Elaboración de la clasificación de los tipos de rodales

In practice, we may have several groups of assets of different types, which we believe deteriorate with similar behaviour or different but related. Model in Section 4.1 is extended with another layer of parameters as a hierarchical BN to learn between groups. Since the failure times are determined by the parameters shapes and scales, we assume the distribution’s parameters learnt for one type of asset share some similar deterioration characteristics with differences with other types. This section introduces how to learn between groups and individualise the deterioration learning with similar or different but a related rate of decay.

4.2.1

Assets with Similar Deterioration Rate

This subsection shows how to use the learned deterioration of a dominant group (data rich) to infer the learning of weak groups (data poor). It is suitable for a situation where assets deteriorate similarly, and experts are confident about the difference between groups. For example, it can be used for a case where there are only a few influencing factors on asset deterioration, or there is a dominant feature has been identified with a significant impact on the rate of deterioration. We do this by assuming the parameters learned for one group approximate those of the other similar types, with differences inferred by their hyperparameters.

Figure 4.6 Multiple asset groups with similar deterioration.

Assume assets in Group A (assets 1 to 8 in this example), Group B (asset 9 to 11) and Group C (asset 12 and 13) have similar deterioration rates resulting from some shared characteristics (such as similar designs with different materials). Group A has more failure data (dominant group) than Group B and C (weak groups). In the case that one group has more failure data than the other groups, we may leverage the deterioration learning from one group to other groups. Figure 4.6 shows a hierarchical BN for these three groups of assets. Shapes (node Group B: shape and Group C: shape) and scales (node Group B: scale and Group C: scale) of these assets were governed by the Group A’s shape (node Group A: shape) and Group A’s scale (node Group A: scale) variables, whose prior probability distributions are using triangular distributions as suggested in Section 4.1.

Experienced experts may have knowledge about the typical deterioration behaviour of assets in Group A since it has more failure data to observe. Assume assets in Group A are concrete-based bridge decks, with the knowledge elicitation as discussed in Section 4.1.3, the experts can express the prior knowledge for parameter shape β and scale η. For example, for scale η in Transition 1 of a typical concrete-based bridge deck, experts estimate it follows a triangular distribution with a lower bound of 10 years, a middle of 23 years and an upper bound of 30 years. This knowledge becomes the prior of the hyperparameter in this model, which is learned from all assets in Group A.

At the same time, experts may know how similar two groups of assets are. For example, experts know about the deterioration of a concrete-based deck (Group A) is more similar to a

stone-based deck (Group B) compares to a timber-based deck (Group C). This knowledge leads to a higher similarity degree (lower variance) between stone-based (Group B) and concrete-based structure (Group A). A doubly truncated normal (TNormal) distribution (its expression can be found in Fenton and Neil [45]), a normal distribution bounded by lower and upper limits is used to model the relationship between the local parameters and the global parameters (hyperparameters):

Group B (C): shape (scale) ∼ TNormal(µ, σ2, L,U )

The mean µ of this distribution is the shape or scale from the rich data group, and variance σ2representing the degree of similarity between these two groups, which is given by experts. The lower bound L and upper bound U of the distributions are also evaluated by experts about extreme values.

Take node Group B: shape as an example: assumes it has a conditional probability distribution given by TNormal (Group A: shape, 0.5, 1, 3). Its mean is given by the distribution of node Group A: shape, which inherits the typical behaviour of the shape between assets in Group A. Its variance is 0.5 – a smaller variance means a higher similarity, representing a high degree of similarity between these two groups (Group A and B). The distribution is restricted to the region between 1 and 3, indicating it has an increasing failure rate (because the shape value is higher than 1 as discussed in Section 4.1.3) with values between 1 and 3. Similarly, for shape in Group C, experts know that a timber-based deck (Group C) may deteriorate faster than a concrete-based deck (Group A), we can assign the µ of the TNormal distribution with an additional arithmetic value to indicate this behaviour. By extracting information from experienced experts about the degree of similarity and the differences between groups, the model offers to reason parameters of a group with only a little data (Group B and C) using data from another group (Group A) that are judged to share a similar deterioration rate.

4.2.2

Asset with Different but Related Deterioration Rate

As discussed in Section 2.2.2, asset feature can be used as an indication of their deterioration characteristics. This subsection introduces the use of feature space to distinguish individual asset into groups for individualised deterioration learning. It also shows how to aggregate the feature values as an indication on the deterioration rate, which can be used to quantify the different but related deterioration between groups, and further extend the model from the last subsection to learn between groups. This model is suitable for situations when experts find it

difficult to quantify the relationship of asset deterioration behaviour between groups directly by themselves.

The deterioration rate may be affected jointly by different features, like heavy loading and aggressive environment conditions (see an example in Yianni et al. [188]). Ideally, the maintainers’ knowledge of these effects could be combined with statistical failure data gathered from a population where the loading and environment vary. From a decision support perspective, this will allow specific assets to be distinguished. For example, Marsh et al. [109] outlined a Bayesian architecture to integrate multiple factors, such as loading and environmental stress, to support decision but it did not show how failure data could be included.

Figure 4.7 Aggregated influence on deterioration from features.

The degree of influence of each feature should be rated so that we can distinguish individual members of assets into suitable groups. Ranked node (see Section 3.2.2) is applied to model each feature. Its rating is ranging from low, medium to high. The higher degree it is rated, the faster it deteriorates, therefore, the shorter the transition time is. As shown in Figure 4.7, two features are modelled for illustration purpose (here, for example, loading as Feature 1 and environmental condition as Feature 2). An example of how to estimate their

states can be adopted from Yianni et al. [188]: take loading of railways bridges as an example, track data of Equivalent Million Gross Tonnes Per Annual (EMGTPA) passes over the bridge can be used to estimate the level of loading. For loading less than 3.5 EMGTPA, they are rated as low, between 3.5 and 12 EMGTPA are classified as medium, and over 12 EMGTPA are defined as high. In Figure 4.7, a medium loading is observed for Feature 1 in Group X because the EMGTPA of the line passes over this bridge is between 3.5 EMGTPA and 12 EMGTPA. Assets with the same combination of feature values can be grouped into the same group assuming they have the same deterioration characteristics. While for assets within different groups, their feature values can be an indication of how similar their deterioration is. By doing this, we can separate assets into two groups in this example: for assets with medium rating in Feature 1 and low in Feature 2, they are grouped as Group X (asset 1 to 3); for assets with high rating in Feature 1 and medium in Feature 2, they are grouped as Group Y (asset 4 and 5).

At the same time, different features may have different strength in influencing the deterioration of assets. For example, the environmental condition may have more influence on the deterioration of a metal bridge than its service type. Experts could have knowledge about the weights of these influence factors and assign them directly. In the case where we lack this type of knowledge, we can assign a hyperparameter for each feature represents its weight (node weight 1 and weight 2). It is modelled with a prior of a uniform distribution with a lower bound of 0 and upper bound of 1. All the weights of the feature are linked together with a node that sums the weights to 1 (node sum). The weights converged and learned where there are many different groups of assets (with varying combinations of the features) with failure data. However, if we do not have many groups, due to the convergence, it is better to rate weights by experts or to give more informative priors for the weight nodes, rather than learn them. Note that in Figure 4.7 the weight nodes are created for demonstration purpose to show how they are built and linked. But in this case, it is better to assign weights directly from experts since there are only two groups.

The degree of influence factors (node: Aggregated Influence) is modelled by a TNormal distribution combined using a weighted mean (wmean, equivalent to a linear model) of the influence factors (node Feature 1 and node Feature 2), and variances σ2are given by experts regarding their confidence level of the weights:

Aggregated Influence ∼ TNormal(wmean, σ2, 0, 1)

For example, experts can assign the weight of loading with 0.3 and environmental stress with 0.7, with a variance of 0.2 as a slightly high uncertainty about these weights so that the combined influence of these factors is only slightly closer to the value of environmental

stress than the value of loading. Since there are only two groups built in the example in Figure 4.7 with little failure data, the posterior of weights will not converge, resulting in both features carrying approximated equal weights. In Group X, though neither features were rated as high, there is still a 22.97% in high influence since the variance is set as 0.2 with high uncertainty.

Developed from the model in the last subsection, here a case where both groups have little data is presented. Therefore, instead of using the parameters of a dominant group as hyperparameters for weak groups, hyperparameters of the overall population that govern all the subgroups are used. These hyperparameters represent the typical deterioration behaviours of assets, for example, the typical rate of bridge deterioration. Meanwhile, the parameters within each group represent the deterioration behaviours of each subgroup adjusting by their aggregated influence resulting from features. The parameter is partitioned modelled by three TNormal distributions (since there are three states in this example: low, medium, high, each state is modelled by a TNormal distribution), with variance and the bounds are given by experts (same as the last subsection). The only difference is the mean, which is adjusted from the typical hyperparameter and aggregated influence. The mean based on the states of the aggregated influence degree: a low influence degree means a longer transition time, therefore it has a higher probability in the high-value region of the hyperparameter; in contrast, a high influence degree is mapped to the low-value region of the hyperparameter. The evaluation of regions is given by experts regarding how easy the assets can be influenced by external factors.