Multiple linear regression and fuzzy logic models applied to the functional service life prediction of cultural heritage

Journal of Cultural Heritage 27 (2017) 20–35 Available online at ScienceDirect www.sciencedirect.com Original article Multiple linear regression and fuzzy logic models applied to the functional service life prediction of cultural heritage Andrés Jose Prietoa, Ana Silvab,∗, Jorge de Britoc, Juan Manuel Macías-Bernald, Francisco Javier Alejandred aETSIE, University of Seville, avenue Reina Mercedes, 4A, Seville, Spain bCERIS-ICIST, Instituto Superior Técnico (IST), Universidade de Lisboa, avenue Rovisco Pais, 1049-001 Lisbon, Portugal cCERIS-ICIST, Department of Civil Engineering, Architecture and Georresources, IST, Universidade de Lisboa, avenue Rovisco Pais, 1049-001 Lisbon, Portugal dDepartment of Architectural Construction II, ETSIE, University of Seville, avenue Reina Mercedes, 4A, Seville, Spain a r t i c l e i n f o Article history: Received 16 September 2016 Accepted 22 March 2017 Available online 14 April 2017 Keywords: Service life prediction Fuzzy inference system Multiple linear regression analysis Built heritage Preventive maintenance a b s t r a c t In this research, a proposal for the assessment of the functional service life of built heritage applying statistical tools is described. A fuzzy inference system is applied in order to establish a ranking in terms of functional service life for the built heritage, thus allowing prioritizing the maintenance and preventive conservation actions in homogeneous groups of buildings, and optimizing the costs involved in maintenance operations. The functionality of a sample of 100 parish churches was evaluated. However, the selection of maintenance strategies for buildings is usually a multiple criteria decision-making problem, encompassing various variables and constraints. Therefore, a multiple linear regression analysis is applied in order to rank the variables in terms of influence in the serviceability estimation of heritage buildings. Currently, social, environmental and economic reasons are raising concern about the durability and functional service life of heritage sites. The results obtained in this study are useful to researchers and stakeholders responsible for the maintenance of historical buildings, since they allow reducing their probability of failure. The preventive maintenance programs can be considered as a cost-effective and environmentally sustainable option to extend the serviceability of heritage buildings. © 2017 Elsevier Masson SAS. All rights reserved. 1. Introduction Cultural heritage buildings are an important economic and cultural capital of European countries [1]. A monument is more than just the construction itself [2], being part of the local identity and a source of memory of historical events [3,4]. National governments and European institutions increasingly recognise the importance of the conservation of cultural assets [5]. In the last decades, there have been an important evolution in policies and practices regarding the preservation of heritage buildings. The Council of Europe proposed, in 2005, a new covenant for ratification concerning the cultural heritage [6]. The degree of involvement of nowadays societies is due to the consciousness of the need for a sustainable management of scarce resources [7]. Currently, around 50% of all building refurbishments in European cities are related in some way to heritage preservation [8]. ∗Corresponding author. Tel.: +351964186538. E-mail addresses: [email protected] (A.J. Prieto), [email protected] (A. Silva), [email protected] (J. de Brito), [email protected] (J.M. Macías-Bernal), [email protected] (F.J. Alejandre). The concept of conservation of cultural built heritage has evolved over the recent decades at the international level, in order to define multidisciplinary approaches to intervention in these buildings, thus leading to their maximum preservation [9,10]. In the conservation of built heritage, the building should be seen as a whole, thus protecting its constructive system and typological characteristics, maintaining its social function, responding to current lifestyles, avoiding its obsolescence and deterioration [11,12]. The conservation of these buildings is particularly complicated and is usually based on a more comprehensive analysis than just aesthetic or historical criteria. In fact, as referred by UNESCO [2], the conservation of built heritage requires the evaluation of several uncertain factors, demanding a thoughtful knowledge of history, a true understanding of the present and an ability to anticipate the future. The management of the conservation operations is usually a difficult task, conditioned by technical, financial and legislative issues [13]. Moreover, as mentioned by Alshweiky and Ünal [14], there is a lack of know-how, experts on conservation and funding for maintenance operations. Kim et al. [15] refer that the lack of accurate decision tools and the use of decision support systems for determining restoration priorities (usually, based only http://dx.doi.org/10.1016/j.culher.2017.03.004 1296-2074/© 2017 Elsevier Masson SAS. All rights reserved. A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 21 on the severity of damage) lead to ineffective tools for prioritizing execution needs and consequently to inefficient rehabilitation and maintenance operations. 2. Research aim One way to improve the planning of the maintenance actions in heritage buildings, optimizing the resources applied in these interventions, is by understanding the degradation of their elements and identifying the instant beyond which they must be intervened [16]. Consequently, the preservation of architectural assets requires the development of methods, strategies and planning of maintenance operations [17]. Furthermore, it is necessary to analyse the heritage buildings context, proposing accurate strategies, establishing effective tools to aid the decision process for the definition of priorities for intervention [18]. Maintenance activities must be seen as an investment opportunity, adopting a series of measures to prevent both material and functional degradation. As mentioned by Vanier and Lacasse [19], the stakeholders must deal with difficult decisions concerning when and how perform maintenance and repair actions in the built environment. These difficulties are due to the lack of knowledge related to the service life prediction and the absence of methods to assist the asset manager in the definition of a proper maintenance, repair or replacement choices [20]. Currently, the service life prediction of the materials and components of the built heritage is an especially important issue to achieve their maximum longevity, avoiding the possible failure of the building and future extremely costly interventions [21]. Therefore, new methodologies are necessary in order to establish the instant in which it is necessary to intervene, thereby allowing a more rational definition of maintenance plans, adopting “just-in- time and fit-for-purpose” operations [22,23]. Therefore, in this study, statistical and fuzzy models are established to describe the functional service life of heritage buildings. Macías-Bernal et al. [24] initially proposed a fuzzy inference system (FIS) to model the functional service life of heritage, cultural and religious buildings. They identified 17 vulnerability and risks factors that affect the buildings’ service life, establishing a mathematical model to determine the functionality index of the buildings analysed, which was called as Fuzzy Building Service Life (FBSL). This model was previous defined to solve a real problem, i.e. the Archdiocese of Seville intended to maintain and rehabilitate a set of religious buildings, and needed to know which buildings should be maintained and rehabilitated first and the sequence of intervention. This methodology has been applied and improved by Prieto et al. [25] and Prieto et al. [26], intending to provide a priority ranking of maintenance actions based on the sample’s performance. Prieto et al. [25] and Prieto et al. [26] applied the same inference rules, intending to validate the model by applying it to different sets of buildings with homogenous characteristics. This method, despite efficiently allowing the ranking of the maintenance needs of heritage buildings, is relatively complex, requiring the knowledge regarding the fuzzy inference system (which was established based on an expert survey) and the application of a specific software to obtain the functionality index of the case studies analysed. Therefore, this study intends to improve the methodology previously proposed by the authors, establishing a new method simplifying the way in which the functionality of the buildings analysed is estimated. A multiple linear regression (MLR) analysis is used to describe a simplified model to predict the serviceability of the built heritage, identifying the variables that most contribute to the functional degradation phenomena of the religious buildings. For that purpose, a sample of 100 parish churches (with homogeneous constructive characteristics but different levels of serviceability) was examined. These models will support decision-makers in developing the most appropriate strategy for the future use of heritage buildings, considering the most relevant factors involved, and applying efficient maintenance strategies. 3. Functional service life model 3.1. Fuzzy set theory and model assumptions The fuzzy set theory has been widely applied as a support tool for decision-making processes and in performance evaluation in engineering [27–29], and specifically in the decision-making support for the restoration and maintenance of historical buildings [30,31]. As mentioned by Kutut et al. [32], in the majority of the real-life situations, human judgements are vague and cannot be translated to numerical values, since human reasoning and decision-making are always associated with some degree of subjectivity. Fuzzy sets are able to deal with uncertain, imprecise and vague data, which is usually the information available for modelling real world phenomena [33–35]. Therefore, in this study, to deal with the uncertainty and vagueness associated with the evaluation of the functional condition of the churches analysed based on expert opinions, the fuzzy logic principles established by Zadeh [36] were used. Unlike Boolean logic, in fuzzy logic, an element can belong to more than one set, with a given degree of membership [37]. Analysing for example the age of the building, classic logic only accepts extreme values, i.e. a building can only be “old” or “new”; on the contrary, in fuzzy logic, each proposition can be partially true and partially false, with a given degree of or membership to each of the conditions [38], i.e. a given building can be 30% “old” and 70% “new”, belonging simultaneously to the two conditions. A membership function  allocates to each element a membership degree in the fuzzy set A, ranging from 0 to 1 [39]. The inputs variables are fuzzified in membership functions A, U is universe of discourse, in which a fuzzy set can take any value in the range of [0,1], as described in equation (1). A: U →[0, 1](1) In this fuzzy inference system (FIS), Gaussian-type membership functions are generally used in the input parameters, as they are considered the most appropriate, reaching a non-zero values at all points, except for the v1input variable, in which a trapezoidal membership function is applied. The system uses the fuzzy operator “and” as connector [26], i.e. “and” represents the intersection between two fuzzy sets. The intersection between fuzzy sets A and B is given by a fuzzy set Aˆ B, which membership function as defined in Eq. (2) ([40], [26]): ␮A∧B(x,y)= T(␮A(x), ␮B(y)) (2) T(x, y)= min(x, y)(3) Where T(x, y) is a T-norm that complies with the commutative, associativity and monotony properties [41], as seen in Eq. (3). The main stage of a fuzzy system is the base of knowledge, which contains a set of natural language rules and it consists of conditional statements. As mentioned by Vesely et al. [37], there is one conditional statement R describing each known case, while individual fuzzy sets A, B, C, etc. within that statement R refer to the values of variables, Eq. (4): R = If A and B then C (4) As an example, in the fuzzy inference system described in the manuscript, if “the drainage of water in the roof occurs rapidly” (A) and “the constructive system is adequate” (B) then “the durability of the roofing system is high” (C). The knowledge base, fuzzy rules and hierarchical structure are formed using a professional expert’s survey (which is extensively described in Section 3.4). The fuzzy 22 A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 system applies the Mamdani inference system [42]. Fuzzy models can be considered as transparent models (Babuˇ ska 1998), describing the relationship between the input and output parameters by means of if-then rules. To illustrate the inference mechanism system applied, two examples of the inference rules established are presented in Eq. (5). (5) IF v1-Geological location is VG-Very Good; AND v3- Environmental conditions is VG-Very Good; AND v4-Constructive system is a medium estimation; THEN the output (Vulnerability) is VG-Very Good. IF r6-Load state modification is R-Regular; AND r9-Facilities is R Regular; AND Vulnerability is VB-Very Bad; THEN the output (Static-structural risks) is VB-Very Bad. Finally, in the defuzzification step, crisp numerical predictions result from the fuzzy predictions using the “centre of gravity” algorithm [37]. Several alternative defuzzification methods are reviewed but centre of gravity is one of the most widely used, and is adopted in this study. The discrete case can be interpreted by a Riemann sum [26,43]. Eq. (6) presents the mathematical expression of the proposed model [24–26], allowing estimating the functional index — designated Fuzzy Building Service Life (FBSL) — of the heritage buildings. In other words, Eq. (6) presents the output of the model based on the inference rules and the fuzzy system proposed. As mentioned by Prieto et al. [26], the model presents a continuous property, i.e. the input membership functions are continuous, the universe of inference is continuous and the fuzzy sets are defined by continuous membership functions, instead of using membership values (discrete universe) [44,45], which implies that small changes in the inputs do not lead to a significant change in the output. FBSL =˙iyi•B(yi) ˙iB(yi)(6) The FBSL can be used to establish the probable loss or gain in functional service life of a building’s elements for different lev els of maintenance. The computational application of the fuzzy model used in this study was implemented in fuzzy logic software developed at IMSE-CNM — Seville Institute of Microelectronics and National Microelectronics Centre and R&D&I centre belonging to CSIC — Spanish National Research Council [66]. It is named Xfuzzy3.0. 3.2. Variables included in the model The deterioration, failure and potential collapse of historical buildings usually occur due to [46,47]: •rain water runoff action associated with the performance and design of the roof structure; •soil settlement and relative movement of the foundation; •inadequacy of the load-carrying structural system; •lack of strength of the materials applied; •design problems; •local and environmental effects associated with the geographic location of the building. Various authors [48–50] refer that the cultural heritage is usually subjected to: •static-structural hazards, such as floods, geotechnical problems, seismic actions, among others; •environmental hazards, such as the weather conditions, pollution, among others; •and anthropogenic factors, such as fires, population and tourism. Furthermore, it is important to analyse the combined effect of the probability of occurrence of a threatening event (hazards) and its potential consequences (loss of life or injury, property deterioration, social and economic obsolescence), which is directly related with existing vulnerabilities (e.g. physical or environmental vulnerabilities) [51,52]. Therefore, this study intends to contemplate all the relevant variables, including risks and vulnerabilities, for the description of the loss of functionality of heritage buildings. The proposed model was defined based on the specifications established in the international standard ISO 31000: 2009 [69], which establishes the basic concepts to manage and assess the risks of the buildings analysed, contemplating the nature and type of the consequences, estimating the probability of future occurrence, and therefore allowing establishing the necessity of intervene in each building considered, as well the priority of intervention in a set of heritage buildings based on the estimated functional service life of each case study analysed [25]. Consequently, 17 input parameters were included in the fuzzy model [24,26]: •vulnerabilities: v1: geological location; v2: roof design; v3: built context; v4: constructive system; v5: preservation; “vulnerabilities” include the variables related with the characteristics of the building, i.e. that portray the physical reality of the building at the inspection time, which usually remains unchanged over time; •static-structural risks: r6: load state modification; r7: live loads; r8: ventilation; r9: facilities; r10: fire; r11: inner environment; static-structural risks include all the variables that are related with the safety of the building (structural, in case of fire, in use) and the factors that vary in intensity and magnitude throughout the life cycle of buildings, depending on each specific circumstance; •atmospheric risks: r12: rainfall; r13: temperature. The rainfall conditions were evaluated based on the data of the Municipality board for Environmental issues of Andalusia, establishing four levels of risk according to the intensity of rainfall in the different areas of Seville’s province. The variable “temperature” was defined comparing the maximum and minimum average annual temperatures in each municipality of Seville, assuming that the most unfavourable situation corresponds to buildings located in an area with more days per year with the highest temperature variation. The environmental conditions were established in relation with the more extreme conditions of the Seville Archdiocese, in order to compare the most unfavourable situations with the real conditions that affect each specific building considered; •anthropic risks: r14: population growth; r15: heritage value; r16: furniture value; r17: occupancy. Table 1 presents the numerical translation of the variables considered, explaining the different possibilities for each variable. The evaluation of the variables analysed shown in Table 1 contemplates three options, the most favourable, the current situation (represented by the middle point) and the most unfavourable situation. However, intermediate situations can also be evaluated by the model; for that, an in-between value should be adopted. 3.3. Characterization of the sample This study is based on 100 heritage buildings located in the province of Seville, Andalusia region, Spain (Fig. 1), corresponding to an area of 14,000 km2. The geographical territory ranges from the southeast, Atlantic Ocean to the northwest close to Sierra Norte Natural Park, which is categorised as a warm Mediterranean weather with an annual average temperature of 18.5◦C; winters are usually mild while summers are remarkably hot. The majority of the historic buildings analysed were built between the 14th and 16th centuries, including some constructions built in the 17th–18th centuries, in a Baroque style. Consequently, the parish A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 23 Table 1 Descriptive variables valuation of the model. Variables Variables designation Qualitative valuation Quantitative valuation Descriptive valuation v1Geological location Good 1.0 Optimum ground conditions (very stable soil – rock bottom) Average 2.5 Ground conditions within acceptable limits of stability Bad 4.0 Very unfavourable ground conditions (clay soil) v2Roof design Good 1.0 Easy and fast evacuation of water on deck (ideal situation – semi-spherical dome) Average 4.5 Good conditions in terms of evacuation of rainfall Bad 8.0 Complex and slow evacuation of water v3Environmental conditions Good 1.0 Building without constructions around it Average 4.5 Building between constructions Bad 8.0 Building between complex constructions v4Constructive system Good 1.0 Uniform characteristics of constructive system Average 4.5 Heterogeneous characteristics of constructive system Bad 8.0 Intermingled different constructive system v5Preservation Good 1.0 Optimal state of conservation Average 4.5 Normal state of conservation Bad 8.0 Building in a neglected state of conservation r6Load state modification Good 1.0 Without any apparently modification Average 4.5 Symmetric and balanced modifications Bad 8.0 Disorderly modifications without any pattern r7Live loads Good 1.0 Live loads below the original level Average 4.5 Live loads equal to the original level Bad 8.0 Live loads higher than the original level (warehouse) r8Ventilation Good 1.0 Natural cross-ventilation in all or in several areas Average 4.5 Natural cross-ventilation in some areas Bad 8.0 No natural cross-ventilation r9Facilities Good 1.0 All facilities are in use and under standards conditions Average 4.5 Some facilities are in use Bad 8.0 Facilities are not ready to be used r10 Fire Good 1.0 Incombustible structure and low fire load Average 4.5 Combustible structure and medium fire load Bad 8.0 Combustible structure and high fire load r11 Inner environment Good 1.0 Low level of health, cleanliness and hygiene of the building’s spaces Average 4.5 Medium level of health, cleanliness and hygiene of the building’s spaces Bad 8.0 Maximum level of health, cleanliness and hygiene of the building’s spaces r12 Rainfall Good 1.0 Area with low annual rainfall Average 4.5 Area with medium annual rainfall Bad 8.0 Area with maximum annual rainfall r13 Temperature Good 1.0 Area with low temperature differences Average 4.5 Area with medium temperature differences Bad 8.0 Area with maximum temperature differences r14 Population growth Good 1.0 Population growth greater than 15% Average 4.5 Population growth 0% Bad 8.0 Population growth less than 5% r15 Heritage value Good 1.0 Properties with great historical value Average 4.5 Properties with average historical value Bad 8.0 Properties with low historical value r16 Furniture value Good 1.0 Social, cultural and liturgical appreciation (high value) Average 4.5 Social, cultural and liturgical appreciation (average value) Bad 8.0 Social, cultural and liturgical appreciation (low value) r17 Occupancy Good 1.0 High activity in the building (high occupancy) Average 4.5 Media activity in the building (average occupancy) Bad 8.0 Low activity in the building (low occupancy) churches have homogeneous constructive, cultural, political and regulatory features. However, the chronology and stylistic characteristics of Mudejar-Gothic buildings in the region have features that are unique to this kind of construction [50]. The Mudejar- Gothic churches in the province of Seville are morphologically characterized by this stylistic dualism: a vaulted Gothic apse and a body of three naves with a timber roof of Moorish origin [53]. As described by Zamarre˜ no et al. [53], its brick walls are complemented with quadrangular, and in some cases, octagonal pillars with raised brick mouldings as decoration. 3.4. Condition survey The condition survey is quantified and validated by a group of 15 professionals with expertise in the management of built heritage [24]. The expert survey is performed in the model’s design stage. The expert profiles are related with the areas of rehabilitation and pathology on construction; direction of accredited laboratories of building materials; restoration; architecture; archaeology; fireman commanders; heads of building maintenance of a municipality provincial capital of 700,000 inhabitants; the direction of a World Heritage conservation building; conservation of a Port Authority; management of an insurance company at international level; and experts in quality management in buildings [26]. The 17 input variables considered were identified based on the analysis of the different factors that influence the functionality and service life of religious buildings, after review the following documents: Spanish Technical Building Code; National Cathedral Plan; Law on Construction Planning; Heritage Conservation Network; UNE 41805:2009 IN [71]; ISO 15686 [68]. Additionally, a Delphi methodology (applied in two rounds), through the Opina software property of University of Seville [70], was used to obtain all the expert’s survey. In the expert survey, two questions are asked related with the 17 factors proposed. The first one intends to understand whether the expert agrees that the factors considered were relevant in determining the service life of the building. The second question intends to 24 A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 Fig. 1. Map of the study’s scope by municipalities, Province of Seville, Andalusia, Spain. understand the relevance of each factor in the overall functionality and durability of the building, adopting a numerical scale ranging between 1 (low influence) and 10 (high influence). Table 2 presents the 17 input factors ranked by the degree of relevance that the experts have assigned to each one of the factors, indicating the average value obtained in a scale of one to ten. Table 2 17 input parameters ranked by the degree of relevance that the experts assigned to each of the factors (data sourced from [24]). Input variable Description Factor incidence level in functional durability. Average valuea Factor incidence level in functional durability. Standard deviation v5Preservation 8.64 2.157 r10 Fire 8.29 1.567 r6Load state modification 8 1.748 v2Roof design 8 1.804 r9Facilities 7.43 1.804 r15 Heritage value 7.11 1.748 r17 Occupancy 7.09 1.567 v1Geological location 6.92 1.286 v4Constructive system 6.83 1.567 r12 Rainfall 6.62 1.489 r7Overloads 6.54 1.859 r14 Population growth 6.38 1.690 r13 Temperature 6.38 1.433 r8Ventilation 6.29 1.502 v3Built context 6.27 1.567 r16 Furniture value 6.25 2.063 r11 Inner environment 5.86 1.120 aThe influence of each variable is quantified adopting a numerical scale ranging between 1, low influence, and 10, high influence. 4. Application of multiple linear regression to identify the relevant parameters in the serviceability of historical buildings Regression analysis is one of the methods most commonly used to measure the relationship between two or more variables [54], thus predicting the behaviour of a dependent or endogenous variable, according to one or more independent or explanatory variables. Multiple linear regression is frequently used as an empirical model or approximating function, which allows establishing a mathematical model to describe a given real-world phenomenon [55]. Generally, the relationship between the dependent and the independent variables is given as presented in Eq. (7) [56]. yi= ˇ0+ ˇ1x1i+ ˇ2x2i+ · · · + ˇpxpi + εi(7) Where yiis the dependent variable; ˇ0, ˇ1i, ˇ2i, . . ., ˇpithe regression coefficients, x1i, x2i, . . ., xpi, the independent variables and εithe random errors of the model. The linear regression coefficients (B) are obtained by the method of least squares. Recently, various studies [38,56,57] address the application of this statistical technique for the service life prediction of claddings and as a tool to maintenance decision support. As mentioned in the previous section, in this study, the serviceability of the historical monuments analysed is obtained based on an expert survey. This model, which applies a fuzzy logic inference system, leading to a functionality index (FBSL), is very complex and timeconsuming, requiring specific software for its practical application. Furthermore, the model encompasses all the relevant variables, considering 17 input parameters. Therefore, in this study, the multiple linear regression (MLR) analysis is applied to model the functionality of built heritage, proposing a new model, functioning as a simplification of the FBSL model, thus reducing the complexity of the model and the number of variables included in the model. Therefore, the first stage of the definition of the MLR model is the identification of the variables that most contribute to the explanation of the dependent variable, i.e. the functionality of the cultural heritage buildings. As mentioned by Martin et al. A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 25 Table 3 Summary of the multiple linear regression model obtained for the prediction of the FBSL index, with 11 variables. Model R R2R2 adjusted Square of the mean square error 1 0.703a0.495 0.489 9.839 2 0.852b0.725 0.719 7.293 3 0.898c0.807 0.801 6.145 4 0.920d0.846 0.839 5.517 5 0.937e0.878 0.872 4.928 6 0.947f0.898 0.891 4.546 7 0.954g0.911 0.904 4.267 8 0.960h0.923 0.916 3.997 9 0.965i0.931 0.924 3.803 10 0.969j0.938 0.931 3.604 11 0.970k0.942 0.934 3.525 aPredictors: (Constant), v2. bPredictors: (Constant), v2, r17. cPredictors: (Constant), v2, r17, r7. dPredictors: (Constant), v2, r17, r7, v3. ePredictors: (Constant), v2, r17, r7, v3, r14. fPredictors: (Constant), v2, r17, r7, v3, r14, v5. gPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10. hPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10, v1. iPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10, v1, r12. jPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10, v1, r12, r15. kPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10, v1, r12, r15, v4. [58], an efficient service life prediction method should be able to define the relevant variables. In this study, a stepwise technique is applied intending to identify which variables are most significant. In this method, only the statistically significant independent variables are included in the model, analysing all the hypotheses of the regression, thus eliminating the effects of multicollinearity, i.e. pseudo-independent variables that are interrelated, thus jeopardizing the multiple regression analysis [59]. In the stepwise technique, the independent variables were introduced in the model from the most significant variable, with high impact in the correlation coefficient of the model, to the less significant variable [59]. For the definition of the MLR models, the SPSS (Statistical Package for Social Sciences) software was used. In the definition of the MLR model, the functionality index is considered the dependent variable, analysing the 17 input variables from the fuzzy inference system (FIS) model. Eq. (8) presents the MLR model obtained, allowing predicting the functionality of the heritage buildings according to the 11 explanatory variables considered. FBSL = −1.902v1− 3.396v2− 0.629v3− 1.160v4− 3.373v5 − 2.173r7− 2.297r10 − 1.318r12 − 1.286r14r14 − 1.531r15 − 1.077r17 + 115.257 (8) Where v1is geological location, v2roof design, v3environmental conditions, v4constructive system, v5conservation state (preservation condition), r7live loads, r10 risk of fire, r12 exposure to rainfall, r14 population growth, r15 heritage value and r17 conditions of occupancy. To apply this model, the explanatory variables must be replaced by their numerical value, shown in Table 1. Table 3 presents the summary of the multiple linear regression model obtained for the prediction of the FBSL index. This model presents correlation and determination coefficients of 0.960 and 0.923, respectively; thus revealing a very strong correlation between the FBSL obtained by the expert survey and the FBSL values predicted by the MLR model. The determination coefficient obtained reveals that 92.3% of the variability of the FBSL index can be explained by the 11 variables included in the model, while the remaining 7.7% is due to other causes that were not analysed in this study. In Table 4, the ANOVA table of the model obtained for the prediction of the FBSL index is presented. The F test is usually used to determine the explanation capacity of the proposed model [56]. For a 5% level of significance, 11 numerator degrees of freedom and 88 denominator degrees of freedom, a critical value of F of 1.900 was obtained, significantly lower than the value obtained in the ANOVA analysis (129.330), which reveals that the model has a statistically significant explanation capacity, and there is at least one statistically significant independent variable. Table 5 shows the regression coefficients used in the mathematical equation (8), which expresses the functionality of the built heritage according to the independent variables considered. The t-test allows evaluating whether each variable is useful to predict/explain the dependent variable. In this study, for a level of significance of 5% and for a sample of 100 case studies, the critical value of t is 1.660, significantly lower (in absolute value) than the t values obtained for the eleven independent variables included in the model. Also, the significance values present in Table 5 are Table 4 ANOVA table of the model obtained for the prediction of the FBSL index, with 11 variables. ModelaVariation – Sum of the squared deviations Freedom degree (FD) Mean squared deviation F Significance 11 Regression 17677.911 11 1607.083 129.330 0.000b Residual 1093.509 88 12.426 Total 18771.420 99 aPredictors: (Constant), v2, r17, r7, v3, r14, v5, r10, v1, r12, r15, v4. bDependent variable: FBSL index. 26 A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 Table 5 Regression coefficients of the model obtained for the prediction of the FBSL index, with 11 variables. ModelaLinear regression coefficients Standardized coefficients t Significance ˇ Standard deviation associated to the regression coefficients Beta 11 (Constant) 115.257 2.335 49.359 0.000 v2−3.396 0.359 −0.378 −9.463 0.000 r17 −1.077 0.386 −0.110 −2.793 0.006 r7−2.173 0.680 −0.117 −3.198 0.002 v3−0.629 0.327 −0.076 −1.921 0.058 r14 −1.286 0.315 −0.144 −4.083 0.000 v5−3.373 0.645 −0.179 −5.226 0.000 r10 −2.297 0.406 −0.197 −5.653 0.000 v1−1.902 0.677 −0.101 −2.811 0.006 r12 −1.318 0.333 −0.147 −3.955 0.000 r15 −1.531 0.424 −0.117 −3.611 0.001 v4−1.160 0.517 −0.087 −2.243 0.027 aDependent variable: FBSL index. always lower than 5%, thus revealing that all the variables are statistically relevant, and the model presents a good adjustment to the reality that it intends to model. The proposed model presents a strong correlation between the FBSL predicted and the values obtained by the fuzzy inference system (FIS) model (based on the real condition of the churches inspected, according to the expert survey performed). However, the proposed model is still relatively complex, with a large number of predictors. Therefore, a simplified model is proposed, easier to apply in practice, with only six variables. The simplified or reduced model was established ensuring that the new model is able to portray adequately the functionality index of the churches under analysis and does not imply a reduction in the accuracy of the overall MLR model higher than 5%. The main drawback of the simplified model is that, as usually occurs in mathematical modelling, the simplification of the model leads to a slight reduction of their accuracy. This model corresponds to model 6 in Table 3, with a determination coefficient of 0.898. When compared with the global model, the simplified model only presents a decrease of its statistical significance of 4.4%, which is a residual value given the much higher simplicity obtained. The simplified model shows a very strong correlation between the predicted and the observed functionality index, revealing that 89.9% of the variability of the functionality of the churches analysed can be explained by the six variables included in the model. Eq. (9) presents the numerical equation of the relationship between the FBSL index and the explanatory variables. FBSL = −3.954v2− 2.247v3− 3.076v5− 3.241r7− 1.876r14 − 2.675r17 + 109.320 (9) Where v2is roof design, v3environmental conditions, v5conser- vation state (preservation condition), r7live loads, r14 population growth and r17 conditions of occupancy. For the application of the simplified model, the explanatory variables must be replaced by their numerical value, presented in Table 1. Table 6 shows the ANOVA table of the simplified model. For this model, it is also concluded that the model has a statistically significant capacity, since the value of F (135.913) is much higher than the critical value (equal to 2.183, obtained by the Fisher–Snedecor tables, adopting a 5% significance level, six numerator degrees of freedom and 93 denominator degrees of freedom). Table 7 presents the regression coefficients. All the variables included in the simplified model are statistically relevant, presenting an absolute t value of the regression higher than the critical value of t (equal to 1.660, obtained by the t-Student distribution tables, for a significance level of 5% and for a sample of 100 case studies). The analysis of the statistical parameters analysed, reveal that the model is well adjusted to the dataset, and is able to describe the variability of the serviceability of the cultural buildings analysed. 5. Discussion of the results The method presented in this study allows evaluating the functionality of the churches analysed. This methodology considers the consequences of environmental, static-structural and anthropogenic conditions on the functional service life of cultural heritage (given by the serviceability index — FBSL). The application of the FBSL model implies the knowledge regarding the fuzzy logic inference system adopted, as well the use of the computational application or the fuzzy model software developed. This model is more complex but also more accurate, encompassing all the relevant variables. Therefore, the models proposed by the MLR allow simplifying the fuzzy model, estimating the functionality level and the ranking of the maintenance needs of the buildings analysed, applying a simple equation, easily applied by the different stakeholders (managers, designers, architects, building pathologists, engineers, among others), not requiring expert knowledge regarding fuzzy inference systems or complex software. Necessarily, the gain in simplicity leads to a loss in the accuracy of the model, even though, as mentioned by different authors [60,61], the decisions to be made Table 6 ANOVA table of the model obtained for the prediction of the FBSL index, with six variables. ModelaVariation – Sum of the squared deviations Freedom degree (FD) Mean squared deviation F Significance 6 Regression 16849.806 6 2808.301 135.913 0.000b Residual 1921.614 93 20.663 Total 18771.420 99 aPredictors: (Constant), v2, r17, r7, v3, r14, v5. bDependent variable: FBSL index. A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 27 Table 7 Regression coefficients of the model obtained for the prediction of the FBSL index, with six variables. ModelaLinear regression coefficients Standardized coefficients t Significance ˇ Standard deviation associated to the regression coefficients Beta 11 (Constant) 109.320 2.678 40.820 0.000 v2−3.954 0.403 −0.440 −9.802 0.000 r17 −2.675 0.375 −0.274 −7.132 0.000 r7−3.241 0.778 −0.175 −4.167 0.000 v3−2.247 0.328 −0.271 −6.855 0.000 r14 −1.876 0.372 −0.210 −5.043 0.000 v5−3.076 0.736 −0.163 −4.181 0.000 aDependent variable: FBSL index. on maintenance actions affecting the built heritage should be aided by user-friendly tools that can be easily applied by the stakeholders. Concerning the ranking of the maintenance and restoration activities, Prieto et al. [26] proposed a numerical scale to correlate the FBSL index and the time until the next maintenance action. Adopting these criteria, a church with a FBSL index under 34 requires an immediate intervention; in the sample analysed, 23 churches require an immediate intervention according to the fuzzy logic model, and the MLR model with 11 variables identify correctly 91% (21 of the case studies) that actually require immediate intervention. The simplified model with only six variables, besides being simpler, loses some accuracy, only identifying correctly 74% of the case studies that require immediate intervention. On the other side of the hierarchical scale, there are 20 churches in a very good conservation state that do not require interventions in the next 50 years. Both MLR models identify correctly 90% of these case studies. Therefore, although the MLR models are not perfect, the results reveal that both models present high levels of accuracy and an adequate behaviour in the description of a complex phenomenon such is the functionality condition of heritage buildings. The MLR models reveal that, as expected, there is an inverse proportional relationship between the functionality of the churches and their vulnerability and risks, i.e. buildings subjected to higher risks and with higher vulnerability conditions, present lower functionality indexes. To illustrate this conclusion, Fig. 2 presents the correlation between the intrinsic vulnerability of the buildings and the external risks, and the influence of these parameters in the functional performance of the buildings. Fig. 2 presents a theoretical analysis, similar to a sensitivity analysis, evaluating in which way the FBSL index varies when the risks and vulnerabilities of the building also vary. Therefore, a theoretical functionality index was defined, given by the following equation: Theoretical FBSL = 1/(V − (V*R)) [24]. In this analysis, the “x” axis shows the risks of the building considered as a whole, the “y” axis presents the theoretical functionality of the building and different categories of vulnerability are analysed. The vulnerability of the building is also analysed as a whole, i.e. when the five variables considered as “vulnerability variables” are quantified by 1 means that the building presents an extremely low vulnerability level (e.g. the building presents optimum ground conditions (v1 — geological location), the drainage system of the roof is extremely efficient (v2 — roof design), the building is in an area without constructions around it (v3 — environmental conditions), it presents uniform characteristics of the constructive system (v4 — constructive system) and is in an optimal conservation state (v5 — preservation)). On the opposite, when the vulnerability of the building is extremely high, the vulnerability value is equal to 8. The evaluation of these variables was discussed in Table 1. In this theoretical analysis, the theoretical vulnerability varies between 1 (extremely low vulnerability level) to 8 (extremely vulnerable) and the same occurs to the theoretical risks, in which 1 means a building subjected to very low risks and 8 a building extremely exposed to risks. The analysis of Fig. 2 reveals that when the risks are low, the building’s functionality depends on its vulnerability, presenting a wide range of FBSL values for different vulnerability values. As mentioned by Rashed and Weeks [62], the concept of vulnerability differs from that of risk, since it is not dependent from any certain magnitude of a given event, but dependent on the context in which that event occurs. In fact, the results show that, as the values of the risks increase, the building’s vulnerability is no longer so relevant, since against very serious hazards, such as an earthquake, there are no 100% safe buildings, and the functionality of the building decreases significantly, regardless of their degree of vulnerability. The multiple linear regression models proposed allow identifying the more relevant variables for the description of the functionality of the churches analysed. Fig. 3 presents the weight of the different variables for the explanation of the variability of the serviceability of the buildings analysed. On the left, the relative weight of the 11 variables considered in the overall MLR model is analysed and, on the right, the six variables included in the simplified MLR model are analysed. Fig. 3 presents for both models the relative importance of each variable to the estimation or for the description of the variability of the functionality index of the heritage buildings analysed. For the multiple linear regression models proposed, the roofs’ design is the most relevant variable, which explain by itself more than 50% (exactly 52% for the model with 11 variables and 55% for the model with six variables) of the variability of the functionality level of the churches, since in several cases, the decision to intervene is exclusively based on the roof’s degradation condition. In the proposed models, occupancy is also a very relevant parameter, followed by live loads, environmental conditions, population growth and preservation condition. Macías-Bernal et al. [24] have discussed the weighing of the 17 input parameters according to the expert opinion, concluding that the preservation condition, risk of fire, load state modification and roof’s characteristics are the most relevant factors (Table 2). In this study, the sequence or the order of relevance of each variable is different from the study of Macías-Bernal et al. [24]. Naturally, different models lead to different conclusions, and the evaluation of the relative importance of each variable is subjective, since there are different possible outcomes to the question asked during the expert survey “which is the most relevant factor for the loss of functionality of cultural heritage”. Moreover, these models are based on the real evaluation of the 100 churches inspected by experts, which always involve complex reasoning perceptions and subjective evaluations of the observed reality. For clarity’s sake, a case study is analysed, in order to discuss the relevance of the various variables evaluated in this study. Fig. 4 presents the Los Sagrados Corazones church, located in San Juan de Aznalfarache, and several pictures, related with the different factors involved in the serviceability of the built heritage, are shown. 28 A.J. Prieto et al. / Journal of Cultural Heritage 27 (2017) 20–35 Fig. 2. Relationship between functionality and risks. Fig. 3. Relative weight of the variables included in the global model (left) and in the simplified model (right). Table 8 Practical application to one heritage building analysed. Variables involved in the definition of the functional service life Valuation Parish Church Municipality Functionality Functionality obtained by the MLR model Vulnerability v13Los Sagrados Corazones San Juan de Aznalfarache 37.981 44.695 v22 v36 v44 v52 Risks r64 r74 r84 r94 r10 6 r11 3 r12 4 r13 3.5 r14 2 r15 4 r16 4 r17 6 A.J. 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