Las Normas Internacionales de Información Financiera

This chapter evaluated optimal-control and MCMC methods on the eflornithine dataset. For the optimal-control methods, underprediction of the initial peak is

0 2 4 6 8 Time (h) 0 5 Rate ( µ mol/h)

Input rate - dense test data Estimated rate True rate 0 2 4 6 8 Time (h) 0.00 0.01 0.02 0.03 Conc. ( µ mol/mL)

Concentration - dense test data Estimated conc. Measurements True conc. 0 2 4 6 8 Time (h) 0 5 Rate ( µ mol/h)

Input rate - sparse test data Estimated rate True rate 0 2 4 6 8 Time (h) 0.00 0.01 0.02 0.03 Conc. ( µ mol/mL)

Concentration - sparse test data Estimated conc. Measurements True conc. 0 2 4 6 8 Time (h) 0 5 Rate ( µ mol/h)

Input rate - real low-dose data Estimated rate 0 2 4 6 8 Time (h) 0.00 0.01 0.02 0.03 Conc. ( µ mol/mL)

Concentration - real low-dose data Estimated conc. Measurements 0 5 10 15 20 25 Time (h) 0 100 200 Rate ( µ mol/h)

Input rate - real high-dose data Estimated rate 0 5 10 15 20 25 Time (h) 0.00 0.25 0.50 0.75 1.00 Conc. ( µ mol/mL)

Concentration - real high-dose data Estimated conc. Measurements

Figure 4.16: Estimates for the noisy test and real datasets, using penalisation of the second derivative in the log domain. Note that the last dataset has a different scale, since rates and concentrations are considerably higher for this dose, and measurements were taken over a longer time. These estimates were performed using MALA. The results for block RWMH and SMMALA are practically identical.

a common problem. However, this is difficult to avoid without making stronger modelling assumptions. In this and other PK examples, it may be reasonable to assume that the input function is0at time0, which may help in capturing the peak. If it is important to accurately estimate the initial dynamics without additional assumptions, it appears necessary to have a higher initial sampling rate. However,

when the bioavailability is the main quantity of interest, the nonzero initial value and the lower peak partly offset each other, making the estimated bioavailability insensitive to these assumptions.

It is important to have an optimisation algorithm that handles evaluation errors gracefully. Such errors can occur when the optimiser attempts to evaluate the cost function for values that cause the ODE solver to fail. The optimiser used here, IpOpt, does perform well in this regard.

Collocation methods are one order of magnitude faster than the other methods, but they can have robustness issues in some situations, resulting in convergence failures. However, these failures can be automatically detected. Thus, one possible strategy is to first attempt to use collocation methods, and to use shooting methods as a fallback in case the collocation methods fail. For sparse data, which is the usual real-world case, multiple shooting is generally faster than single shooting.

While the shooting methods use an ODE solver that automatically selects suitable integration options, taking stiffness issues into account, the collocation methods as used here require the user to choose interpolating polynomials and discretisation step size. For future work, it would be useful to investigate how these choices affect the robustness. Better settings may potentially make the collocation methods more widely applicable.

The choice of prior greatly influences the result, although it is far from obvious which one should be preferred. The accuracy in terms of RMSE values is similar. Figure 4.8 suggests that penalisation of the first derivative in the log domain may be more likely to produce a peak similar to the one present in the true input function. However, these kinds of considerations are subjective.

The example of using the L-curve method to choose a value ofτ shows that this method offers no clear advantage to the discrepancy criterion. Since this method requires the user to do a lot of work, it cannot be recommended in the form presented here.

All MCMC methods struggle with data that have almost zero measurement noise. When the data are noisy, as they usually are in real data, the block RWMH, MALA and SMMALA methods combined with Karhunen-Lo`eve basis functions are generally able to sample from the posterior reasonably efficiently. This conclusion is strengthened by the observation that both sampling methods provide virtually identical results for the same prior and data. While the SMMALA compares favourably to MALA, in terms of median running time, the large variability of the results makes it difficult to make any definite conclusions. Since the proposal covariance matrix in block RWMH and the metric tensor in MALA are fixed from the start, it is possible

that they could be more sensitive to their initial conditions than SMMALA. For this reason, using SMMALA may be advisable in order to have enhanced robustness. The required running time per sample is similar for MALA and SMMALA. This may seem surprising, as SMMALA recomputes the metric tensor in each iteration. However, the most expensive step in the metric tensor computation is to compute the gradients of the plasma concentration predictions at the measurement time points. This step is also required in MALA in order to compute the gradient of the log-posterior. Total running time depends on the time per sample as well as the total number of samples required, which depends on the sampling method’s ability to generate approximately uncorrelated samples.

For a given amount of time, the block RWMH method generated approximately one order of magnitude more samples than MALA or SMMALA. The main reason for this is that RWMH does not require gradient computations, which can be a computational bottleneck. Despite this, the ESS is similar for all three methods. This shows that the gradient-based methods do produce higher-quality samples, with lower autocorrelations in the Markov chains, at the expense of requiring more computation per sample.

In many cases, lack of nonnegativity constraints results in implausible results, and can also unnecessarily increase uncertainty. This is particularly pronounced in the sparsely sampled later parts of the time series, where concentrations are close to zero. With no constraints in place, there is a large number of candidate functions that could explain the data. When nonnegativity constraints are added, only functions that stay close to0can explain the data. For this reason, modelling the input function in the log domain should be preferred.

While the parameterisation should ideally not affect the results, in this case it did. Few sampling procedures involving B-splines were successful. However, for the ones that were successful, estimation errors were higher than for the corresponding procedures using Karhunen-Lo`eve functions.

When starting with a new model and dataset, it is important to investigate typical values of the sum of squared residuals, so that the prior over τ does not influence the estimate in ways that were not intended. A poor choice can cause τ to be underestimated, allowing sample paths to make large excursions between measurements and causing the estimation uncertainty to increase.

Computing the likelihood, and especially the gradient of the likelihood, is expensive as it includes running an ODE solver. For this reason, methods that update all parameters influencing the likelihood simultaneously have an advantage in terms of speed. The single-component RWMH methods are much slower than block RWMH

for this reason — to update20coefficients, the ODE solver needs to run 20times. In contrast, updatingτ independently using Gibbs sampling is practically free, as updatingτ does not require updating the likelihood.

Finally, it is important to stress that this should be considered to be an exploratory study. Many combinations of optimisation or sampling methods, priors, and function parameterisations are applied to a relatively small dataset. This allows for a multitude of possible comparisons, resulting in multiple hypothesis testing issues. For this reason, only major results are reported here. More extensive tests would be required to conclusively tell whether multiple shooting is generally faster than single shooting, or whether MALA is faster than SMMALA. Testing the methods on other problems, as done in the later chapters, helps increasing the confidence in the conclusions.