An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic

water Article An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic Jan Rucka 1, Jan Holesovsky 2, Tomas Suchacek 1,* and Ladislav Tuhovcak 1 1Institute of Municipal Water Management, Faculty of Civil Engineering, Brno University of Technology, Zizkova 17, 602 00 Brno, Czech Republic; [email protected] (J.R.); [email protected].cz (L.T.) 2Institute of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Zizkova 17, 602 00 Brno, Czech Republic; holesovsky[email protected].cz *Correspondence: [email protected].cz; Tel.: +42-060-533-4365 Received: 18 December 2017; Accepted: 30 March 2018; Published: 4 April 2018   Abstract: Pressure management is the basic step of reducing water losses from water supply systems (WSSs). The reduction of direct water losses is reliably achieved by reducing pressure in the WSSs. There is also a slight decrease in water consumption in connected properties. Nevertheless, consumption is also affected by other factors, the quantification of which is not trivial. However, there is still a lack of much relevant information to enter into this analysis and subsequent decision making. This article focuses on water consumption and its prediction, using regression models designed for an experiment regarding an administrative building in the Czech Republic (CZ). The variables considered are pressure and climatological factors (temperature and humidity). The effects of these variables on the consumption are separately evaluated, subsequently multidimensional models are discussed with the common inclusion of selected combinations of predictors. Separate evaluation results in a value of the N 3 coefficient, according to the FAVAD concept used for prediction of changes in water consumption related to pressure. The statistical inference is based on the maximum likelihood method. The proposed regression models are tested to evaluate their suitability, particularly, the models are compared using a cross-validation procedure. The significance tests for parameters and model reduction are based on asymptotic properties of the likelihood ratio statistics. Pressure is confirmed in each regression model as a significant variable. Keywords: water consumption; pressure; regression; analysis of variance; maximum likelihood 1. Introduction The volume of water consumed per unit of time is the operating parameter of each water supply system. Great attention is currently paid to optimizing the pressure conditions in water supply systems. However, a major part of the optimizing approaches is dedicated to reduction of both water loss and energy cost [ 1 , 2 ]. The first step in reducing water losses is usually the revision of pressure conditions and their subsequent adjustment, enabling the optimum scope of pressure conditions while maintaining the hydraulic capacity of the mains. In most cases, pressures in the network are decreased to achieve reduced water loss and lower failure rates of the mains. Yet, evidently, with lowered pressure there is also a decrease in the volume of water billed to the individual consumers. Therefore, lowering the pressure in the network has a positive effect on water loss while having a negative effect on the volume of billed water and the economic performance of the company. In our experience in regard to this solution of optimization, waterworks operators stress the following criteria, listed in a decreasing order according to priority: (1) the economic aspect—the problem of water loss is solved strictly from an economic viewpoint. The effectiveness of each proposed measure is Water 2018,10, 424; doi:10.3390/w10040424 www.mdpi.com/journal/water Water 2018,10, 424 2 of 20 always calculated from the economic perspective of the company. Measures with negative economic effects are only implemented under the condition that they generate a significant secondary benefit other than economic (for example, environmental); (2) the environmental viewpoint—in cases where it is a relevant problem within the given situation, the capacity of the water source is considered. Reduction of water loss presents a decreased burdening of the source and the ecosystem as a whole; (3) quality of delivered water—reduction of water loss causes a decreased consumption of water, increasing the age of water in the network and, in specific cases, this may result in a higher frequency of complaints regarding its quality. This point of view is always considered individually with regard to the specific situation; (4) company prestige—last in terms of priorities, yet still a relevant criterion. Performance indicators of water companies are annually published in a country-wide professional almanac. Achieving outstanding results in the category of water loss is a matter of prestige. In order for such water consumption to be included as a criterion in the optimization of pressure conditions, it is necessary to express it mathematically as a function of pressure. This mathematical description complicates its stochastic nature, as is apparent, for example, from [ 3 ]. It has been proven that many more factors aside from pressure conditions have an effect on water consumption. Several previous studies have shown that water consumption is affected not only by customer price [ 4 – 6 ], but also by climate factors [ 6 – 8 ] and pressure in the water mains [ 2 , 5 , 9 , 10 ]. For example, in [ 11 ] an analysis of the dependency of water produced for the city of Brno on meteorological factors was conducted and a certain dependency was indicated. This analysis was carried out for the entire water supply system. On the other hand, it should be noted that for each type of user the dependency on factors influencing consumption always varies (see e.g., [ 12 ]). According to [ 13 ] water consumption is divided for the purposes of simulating the changes in water consumption with pressure changes into “inside-the-house” and “outside-the-house”. Both areas of consumption then have different coefficients expressing their dependency on pressure. Subsequently, the average coefficient expressing dependency of pressure for the entire building and for each consumer is calculated. Subsequently, this cumulative coefficient per user is implemented into the FAVAD (Fixed and Variable Discharge) equation according to [14], or into its simplified form according to [13]. Nowadays, there are not many N 3 set coefficients within the meaning of [ 13 ] based on real studies. For example the value of the “inside-the-house” consumption coefficient was set at 0.2 for the Johannesburg student campus at the University of Johannesburg in [ 15 ]. Also worth mentioning is the principle of minimum pressure, which must be ensured in the water supply network so that the water supply can be realized. If this value is under-stepped, the volume of water supplied is considered zero [ 9 ]. In this study, the results of which are presented here, this limit was defined by valid legislation at 0.15 MPa and was not broken during the experiment. This paper provides information regarding the effect of water pressure and selected climatological factors on water consumption in an administrative building. It had three goals. The first was to establish the value of the N 3 coefficient used in the FAVAD concept. In the past, a sufficient number of studies used the FAVAD concept, covering the entire spectrum of various types of end users. A new regressive model was not created for this purpose, but an existing model used according to [13]. The second goal was to verify whether it is possible to predict water consumption under stable pressure conditions (albeit with various levels of pressure) on the basis of climatological factors, using linear regression. The third goal was to extend the FAVAD concept into the area of water consumption prediction, according to [ 13 ] and stipulate a regressive model with a newly added variable of climatological factor. All regression models were subject to statistical testing to confirm or reject their significance. Water 2018,10, 424 3 of 20 2. Materials and Methods 2.1. Facility Details, Measuring Campaign and Measuring Equipment 2.1.1. Facility Details The case study was conducted on an office-type building, which is a very common kind of office facility in the CZ. The building under study has three above-ground floors and the maximum number of workers is 35. The detailed layout of the building with distribution of workers within the floors is well apparent from Figure A1. The workers are evenly allocated on all floors, each of them equipped with similar fittings. The toilets are fitted exclusively with volumetric flushing systems with a pre-stored storage water tank. There are no pressure flushers in the building. There are also no showers. The survey determined that, in the meaning of [ 13 ], all measured water was consumed “inside the house”. Characteristic flow rates for this building are shown in Table A1. Working hours of one five day working week had three different periods. The period of working hours in the individual days for Mondays and Wednesdays is from 7:00–17:00, on Tuesdays and Thursdays 7:00–15:00 and on Friday 7:00–13:30. The working day lengths for individual days remained the same over the course of the entire duration of the measuring campaign. 2.1.2. Measuring Campaign The main part of the measuring campaign study was divided into several shorter periods of time corresponding to the measurement cycles. Within one measuring cycle, the pressure reducing valve (PRV) setting was set constant. The length of the measurement cycle was approximately 14 days (i.e., 10 working and 4 weekend days), with possible fluctuations due to specific circumstances such as public holidays etc. The length of the cycle was determined with respect to allowing very low or very high temperatures or humidity to occur in multiple cycles, as these extremes occurred in the shorter term. Because it was not possible to change the pressure conditions in the entire pressure zone of the water network, the measuring devices were attached to the water hook-up in the immediate vicinity of the water mains. These devices included (in the direction of the water flow) flow-rate measurement, pressure reducing valve “PRV”, and pressure sensor (see Section 2.1.3). The diagram of the connection of these measuring devices is apparent from Figure A2. The installation in the immediate vicinity of the water mains ensured analogous conditions to a change of pressure conditions in the entire network. The output pressure of PRV was determined individually for each measuring cycle. The value of the output pressure was changed in leaps. The limiting conditions to the minimum output pressure of the PRV valve were taken into account, in order to remain consistent with the legislation of the CZ, e.g., to ensure minimal pressure or normal operation in the building. According to [ 16 ], the minimal pressure in the hook-up in the CZ is 0.15 MPa, however due to the complaints by workers it was not possible to reduce the pressure to this value. There was an attempt to set the output pressure in a manner that the average pressure during working hours was in the middle interval in the given category. However, this was not always achieved, particularly due to unplanned interventions with the measuring devices. 2.1.3. Measuring Device The following devices were used for measuring: • Change of pressure conditions—a spring-based PRV with the output pressure range of 0.15–0.60 MPa was used. The hydraulic losses caused by the PRV were low even at the maximum hourly flow rate. PRV dimension was chosen with respect to the hydraulic losses and characteristic flow rates through the PRV. The characteristic flow rates in the given building are presented in Table A1 and the PRV head loss diagram is shown in Figure A3. • Flow volume measurement—The volume of water flowing was measured using a water meter with a pulse generator and the pulse value of 1 liter. The water meter corresponded to the “C” Water 2018,10, 424 4 of 20 level of precision in the sense of [ 17 ]. Nominal flow rate of the water meter in the sense of [ 17 ] is 2.5 m 3 h −1 . In order to sustain the guaranteed precision, undisturbed spacing lengths were maintained both upstream and downstream of the water meter. The value of the pressure was recorded along with the water meter value every 15 s. This was an interrupted measurement with a relatively short time period. • Pressure measurement—The pressure was measured with an integrated pressure sensor with a range of 0.0–1.0 MPa, and measurement accuracy of 0.25% of the range (i.e., 0.0025 MPa). 2.2. Data and Its Verification The data set consists of daily observations of the total volume of water consumed and the pressure on connection in the period from September 2016 to September 2017. Aside from the variables of interest to be further embedded into a model (see Section 2.3), the observations were also accompanied by other metadata. Those included the actual number of employees at the workplace, their working time (varying daily between 6.5 and 10 h, see Section 2.1.1), and indication of extraordinary events that could significantly affect water consumption. The climatological covariates, i.e., temperature and humidity, were obtained from the Czech Hydrometeorological Institute in the form of a series with resolutions of one hour. The relative daily values of the meteorological covariates were observed by averaging measurements corresponding to the working time during the current day. First, in order to compare the water consumption under specific day conditions, the normalized consumption was determined with respect to the actual number of employees and the working-time duration. The total consumption may be affected by fluctuation of employees during each particular shift. However, an initial survey found that employees spent the vast majority of their working time in the office. On account of that, a simplifying assumption of constant number of workers was made. Thus, for a given i -th day the normalized water consumption Ci ( l·person−1·hour−1 ) was evaluated as follows Ci=Vi pihi , (1) where Vi denotes the total water consumption, pi is the number of employees, and hi is the number of working hours during the i -th day, respectively. This standardized quantity is considered to be independent of current circumstances in the office. Since any further analysis is based purely on this variable, from here on we refer to normalized consumption simply as consumption. As the observation period lasted one year, usually certain long-term impacts had to be taken into account, i.e., an inter-annual trend or cyclic components commonly related to a period of the year. In the CZ there has been a downward trend in both total water consumption and water consumption per capita since 1989. Nevertheless, as discussed e.g., in [ 18 , 19 ], in the last three years the total consumption for the CZ is considered rather stable from a global perspective, particularly in the Zlin region (in which this case study was performed). As is apparent from Figure A4, consumption for the last three years is practically stable for the “other users” group to which this case study belongs. The overall annual consumption in the given administrative building in the last three years only differs in the range of tens of a percent (239.6 m 3 –238.3 m 3 –241.1 m 3 ), while the maximum number of workers per shift continued to be the same. In Figure 1the time dependency of the consumption is visualized in the period of interest. As is evident from Figure 1, as well as from the plot of first differences (not shown) there is no significant trend to be embedded into a model. Hereby we follow the results obtained in [ 18 , 19 ] and put no specific emphasis on any trend component. Moreover, especially due to focus of the study on an office building, the data exhibits no seasonal effects. This agrees with the authors’ experiences within similar facilities in the area of CZ. All consumption during the working hours is “inside-the-house” in the sense of [13] and no irrigation or gardening is performed. In general, the latter two usually represent the main part of the seasonal increments in consumption. Hence, the mean water consumption is considered to remain stable over the year. Water 2018,10, 424 5 of 20 In order to exclude unreliable measurements observed under extraordinary conditions, the metadata was closely investigated. The outliers could seriously harm any further analysis and lead to significant bias in estimates of the dependency structure. There were several types of uncommon events identified. Particularly, from the data set all days labelled as anomalous were excluded. This includes mostly situations where significant volumes of water were used for other purposes than the routine needs of the office (see e.g., the peak at the end of year 2016 in Figure 1). Further, we removed those days in which the number of workers dropped below 10% of the office capacity, i.e., typically the holiday season. Moreover, data have been excluded from the days with uncommon behavior of the consumption aggregation monitored during the day. This typically indicates a failure of the measuring equipment. Altogether about 9% of available data was discarded, whereby we assumed the accuracy of any statistical inference should not be violated. A closer description of the excluded data is shown in Table A2. Figure 1. Time series of observed consumption; only working days plotted. Values labelled as invalid are highlighted by red circles. By a stability assessment of the designed regression models (discussed below in Section 2.3), we identified significantly outlying observation at 10 January 2017 (the first invalid observation of the year 2017 in Figure 1). Although no extraordinary event was indicated in the metadata, this observation was additionally excluded. This decision is based both on visual inspection of the neighboring measurements in the consumption series, as well as on statistical criterion. For both models presented in Section 2.3, the corresponding standardized residual is more than three standard deviations from the residual mean. Moreover, the significance of the outlier was also indicated by Grubbs’ test [20] at significance level 0.05. Details are more closely discussed in Section 3. 2.3. Statistical Inference for Pressure The main object of study is the assessment of dependence relations between consumption and monitored covariates, i.e., primarily the water pressure and climatological factors. The analysis was performed in two stages. First, the influence of the pressure was studied separately. From the practical point of view, unlike the meteorological inputs, the water pressure is the only operational variable that can be managed. Second, we considered all the factors and built a model for evaluation of their contribution to water consumption. With respect to the experiment setup, the pressure observations were categorized as follows: category A (less than 0.25 MPa); category B (0.25 to 0.30 MPa); category C (0.30 to 0.35 MPa); category D (0.35 to 0.40 MPa) and category E (over 0.40 MPa). The changes in measurement cycles over time are shown in Figure 2. Despite some variation of the pressure within a measurement cycle, the above mentioned categorization should preserve the classification of the PRV settings. Due to measurement accuracy discussed in Section 2.1.3, the observations are unlikely to overleap between diverse classes. Water 2018,10, 424 6 of 20 Figure 2. Time series of average pressure and its categorization (only valid observations plotted). The statistical inference is based both on a linear model and its derivations, as well as nonlinear analysis. Initially, one-way analysis of variance (ANOVA) was performed to assess the significance of the dependency between the observed consumption and the water pressure categories. The necessary preconditions to ANOVA were verified using the Bartlett and Levene test for variance equality, the normality of the data was checked by goodness-of-fit test [ 21 ] (Pearson’s χ2 , Lilliefors, and Anderson-Darling (A-D) test was applied, respectively). To evaluate the effect of changes in pressure, a regression curve was fitted to the data by the means of the Least Squares (LS) method. For this purpose we consider in the first case a simple linear model of the form Ci=β0+β1Pi. (2) where Ci denotes consumption in the meaning of (1) obtained under the pressure Pi ( i= 1, · · · , n ; n is the number of observations). The model (2) is particularly meant as a possible local approximation of the relation between consumption and pressure. It is used primarily for performance comparison with a non-linear model discussed below. The regression parameter estimates are obtained as a common solution to the system of normal equations [22]. Usually, a non-linear dependence between consumption and pressure is considered. Hereby we follow the study [11] in which the authors introduced a relationship in the following form: Ci=C0Pi P0N3 , (3) where C0 and P0 denote the consumption and the pressure corresponding to the highest pressure category E, respectively. A curve similar to (3) was also applied in [ 11 ]. An advantage of the relation (3) stands in the simple interpretation of the coefficient N 3 as a measure of the pressure change to the water consumption. In our case, the coefficients N 3 and C 0 were considered as unknown regression parameters to be estimated. The pressure P 0 is assumed fixed, equal to the average of category-E pressures. The estimates of the regression parameters are obtained by the Maximum Likelihood (ML) method. Hence we obtained the values Ci=C0·piN3+εi, (4) where pi=Pi/P0 is the normalized pressure, and εi are independent and normally identically distributed random variables following the distribution N0, σ2 , where σ2> 0 is unknown. Given the observation pi , the conditional density of Ci is again normal with the expected value C0·piN3 and variance σ2 . Thus, it can be derived that the logarithmic likelihood function lC0,N3,σ2 takes the form lC0,N3,σ2=−n 2log(2π)−n 2log σ2−1 2σ2 n ∑ i=1Ci−C0piN32. (5) Water 2018,10, 424 7 of 20 The ML estimates ˆ C0 , ˆ N3 , ˆ σ2 of the parameters C0 , N3 , and σ2 are obtained in order to maximize the relation (5). Note that ML estimates for the model (2) are the ordinary LS estimates. The use of the ML method, in general, gains a particular advantage in native estimation under the assumption of the non-linear model (4). The estimates follow good asymptotic properties and are suitable for further extensions of the model addressed later. The significance test and confidence interval estimation for the parameters are based on likelihood ratio (LR) statistics LR =2hl(ˆ C0,ˆ N3,ˆ σ2)−l(C0 ∗,N3 ∗,σ2∗)i, (6) where the starred variables denote fixed values of the corresponding parameters to be tested against. Testing of a particularly selected parameter is accomplished with the LR test with nuisance parameters [ 22 ]. The LR statistics follows, under some regularity conditions, asymptotically χ2 distribution with degrees of freedom equal to the number of target parameters (or to the number of parameters if no nuisance parameters are present) [ 22 ]. Confidence boundaries for the parameter estimates are determined from the profile likelihood function [ 23 – 25 ]. In a rough description, for a given significance level α , the values of the particular parameter are determined for which the LR test does not indicates significant deviations. Hence, the intersections between the logarithmic likelihood function need to be identified (in which the remaining parameters are held fixed and treated as nuisance parameters), and the threshold hl(ˆ C0,ˆ N3,ˆ σ2)−χ2 1(1−α)i , where χ2 1(1−α) denotes the (1−α)quantile of the χ2distribution with one degree of freedom. In order to assess the performance and verification of both models (2) and (3), a k -fold cross validation was applied [ 26 – 28 ]. In order to assess the regression performance and verification a k-fold, cross validation was applied. Hence, the whole data set was partitioned into k sub-samples comparable in size. In kruns, the regression curve was evaluated using one of the k sub-samples as a testing set, while the remaining sub-samples served as a training set to fit the regression. Namely k= 10 was chosen with respect to available data size. This choice has commonly been made for similar purposes. Given a regression curve estimated from the training set, as the performance criterion is taken the mean square error (MSE) of the testing subsample. 2.4. Statistical Inference for Climatological Factors In the following, the contributions of selected climatological factors to the change in consumption are investigated. For the study, a series of daily averages of temperature and humidity are available. A procedure of variable addition into a model was discussed in [ 28 ]. Hereby the authors apply the forward selection of predictors with respect to correlation-based relations to the explained variable. In our case, however, it could be expected that the two climatological variables are physically linked to each other. If it were so, it would be inappropriate to include both predictors simultaneously since such a mis-specified model may be harmed by a large bias in parameter estimates. Due to verification of such an assumption, attention was paid to the characteristics of possible correlation between them and the pressure. The pairwise Spearman’s rank correlation coefficients were evaluated and this was followed by their significance test. As discussed further in Section 3.3, both climatological covariates are inconsiderably related. The problem of inclusion of redundant covariates has been well discussed, for example, in the study [ 29 ]. Here the authors consider an evaluation framework for determination if a variable should be embedded to a model with respect to its explanation capability. This is especially suitable if large sets of variables are taken into account. In the case of our study, particularly because of evident relation between both climatological covariates, the inference is limited only to the instances, where the contribution of both factors is evaluated separately. This was done in two ways. First we evaluated the influence of a selected covariate on consumption separately corresponding to particular categories of pressure. Here we assume a simple linear regression model of the form equal to (2) with the pressure Pi replaced with other predictors Water 2018,10, 424 8 of 20 of interest, i.e., either the temperature or the humidity. Inferences concerning the fit and estimated regression parameters are based on the usual properties of the linear model. Subsequently we considered two extensions of the model (3) with embedded climatological covariates. Formally, we assume the relation Ci=C0Pi P0N3 +βTi, (7) where C0 , N3 , and β are regression parameters to be estimated. The variable Ti corresponds either to temperature or humidity, respectively, observed simultaneously with the consumption Ci. The estimates of the parameters above are obtained again by the ML method. Following the assumptions of the model (4), the corresponding logarithmic likelihood function can be obtained in a form very similar to (5); the summands Ci−C0piN32 in the last term are merely replaced by Ci−C0piN3−βTi2 . The use of the ML method is particularly suitable for hypotheses testing. We should be particularly interested in the determination of sub-model significance, i.e., whether the model (7) can eventually be reduced to (3). This is provided by the LR test where all the parameters from the model (4) are considered as nuisance parameters and the newly included parameter β is tested for significance against a zero value. 3. Results 3.1. Outlier Identification On account of available metadata, obviously invalid measurements were initially removed from the data indicating records of extraordinary events. A first approach to the reduced observations was made by a direct fit of the regression curve (3) to consumption measurements. Here we omit the suitability discussion of the curve and this will be discussed later in Section 3.2. In Figure 3 the standardized residuals of the fit are visualized along with their mean and confidence bounds of half-width equal to three standard deviations estimated from the residuals. Figure 3. Standardized residuals of the fit by regression curve (3). Horizontal lines show the residual mean and interval of half-width equal to 3 standard deviations. Isolated observations beyond these bounds on the right indicate the presence of an outlier. The behavior of the series in Figure 1(at corresponding date 10 January 2017) agrees with this conclusion. The neighboring values report no significant difference from the remaining flow of the process. In order to achieve other than only visual arguments, the one-sided Grubbs’ test for outliers was applied. As expected, at a significance level of 0.05 the observation was determined as a significant Water 2018,10, 424 9 of 20 outlier, with the Grubbs’ statistics equal to 4.81 (i.e., the value is almost five standard deviations distant from the residual mean). The presence of an outlier in the series could seriously corrupt any further statistical inference leading to bias. Because of the presented results the corresponding observation was additionally removed from the data set and thus will not be taken into account in any further considerations. 3.2. Dependency of Consumption on Pressure The remaining n= 224 observations of pressure were categorized relative to the description in Section 2.2. Figure 4illustrates the frequencies of the particular categories. As a consequence of measurement design, the categories are rather uniformly represented in terms of the frequencies. The boxplots of consumption are also visualized corresponding to specific pressure categories. The whiskers show, as is common, the 1.5 inter-quantile range. Figure 4. Histogram of frequencies in pressure categories; ( a ) boxplots of consumption observations within the categories; (b) whiskers show 1.5 inter-quantile range. In the first step a one-way ANOVA was performed. Primarily, the necessary preconditions need to be verified. The normality of consumption within each category is checked using goodness-of-fit tests. Particularly the Pearson’s χ2 test, Lilliefors test (generalization of Kolmogorov-Smirnov test for not completely specified distributions), and the Anderson-Darling test were applied. The latter, especially, is often used in hydrology [30,31]. The determined p-values are summarized in Table 1. At a significance level of 0.05 the assumption of normality is not rejected in all cases except the A category. By visual inspection of the data it should be concluded that the histogram of the observations in category A is rather symmetric, with no evident deviation from a normal distribution. The reason for rejecting normality can probably be found in two slightly outlying values (although no other outliers were identified by the Grubbs’ test) visible on the boxplot in Figure 4. These indicate a heavier right tail of the distribution. Although the results of one-way ANOVA should not be significantly harmed by such violation, we later also present results obtained by non-parametric methods. Table 1. Determined p-values of the goodness-of-fit tests. Statistical Criterion Category of Pressure A B C D E χ2test 0.0084 0.5049 0.6747 0.4959 0.3735 Lilliefors test 0.0018 0.2961 0.4133 0.5000 0.3215 A-D test 0.0011 0.1239 0.1991 0.4207 0.3382 Water 2018,10, 424 16 of 20 Figure A1. Layout of the selected administrative building (a) 1st floor and (b) 2nd and 3rd floor. Water 2018,10, 424 17 of 20 Table A2. Days excluded from the statistical analysis of data sets. No. Date Day of Week Problem Description Explanation of Problem 1 26.09.2016 Monday Zero consumption during 8–11 Technical problems with measuring device 2 03.10.2016 Monday Unknown consumption during 9–11 3 16.11.2016 Wednesday Very low consumption per person Failure of water supply—water mains breakage—failures duration 6 h 4 21.12.2016 Wednesday Very high consumption per person Very low number of workers—Christmas holidays + 7 visitors over the course of approximately 4 h 5 22.12.2016 Thursday Very low number of workers—Christmas holidays + 15 visitors over the course of approximately 2 h 6 27.12.2016 Tuesday Very low number of workers—only accountants present + creation of an ice rink 7 28.12.2016 Wednesday 8 29.12.2016 Thursday Very low number of workers—only accountants present + some workers outside the evidence 9 30.12.2016 Friday 10 10.01.2017 Tuesday Excluded on the basis of Grubbs’ test results 11 26.01.2017 Thursday Low consumption per person 8 left the building for an unknown number of hours 12 20.04.2017 Thursday 21 workers left the building for 5 h 13 10.07.2017 Monday Negligible consumption per person Very low number of workers—summer holidays—only workers in the reception—smaller number of workers in the complex than in evidence 14 11.07.2017 Tuesday 15 12.07.2017 Wednesday 16 13.07.2017 Thursday 17 14.07.2017 Friday 18 17.07.2017 Monday 19 18.07.2017 Tuesday 20 19.07.2017 Wednesday 21 20.07.2017 Thursday 22 30.08.2017 Wednesday Very low consumption per person 19 workers left the building for 3.5 h 23 31.08.2017 Thursday 23 workers left the building for 4.5 h Figure A2. 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