# Analyse de la qualité d'ajustement d'une base de données de dégradation

Peter B. El Hajj, Bruno Castanier, Franck Schoefs

► To cite this version:

Peter B. El Hajj, Bruno Castanier, Franck Schoefs. Analyse de la qualité d'ajustement d'une base de données de dégradation: Exploitation d'une base de données de suivi d'exploitation. Journées Fiabilité des Matériaux et des Structures (JFMS2016), Mar 2016, Nancy, France. hal-02517089

HAL Id: hal-02517089

https://univ-angers.hal.science/hal-02517089

Submitted on 24 Sep 2020

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Analyse de la qualité d'ajustement d'une base de données de dégradation – Exploitation d'une base de données de suivi d'exploitation.

Boutros EL HAJJ * — Bruno CASTANIER ** — Franck SCHOEFS *

* Institut de Recherche en Génie Civil et Mécanique GeM – Université de Nantes
Boutros.el-hajj@etu.univ-nantes.fr

** Laboratoire Angevin de Recherche en Ingénierie des Systèmes LARIS – Université d'Angers

RÉSUMÉ. La qualité d'un processus d'estimation statistique est directement liée à la richesse de l'échantillon disponible. Dans un contexte de Génie Civil, il est rare de pouvoir considérer que les structures formant cet échantillon soient homogènes. Nous sommes donc toujours dans le compromis entre un nombre suffisant d'observations et l'homogénéité de ce même échantillon. Nous proposons dans cette communication de nous intéresser à la relation entre la caractérisation spécifique du comportement d'un individu parmi une population donnée et la qualification d'une base de données en fonction de son niveau d'hétérogénéité. Pour cela, nous proposons d'analyser les performances en termes d'adéquation statistique au modèle de dégradation sur la base de données simulées présentant des niveaux d'hétérogénéité différents. Cette étude permet de dessiner les premiers points pour une méthodologie de qualification d'une base de données en terme d'hétérogénéité orientée aide à la décision dans un contexte de gestion des risques et de maintenance. Le modèle de dégradation étudiée est basé sur des processus stochastiques, dépendant de l'état, non stationnaire, et ayant une distribution gamma.

ABSTRACT. The quality of a statistical estimation process is directly linked to the richness of the available database. In Civil Engineering, it is rare to consider that the structures forming a database are homogeneous. We are always in the compromise between a sufficient number of observations and the homogeneity of the database. In this paper, we are interested in the relationship between the specific characterization of the behaviour of an individual from a given population and the qualification of a database according to its level of heterogeneity. To this aim, we propose to analyse the performance of the estimation process in terms of statistical relevance to degradation model using simulated databases with different levels of heterogeneity. This study aims to draw the first points for methodological qualification of a heterogeneity in database decision support in a context of risk management and maintenance. The studied degradation model is based on a bi-variate, state-dependent stochastic processes with a gamma distribution.

MOTS-CLÉS: modélisations de dégradation, processus d'estimation, processus gamma dépendant de l'état, processus stochastique non-stationnaires, hétérogénéité de base de données.

KEYWORDS: degradation modelling, state-based gamma processes, non-stationary stochastic processes, heterogeneity in databases.
# 1. Introduction

The construction of predictive degradation models for mechanical and civil engineering that takes into account uncertainties remains a real challenge for the scientific community. The most common approaches in this context are the integration of the effects of the uncertainties in models by means of randomization of their parameters in ranges that are representative of the degree of uncertainty, the variability of the initial composition of materials and the environment.

In this uncertain context, the parameters of a predictive degradation model must be updated from newly available data by approaches such as "inverse problems" or "Bayesian approaches". However, these approaches suffer from several difficulties, for example, the hypothesis of homogeneity of materials and behaviours does not ensure the generalisation of a local analysis of degradation to the assessment of the overall performance of the structure; the effects of this hypothesis evolve with the course of the operation. We can also mention problems associated with the identification of the parameters of the degradation model, especially when often their relationship to observed effects is the result of complex physico-chemical couplings, not to mention the associated numerical difficulties.

The literature is rich with degradation models that are specific to different modes of mechanical degradation (corrosion, wear, fatigue, etc.). These models aim to predict the degradation of the structure and can also be used for calculations such as life expectancy, reliability and maintenance decisions. In this scope, we can see the importance of an adaptive degradation model to different pathologies (Si et al. 2011). Approaches based on stochastic processes display a statistical and mechanistic relevance for modelling of accumulative degradation encountered in civil engineering, especially the gamma process (Van Noortwijk 2009). Here, we propose to model degradation using non-stationary state-dependant approaches; calibrated and updated from NDT inspection data.

In this study, we are interested in cases where databases are poor in size. The estimation and calibration of a model from such databases can result in large uncertainty associated with the estimated parameters. Therefore, an aim of this work is to propose a procedure to narrow down the uncertainty surrounding the estimation of the parameters. One solution is by considering the association of heterogeneous databases (e.g., two close buildings having the same design). The ultimate objective is to increase the confidence in a maintenance plan applied to a farm of similar structures by increasing the number of samples or to know if it is required to consider sub-families. In this context, we consider heterogeneous structures as structures built with identical material and procedures, and situated in similar environments (same reliability classes (Eurocode 2005)).

In this paper, we aim to carry out a preliminary investigation and analysis on the trade-off between the quality of the data in terms of homogeneity and the sufficient sample size to ensure the convergence of the estimation process. In order to identify the effects of the heterogeneity of the database on the quality of the estimate, we will use a simulated database discussed within an application to the cracking of concrete due to corrosion.

We are interested in discrete-in-time inspection, and we will consider only homogeneity related the material variability. To our knowledge, there are no studies applied in these cases, however, we found papers dealing with heterogeneity related to data collection procedures (Gangone, Whelan, and Janoyan 2011) (such as different physical indicators, combined dynamic and static assessment), and studied for SHM in (Glisic et al. 2014).

This paper is organised as follows. In section 2 the degradation model and the estimation algorithm are briefly explained. In section 3 we discuss methods to model heterogeneity in the model. In section 4 an illustration of utilising heterogeneous databases to ameliorate the calibration of the model. Finally, in section 5 conclusions and perspectives work are drawn.

# 2. Degradation model

The use of degradation meta-models based on stochastic processes such as the gamma process seems promising for maintenance management and optimization in civil engineering (Van Noortwijk 2009). It allows modelling the evolution of the degradation using observations via NDT (inputs) while maintaining the most critical aspects of the degradation mechanism (outputs) in the model for the decision.

In this section, we present the multi-variate non-stationary state-based degradation meta-model, used to model the degradation in this study. The meta-model represents the degradation process by modelling chosen physical indicators using state-based stochastic processes. The choice of the physical indicators is grounded on their weight in representing the degradation process, their accessibility through NDT, and on their importance in decision making. The idea of modelling the degradation using more than one variable can be very rewarding in terms of degradation modelling and maintenance management (El Hajj et al. 2015a). For more details on the investigated degradation model, please refer to (El Hajj et al. 2015b).
## 2.1. Pathology and choice of indicators

In this paper, we propose to treat the case of crack propagation due to corrosion. This deterioration is dominated by the excessive accumulation of rust from the ongoing corrosion process resulting in crack propagation until reaching the ultimate point of rupture.

The parameters of importance in this phase are the corrosion current density and the width of the crack (Li et al., 2006). These parameters are important to decision making and are accessible through NDT. The corrosion current density $i_{corr}$ is an instantaneous rate of corrosion measured using NDT, expressed in $(\mu A/cm^2)$. The crack is considered reachable and its width is easily measured (Gauge block or image analysis: (O'Byrne et al., 2013; O'Byrne et al., 2014)).

## 2.2. Degradation model

The cracking phase due to corrosion is then characterized by two state-dependent gamma processes:

* $(\rho_t)_{v_t \ge 0}$ modelling the width of the crack « a » (mm)

* $(\theta_t)_{v_t \ge 0}$ modelling the corrosion current density « $i_{corr}$ » $(\mu A/cm^2)$

The evolution of degradation over a period of time $\tau$ is given by positive increments for the degradation processes respectively $(\Delta\rho, \Delta\theta)$ which are gamma distributed with two time independent parameters each, these are respectively: A state-based shape function $\alpha \cdot \tau$, and a constant scale parameter $\beta$.

In order to simulate the processes, we first seek to characterize the evolution in terms of the $\theta$-process (i.e., corrosion current density and can be seen as the original causal process), then doing so for the respective effect $\rho$-process (i.e., crack width and can be seen as an effect process).

Therefore, the process is sequential and we can then write, $\forall (\rho, \theta) > 0$ :

$$ \Delta\theta(\tau; \rho, \theta) \sim g(\alpha_{\theta}(\rho, \theta) \cdot \tau, \beta_{\theta}) \quad (1) $$

$$ \Delta\rho(\tau; \rho, \theta, \Delta\theta) \sim g(\alpha_{\rho}(\rho, \theta, \Delta\theta) \cdot \tau, \beta_{\rho}) \quad (2) $$

With the respective shape functions:

$$ \alpha_{\theta}(\rho, \theta) = (a_3 \cdot \rho + a_4) \cdot e^{\frac{-(\theta - a_1)^2}{a_2}} \quad \text{and} \quad \alpha_{\rho}(\rho, \theta, \Delta\theta) = (a_6 \cdot (\theta + \frac{\Delta\theta}{2}) + a_7) \cdot e^{-a_5 \cdot \rho} \quad (3) $$

Discussions for the construction of the respective parametric shape functions $\alpha_{\theta}(\rho, \theta)$ and $\alpha_{\rho}(\rho, \theta, \Delta\theta)$ can be found in (El Hajj et al. 2015a). For the estimation process, we have proposed an estimation procedure based on the maximum likelihood method (MLE). Details can be found in the same reference.

Hereafter, we denote $\theta$ as a parameter vector formed of the parameters $\theta = [a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, \beta_{\rho}, \beta_{\theta}]$.

## 3. Heterogeneity description

The study of heterogeneity, or homogeneity, falls into what is known as meta-analysis, or analysis of analyses (Glass 2012), that aims to combine results from different and independent studies. There is a trend among some researchers that attempt to homogenize the *a priori* heterogeneous database by seeking to identify covariates (marginal law analysis). This leads to analysing the behaviour of these covariates (per elsewhere stochastic), resulting in more and complex evolution laws. The validity of any distribution assumption remains heavily criticized, and generally for most meta-analysis, it appears that heterogeneity is being consistently underestimated in most meta-analysis (Kontopantelis, Springate, and Reeves 2013). Even though the study of heterogeneity is old (Simpson and Pearson 1904) and extensive (Hunter and Schmidt 2004), Kontopantelis et al. have emphasized on the complexity of assessing heterogeneity and the necessity for more developed methods (Gavaghan, Moore, and McQuay 2000).

For example, a classical approach to measure heterogeneity is the use of Cochran's Q test, where the null hypothesis that all studies are evaluating the same effect is examined. Cochran's Q is computed as the weighted sum of squared differences between each study's estimate and the overall estimate across studies, weighting each study's contribution in the same manner as in the meta-analysis. Q is distributed as a chi-square statistic with k (number of studies) minus 1 degrees of freedom (Cochran 1954).

However, to study heterogeneity in databases, we first need to have one. Since we lack of real databases, we propose a simple method to simulate heterogeneous databases using the degradation meta-model introduced in section 2. Then, using the simulated databases, we investigate the stability/accuracy/quality of the estimation process.
Therefore, first we consider the existence of an original (basic) homogeneous database formed of 5 realisations, generated using the degradation-meta model using the following parameters:

$$a_1 = 1, a_2 = 1, a_3 = 1, a_4 = 1.2, a_5 = 0.8, a_6 = 1.8, a_7 = 2, \beta_{\rho} = 0.3, \beta_{\theta} = 0.3.$$

The 5 realisations are considered from 5 statistically independent but identical structures.

Then, to the original (homogeneous) database, we add generated heterogeneous realisations that accounts for new added heterogeneous structures. The generated heterogeneous databases are generated using the same degradation meta-model with modification of the parameters.

The heterogeneity in a generated databases is controlled by a quantified modification of the parameters. In the next section, we explain the proposed approach to simulate heterogeneous realisations.

## 3.1. Generating the heterogeneous realisations

The simulation of heterogeneity is carried out in two step:

**Step-1: Introduce a jump – hereafter denoted h – on the parameters of the bivariate model:**

The jump on the parameters h which also can be seen as the heterogeneity level of the simulated database, is expressed in terms of percentage over the original parameters (for example h = +10%).

**Step-2: Apply a small variability of $\mp 2.5\%$ on the jump**

The added variability of $\mp 2.5\%$ aims to cover a family of structures that have a very close variation around the original jump h. A family of structures is a group of structures that are supposed to be identical but due to uncertainties, relate to civil engineering (conception, design, etc.), are slightly different.

In this study, we consider the possibility of adding N structure(s) or realisation(s) to the original databases, where N $\in$ [1,15]. The aim here is to add one realisation at a time and investigate its effect on the estimation process.

To summarize, the simulation of heterogeneous databases is done by modifying the parameters of bi-variate model as follow:

$$a_i = a_i \times (1 + \text{unirnd}(h - 2.5\%, h + 2.5\%)) \quad \text{and} \quad \beta = \beta \times (1 + \text{unirnd}(h - 2.5\%, h + 2.5\%)) \quad (4)$$

Where unirnd is a uniform distribution, $a_i$ and $\beta$ are the parameters of the degradation meta-model (Eq. 1, 2 and 3).

The value of h accounts for the family of structures in the interval [h - 2.5%, h + 2.5%]. The considered values for h are: 0%, 10%; 20%, 30%, 40% and 50%. These values generates realisation with faster degradation than the original ones as it is illustrated in Figure 1 where the estimated PDFs of the times of failure are summarized.

<figure><img src="image_2.png" /><figcaption>Figure 1. PDF of failure for different sent of parameters</figcaption></figure>
## 3.2. Effect of heterogeneous databases on the estimation process

We propose to analyse the effect of including heterogeneous databases for the estimation process using the Mean Squared Error (MSE) as a comparison measure. The MSE is used as the performance criteria for the estimation process. To illustrate the performance of the estimation process and its convergence, we propose to evaluate the MSE of the estimated parameter vector $\hat{\Theta}$ based on the simulated database given the set of the true parameter values $\Theta^*$:

$$MSE = trace\{E\{(\hat{\Theta} - \Theta^*)(\hat{\Theta} - \Theta^*)^T\}\} \quad (5)$$

The parameters of the model are estimated every time a new structure is added. Then, the MSE is calculated using 2000 simulations. Results are plotted in Figure 2, where every point on this curve is the MSE for N added structure(s) and their associated level heterogeneity h. To study the effect of added heterogeneity, we propose to compare each curve with the one issued from the homogeneous databases, i.e., the bottom one (in red). To that aim, in Figure 3 we illustrate the evolution of the relative difference, i.e. $MSE_h - MSE_{0\%}$.

<figure><img src="image_3.png" /><figcaption>**Figure 2. MSEs for different level of heterogeneity function of the number of added structures on the 5 original structures**</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3. Evolution of the difference between MSE relative to MSE$_{0\%}$**</figcaption></figure>

In Figure 2, we can see that the curves share the same decreasing tendency, illustrating a convergence of the estimation toward the MSE of the heterogeneous set only. However, we notice that the evolution of the inter-distances between the six curves is not always the same (Figures 2 and 3). For N = 1, the inter-distances between the curves are non-linear. Then, for N = 8, we can see that there is a certain alteration to this pattern where the distances between the curves tend to become linear, before starting to diverge as we can see for N = 15 where the curves illustrate non-linear inter-distances.

This variation is particularly observable for the h = 20% (green curve) where for N = 8 we can see a lump. For h = 10%, we can see that the MSE curve is close to the homogeneous curve and retains the same distance from it.

We can theorize that for N higher than the original number of structure (i.e. 5), the estimates start to drift from the original parameters towards the parameters of the added heterogeneous databases. Here we can talk of a first equilibrium.

This theory was shown to be somehow true in the estimation code built under Matlab where the code tends to take longer time and become less efficient for N ≈ 5. To further explain the last point, we propose to compare the efficiency of the estimation processes.

The estimation algorithm aims to maximize the likelihood function by varying the parameters of the model. However, this is a mechanical problem, therefore, constraints needs to be defined in order for the solution to be mechanically acceptable. In this case, we define the following ]0.2, 20[ and the estimated parameters. An estimation process is said to be successful if the estimates $\hat{\Theta}$ are all within this interval.

In Figure 4, we illustrate the efficiency of the estimation process. Efficiency is defined as the number of successful estimations, i.e. an estimation in the acceptable interval.
<figure><img src="image_5.png" /><figcaption>**Figure 4. Efficiency of the estimation process**</figcaption></figure>

First of all, we can notice the constant increase in efficiency for the homogeneous case. This is evident since by adding more structures to the database, the estimation process tends to be more efficient. However, this tendency is not the same for the other cases. The efficiency for these cases decrease to reach a minimum at $N = 4$, then start to increase.

The decrease in efficiency is due to the fact that sometimes it is impossible to estimate one set of parameters $\hat{\Theta}$ capable of generating such a disperse database. On the other hand, the increase in efficiency is due to the fact that the estimation process starts to converge towards the parameters of the heterogeneous database.

Furthermore, it is interesting to point out that the minimum efficiency is for $N = 4$ and not for $N = 5$, i.e., the size of the original database. A possible cause is that the added database has more variability on its parameters (±2.5%), hence, will destabilize the equilibrium of the estimation process.

**4. Conclusions and perspective**

In this short paper, we proposed a preliminary study regarding heterogeneity in databases for the calibration of a degradation meta-model. The degradation meta-model is based on bi-variate state-based gamma processes (El Hajj et al. 2015b). There remain a difficulty in considering, defining and analysing heterogeneity.

A simple numerical example was proposed and portrays a potential numerical benefit of including heterogeneous databases, however, the study cannot be generalised. To that aim, in future works we will consider three original sets of parameters that are representative for three speeds of degradation: Normal, fast and slow; Normal being for the case presented in this paper (Figure 5).

<figure><img src="image_6.png" /><figcaption>**Figure 5.** *Figure Three original databases to be studied*</figcaption></figure>
Other future works can include the investigation of the effect of the estimation process on the decision making process and the assessment of heterogeneity in a database, i.e. to know when to split a databases or a model in two.

## 5. Acknowledgments

The authors of this paper would like to thank the Region Pays de la Loire (France) for its support to the SI3M project.

## 6. References

Cochran, William G. 1954. "The Combination of Estimates from Different Experiments." *Biometrics* 10 (1): 101–29.

El Hajj, Boutros, Bruno Castanier, Franck Schoefs, and Thomas Yeung. 2015a. "A Maintenance-Oriented Degradation Model for a Reinforced Concrete Structure Subject to Cracking." *Part O: Journal of Risk and Reliability* (under Final Review).

———. 2015b. "A Condition-Based Deterioration Model for the Stochastic Dependency of Corrosion Rate and Crack Propagation in a Submerged Concrete Structure." *Computer-Aided Civil and Infrastructure Engineering* (under Final Review).

Eurocode. 2005. *BS EN 1992-2:2005 - Eurocode 2: Design of Concrete Structures - Part 2: Concrete Bridges - Design and Detailing Rules. Eurocode 2*. Vol. 2.

Gangone, Michael V., Matthew J. Whelan, and Kerop D. Janoyan. 2011. "Wireless Monitoring of a Multispan Bridge Superstructure for Diagnostic Load Testing and System Identification." *Computer-Aided Civil and Infrastructure Engineering* 26 (7): 560–79

Gavaghan, David J., R. Andrew Moore, and Henry J. McQuay. 2000. "An Evaluation of Homogeneity Tests in Meta-Analyses in Pain Using Simulations of Individual Patient Data." *Pain* 85 (3): 415–24. doi:10.1016/S0304-3959(99)00302-4.

Glass, Gene V. 2012. "Primary, Secondary, and Meta-Analysis of Research'." *American Educational Research Association* 5 (10): 3–8.

Glisic, Branko, Matthew T. Yarnold, Franklin L. Moon, and A. Emin Aktan. 2014. "Advanced Visualization and Accessibility to Heterogeneous Monitoring Data." *Computer-Aided Civil and Infrastructure Engineering* 29 (5): 382–98.

Hunter, John E, and Frank L. Schmidt. 2004. *Methods of Meta-Analysis: Correcting Error and Bias in Research Findings*. https://books.google.fr/books/about/Methods_of_Meta_Analysis.html?id=kImkui18i94C&pgis=1.

Kontopantelis, Evangelos, David A Springate, and David Reeves. 2013. "A Re-Analysis of the Cochrane Library Data: The Dangers of Unobserved Heterogeneity in Meta-Analyses." *PloS One* 8 (7): e69930. doi:10.1371/journal.pone.0069930.

Li, Chun-Qing, Robert E. Melchers, and Jian-Jun Zheng. 2006. "Analytical Model for Corrosion-Induced Crack Width in Reinforced Concrete Structures." *Structural Journal* 103 (4): 479–87.

O'Byrne, Michael, Bidisha Ghosh, Franck Schoefs, and Vikram Pakrashi. 2014. "Regionally Enhanced Multiphase Segmentation Technique for Damaged Surfaces." *Computer-Aided Civil and Infrastructure Engineering* 29 (9): 644–58.

O'Byrne, Michael, Franck Schoefs, Bidisha Ghosh, and Vikram Pakrashi. 2013. "Texture Analysis Based Damage Detection of Ageing Infrastructural Elements." *Computer-Aided Civil and Infrastructure Engineering* 28 (3): 162–77.

Si, Xiao-Sheng, Wenbin Wang, Chang-Hua Hu, and Dong-Hua Zhou. 2011. "Remaining Useful Life Estimation – A Review on the Statistical Data Driven Approaches." *European Journal of Operational Research* 213 (1). Elsevier B.V.: 1–14.

Simpson, R. J. S., and Arl Pearson. 1904. "Report On Certain Enteric Fever Inoculation Statistics on JSTOR." *The British Medical Journal* 2 (2288): 1243–46. http://www.jstor.org/stable/20282622?seq=1#page_scan_tab_contents.

Van Noortwijk, J.M. 2009. "A Survey of the Application of Gamma Processes in Maintenance." *Reliability Engineering & System Safety* 94 (1): 2–21. doi:10.1016/j.ress.2007.03.019.