Capítulo IV: Evaluación Interna
4.1. Análisis Interno AMOFHIT
4.1.2. Marketing y ventas (M)
For this application we use the same model as that which is presented by Solonen et al. (2012), but have changed the design problem slightly.
A glass of liquid that has an initial temperature of T (0) is cooled or heated by water that is at a temperature of Tw. The surrounding air has a temperature of Ta. The model for the liquid temperature is given by
dT dt = − kwAw M c (T − Tw) − kaAa M c (T − Ta), (3.7)
where kw and ka are the heat transfer coefficients through glass and through the air- liquid interphase, respectively; Aw and Aa are the areas of water-liquid and air-liquid interfaces, respectively; c is the specific heat capacity; and M is the mass of liquid. We placed a restriction on the prior design space where T (0) > Tw so that the mean response function decreased with time (i.e., was a cooling curve). The observations are given by the solution to equation (3.7), T (t), which is given by equation (3.8), plus normally distributed, independent error with a constant variance σ2 = 0.33:
T (t) = AaTaka− C1e
−t(Aaka+Awkw)/M c) + A wTwkw Aaka+ Awkw
, (3.8)
where C1 = −T (0)(Aaka+ Awkw) + AaTaka+ AwTwkw. The value of σ2 = 0.33 is the same as that which was used by Solonen et al. (2012).
The purpose of the experiment is to estimate θ = (kw, ka). In Solonen et al.’s (2012) work, the design problem was to determine the optimal T (0) and Tw values. Measurements of the temperature were taken every two minutes, stopping at 20 minutes.
We extend the work of Solonen et al. (2012) by looking for ‘closer to optimal’ times at which to take the 10 temperature measurements, rather than simply taking the mea- surements at two minute intervals. Here the design parameters to be optimised are d = (T (0), Tw, d1, δ), or d = (T (0), Tw, a, b), depending on the lower dimensional param- eterisation that is used. The design space for the sampling times was extended from [0, 1200] seconds to [0, 2000] seconds. The 10 sampling times were generated using the
lower dimensional parameterisation schemes mentioned in Section 3.2 (geometric, even spacing, and beta).
It was assumed that one set of measurements had already been taken at (T (0), Tw) = (23, 5) and the parameter estimates that resulted from the MCMC simulations that fitted the model to the data were used to construct the prior distribution for θ.
For this example, we investigated all three Bayesian utility functions discussed in Section 3.2. We used 20000 particles to estimate our utility functions via importance sampling. The run time for these examples was approximately 0.5 hr when the mean response variance utility was used (for each of the proposal schemes) and 1.5 hrs for when the KLD and inverse of the determinant of the posterior variance-covariance matrix were used as the utility functions.
For all three design criteria, and all three proposal schemes, the optimal temperatures were found to occur at either extreme of the design space: (T (0), Tw) = (60, 4) (i.e., hot liquid, cold water). To reduce the computational time, we set (T (0), Tw) = (60, 4) and re-ran the MCMC simulations to find the optimal designs for two design variables (d1, δ) or (a, b), depending on the scheme that was used. Two-dimensional contour plots of the thinned posterior samples of the design variables for the various lower dimensional parameterisations that were generated using the mean response variance as the utility function are displayed in Figure 3.6.
The sampling times generated by the different utility functions and proposal schemes are given in Figure 3.7 (online figures are in colour), and the utility function values for the different proposal schemes are given in Table 3.5. The utility function values for the optimal designs (Table 3.5) were calculated using Monte Carlo integration (equation (3.2)) with M = 20000.
For all three utility functions, the geometric scheme produced designs which gave the highest utility values, followed by the even spacing scheme, and the beta scheme (which gave the lowest utility values) (Table 3.5). The designs that resulted from each of the lower dimensional parameterisation schemes were found to be quite similar across the different utility functions, and produced higher utility function values than the original sampling times used by Solonen et al. (2012) (where samples were taken every 2 minutes from 0 to 20 minutes). Within each of the utility functions, the designs were somewhat similar across the different lower dimensional parameterisations in that they were fairly clustered and occurred around a similar region of the design space, with the beta scheme giving a wider coverage than the other two schemes. None of the parameterisation schemes generated designs that were spread over the entire design space.
A quantitative comparison of the optimal designs and utility functions was carried out in the same manner as for Section 3.4.3.
For all of the utility functions, the designs that were optimal for one utility function did not perform as well when input into another utility function (compared to that utility function’s own optimal design). This is most noticeable when the optimal designs from
(a) (b)
a
b
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18d
1δ
200 400 600 800 1000 1200 20 40 60 80 100 120 140 160 180 200 (c)d
1δ
200 400 600 800 1000 1.1 1.2 1.3 1.4 1.5 1.6Figure 3.6: Two-dimensional contour plots of the expected utility surface (mean response variance) for the various lower dimensional parameterisations for the cooling example (a) Beta scheme, (b) Even spacing scheme, and (c) Geometric scheme.
Design criterion Proposal
scheme
Log utility function value (95% CI)∗∗ Optimal design Even spacing 0.50 (0.50, 0.51) (d∗1, δ ∗ ) = (271.97, 34) KLD Beta 0.45 (0.45, 0.46) (a∗, b∗) = (4.73, 14.14) Geometric 0.54 (0.54, 0.55) (d∗1, δ∗) = (205, 1.11) Original design 0.35 (0.35, 0.36) NA Even spacing 60.55 (60.55, 60.55) (d∗1, δ∗) = (267.75, 36.44) 1/det(posterior var-cov) Beta 60.46 (60.46, 60.47) (a∗, b∗) = (4.72, 14.83) Geometric 60.75 (60.75, 60.75) (d∗1, δ∗) = (196.72, 1.12) Original design 57.41 (57.40, 57.42) NA Even spacing 3.45 (3.45, 3.45) (d∗1, δ∗) = (288.89, 31.29) Mean response variance Beta 3.34 (3.34, 3.34) (a∗, b∗) = (4.59, 14.14) Geometric 3.73 (3.73, 3.73) (d∗1, δ ∗ ) = (219.58, 1.07) Original design 3.34 (3.34, 3.34) NA
∗∗The proposal schemes which produced the designs that gave the highest utility values have been boldfaced for each utility
function.
Table 3.5: Utility function values for the various proposal schemes and the original design used by Solonen et al. (2012) for the cooling example.
Utility function 1 U1(d, y) Utility function 2 U2(d, y) U2(dU1)/U2(d∗)
KLD 1/det(posterior var-cov) 0.94 (0.94,0.94)
Mean response variance 1/det(posterior var-cov) 0.95 (0.95, 0.95)
1/det(posterior var-cov) KLD 0.98 (0.98, 0.98)
Mean response variance KLD 0.88 (0.88, 0.89)
KLD Mean response variance 0.77 (0.77, 0.77)
1/det(posterior var-cov) Mean response variance 0.76 (0.76, 0.76)
Table 3.6: Comparison of the optimal designs for each utility function (U1(d, y)) evaluated at the other
(a) (b)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Sampling times (sec)
KLD
Geometric scheme Even spacing scheme Beta scheme
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Sampling times (sec)
1/det(posterior var−cov) Geometric scheme Even spacing scheme Beta scheme
(c)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Mean response variance
Sampling times (sec)
Geometric scheme Even spacing scheme Beta scheme
Figure 3.7: Sampling times (in seconds) for the cooling example generated under the three lower dimen- sional parameterisation schemes using (a) the KLD, (b) the (inverse of the) determinant of the posterior variance-covariance matrix, and (c) the mean response variance as the utility function.
the KLD and the inverse of the determinant of the posterior variance-covariance ma- trix utilities were input into the mean response variance utility. This suggests that the mean response variance utility was not as efficient at designing for precisely estimating parameters as the other utilities were.