Stochastic Petri-net models to predict the degradation of ceramic claddings

ABSTRACT A stochastic Petri-net formalism is proposed to predict the degradation of ceramic claddings over time in order to understand how different environmental exposure conditions contribute to the overall degradation of these claddings. For that purpose, the degradation condition of 195 ceramic claddings located in Lisbon, Portugal, is evaluated through in situ visual inspections. In the first part of the study, a stochastic deterioration Petri-net model is proposed for the entire sample. In the second part, the original sample is divided according to the environmental exposure conditions, evaluating the influence of these conditions on the deterioration process of ceramic claddings. Four main degradation agents are analyzed: exposure to moisture; distance from the sea; orientation; and wind–rain action. The results reveal that Petri nets can accurately describe the deterioration process of ceramic claddings, providing relevant information regarding the performance of these claddings through their life cycle and according to the environmental exposure conditions to which they are subject. These results are extremely relevant for different practitioners: the approach allows the adoption of more sustainable and durable solutions at the design stage, as well as improving the durability of the ceramic claddings by performing optimized maintenance plans and strategies.


Introduction
The ageing of the European building stock and the high burden related to their maintenance and repair has increased the relevance of adopting condition-based maintenance plans and strategies (Beer et al., 2011;Hovde, 2002;Paulo, Branco, & de Brito, 2014;Silva, de Brito, & Gaspar, 2016). The decision-making surrounding the maintenance and rehabilitation of the built heritage requires more economic and sustainability awareness, particularly the adoption of technical criteria in the decision process of whether to intervene. However, this rational approach is only possible if a reliable understanding of the durability and service life of building materials and components exists (Aikivuori, 1999). Therefore, in the past two decades various authors (Gaspar & de Brito, 2008;Lacasse & Sjöström, 2004;Shohet & Paciuk, 2004;Shohet, Puterman, & Gilboa, 2002;Silva et al., 2016) have developed several service life prediction methodologies. These were intended to improve the knowledge regarding the deterioration processes and mechanisms of building and components, as well as to provide relevant information concerning 'how' and high level of randomness and uncertainly, and their stochastic nature should be recognized in the building management system research.
Currently, Markov-based models are the most common stochastic technique applied to modelling the deterioration of assets (Butt, Shahin, Feighan, & Carpenter, 1987;Cesare, Santamarina, Turkstra, & Vanmarcke, 1992;. These models are considered intuitive and computationally inexpensive due to the memoryless property (i.e. lack of maintenance and other records), which allows an estimate of the future performance only based on the current performance, which is particularly relevant when limited information is available (Ferreira, Neves, Silva, & de Brito, 2018). In this study, a Petri-net modelling technique is used to model the deterioration of ceramic claddings, considering the isomorphism between Markov chains and stochastic Petri nets. Petri-net models have been successfully applied to model dynamic systems in several areas due to their flexibility and high accuracy in modelling complex phenomena (Al-Ahmari, 2016;Chen, Li, & Barkaoui, 2014;Cheng, Li, Wang, Skitmore, & Forsythe, 2013;Rinke et al., 2017;Tang, Guo, Dong, Li, & Guan, 2008;Uzam, Gelen, & Saleh, 2016;van der Aalst, 2002;Yianni, Rama, Neves, Andrews, & Castlo, 2016).
The durability and service life of buildings, and especially external claddings, are strongly influenced by the environmental exposure conditions. In this study, a stochastic Petri-net formalism is also applied to analyze the influence of different environmental degradation agents in the overall degradation of ceramic claddings over time. For that purpose, a sample of 195 ceramic claddings, located in Lisbon, Portugal, is analyzed using in situ visual inspections to quantify deterioration. The deterioration state of each cladding analyzed is defined based on a discrete scale divided into five condition levels, ranging between condition A (with no visible degradation) to condition E (most serious degradation condition, corresponding to generalized degradation).
This study can be divided into three parts. In the first, a deterioration stochastic Petri-net-based model is proposed and calibrated using the overall sample. In the second, the original sample is divided according to the environmental exposure conditions, evaluating the influence of these conditions on the deterioration process of ceramic claddings. For that purpose, four environmental exposure factors are analyzed: exposure to damp; distance from the sea; orientation; and the combined action of rain and wind. Finally, in the third, an independence analysis is performed using a one-way analysis of variance (ANOVA) test.
In the proposed stochastic Petri-net model, the degradation phenomenon is modelled as a sequential dynamic system (in which the maintenance actions are not considered, assuming that each cladding only transits from a given condition to a more unfavourable one) and a set of probabilistic distribution functions are used to model the transitions times between degradation condition states. Compared with other stochastic modes, Petri nets have greater flexibility under a consistent formalism. On the one hand, the flexibility allows for the modelling of all aspects of the lifecycle (e.g. inspection, maintenance and repair) in a consistent framework. The use of a consistent formalism allows for faster development and simpler maintenance of software. This method allows the use of several distribution functions to model the time of transitions between condition states, which is a major advantage when compared with Markov chains, which only allows one to use the exponential distribution. The optimization of the models' parameters is performed using a Monte Carlo and genetic algorithm optimization framework in which the fitness function is defined based on the concept of maximizing the likelihood. The results obtained show the relevant impact of the environmental exposure conditions in the service life of external ceramic claddings. This information allows the identification of the deterioration rate of ceramic claddings according to their location and exposure conditions, thus improving the decision-making regarding the design, execution, maintenance and repair of these claddings.
The paper is structured as follows. A basic background concerning the Petri-net formalism is described is the next section. The method proposed to predict the life cycle performance of ceramic claddings is described. A one-way ANOVA test is performed. Finally, a discussion of the results is presented and the main conclusions are provided.

Background
The use of Petri nets has increased in the last years, essentially due to their flexibility at modelling dynamic systems in different areas of knowledge (Al-Ahmari, 2016;Chen et al., 2014;Cheng et al., 2013;Rinke et al., 2017;Tang et al., 2008;Uzam et al., 2016;van der Aalst, 2002;Yianni et al., 2016). Petri-net models provide a graphical and mathematical formalism; the graphical representation allows the intuitive description of the problem, while the mathematical formalism allows an efficient algorithmic to be developed to solve complex problems. Furthermore, the different variants of Petrinet models that have been developed over the years are all related through the basic net formalism upon which they are built. This flexibility allows one to meet the needs in different application domains and the incorporation of many rules in the model to accurately simulate complex situations, still keeping the model size within manageable limits (Girault & Valk, 2013).
The original concept of Petri nets was developed by Carl A. Petri (Petri, 1962). A Petri net is a particular kind of graph with an initial state called the initial marking, M 0 (Murata, 1989;Peterson, 1977;Schneeweiss, 2004). A Petri net comprises two kinds of nodes: (1) places, which represent resources or particular states of the system; and (2) transitions, which represent an action or an event changing the state of the system. Both nodes are linked by directed edges (arcs). An arc connects a place with a transition, or vice versa. Each place has the ability to store a varying number of tokens. Tokens represent the present state of the system, and their distribution over the places is called its marking. The movement of the tokens between places and the evolution of the system from one state to another is governed by the firing of transitions. A transition is enabled to fire when the number of tokens in all input places is greater than or equal to the weight of the precondition arcs (directed edges that link the input places to the transitions). These two elements (tokens and transitions) are responsible for modelling the dynamic behaviour of the system. Figure 1 shows a basic example of a Petri net and illustrates it before and after a transition fires. The net is composed of three places (denoted by circles), one transition (a rectangle) and tokens (black dots) within the places. Places and transitions are linked by arrows that indicate the direction the tokens should move. Places P 1 and P 2 are the input places of transition T 1 , while place P 3 is the output of transition T 1 . In Figure 1(a), transition T 1 is considered ready to fire once the preconditions are verified. After the transition fires, the number of tokens of the preconditions are removed from the respective input places and the number of tokens of the post-conditions (arrows that link the transition to the output places) are added to the respective output places (Figure 1(b)).

Research objectives
The aim of this work is to develop probabilistic deterioration models for building facades that incorporate the impact of local environmental factors. To achieve this, the following objectives were defined: (1) propose a flexible deterioration model based on Petri nets; (2) evaluate the dependence between performance and a set of environmental conditions using the ANOVA; and (3) propose updated deterioration models by taking into account the results of the ANOVA. Finally, the proposed models are applied to the facade data set to predict the future performance of these facades and the impact of different exposure classes on their degradation condition over time.

Deterioration Petri-net model
The deterioration process can be modelled using Petri nets through the definition of a linear sequence of places and timed transitions, where each place represents a condition state of the classification system adapted and the timed transitions define the movement between conditions states (Andrews, 2012;Yianni et al., 2016). The position of the token in the network indicates the present condition of the element. Figure 2 illustrates the Petri-net structure used to model the deterioration process of ceramic claddings over time.
The model is composed of five places (P 1 -P 5 ) and four transitions (T 1 -T 4 ). Place P 1 represents the best condition state and place P 5 the worst condition level. Since maintenance is not considered in this study, the condition level of the cladding deteriorates continuously over time until it reaches the worst condition level defined in the performance scale. Timed transitions are used to model the sojourn time in each condition level (i.e. the time the element spends in condition level i before moving to condition level i + 1).

Timed transitions
Although not originally included in Petri nets, timed transitions allow one to include the time dependence in these models. The methodology used here is based on the stochastic Petri nets formalism proposed by Dugan, Trivedi, Geist, and Nicola (1984). A firing rate is assigned to each transition that can be modelled by any probabilistic distribution.
Since there is limited information on the statistical properties of the deterioration process of ceramic claddings, a set of five probabilistic distributions (exponential, Weibull, log-normal, Gumbel and normal distributions) are used to model the sojourn time in each condition level. Molloy (1982) has proven there is an isomorphism between a bounded Petri net with exponentially distributed transition rates and a finite Markov process. This property can be used to validate the deterioration Petri-net model proposed, since Markovbased models are the most commonly used stochastic technique in modelling deterioration. A normal distribution was used since it is usually adequate to model the average of several independent random variables. If the ceramic cladding is seen as a set of areas deteriorating independently, then the average condition can be modelled as a normal distribution. However, a significant drawback of the normal distribution is the non-null probability of negative values which, in the present context, is not physically possible. Alternatively, the log-normal distribution can be used to model the times of transition avoiding negative values. If, on the other hand, the condition is mostly characterized by the deterioration of the most critical areas of the ceramic cladding, extreme distributions such as the Weibull and Gumbel might be more adequate.
This modelling technique has several advantages when compared with the Markov chains. The graphical representation can be used to describe the problem intuitively. It is more flexible and has more capabilities than the Markov chains; it allows the incorporation of more rules into the model to simulate accurately complex situations and it keeps the model size within manageable limits. Moreover, this modelling technique is not restricted to the exponential distribution to simulate the time in the system. As a main disadvantage, there are no closed-form expressions for the probability distribution used and simulation techniques are required.

Parameter estimation
The probability density function that best describes the deterioration process is that resulting in higher probabilities of occurrence of the observed transitions. In order to identify the probability distribution that provides a best fit, parameter estimation is required. The parameters of the probability density function are fitted to history based on the concept of maximum likelihood used in the formulation proposed by Kalbfleisch and Lawless (1985). The likelihood is defined as the predicted probability of occurrence of the observed transitions: where i is the condition level in the initial instant; j is the condition level in the final instant; n is the number of elements of the historical database; k is the number of intervals between inspections; and p ij is the probability of transition from condition level i to condition level j. However, for numerical convenience, it is easier to work with the logarithm of the likelihood (equation (2)). The log-likelihood is used to simplify the computations and improve robustness of the algorithm: The probability of occurrence of the observed transitions, p ij , is estimated by Monte Carlo simulation. This method allows one to obtain good numerical approximations in situations where it is not feasible to obtain analytical solutions and can be used to consider the propagation of uncertainties during the claddings' service life. Therefore, using Monte Carlo simulation, the distribution of the final condition can be obtained and, consequently, the probability of the observed transition occurring can be computed. The procedure used to compute the probability of occurrence of the observed transitions is illustrated in Figure 3. To each transition observed in the historical database, the algorithm has as input: time interval between observations, Dt; condition level in the initial instant, i; and condition level in the final instant, j. The condition level in the initial instant, i, is used to define the initial marking, M 0 , of the Petri net; the time interval between observations, Dt, is the time horizon of the analysis; and the condition level in the final instant, j, is used to compute the probability of occurrence in the end of the procedure. The identification of the first transition to fire is performed through the initial marking, M 0 , since once the Petri net defined for the deterioration model ( Figure 2) is arranged in a sequential manner, each condition level can only enable one transition. In P 1 T 1 T 2 T 3 T 4 P 2 P 3 P 4 P 5 Figure 2. Petri-net scheme of the deterioration process. the next step, the random sojourn time in the condition level is generated from the inverse cumulative density function of probability distribution and the Petri net marking and time are updated. The process is repeated until the time horizon is reached. The procedure depicted is repeated for each transition observed in the historical database and the output of each sample is the condition index at the time horizon.

Optimization
The optimization of the parameters of the probability distributions is performed using genetic algorithms (GAs). They are an adaptive heuristic search algorithm inspired by natural evolution, such as inheritance, mutation, selection and crossover (Holland, 1975). Nowadays, GAs are widely used in optimization problems. This technique is very robust and efficient for objective functions computed using Monte Carlo simulations. GAs use only information of the objective function, not requiring the computation of gradients, which greatly simplifies the problem and avoids numerical errors (Man, Tang, & Kwong, 1999;Morcous & Lounis, 2005). In this study, the optimization of the parameters was performed using a GA available in MATLAB (MatLab, 2016). The aim of the optimization algorithm is to find the parameters of the probability distributions that maximize the fitness function: The parameters used in the GA are: . size of the population: 50 individuals when the number of optimization variables is less than or equal to five; and 200 individuals otherwise . stopping criteria: the algorithm stops if the average relative change in the best fitness function value over 50 generations is less than or equal to 10 −6 The mutation procedure was performed using the Gaussian algorithm implemented in MATLAB, where the mutation is generated by adding a random number taken from a normal distribution with mean zero.

Statistical analysis
To ensure the validity of the models proposed in this study, it is relevant to examine the independence of the variables considered. For that purpose, a statistical comparison of the results is performed using an ANOVA, and subsequent post-hoc comparisons are made using a Tukey multiple comparison test. The one-way ANOVA test (Neter,   compares the means between three or more of independent variables, determining whether any of those means are significantly different from each other, where the hypothesis is defined by: H 0 : All group means are equal.
H 1 : At least one group mean is different from the others.
The statistical test can be validated through the comparison of the critical value of the F-ratio with the table Fvalue. The critical F-value is given by: where SSR is the between-groups sum of squares; SSE is the within-groups sum of squares; MSR is the betweengroups mean squares; MSE is the within-groups mean squares; k is the number of group tested; and n is the number of observations in each group. If the critical F-value is greater than the table F-value or if the p-value is smaller than the significance level considered, the null hypothesis should be rejected. The table F-value has a Fisher distribution with k − 1 and n × k − k degrees of freedom. When the null hypothesis is rejected, it is concluded that at least the mean of one group tested is statistically different from the others. However, the ANOVA test does not indicate which groups are statistically different from each other. To determine that, a Tukey multiple comparison test (Hochberg & Tamhane, 1987) must be performed. This test is based on the studentized range distribution and it is optimal for procedures with equal sample sizes. The Tukey test compares all pairwise possible between groups and it can be stated that the groups compared are statistically different and the null hypothesis is rejected if the following relationship is verified: where y i and y j are the mean of the groups i and j respectively; MSE is the within-groups mean squares obtained from the one-way ANOVA test; n i and n j are the number of observations in groups i and j respectively; and q(a; k; n × k − k) is the upper 100 × (1 -a)th percentile of the studentized range distribution with parameter k and n × k − k degrees of freedom and a significance level of a.

Results and discussion
The deterioration Petri-net-based method described in the fourth section is applied to model the deterioration process over time of a sample of 195 ceramic claddings located in Lisbon, Portugal. The degradation state of each cladding in the data set was analyzed based only on in situ visual inspections. Each place illustrated in Figure 2 represents one of the five deterioration conditions of ceramic claddings proposed by Bordalo, de Brito, Gaspar, and Silva (2011) and Silva et al. (2016). The application of the deterioration Petri-net model is performed in two phases. In the first, the complete sample is analyzed, evaluating among the five probabilistic distribution functions mentioned above. In the second, the original sample is divided according to the environmental characteristics, and the most relevant characteristics to explain the degradation of ceramic claddings, namely exposure to damp, distance from the sea, facades' orientation and wind-rain action, are analyzed. Finally, an inference analysis is performed to investigate the independency between the variables analyzed (i.e. the environmental characteristics of the ceramic claddings).
In this study, for each cladding considered in the historical database, only the initial and the final condition levels are known. For the first, it is assumed that, at time zero, the cladding is in the most favourable condition level (level A, with no visible degradation) and the final condition level corresponds to the degradation condition observed at the inspection time.

Classification of the degradation condition of ceramic claddings
Various authors (Gaspar & de Brito, 2008;Shohet et al., 2002;Shohet & Paciuk, 2004) developed degradation scales to assess and characterize the deterioration condition of the buildings' facades. In this study, the deterioration condition of ceramic claddings is established by a discrete scale, with five condition levels, ranging between A (most favourable condition, with no visible degradation) to E (worst case scenario, with generalized degradation) (Bordalo et al., 2011). In this scale, level D corresponds to the end of service life of ceramic claddings, considering that, once the cladding reaches this state, a maintenance action must be performed. The scale adopted is based on the evaluation of the physical and visual condition of the sample analyzed, according to the defects observed, their extent and severity. In this study, the defects observed in ceramic claddings are divided into four categories: visual defects; cracking; joint deterioration; and adhesion failure. The overall degradation level of ceramic claddings is evaluated through a numerical index (Gaspar & de Brito, 2008, called severity of degradation, which is obtained through the ratio between the extent of the facades' degradation, weighted as a function of the degradation level and the severity of the defects, and a reference area, equivalent to the whole area of the facade degraded to the maximum possible level: where S w is the degradation severity of the coating, expressed as a percentage; k n is the multiplying factor of anomaly n, as a function of their degradation level, within the range K = {0, 1, 2, 3, 4}; k a,n is a weighting factor corresponding to the relative weight of the anomaly detected (k a,n R + ); A n is the area of coating affected by an anomaly n; A is the facade's area; and k is the multiplying factor corresponding to the highest degradation level of a coating of area A.
The coefficient k n allows one to weight the severity of the defects detected in the cladding, where the value zero is adopted for defects in condition A, 1 for defects in condition B and so on until 4 for condition E. The coefficient k a,n corresponds to the weighting coefficient associated with the relative importance of each defect, according to their severity, repair cost, influence in the overall degradation of the cladding, propensity to cause other defects, and risk for the owners' and users' safety. Table 1 shows the defects considered in this study, the qualitative and quantitative scales adopted, and the weighting coefficients applied in equation (5). Figure 4 provides some illustrative examples of the visual appearance and some defects observed in ceramic claddings in the different degradation levels.
Probabilistic analysis of the degradation of ceramic claddings over time Table 2 shows the optimized estimation parameters obtained for the distribution functions analyzed, as well as the respective likelihood value. Table 3 presents the number of observed and predicted ceramic claddings in each condition level for each probability distribution function, with the relative error associated with the estimates. The light and dark grey cells indicate the distributions with lower and higher relative errors respectively for each condition level.
One main conclusion that can be drawn, based on these results, it is that two-parameter distributions (Weibull, log-normal, Gumbel and normal) show a higher agreement to the data set than the exponential distribution ( Table 2). The accuracy of the five distribution functions in the description of the behaviour of the historical data can be evaluated through the results in Table  3, which compare the number of observed and predicted claddings in each degradation condition. The quality of fit of the proposed model can be evaluated by comparing the number of facades observed and predicted in each condition state ( Figure 5). These results show that a reasonable fit is attained with all distributions, but also that larger relative errors might result for levels A and E, due to the smaller sample size.
Regarding the relative error (Table 3), in 76% of the cases the error is lower than 10% (and in 40% of the cases is below 5%). The extreme condition levels (A and E) are those with the highest errors (between 4% and 30%); in level A, the highest errors occur for the normal (24.3%) and Weibull (11.2%) distributions; in level E, the highest errors occur for the normal (29.1%), Weibull (23.0%) and Gumbel distributions (21.3%). Note that in the sample analyzed only 8% of the cases studied belong to level A and only 2% to level E. Therefore, the model is naturally less accurate for the conditions for which less information exists. Analyzing the mean relative error for all condition levels, the log-normal distribution shows the lowest overall mean relative error (5.4%), followed by the exponential distribution (7.7%) and Weibull distribution (with an overall mean relative error of 8.9%).  applied Markov chain models to analyze in a probabilistic way the degradation of ceramic claddings over time, based on the same data set used in this study. The values of the parameters obtained by Markov chains and with the exponential distribution through Petri nets are rather similar (Table 4). The highest differences occur for the mean and standard deviation of the parameter T 4 (concerning the transition between levels D and E). This occurs due to the low number of elements in level E. The differences obtained in the other parameters are due to sampling errors associated with the Monte Carlo simulation. The results obtained by Petri nets confirm that there is an isomorphism between both models and the Petri-net formalism is suitable to evaluate and describe the degradation of ceramic claddings. Figure 6(a) presents the mean future condition profile over time for all probability distribution functions analyzed. The condition profile obtained for the exponential distribution is quite distinct from the other four profiles; this distribution shows a profile with a simple parabolic shape without inflection points. The probabilistic curves obtained by Weibull, Gumbel and normal distributions are very similar throughout the time horizon considered, and small changes of the concavity are visible when there is a transition in the condition level. The profile obtained when considering a log-normal distribution is a mixture of the two types of profiles referenced above. The profile of the log-normal distribution is very similar to the profiles of the Weibull, Gumbel and normal distributions during the first transitions (levels A to B, B to C, and C to D). The main difference lies in the transition between levels D and E, where the profile shows more likeness to the profile of the exponential distribution. In terms of the dispersion of the results (Figure 6(b)), the values obtained are low, and the two-parameter distributions present lower dispersion values over the time horizon than the exponential distribution. Figure 7 shows the probability distribution of all degradation condition levels over time. The differences between the probability distribution functions, described above, are also visible in these plots. Based on the results obtained, the following conclusions can be drawn: . For level A, the predicted probabilities for all distribution functions are similar, beginning with a probability equal to 1 in instant zero, decreasing over time, and at year 30, the probability of a ceramic cladding belonging to level A is close to zero (Figure 7(a)). Furthermore, between years 5 and 7, the probability of a ceramic cladding belonging to either level A or B is Notes: a With leakagethe degradation level is increased by one. b Cracking: detectable at a distance greater than 5 m only if binoculars are used. c Tenuous cracking line: easily detectable at a distance greater than 5 m using binoculars. d Well-defined cracking: visible from a distance of more than 5 m without using binoculars.   practically the same, i.e. whatever the distribution chosen to describe the process of deterioration, the results show that the transition between levels A and B occurs during this time interval. . For level B, the differences between the distribution functions analyzed is more visible; the maximum probability of a ceramic cladding belonging to level B occurs between years 12 and 16 (however, this peak is 15-20% lower when the exponential distribution is used; Figure 7(a)). According to the exponential distribution, the transition from level B to C occurs around year 22, while with the two-parameter distribution functions it lies between conditions B and C occurs six years later (around year 28). . The maximum probability of a ceramic cladding belonging to level C (Figure 7(b)) is 0.50 for the exponential distribution, while for the other distributions this varies between 0.70 and 0.80. Once the maximum probability is achieved, the slope of the exponential distribution is less pronounced when compared with the other distributions. The transition between levels C and D occurs around year 54 for the two-parameter distributions and 13 years later (around year 67) for the exponential distribution. . The higher differences between the results of the various distributions are obtained in levels D (Figure 7 (b)) and E (Figure 7(c)). Exponential and log-normal distribution functions, during the time horizon analyzed, never show a transition between these two condition levels. For Weibull and normal distributions, the transition between these condition levels occur around year 84 and for the Gumbel distribution this transition occurs around year 90.
The results obtained reveal that exponential and normal distributions are not the most appropriate functions for the description of the degradation phenomena of ceramic claddings, especially in the description of the transition between levels D and E. The lack of accuracy of the exponential distribution can be explained by a limitation of the distribution, since it has only one parameter, which can compromise its adequacy to model the Note: Light and dark grey cells indicate the distributions with lower and higher relative errors respectively for each condition level. Figure 5. Comparison between the results predicted by the model (for the five probability distribution functions adopted) and the measured degradation state from the database (observed values).
deterioration process. For the normal distribution, the inadequate adjustment to the data set is related to the low number of claddings in level E (only three elements; Table 3). The results show that the low number of observations is insufficient for the normal distribution to properly estimate the parameters that best describe the deterioration process. However, it is stressed that the time horizon considered in this study, and presented in the analysis of Figure 6, is not realistic for ceramic claddings, since this type of cladding is usually subjected to maintenance actions around year 50 (or at level D, which represents a minimum accepted level of performance); until this time, i.e. the expected end of service life of ceramic claddings, there is a good fit of the normal distribution to the description of the degradation conditions of ceramic claddings. From the available data, the Weibull distribution seems to result in a better fit with the observed results and will be used in the subsequent analysis. Nevertheless, note that the data available at this time are limited, and consequently larger data sets might show that this is not always the most adequate distribution. In particular, both the Gumbel and the log-normal distribution result in a reasonable fit and should be considered in future works.
As stated above, level D corresponds in this study to the minimum accepted performance level, and thus to the end of service life of ceramic claddings. According to the Petri-net-based model proposed in this study   (adopting one of the two-parameter distributions: normal, log-normal, Weibull and Gumbel), the ceramic claddings have a maximum probability of transition to level D around year 54. This age can be considered as the expected service life of ceramic claddings, for the sample analyzed, and according to the proposed model. This value is coherent with the empirical knowledge and the literature related with the durability of this type of cladding: (1) according to Silva et al. (2016), the estimated service life of adhesive ceramic wall claddings varies between 46 and 50 years; (2) Galbusera, de Brito, and Silva (2015) proposed an average service life around 51 years for ceramic claddings; (3) BMI (2001) suggested that the estimated service life of ceramic claddings varies between 25 and 55 years (with an average of 45 years); and (4) Gaspar (2017) proposed an estimated service life of 49 years for external ceramic claddings.

Independence analysis
A one-way ANOVA was performed in order to check whether the complete sample and the 12 variables into which it was divided are independent, i.e. whether for a determined time horizon the final condition level obtained is influenced by the different variables in which the complete sample was divided. The ANOVA test results are presented in Table 5. The statistical test was performed for five different time horizons using 50,000 samples in the simulation process.
As mentioned above, the test can be validated through the comparison of the critical value of the F-ratio (Table  5) with the table F-value or through the comparison of the p-value with the established significance level. For a significance level of 0.05, the table F-value is 1.752.
Through the analysis of the results obtained, it is possible to see that for the five time horizons tested the null hypothesis is rejected. This result shows that at least one group mean is statistically different from the others, but it does not tell which groups are statistically different from the others. To determine that, a Tukey test was performed.
The Tukey test compares all pairwise possible between groups and it can be stated that the groups compared are statistically different and the null hypothesis is rejected if the relationship of equation (4) is verified. The value of the right-hand side of equation (4) is constant for all analyses and is equal to 3.12. The values of the left-hand side are presented in Table 6. Values in italics identify the pairwise where the null hypothesis is not rejected.
As in the probabilistic analysis, the results obtained in the statistical test can be divided into two parts. In the first part of the analysis, the complete sample was compared with each subset. The results show that the mean of intermediate situationsintermediate distance from the sea (SI) and moderate wind-rain action (WM)are statistically closer to the overall sample. In the other situations, the differences between the mean of the groups are statistically significant showing that the variables are independent.
In the second part, a comparison with each family of variables was performed. The results show that for year   ); SF, far from the sea (> 5 km); SI, intermediate position from the sea (between 1 and 5 km); ON, orientation north; OE, orientation east; OS, orientation south; OW, orientation west; WL, low exposure to wind-rain action; WM, moderate exposure to wind-rain action; and WH, high exposure to wind-rain action.
10 the mean of intermediate distance from the sea (SI) and close to the sea (SC) are statistically close, and for year 40 there are significant dependence between variables orientationeast (OE) and orientationnorth (ON), and orientationwest (OW) and orientationsouth (OS). In the remaining situations, the differences between the mean of the groups are statistically significant showing that the variables are independent.
The different environmental factors have significantly different impacts on the performance of the facades during their life. The results of the ANOVA seem to indicate that the greater impact of environmental conditions is in the first 20 years of the life of the facade, except for orientation, which has a relatively uniform impact during the entire life. This is consistent with the expected deterioration mechanics, as orientation mostly affects the slow evolving damage associated with exposure to ultraviolet light (UV), while the other exposures influence water and chloride content which impact deterioration much faster. The impact during the first years of life is particularly relevant under aggressive exposures (e.g. high exposure to damp, less than 5 km from the sea, high exposure to wind-rain), showing that for these facades the change from conditions A to B and C is much faster than for the entire population.

Impact of environmental conditions on deterioration
The typical degradation mechanisms that lead to the premature degradation of external claddings include climatic or atmospheric weathering factors, such as UV radiation, temperature, moisture, air constituents and marine environments (Lewry & Crewdson, 1994;Shohet & Paciuk, 2006). Naturally, ceramic claddings are subjected to different environmental exposure conditions, which strongly influence their deterioration rate and their expected service life. In this sense, this study intends to evaluate the influence of four environmental conditions (exposure to moisture, distance from the sea, orientation, and UV radiation and wind-rain action) in the description of the overall degradation of ceramic claddings over time.
The results obtained in the analysis, according to the various environmental characteristics, are shown in Table 7 and Figure 8. Table 7 shows the probability of belonging to each degradation condition, according to the variable considered; Figure 8 presents the predicted future condition over time for these variables. The predicted future condition profile for each variable is obtained through the distribution that presents the minimum normalized log-likelihood value ( Table 7). The normalized log-likelihood is obtained as the ratio between the likelihood and the number of elements in the sampleequation (6). The obtained value represents where n is the number of elements in the data set.
Distance from the sea (exposure to marine environments and sea salts) The interaction between the salt spray from the sea and building materials has been extensively analyzed in the literature (Brocken & Nijland, 2004;Lubelli, Hees, & Groot, 2004). The presence of salt, usually on the external surfaces of the facades due to the action of the wind containing the spray (Hossain, Lachemi, & Şahmaran, 2009), causes the occurrence of defects and leads to the progressive deterioration of porous materials, such as ceramic claddings. Severe exposure to sea spray, in coastal areas, leads to various defects, e.g. efflorescence, exfoliations or spalling (Sena da Fonseca, Simão, & Galhano, 2013). Figure 9 shows an illustrative example of the degradation condition of a ceramic cladding located less than 2 km from the sea due to the influence of sea spray.
In this study, the ceramic claddings are divided into three categories regarding their distance from the sea: (1) sea front areas (less than 1 km); (2) in coastal areas (between 1 and 5 km from the sea); and (3) away from the sea (more than 5 km). The samples for the variable less than 1 km from the sea, between 1 and 5 km, and more than 5 km include 77, 62 and 56 elements respectively.  Figure 9. Illustrative example of the pathological situation of a case study located less than 5 km from the sea.
As expected, the influence of the sea in the overall condition of the claddings is evident. The claddings located near the sea tend to belong to the most unfavourable conditions levels, levels D and E (P = 71.7%). This tendency decreases with the distance from the sea, and the probabilities of a cladding belonging to the most unfavourable conditions (levels D and E) are 62.8% and 29.9% for the intermediate distance from the sea and for claddings located more than 5 km from the sea respectively. Claddings located more than 5 km from the sea, less prone to the presence of sea spray, have a higher probability of belonging to the most favourable conditions (levels A and B) for a longer period of time.

Exposure to moisture
The exposure to moisture is one of the main causes of several deterioration mechanisms that occur in buildings and components. External ceramic claddings are particularly exposed to this degradation agent, which when associated with salt leaching can cause aesthetic degradation (e.g. efflorescence, glaze cracking, moisture stains and biological growth) and jeopardize the integrity of the claddings system (e.g. cracking, swelling, detachment, spilling and material loosening) (Charola, 2000;Madkour & Khallaf, 2012;Rirsch, MacMullen, & Zhang, 2011;. Figure 10 shows some examples of the deterioration condition of ceramic claddings highly exposed to moisture. Concerning exposure to moisture, the ceramic claddings analyzed in this study are divided into two categories: (1) low, for claddings in buildings located in an urban context, more than 5 km from the sea and without the influence of dominant winds carrying sea salts; and (2) high, for buildings located closer to the sea or other humidity sources, under the direct influence of sea winds, and in areas with an average annual rainfall higher than 500 mm and an average annual relative humidity higher than 75%. In the sample analyzed, 111 case studies (57%) present low exposure to moisture and 84 (43%) high exposure to moisture.
The results obtained reveal that claddings with low exposure to moisture are more prone to remain longer in lower degradation conditions (levels A and B) and none of the claddings belongs to level E. On the other hand, claddings with high exposure to moisture show more tendency to belong to the most unfavourable conditions, i.e. levels D and E, with a probability of 70.6%.
Orientation (exposure to UV radiation) Concerning the facades' orientation, the sample analyzed was divided into the cardinal directions: north, east, south and west. The number of elements of each sample was 58, 40, 41 and 43 respectively.
Ceramic claddings facing west remain a longer period of time in levels A and B, with a probability of 50.2%. Oppositely, ceramic claddings facing south are more prone to belong to level E (P = 4.1%), followed by claddings facing north (P = 1.9%).
Various studies (Gaspar & de Brito, 2008;Silva et al., , 2016 indicate that, in Portugal, facades facing north and west are more prone to degradation, thus presenting higher degradation levels; north is colder (with lower periods of UV radiation) and more exposed to the action of moisture; and west has the highest probability of the combined occurrence of wind and rain. On the other hand, in Portugal, cladding facing south is exposed to a longer period of time to UV radiation, which also contributes to the deterioration of ceramic claddings (Guan, Alum, Liu, & Yang, 1997;Yiu, Ho, & Lo, 2007). Therefore, the results obtained seem coherent, revealing that ceramic claddings facing north and south present higher degradation levels, even though due to Figure 10. Most common defects in ceramic claddings due to the presence of damp. different deterioration mechanisms and with different pathological occurrences.

Wind-rain action
Regarding the exposure to wind-rain action, the data set is divided into three categories, according to the height of the building and the density of ground occupation in the surrounding area : . low exposure, corresponding to low-rise buildings (up to two storeys), in densely populated areas, protected from the prevailing winds by other buildings, adjacent hills or vegetation . moderate exposure, corresponding to medium-high buildings, in populated urban areas, protected from the prevailing winds by other buildings, adjacent hills or vegetation . high exposure, for buildings with more than four stories or in open country or crossroads.
The samples obtained for low, moderate and high exposure are composed by 45, 97 and 53 elements respectively. The results obtained by the Petri-net-based model reveal that claddings with high exposure to the combined action of wind and rain have a lower probability of Figure 11. Illustrative example of the influence of the exposure to the wind-rain action in the degradation condition of ceramic claddings.
belonging to the most favourable conditions levels, i.e. levels A and B (P = 26.3%). In the sample analyzed, claddings with a moderate wind-rain exposure have the highest probability of belonging to levels A and B (P = 50.9%). Ceramic claddings with low and moderate exposures present an almost nil probability of belonging to condition E.
According to the literature, the combined effect of wind and rain affects the way that raindrops reach the cladding's surface, as well as the pattern of runoff flow along the cladding (Choi, 1999). Therefore, as expected, in the sample analyzed, cladding with high exposure to wind-rain action present higher degradation levels. On the other hand, in some situations claddings with low exposure may not correspond to the ideal situation, since claddings protected from the rain are more prone to retain superficial dirt and other surface anomalies due to the deposition of particles. In a preliminary stage of the degradation process in which the ceramic claddings only present visual or aesthetic anomalies, this factor can be relevant to justify the fact that claddings with moderate exposure are more prone to belong to level A (10.9%) when compared with claddings with low exposure (P = 2.3%). However, under current conditions, coatings more exposed to the combined action of wind and rain degrade more rapidly than those protected from this action. Figure 11 shows two case studies with moderate and low exposure to damp. Case study (a) is 62 years old and shows a severity of degradation of 50%, with moderate exposure to wind-rain action; and case study (b) is 63 years old and presents a severity of degradation of 39.8%, with low exposure to wind-rain action, thus with a lower degradation level than case study (a). Note that in this study the influence of the environmental exposure conditions was analyzed individually. However, the degradation process is extremely difficult to model since it has a multidimensional nature, where the deterioration of ceramic claddings occurs due to the simultaneous action of different factors.

Conclusions
In this study, the performance over time of ceramic claddings is analyzed using a deterioration model based on stochastic Petri nets. The use of Petri nets has increased in the last years to model dynamic systems in different fields of knowledge. This modelling technique has several advantages when compared with Markov chains or other stochastic approaches. Graphical representation can be used to describe the problem intuitively. Petrinet models are more flexible and, unlike Markov chains, allow the incorporation of many rules in the model to accurately simulate complex situations and keep the model size within manageable limits. Moreover, this modelling technique is not restricted to the exponential distribution to simulate the time in the system. In some situations, the use of exponential distributions can result in a gross approximation of the system characteristics.
The Petri-net model described in this study does not consider the effects of maintenance actions. In fact, the age of the claddings, i.e. the time interval between observations, corresponds to the period of time since construction or the last maintenance action until the inspection time (final observation). Therefore, the deterioration phenomenon is modelled as a sequential dynamic system and the sojourn time in each condition level is defined as a random variable.
The deterioration rates were estimated from available historical date and five probability distributions (exponential, Weibull, Gumbel, log-normal and normal) were analyzed to examine which distribution has a better fit to the historical date. The likelihood values of the five distributions are quite similar; however, the Weibull distributions shows a minor likelihood value and, consequently, a better fit to the historical data.
Concerning the independence analysis, the division of the original sample by environmental characteristics is important to evaluate the influence of these characteristics on the deterioration of the ceramic claddings over time. From the analysis of the influence of the environmental degradation agents, it is concluded that claddings in coastal areas, with high exposure to damp and wind-rain actions, or facing north, show more tendency to belong to the most unfavourable condition levels, i.e. levels D and E (approximately a probability of 70%). Oppositely, claddings exposed to more favourable conditions (more protected from the aggressive environmental agents) are more prone to remain in lower degradation conditions (levels A and B) for a longer period of time.
The Petri-net model proposed in this study can accurately describe the deterioration process of ceramic claddings, providing relevant information regarding the performance of these claddings according to their environmental exposure conditions during their service life. This study provides some guidance about the relationships between the service life of the ceramic claddings and their environmental exposure conditions. This information is extremely relevant to managers, insurers, owners and users, allow the adoption of more sustainable and durable solutions in the design stage and for the definition and implementation of reliable maintenance policies, enabling a more rational management of the European building stock.