Principales resultados de las estimaciones realizadas

Having successfully identified the NROY space, the question of how to query from such region to refocus the emulator remains a challenging problem. The

Figure 5.2: Samples being generated at each wave for the torus implausibility function of Williamson and Vernon [223]. At the final level, the adaptive sampler correctly identifies the four regions of the zeroth contour level (in red). It should be noted that exploration of the space could be done with samples from previous levels if required.

seminal papers [48] and [49] performed a sequential design for this purpose. The common choice in the literature is to select points greedily from the NROY space to build a new emulator completely focused on that region [191]. This implies that a new emulator is built based on the identified region at every wave. In this dissertation, a different (albeit conservative approach) is followed. Since the simulator is assumed to be computationally very expensive, it might seem unrealistic to expect that the computational budget is kept the same at every wave. Moreover, discarding points might represent a waste of resources and information. In turn, the points must be chosen carefully at each wave to later add them to the set of training runs and build a new Bayesian emulator.

The idea is that the most general information would likely be extracted at the very first iterations while greater accuracy will be pursued at later stages of the history matching procedure. For example, at the first waves a good characterisation of the global trend can be identified. It is therefore appropriate to guide the choice of training points by following a learning criteria. This is done in Bayesian optimisation or in reliability analysis problems [218, 19, 139]. For comparison, three active learning criteria are presented. The first criterion is the expected

contour improvement by Ranjan et al. [184]. The improvement function is defined as

I(x) = 2(x) − min(f (x) − z)2, 2(x) , (5.9) where f (x) is the Gaussian process emulator at configuration x, z the targeted contour level, and (x) = k v(x) the number of predicted standard deviations derived from the uncertainty model, v(x) = pσ2_{(x) + σ}2

md+ σ2me. In history

matching k is typically set to 3, following the three sigma rule. As the emulator is random in nature, the expected value of the contour improvement is used instead of a pointwise estimate. The expected contour improvement (ECI) can be computed as

E[I(x)] =2(x) − (µ(x) − z)2− σ2(x) [Φ(u2) − Φ(u1)]

+ σ2(x) [u2φ(u2) − u1φ(u1)] (5.10)

+ 2[µ(x) − z] σ(x) [φ(u2) − φ(u1)]

where (x) is the k-th multiple of standard deviations, u1 = (z − µ(x) − (x))/σ(x),

u2 = (z − µ(x) + (x))/σ(x), and Φ(·) and φ(·) are the standard Gaussian

cumulative and density functions respectively.

The second criteria to be tested is the expected risk by Echard et al. [73], which was originally was designed for reliability analysis. This implies learning the set {x : g(x) > 0}, with g(x) the performance or limit-state function of a configuration x. The critical level g(x) = 0 is referred to as a transition level, as its correct emulation will allow one to effectively classify a given configuration in the simulation in terms of the system’s performance. In this chapter, the problem is explicitly stated in terms of the contour level z which corresponds to the observed data in the experimental setting. This means that the risk function is defined as

Rz(x) =

(

(f (x) − z)+ if µ(x) ≤ z

(z − f (x))+ if µ(x) > z

, (5.11)

where (·)+ denotes the non-negative part of the argument, and µ(x) denotes the

expected value of the Gaussian process emulator at configuration x. The expected risk is used as a learning criteria due to the random nature of the output of the emulator. The derivation is a straightforward solution of one dimensional

Gaussian integration, which for completeness is included in 5.A. The analytical expression can be written in compact form as

E[Rz(a)] = σ(x) [−sign(¯z) ¯z Φ (−sign(¯z) ¯z) + φ(¯z)] , (5.12)

where ¯z = (z − µ(x))/σ(x) denotes the standardised contour level, sign(·) the sign function, and the pair Φ(·) and φ(·) are the cumulative and density functions of a one-dimensional Gaussian random variable respectively.

Lastly, the third learning criteria to be compared, is a variation of the entropic profile presented by Lv et al. [145]. Originally formulated in the reliability analysis literature, it was designed to measure the entropy of a random variable in a neighbourhood of two standard deviations from the origin. In this chapter the concept has been extended. Again, an explicit solution is presented for a contour level z observed in the experimental data. The entropic profile is defined as

Hz(f (x)) =      Z z+k σ(x) z−k σ(x) − ln π(y) π(y) dy      , (5.13)

where σ2_{(x) denotes the predicted variance from the emulator and π(y) the}

non-standard Gaussian distribution from the emulator response y = f (x) is distributed as a N (µ(x), σ2_{(x)). As shown in 5.B, the entropic profile can be}

written compactly as Hz(η(x)) =    h ln√2π σ(x)  + 0.5 i [Φ(¯z2) − Φ(¯z1)] − 0.5 [¯z2φ(¯z2) − ¯z1φ(¯z1)]   , (5.14) where ¯z1 and ¯z2 denote the standardised contour levels, that is, ¯z1 = (z − µ(x) −

k σ(x))/σ(x) and ¯z2 = (z − µ(x) + k σ(x))/σ(x).

All three of the previously discussed criteria rank the samples from the identified NROY space. It is important to note that these type of learning criterion take into account a one-step-look-ahead pointwise strategy. Other options include A-optimal designs, which incorporate area impacts to the improvement of the emulator’s response surface. In particular, following dynamic programming strategies, one can define a learning criteria with a known number of sequential decisions. As a consequence, this selection of points chooses a batch of candidate runs. This is known as finite-horizon dynamic programming [28].

Given the proposed learning criteria, a natural question is how to choose the points given such ranking. As nearby sample points are likely to be similarly ranked, it would be naive and a waste of computational resources to query the simulator on just the top-ranked NROY samples. Alternatively, to choose uniformly from the sample points would ignore any ranking at all. An adequate leverage between these extremes is achieved by the following procedure.

Firstly, a learning criterion is chosen to rank the samples, as well as a cut off point. That is, a threshold to guarantee that some percentage of the maximum attainable gain can be held. In this work, this percentage is assumed to be 50%. The current setting selects the best first point following an active learning criteria. Although, the rest of chosen points will have a diluted effect from the learning criteria. As commented above, choosing a high percentage would likely concentrate the candidates in narrower neighbourhoods around the best sample. In contrast, a low percentage would not acknowledge the ranking. Secondly, a maximin design is proposed to chose the next batch of training points as follows. The seed of this design is selected to be the top ranking point from the NROY samples, i.e. the point that has the highest expected learning criteria. It is important to note that this type of sampling usually starts with the mean of a cloud of points [see 139, for more details]. Since the NROY space has potentially a complicated topology, the average point might lie outside the region of interest and choosing a point from outside the NROY space would be a poor selection.

After choosing the starting point x0, the maximin design proposes the furthest

sample available, so x1 = arg maxXNROY||x − x0|| is chosen as a successor. This selection takes both points out from the bag of samples and initialise the set of new sample points containing both, X∗ = {x0, x1}. The training points currently

in use for the emulator are also included in X∗, that is X∗ = X∗ ∪ D. This prevents redundancy in the selection of new points. The next step computes the distances to both x0 and x1 for the remaining sample points in the sampled NROY

collection. After this, each sample point in the NROY set has two associated distances to X∗. The next step involves choosing the minimum for each NROY sample. Thus, the distance of each point to the set X∗ is computed. Following this, x3 is assigned as the furthest point observed and is taken out of the bag

of samples and incorporated to the collection of new training runs. That is, the update X∗ = X∗ ∪ {x3} is performed. The selection of new points follows this

as the farthest point. Performing the previous steps allows us to obtain N new samples to train a refocused emulator. As mentioned above, the first point will follow the specific learning criteria chosen. The rest of selected points will followed a space-filling design with a diluted effect from the criteria in use.

In summary, the use of the maximin selection allows us to (i) retain the best possible point; (ii) collect samples from a space-filling design in NROY space; and (iii) restricts the choice of new points following the active learning criteria.