Challenge for planning by using cluster methodology: the case study of the Algarve Region

sustainability Article Challenge for Planning by Using Cluster Methodology: The Case Study of the Algarve Region David Bienvenido-Huertas 1,* , Fátima Farinha 2, Miguel JoséOliveira 2, Elisa M. J. Silva 2 and Rui Lança 2 1Department of Building Construction II, University of Seville, Av. Reina Mercedes 4A, 41012 Seville, Spain 2University of Algarve, Institute of Engineering, Campus da Penha, 8005-139 Faro, Portugal; [email protected] (F.F.); [email protected] (M.J.O.); [email protected] (E.M.J.S.); [email protected] (R.L.) *Correspondence: [email protected] Received: 31 January 2020; Accepted: 17 February 2020; Published: 18 February 2020   Abstract: This study analyses the most appropriate methodology to make similarity classifications among the cities of the Algarve (Portugal) according to 105 sustainability indicators monitored with the Observatory of Sustainability of the Algarve Region for Tourism (OBSERVE). The methodology used to establish the similarities was the cluster analysis with 4 different approaches which reduced the dimensions of the data set: total approach, pillar approach, subject area approach, and indicator approach. By combining the approaches, a total of 620 different cluster analyses were performed. The results reflected that the data analysis approaches with less dimensions were those that performed the best groups among cities. In this sense, the approaches with a high number of indicators (e.g., the total or the pillar approach) were characterised by misclassifying cities in more than 30% of the indicators. Thus, the most acceptable cluster analysis approach was that with a low number of indicators. Through this approach, it was possible to make correct groups of the sustainability level of the cities of the Algarve. These results provided an appropriate methodology for the decision-making regarding the sustainability of a region and could be extrapolated to other regions to assess sustainability or environmental indicators. Keywords: cluster analysis; sustainability indicators; the Algarve region; OBSERVE platform; sustainable tourism 1. Introduction 1.1. Tourism and Sustainability Indicators The sustainable development of societies is an important issue belonging to the governmental policy to achieve a more sustainable development. In this regard, one of the activities of the cities which significantly influences their sustainability level is their tourist activity [ 1 – 3 ]. This activity similarly influences as other activities do, such as urban, agricultural or maritime [ 4 – 6 ], due to the impact of the tourist activity of cities in their social, economic and environmental dimensions. A positive and clear aspect of such activities is the improvement of employability in a region by generating job positions, although some negative aspects such as the alteration of lifestyles, damage in the heritage or the alteration of ecosystems could be a sign of the unsustainable impact of the activity in a region [ 7 ]. In addition, tourist activities affect the sustainability of other sectors, such as infrastructures [ 8 ]. Without an adequate control, the expansion of the tourism market will increase the pressure on the ecosystems on which the livelihood of local communities depends [ 7 ]. Therefore, tourist offers in regions or cities should be managed in a broader sustainability context [ 9 ], also guaranteeing the economic advancement of such activities [ 10 ]. Improving the sustainability usually also leads to an improvement of competitiveness [11,12]. Sustainability 2020,12, 1536; doi:10.3390/su12041536 www.mdpi.com/journal/sustainability Sustainability 2020,12, 1536 2 of 16 Therefore, it is necessary to control the sustainability of the tourist activity of a region, thus making possible to guarantee that the tourist offer could contribute to the economy of the region without affecting its inhabitants. For this purpose, the monitoring of sustainability indicators allows the evolution of a region to be determined and, in this way, appropriate policies to be established towards a more sustainable tourism [ 13 , 14 ]. Sustainability indicators are understood as those tools contributing to the analysis and assessment of the information so that managers could make right decisions [ 15 ]. As general criterion, sustainability indicators should be quantitative to be assessed [ 16 ]. However, there are not defined criteria about which such indicators should be, thus varying among different research studies [ 17 ]. Some examples are as follows: (i) Miller [ 18 ] established 9 indicators for sustainable tourism through a Delphi survey of tourism researchers; (ii) Liu et al. [ 19 ] considered 20 indicators according to the parties interested: tourists, local residents, governmental agencies, and business owners; (iii) Blancas et al. [ 20 ] used a set of 32 indicators to assess the sustainability of coastal tourist destinations based on 3 dimensions: social, economic, and environmental. In a subsequent study [ 21 ], these authors considered a set of 89 indicators to assess the sustainable tourism based on the same dimensions (social, economic, and environmental); (iv) Nestic ò and Maselli [ 22 ] established a set of 23 indicators for the economic evaluation of tourism projects in the islands; and (v) Castellani and Sala [ 23 ] used 20 indicators concerned with: tourism characteristics of the region under investigation; environmental factors; economic and social conditions of local communities; and demographic dynamics. In any case, the use of a set of indicators will allow the behaviour of different aspects of tourism to be evaluated in the different regions of each study. Through this evaluation, the necessary corrective policies may be established by the government of the region. 1.2. Data Analysis of Sustainability Indicators To establish corrective measures, an essential aspect is the treatment of the compiled information of the monitored indicators [ 24 ]. Some of these analyses are based on spatial distribution through maps of the sustainability of a region. In this regard, H é ly and Antoni [ 25 ] developed a grid analysis map of the Besançon region (France) from which the strengths and weaknesses of the territory can be identified in terms of sustainability. In another similar study, Palmisano et al. [ 26 ] analysed the spatial distribution of a set of rural sustainability indicators to establish a common Rural Sustainable Development strategy to allocate the European Agricultural Fund for Rural Development Budget. Through these indicator analyses, decisions could be established using different methods, such as decision-making matrix [27] and fuzzy logic [28]. However, the analysis of sustainability indicators could be complex when various typologies are analysed. In this sense, many studies have evaluated the possibility of conducting a cluster analysis. A cluster analysis is a multivariant statistical technique which allows a set of objects to be classified in a way that, on the one hand, similar objects are in the same conglomeration and, on the other hand, different objects are in several groups, resulting in various homogeneous groups among them. Thus, this analysis has been used in previous research studies to make groups between indicators: (i) Akande et al. [ 29 ] classified 32 indicators of the Smart level and of sustainability of European cities in 5 components through hierarchical clustering; (ii) Yi et al. [ 30 ] assessed the sustainability of the 17 cities of the area of Shandong (China). For this purpose, they used 21 indicators of environmental, social, and economic dimensions and classified the 17 cities in 4 groups through the average values and growth values; (iii) Dang et al. [ 31 ] applied a classification analysis of the indicators included in the certifications of China’s new Assessment Standard for Green Eco-districts (ASGE) and of Leadership in Energy and Environmental Design for Neighbourhood Development (LEED-ND). To do this, authors used k -means to make groups; and (iv) Neri et al. [ 32 ] conducted cluster analyses in an indicator framework of input-state-output to group 83 countries according to their sustainability. A total of 3 indicators (the emergency flow per capita, the Gini index of income distribution and the Gross Domestic Product per Capita (GDP)) and the k-median algorithm were used for the analysis. However, Sustainability 2020,12, 1536 3 of 16 more research studies analysing new methodologies are required to assess the sustainable tourism with the aim to reduce current existing limitations [ 33 ]. Assessment methodologies of regions are included in such new methodologies. Previous research studies did not analyse the limitations of group techniques when the various monitored indicators are individually analysed. Additionally, there are few research works analysing a broad sample of sustainability indicators. 1.3. Aim of This Study For this reason, this study makes different methodologies to group the cities of a region based on their sustainability indicators. In this way, the strategical decision-making of local governments are contributed to be in accordance with the United Nations Sustainable Development Goals [ 34 ]. This study therefore aims to analyse the most appropriate methodology to make similarity classifications among cities of a region according to sustainability indicators. For this purpose, the case study used is the Algarve region due to the importance of tourism in the region [ 35 ]. The sustainability indicators in the region are monitored through the Observatory of Sustainability of the Algarve Region for Tourism (OBSERVE) platform [ 36 ]. Based on the data compiled by OBSERVE in 105 sustainability indicators between 2011 and 2015, this study conducts different methodological cluster analysis approaches. The results determined the most appropriate methodological approach to group the 105 indicators appropriately and guaranteed consistent classifications of the cities to establish appropriate policies in each indicator. Thus, the results of this study also analysed the possibilities of determining similarities among cities when analysing a large number of sustainability indicators. 2. Methodology 2.1. Area of Study: Algarve Region The Algarve Region is in the south of Portugal and is made up of 16 municipalities whose names come from the capital city of each (see Figure 1). This region has a coast of about 200 km long and is the most important tourist destination in Portugal [ 35 ] (with 43.8% of the total overnight stays [ 37 ]) and in Europe [ 38 ]. This aspect becomes important as the incomes from tourism in Portugal corresponded to 7% of the GDP of the country and 6.3% of employment in the year 2016 [ 39 ]. Sustainability is therefore one of the 10 challenges proposed by the Portuguese tourism policy for the next 10 years [ 40 ]. It is important to note that tourism has evolved continuously since the construction of the Faro Airport in 1965 [ 41 ], thus implying a large number of visitors in the present days. In this sense, the region received around 2.7 million international visitors in 2015 [42]. Sustainability 2020,12, 1536 4 of 16 Sustainability 2020, 12, 1536 4 of 17 Figure 1. The Algarve region and municipalities (capital cities are drawn with a circle). The beaches of the region are the main tourist attraction [35] and traditionally the most valued attribute by tourists [43]. Therefore, this destination attracts both national and international tourists [44,45]. It is also important to highlight the relevance of golf facilities in the Algarve region. In this regard, the region has expanded its range of golf since 1990 [46], thus making this type of tourism one of the best counterweights to the strong seasonality of the region [47]. In addition, there is a wide range of existing accommodation, including luxury hotels and hostels [48]. Consequently, the Algarve is the region of Portugal with the greatest tourist activity. 2.2. OBSERVE Platform Controlling the sustainability level of a region through observatories is essential to guarantee the correct development of urban environments [49]. For this reason, the sustainability of the Algarve region can be measured by the OBSERVE platform [36], whose objective is to monitor various sustainability indicators classified in 4 dimensions (also known as pillars) (see Figure 2): environmental, institutional, economic, and sociocultural. The indicators for each of these pillars were chosen based on the consensus of various public bodies and institutions, such as the Algarve Hotels Association (in Portuguese, Associação dos Hotéis e Empreendimentos Turísticos do Algarve) and the Algarve Regional Coordination and Development Commission (in Portuguese, Comissão de Coordenação e Desenvolvimento Regional do Algarve) [36]. Thanks to meetings and surveys, the indicators to be monitored by OBSERVE were determined [36]. Table 1 summarises the OBSERVE platform's sustainability indicators [36]. Figure 1. The Algarve region and municipalities (capital cities are drawn with a circle). The beaches of the region are the main tourist attraction [ 35 ] and traditionally the most valued attribute by tourists [ 43 ]. Therefore, this destination attracts both national and international tourists [44,45] . It is also important to highlight the relevance of golf facilities in the Algarve region. In this regard, the region has expanded its range of golf since 1990 [ 46 ], thus making this type of tourism one of the best counterweights to the strong seasonality of the region [47]. In addition, there is a wide range of existing accommodation, including luxury hotels and hostels [48]. Consequently, the Algarve is the region of Portugal with the greatest tourist activity. 2.2. OBSERVE Platform Controlling the sustainability level of a region through observatories is essential to guarantee the correct development of urban environments [ 49 ]. For this reason, the sustainability of the Algarve region can be measured by the OBSERVE platform [ 36 ], whose objective is to monitor various sustainability indicators classified in 4 dimensions (also known as pillars) (see Figure 2): environmental, institutional, economic, and sociocultural. The indicators for each of these pillars were chosen based on the consensus of various public bodies and institutions, such as the Algarve Hotels Association (in Portuguese, Associaç ã o dos Hot é is e Empreendimentos Tur í sticos do Algarve) and the Algarve Regional Coordination and Development Commission (in Portuguese, Comiss ã o de Coordenaç ã o e Desenvolvimento Regional do Algarve) [ 36 ]. Thanks to meetings and surveys, the indicators to be monitored by OBSERVE were determined [ 36 ]. Table 1summarises the OBSERVE platform’s sustainability indicators [36]. Sustainability 2020,12, 1536 5 of 16 Sustainability 2020, 12, 1536 5 of 17 Figure 2. The 4 dimensions of the OBSERVE platform’s sustainability indicators. Table 1. The OBSERVE platform's sustainability indicators [36]. Pilla r Subject Area Indicato r Environmental Air quality index Air quality index Energy management Consumption of electric energy by inhabitant Consumption of motor fuel by inhabitant Production of electric energy using renewable sources Environmental management Environmental expenditure of municipalities by 1000 inhabitants Number of blue flags, beaches and marinas Number of bathing waters and quality classes Materials and waste management Percentage of municipal waste prepared for reuse and recycling Urban waste selectively collected per inhabitant Urban waste collected per capita Mobility Number of embarked and disembarked passengers in Faro Airport Number of passengers per kilometre carried by enterprises exploring inland transportation Movement of passengers in inland waterways Number of embarked and disembarked passengers of cruise ships in Portimão Port Charging stations for electric vehicles Average daily traffic on main roads and secondary roads Walking and cycling routes Natural capital management Burnt area Investments on protection of biodiversity and landscapes of municipalities Water cycle management Percentage of safe water Fresh water supplied per inhabitant Wastewater sewerage per capita Institutional Governance and citizenship Abstention rate Percentage of capital expenditure Broadband internet access per 100 inhabitants Figure 2. The 4 dimensions of the OBSERVE platform’s sustainability indicators. Table 1. The OBSERVE platform’s sustainability indicators [36]. Pillar Subject Area Indicator Environmental Air quality index Air quality index Energy management Consumption of electric energy by inhabitant Consumption of motor fuel by inhabitant Production of electric energy using renewable sources Environmental management Environmental expenditure of municipalities by 1000 inhabitants Number of blue flags, beaches and marinas Number of bathing waters and quality classes Materials and waste management Percentage of municipal waste prepared for reuse and recycling Urban waste selectively collected per inhabitant Urban waste collected per capita Mobility Number of embarked and disembarked passengers in Faro Airport Number of passengers per kilometre carried by enterprises exploring inland transportation Movement of passengers in inland waterways Number of embarked and disembarked passengers of cruise ships in Portimão Port Charging stations for electric vehicles Average daily traffic on main roads and secondary roads Walking and cycling routes Natural capital management Burnt area Investments on protection of biodiversity and landscapes of municipalities Water cycle management Percentage of safe water Fresh water supplied per inhabitant Wastewater sewerage per capita Institutional Governance and citizenship Abstention rate Percentage of capital expenditure Broadband internet access per 100 inhabitants Innovation and knowledge Gross expenditure on research and development of institutions and enterprises Gross expenditure on research and development as percentage of gross domestic product Sustainability 2020,12, 1536 6 of 16 Table 1. Cont. Pillar Subject Area Indicator Economic Economic impact Gross value added of enterprises Apparent labour productivity in establishments, food and beverage service activities Inflation Per capita purchasing power Number of establishments and economic activity Persons employed of establishments and economic activity Turnover of establishments and economic activity Relative contribution of establishments, food and beverage service activities to the Algarve Economy (GVA of Enterprises) Job Employment by gender and economic sector Seasonality Seasonal employees Establishments open all year Tourist occupation Lodging capacity in hotel establishments Nights in hotel establishments Revenue per available room (Rev Par) of hotel establishments Average stay in hotel establishments Accessibility Number of accessible beaches Culture Number of cultural properties Municipalities’ expenditures on cultural heritage Sociocultural Demography Annual population balances: natural and migratory Resident population Foreign population with status of resident Education Population education level with 15 and more years Health care Health Care Pressure Lodging capacity in hotel establishments by 1000 inhabitants Tourist intensity Regional tourist density Municipal tourist density Safety Crime rate Number of registered crimes Social cohesion Regional development composite index (cohesion) Beneficiaries of social integration income, of social security per 1000 inhabitants in active age Number of secondary houses per 100 houses 2.3. Group Approaches There is much information compiled by the OBSERVE platform. However, it is necessary to establish appropriate procedures to analyse indicators and, in this way, to establish the most appropriate performance patterns to mitigate possible unsustainable values in some zones of the Algarve. One of the first steps should be the classification of the cities of the region according to the values recorded in each sustainability indicator. For this reason, this study assessed the possibilities to group the 16 cities of the Algarve (see Figure 1). A total of 105 indicators from the OBSERVE platform related to the sustainability of the region were selected. Some of the indicators correspond to developments of a particular indicator. For example, the crime rate indicator was divided into different subcategories, such as crimes against people and crimes against heritage. Likewise, some subject areas included in Table 1were not considered as data of each city were not available (e.g., the subject area of mobility). Additionally, indicators were intended to be analysed in a long temporary period. For this study, the period 2011–2015 was considered as a wide sample of indicators with data for where the research is available. A total of 4 various approaches were used for cluster analyses. Such approaches were based on the structure used by the OBSERVE platform to classify indicators: Total approach (TA), Pillar approach (PA), Subject area approach (SAA), and Indicator approach (IA). These approaches suppose that the dataset used in the cluster analysis has a lower number of variables from left to right, so that the TA groups all indicators (i.e., it corresponds to a multidimensional group), whereas the IA corresponds to individual analysis of each indicator (i.e., it corresponds to a 1D cluster). Sustainability 2020,12, 1536 7 of 16 The analysis was carried out for the period 2011–2015, and the approaches were independently analysed in each year. Table 2includes the cluster analysis per approach and year. The results of this research are based on a total of 620 clusters. Table 2. Cluster analyses carried out in the research. Year Approach TA PA SAA IA 2011 1 4 14 105 2012 1 4 14 105 2013 1 4 14 105 2014 1 4 14 105 2015 1 4 14 105 2.4. k-Means The algorithm k -means was used for cluster analyses. k -means is an iterative clustering algorithm based on the centroid concept of a group of individuals [ 50 ]. The method is based on an X sample of n individuals classified in k groups, for which a W partition of such sample with W=(w1,. . . ,wa,. . . ,wb,. . . ,wk) is considered, thus achieving that the total sum of the sums of squares of the Euclidean distances within each group is minimum: argmin W k X a=1 X xi∈wa p X r=1 (xir −µar)2(1) At the performance level, the k-means algorithm’s steps are as follows: •Step 1: the number of kgroups is identified to carry out the analysis. •Step 2: kindividuals from the dataset are randomly selected, constituting the initial centroids. • Step 3: by using the association measurement chosen, the distance of each individual to each k centroid is calculated. •Step 4: kgroups are created by allocating each individual to the closest centroid. •Step 5: the new centroids of each kgroup are identified. • Step 6: steps 3 and 4 are repeated. This step could lead to two situations: (i) going to step 5 if in step 4 some of the individuals change the group, thus repeating the cycle; and (ii) the cluster analysis process is finished when no individual changes the group in step 4. The method is sensitive to initial centroids, so different results could be given by varying the initial values of k . In this sense, the greater the k used in the algorithm, the lower the variation within groups (i.e., more individual groups are usually created, thus losing the main potential of the analysis: to detect similarity patterns among individuals). If the variables have various units (as in this research), a pre-processing to normalise data should be conducted before the cluster analysis (i.e., the variables are rescaled between 0–1 by using a min-max normalisation). To optimally select the number of clusters, a total of 3 different analyses were used in this research. Such analyses were based on the Elbow method, the silhouette index ( s(i) ), and the ratio between the sum of squares and the total sum of squares (BSS/TSS). The Elbow method consists in selecting the optimal number of k by minimising the total within-cluster sum of squares (WSS) [51]. The Elbow method is made up of 4 phases: 1. k-means is applied for different values of k. Sustainability 2020,12, 1536 8 of 16 2. For each k, WSS is calculated: WSS = K X k=1 X i∈Sk p X j=1 xij −xkj2(2) where Sk is the set of instances grouped in the k -th cluster, and xkj is the j-th variable of the cluster center for the k-th cluster. 3. The WSS curve is plot with respect to the number of kgroups. 4. The location of the elbow in the graphic is generally considered as an indicator of the optimal number of groups (see Figure 3). Sustainability 2020, 12, 1536 8 of 17 The Elbow method consists in selecting the optimal number of k by minimising the total withincluster sum of squares (WSS) [51]. The Elbow method is made up of 4 phases: 1. 𝑘-means is applied for different values of 𝑘. 2. For each 𝑘, WSS is calculated: 𝑊𝑆𝑆 =  𝑥 −𝑥    ∈   (2) where 𝑆 is the set of instances grouped in the k-th cluster, and 𝑥 is the j-th variable of the cluster center for the 𝑘-th cluster. 3. The WSS curve is plot with respect to the number of 𝑘 groups. 4. The location of the elbow in the graphic is generally considered as an indicator of the optimal number of groups (see Figure 3). Figure 3. Example of the total within-cluster sum of squares of each cluster conducted in the study. The elbow of the graphic can be clearly seen [51]. This characteristic especially takes place in cases in which there is a gradual and continuous data transition. For these cases, the method does not provide a unique possible solution, but several possible solutions which should be inspected to determine the best. For this reason, this study combines the Elbow method with two indicators: 𝑠(𝑖) and BSS/TSS. The BSS/TSS ratio is a relation of the cluster compactness. It is a percentage relation, with values between 0 and 100%. The greater the ratio value, the greater the compactness of individuals within a group. Likewise, given that TSS=BSS+WSS, by having a greater BSS, WSS will be lower. The ratio formulation is as follows: 𝐵𝑆𝑆 𝑇𝑆𝑆 =∑∑𝑥   −𝑥       ∑∑𝑥   −𝑥       +∑∑∑𝑥 −𝑥     ∈   (3) where 𝑥 is the grand mean of the means of each group. Finally, 𝑠(𝑖) is among the most used indexes in the cluster analysis [52]. The index shows the similarity of an individual with the rest of the individuals of a same group. So, it measures the quality of a group. For this purpose, the following equation is used: 𝑠(𝑖)=𝑏(𝑖)−𝑎(𝑖) 𝑚𝑎𝑥󰇝𝑎(𝑖),𝑏(𝑖)󰇞 (4) where 𝑎(𝑖) is the average distance between the individual (i) and the rest of points of the same group; and 𝑏(𝑖) is the minimum distance between the individual and the rest of groups. The Figure 3. Example of the total within-cluster sum of squares of each cluster conducted in the study. The elbow of the graphic can be clearly seen [ 51 ]. This characteristic especially takes place in cases in which there is a gradual and continuous data transition. For these cases, the method does not provide a unique possible solution, but several possible solutions which should be inspected to determine the best. For this reason, this study combines the Elbow method with two indicators: s(i) and BSS/TSS. The BSS/TSS ratio is a relation of the cluster compactness. It is a percentage relation, with values between 0 and 100%. The greater the ratio value, the greater the compactness of individuals within a group. Likewise, given that TSS=BSS+WSS, by having a greater BSS, WSS will be lower. The ratio formulation is as follows: BSS TSS =PK k=1Pp j=1xkj −xG2 PK k=1Pp j=1xkj −xG2+PK k=1Pi∈SkPp j=1xij −xkj2(3) where xGis the grand mean of the means of each group. Finally, s(i) is among the most used indexes in the cluster analysis [ 52 ]. The index shows the similarity of an individual with the rest of the individuals of a same group. So, it measures the quality of a group. For this purpose, the following equation is used: s(i)=b(i)−a(i) maxa(i),b(i)(4) where a(i) is the average distance between the individual (i) and the rest of points of the same group; and b(i) is the minimum distance between the individual and the rest of groups. The silhouette index could obtain values between -1 and 1. The meaning of such values determines the suitability of the Sustainability 2020,12, 1536 9 of 16 cluster analysis: (i) if the value is between 0 and 1, the observation is correctly grouped, obtaining optimal values those groups closer to 1; (ii) if the value is 0, the individual is between two groups, thus meaning that either the individual has very different characteristics from the rest which could not be grouped with the others or that the cluster analysis has excessively classified the individual groups; and (iii) if the value is between -1 and 0, the individual is placed in the incorrect group. Figure 4shows an example of one analysis of the silhouette index followed in the research. Sustainability 2020, 12, 1536 9 of 17 silhouette index could obtain values between -1 and 1. The meaning of such values determines the suitability of the cluster analysis: (i) if the value is between 0 and 1, the observation is correctly grouped, obtaining optimal values those groups closer to 1; (ii) if the value is 0, the individual is between two groups, thus meaning that either the individual has very different characteristics from the rest which could not be grouped with the others or that the cluster analysis has excessively classified the individual groups; and (iii) if the value is between -1 and 0, the individual is placed in the incorrect group. Figure 4 shows an example of one analysis of the silhouette index followed in the research. Figure 4. Example of the silhouette index of each cluster conducted in the study: (a) evolution of the average silhouette with variations of the number of groups; and (b) distribution of the silhouette in each town. Thus, the control of this value leads us to know whether individuals are correctly grouped. It is important to stress that, in a multidimensional cluster analysis, the silhouette index obtained is an average of the various dimensions. Although the average silhouette value is high, there are erroneous similarity patterns among the different variables. 3. Results and Discussion Firstly, the optimal number of k for each approach was determined in the cluster analysis. For this purpose, the elbow method and the analysis of 𝑠(𝑖) and of BSS/TSS were used. Through this assessment, the optimal number of k was determined in the 620 clusters conducted in the research. After determining the optimal number of k for each approach, the statistical parameters obtained from 𝑠(𝑖) and BSS/TSS were analysed to assess the most appropriate approach for the cluster analysis of the Algarve region’s sustainability indicators. Figure 5 includes the distributions of BSS/TSS obtained among the different approaches in the 5 years analysed with violin-plots. Violinplots are an evolution of box-plots by including information of the kernel density and rotating them to both sides of the box [53]. As can be seen, the values of BSS/TSS obtained were high as all groups obtained rations greater than 70% due to the process of optimal selection of k followed in the research. However, the use of approaches with a lower number of dimensions in the cluster analysis increases the BSS/TSS ratio. In this regard, the use of the 1D approach of indicators allowed average values of BSS/TSS greater than 94% to be obtained, with an increase with respect to the TA between 3.75 and 12.84% in all the years analysed (see Table 3). This approach to reduce the dimensions of the cluster analysis was the only approach obtaining better results in all years, as in the PA and SAA, the behaviour was different depending on the year: (i) in the PA, better results were obtained in 2013 and 2014, whereas in the other years, the BSS/TSS ratio was lower than that of the TA; and (ii) in the SAA, Figure 4. Example of the silhouette index of each cluster conducted in the study: ( a ) evolution of the average silhouette with variations of the number of groups; and ( b ) distribution of the silhouette in each town. Thus, the control of this value leads us to know whether individuals are correctly grouped. It is important to stress that, in a multidimensional cluster analysis, the silhouette index obtained is an average of the various dimensions. Although the average silhouette value is high, there are erroneous similarity patterns among the different variables. 3. Results and Discussion Firstly, the optimal number of k for each approach was determined in the cluster analysis. For this purpose, the elbow method and the analysis of s(i) and of BSS/TSS were used. Through this assessment, the optimal number of k was determined in the 620 clusters conducted in the research. After determining the optimal number of k for each approach, the statistical parameters obtained from s(i) and BSS/TSS were analysed to assess the most appropriate approach for the cluster analysis of the Algarve region’s sustainability indicators. Figure 5includes the distributions of BSS/TSS obtained among the different approaches in the 5 years analysed with violin-plots. Violin-plots are an evolution of box-plots by including information of the kernel density and rotating them to both sides of the box [ 53 ]. As can be seen, the values of BSS/TSS obtained were high as all groups obtained rations greater than 70% due to the process of optimal selection of k followed in the research. However, the use of approaches with a lower number of dimensions in the cluster analysis increases the BSS/TSS ratio. In this regard, the use of the 1D approach of indicators allowed average values of BSS/TSS greater than 94% to be obtained, with an increase with respect to the TA between 3.75 and 12.84% in all the years analysed (see Table 3). This approach to reduce the dimensions of the cluster analysis was the only approach obtaining better results in all years, as in the PA and SAA, the behaviour was different depending on the year: (i) in the PA, better results were obtained in 2013 and 2014, whereas in the other years, the BSS/TSS ratio was lower than that of the TA; and (ii) in the SAA, better results were Sustainability 2020,12, 1536 16 of 16 31. 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