory, exact measurements (f) continuous trajectory, with measurement error

Figure 5.5: Fitted exponential trajectory with and without jump regularization for the cases of the WIENERexponential parameter estimation problem as described in table 5.1. In (a) and (b), the fit originating from the new method with jump regularization is depicted. The discontinuities at the shooting nodes are clearly visible. Though the residual is much smaller in the settings (c) and (d) without jump regularization, the resulting estimates are worthless. The continuous trajectory in (e) and (f), apparently giving a good fit, also delivers unsatisfactory results. See table 5.1 for details.

The new method with jump regularization, shown in (a) and (b), successfully recovers both the kinetic parameter and the initial state value.

Fitted trajectory as blue line (—), shooting nodes marked with a blue dot (•), measurements marked with a red dot (•).

In appendix S.3.2 we show how to set up and solve this problem using the software package :sfit.