NEURO-FUZZY NETWORK MODELING OF DATA CENTRE ENERGY PERFORMANCE

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1 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance CHAPTER 6 NEURO-FUZZY NETWORK MODELING OF DATA CENTRE ENERGY PERFORMANCE 6.1 Introduction An exploratory study of the whole-system based neuro-fuzzy network modeling for data centre energy performance is conducted as part of the efforts to understand an empirical energy use pattern of data centres in Singapore. The models developed integrates energy consuming constituents contributing to the energy consumptions of data centres, and analyzes various factors such as physical parameters of space and environment control facilities, design and operation strategies of HVAC, UPS, IT equipment and lighting systems. Neuro-fuzzy network presents itself a desired and applicable candidate of modeling energy performance of complex systems by taking advantages of mimicking human decision and self-training by existing data sets. A well structured and trained neuro-fuzzy model can serve as evaluation tool kits at both design and post-diagnostic stages of data centres. Energy saving potential can also be estimated quantitatively. FuzzyTech 5.5 is employed as the modeling software in this study. In this study, the neuro-fuzzy network models are established based on the energy performance benchmarking of the six data centres studied. Due to the small data base, these models cannot provide highly accurate evaluation of data centre energy performance and prediction of data centres energy saving potential. However, they can serve as efficient and effective tools, prior to the further and detailed measurement and verification (M&V) or energy simulation for the evaluation of data centre energy 146

2 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance performance and estimation of energy saving potential. 6.2 Application of neuro-fuzzy network in modeling data centre energy performance Fuzzy logic is a technology that enhances model-based system design by using both intuition and engineering heuristics. This provides an efficient way of designing, optimizing, and maintaining highly complex systems and results in robust and fault tolerant systems. The use of linguistic modeling, instead of mathematical modeling, greatly enhances system transparency and modification potential. It leads to quick development cycles, easy programming and accurate control. Fuzzy logic principle is especially advantageous for problems that cannot be easily represented by mathematical modeling because data is either unavailable, incomplete, or the process is too complex to use a traditional approach [FuzzyTech, 2001]. There are a number of reasons for neuro-fuzzy network to be employed as modeling approach in this study. Firstly, data centres energy consumption is a complex system. A typical data centre contains various energy consuming systems including IT equipment, UPS, HVAC system, lighting, office equipment, building control system, fire suppression system, current transformer and stand-by power generator. The interactions among energy consuming systems also make this system more complex. Energy performance of data centres is also affected significantly by physical variables such as space usage, indoor temperature, humidity, as well as the design and operation strategies. While it is possible to develop a deterministic--namely precise mathematical or empirical--model, the complexity of the data centre energy performance makes it 147

3 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance difficult to be modeled using conventional modeling approach such as multiple regression. Energy simulation, the popular tool of modeling building system and predicting energy consumption, is another possible solution for modeling data centre. However, simulation is highly time and cost consuming. By comparison, neuro-fuzzy network with the capability of modeling complex system can be a suitable solution in this study. Secondly, the term energy performance is by nature a fuzzy concept. It not only associates with the energy consumption and power demand, but also the operational energy efficiencies of various systems and total data centre. A precise definition of energy performance does not exist. Traditional indicators such as annual energy consumption (kwh/m 2 /year) and power demand density (W/m 2 ) per unit gross floor area are not sufficient for describing data centres energy performance. For example, the total data centre power demand densities of data centre 4 and data centre 6 are 348 W/m 2 and 231 W/m 2 respectively. This does not mean that data centre 6 is more energy efficient than data centre 4, as data centre 6 may have a sparse equipment distribution over a large area, and also resulting in significant amount of energy spent on space cooling. Besides, it is found that the UPS system in data centre 6 is the least efficient among all the data centres studied. It is evident that energy efficiencies of HVAC and UPS systems, area occupancy, over-designing of the supporting systems, designed power demands, and environmental setting put significant impact on data centres energy performance. For the reasons above, fuzzy logic which mimics human logic by combining various parameters or rules for making description or decision is suitable for the application of modeling data centre energy performance. 148

4 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Thirdly, classification of energy performance involves the usage of fuzzy concept. For example, assuming the actual total data centre power density is categorized into 3 classes, which are less than 300 /m 2, 300 W/m 2 to 410 W/m 2 and higher than 410 W/m 2 respectively, a data centre of 301 W/m 2 will be classified under Class 2, and a data centre of 299 W/m 2 will be classified under Class 1. This could be an unreasonable classification since the difference of power demand between these two data centres is very slight. This reflects the fact that the defined power demand range of each class should not be a crisp range. There should be a buffer area between each class. In this case, fuzzy logic is particularly suited for addressing this issue. 6.3 Knowledge background of neuro-fuzzy network Structure of fuzzy logic model A typical fuzzy logic based approach involves three modules: a fuzzifier, a fuzzy interference engine, and a defuzzifier as illustrated in Figure 6.1. Input linguistic variables Fuzzy inference Output linguistic variables Linguistic Level Technical Level Fuzzification Input real variables Defuzzification Output real variables Figure 6.1 Typical structure of rule-based fuzzy logic model (Source: FuzzyTech, 2001; Gouda et al., 2001; Krarti, 2003) A fuzzifier is used to map input vectors into fuzzy sets defined by linguistic variables 149

5 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance such as Bad, Average and Good and membership functions. The inference engine associates fuzzy sets to other fuzzy sets using a rule library of IF-THEN fuzzy rules that highly intuitive and easily understood by human users. Finally the defuzzifier is used to map fuzzy sets to output vectors. Further information of fuzzy logic is provided following in section Principles of neural network For the purpose of enhancing fuzzy logic systems with learning capacities, neural network (NN) is introduced. The objective of a neural net is to process information in a manner that has been previously trained into the net. Training uses either sample data sets of inputs and corresponding outputs, or an expert who rates the performance of the neural net. A neural network can be any model in which the output variables are computed from the input variables by compositions of basic function or connections. However, one of the most commonly used neural network models is the multilayer perception [Krarti, 1998]. Figure 6.2 shows a schematic diagram of the structure of a neural network. Figure 6.2 Neural network schematic diagram (Source: Krarti, 2003) 150

6 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance The first and last layers of neurons are called input and output layers; between them are one or more hidden layers. The neuron depicted by the small circles is the fundamental building block of a network. Each element of the input set, INPUT i, is multiplied by a weight, W ij, and the products are summed to provide the output at the neuron, OUTPUT j : OUTPUT j = Σ i INPUT i W ij +B j Eqn (6.1) where, B j is the bias at the neuron j. The bias is the activation threshold for the neuron j. This bias avoids the tendency of an activation function to get stuck in the saturated, limiting value area of the activation function described below. After each OUTPUT j is calculated, an activation function is applied to modify it. The activation function is typically a bounded monotonic function. Sigmoid function is the most common one. ƒ(x)= 1/ (1+exp(-x)) Eqn (6.2) The weights W ij of the neural network are adjusted iteratively so that application of a set of inputs produces the desired set of outputs. If the computed outputs do not match the known values during network training, the NN model is in error. The error, E, is typically calculated as the sum of squared differences between computed and target values: E= Σ j (OUTPUT j -TARGET j ) 2 Eqn (6.3) 151

7 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance If the error E is large, then the neural network weights, W ij are adjusted to reduce this error. The most commonly used approach to adjust the weights is the gradient descent method. Using this approach, the weights is changed from W ij to (W ij a de/dw ij ), where parameter a is called the learning rate. This procedure of weight adjustment is called back-propagation. A simplified procedure for the learning process of neural network is summarized below: Provide the network with training data consisting of patterns of input variables and target outputs. Assess how closely the network output matches the target outputs. Adapt the connection strengths (i.e., weights) between the neurons so the network produces better approximations of the desired target outputs. Continue the process of adjusting the weights until some desired accuracy level is achieved. Analyse the results to assess the validity of the parametric relationships Neuro-fuzzy network [FuzzyTech, 2001] The key benefit of fuzzy logic is that it allows design consultant to describe desired system behaviour with simple IF-THEN relations. However, this is at the same time its major limitation. In many applications, knowledge that describes desired system behaviour is contained in data sets. The design consultant must derive the IF-THEN rules from the data sets manually. In this case, a neural network presents a solution since it can train the model itself from the data, while the fuzzy logic model provides 152

8 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance the rules which governs the paradigm relationships. On the other hand, neural network itself is limited by a number of reasons. Firstly, neural network solution stands as a black box. Design consultant can neither interpret what derive a certain behaviour, nor modify a neural network manually to influence a certain behaviour. Secondly, neural network require prohibitive computational effort, and numbers of sample data during the modeling process. Thirdly, selection of the appropriate net model and setting the parameters of the learning algorithm is still a tough job which requires extensive experience. Among the mentioned reasons, inadequacy of data base and experience, as well as the lack of an easy way to verify and optimize a neural network solution are probably the major limitations of neuro network. For this application, combined use of neural network and fuzzy logic might provide a powerful technique which compensates individual method s weakness. Neural network can learn from data sets, while fuzzy logic solutions are easy to verify and optimize. Neuro-fuzzy network combines the advantages of fuzzy systems, the explicit knowledge representation of knowledge, and the ability to cope with uncertainties with the learning power of neural network Method of neuro-fuzzy network modeling using FuzzyTech 5.5 FuzzyTech 5.5 is powerful software using neuro-fuzzy technology, with which models describing complex systems can be established easily and clearly. As shown in Figure 6.3, the basic neuro-fuzzy network model typically consists of input variables, rule blocks and output variables. 153

9 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Fuzzification Fuzzy Inference Defuzzification Figure 6.3 Basic structure of fuzzy logic model Fuzzification The fuzzification interface consists of the following operations [Gouda et al., 2001]: 1) Perform a scale mapping that transfers the input variable ranges into a corresponding universe of discourse (quantization/ normalization). 2) Performs the fuzzification strategy that converts crisp input data into suitable linguistic variables, which may be viewed as labels of fuzzy sets. Firstly, in the fuzzification interface, real values are transformed into linguistic variables. In this study all the input and output variables are defined correspondingly to three classes within the full range of observed values. For example, Total data centre energy consumption is converted into fuzzy sets such as low, medium and high. 154

10 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance The degree to which the crisp value belongs to a set is represented by a value between 0 and 1, called the degree of membership µ. Fuzzy logic describes degree of membership by a function known as a membership function (MBF). Standard membership functions are used in most practical applications with types of Z, Lambda, Pi and S, as shown in Figure 6.4. Figure 6.4 Standard membership function types Figure 6.5 below shows the fuzzification of term total data centre energy consumption. Based on the data collected during the measurements in the 6 data centre studied, the details of the ranges of defined input and output variables are given in Appendix H. Degree of Membership Range of real value 4100 kwh/m 2 /year Figure 6.5 Fuzzification of variable 155

11 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance For example, energy consumption at 4100 kwh/m 2 /year is a member of the fuzzy sets for the terms: Low to the degree of 0.00 Medium to the degree of 0.30 High to the degree of Fuzzy inference Once all input variable values are translated into the respective linguistic variable values, the so-called fuzzy inference step evaluates the set of IF-THEN rules that define system behaviour. IF-THEN rules always describe the reaction to a certain situation as: IF <situation> THEN <action>. The inference is a calculus consisting of the steps: aggregation, composition and, if necessary, result aggregation. First, aggregation determines the degree to which the complete IF part of the rule is fulfilled. Special fuzzy operators are used to aggregate the degrees of support of the various preconditions. There are in Min operator, which corresponds with the linguistic AND, is most common and has been employed for the study. Equation below shows the principle of this method. Figure 6.6 shows the dialog of input fuzzy operator. AND: µ (A^B)=min {µ(a), µ(b)} Eqn (6.4) 156

12 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Figure 6.6 Operator of input aggregation Each rule defines an action to be taken in the THEN part. The degree to which the action is valid is given by the adequacy of the rule to the current situation. This adequacy is computed by the aggregation as the degree of truth of the IF part. In the composition step, the degree of truth of the IF part of the rule is multiplied by a weighting factor, named Degree of Support (DoS). This factor represents the individual significance of the rule. By this way, rules themselves can be fuzzy, with a support between 0 and 1. The support of a consequence is calculated by linking the support of the entire condition with the DoS by composition operator. See the equation below. µ THEN =µ IF * DoS Eqn (6.5) 157

13 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Figure 6.7 shows an example of the defined rules. Input variables Degree of support (DOS) Output variables Figure 6.7 Definition of fuzzy rules A result aggregation step determines the maximum degree of support for each consequence which is used for all further processing. Also shown in Figure 6.6, there are two methods of result aggregation, the maximum method (MAX) and the bounded sum method. If more than one rule produces the same consequence, an operation must aggregate the results of these rules. MAX operator takes the maximum of the Dos values as final result. The BSUM method takes the bounded sum. In this study, MAX method is employed to aggregate the results of these rules, as presented by the equation below. MAX result aggregation: µ RESULT = max i (µ THEN, RULEi ) Eqn (6.6) With the application of neuro-fuzzy network in this study, all the fuzzy rules were formulated automatically from the training data. Firstly, a full range of possible rules is generated. For example, say, there are two input variables and one output variable; and each variable has three classes low, medium, high. The total number of possible 158

14 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance rules is 3x3x3=27. After the neural network training, weight of each fuzzy rule is determined through Dos. The insignificant rules will be removed from the rule block. Further discussion of neuro-fuzzy training is provided in Section Defuzzification The result produced from the fuzzy rules are linguistic values, namely fuzzy, for the output variables. The defuzzification step translates the linguistic results into real values. There are several methods of defuzzification. Centre-of-Area (CoA), sometime also called Centre-of-Gravity (CoG) is the most frequently used defuzzification method. This method is based on taking the aggregate of the fuzzy outputs from each rule, weighted by their grades of fuzzy input set membership, see Figure 6.8. Figure 6.8 Deffuzification with Centre-of-Area The Centre-of-Maximum (CoM) is another commonly used method of defuzzification. It computes a crisp output as a weighted average of the term membership maxima, weighted by the inference results (Figure 6.9). The locations of the individual term membership maxima are indicated by the gray arrows and the inference result is shown 159

15 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance by the height of the black bar in the arrows. Figure 6.9 Deffuzification with Centre-of-Maximum Most CoA implementations are only approximations since they neglect overlapping and can be represented with methods such as CoM. Note that, since the quality of approximation has proved to be fair enough for the vast majority of applications, and CoA defuzzification is significantly slower than the approximated CoA methods, most hardware and software implementations of fuzzy logic algorithm only support CoM and others rather than the real CoA [FuzzyTech, 2001]. For this reason, CoM is employed in this study Neuro-fuzzy training i) Training of fuzzy rules The basic idea of neuro-fuzzy approach is to set up a complete fuzzy rule base, not by entering the rules, but by presenting a set of training samples and selecting the set of rules that represent the samples [FuzzyTech, 2001]. In the neuro-fuzzy model, neural network training function is used to evaluate the significances or weights of the 160

16 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance predefined fuzzy rules. Unlike early neuro-fuzzy network approaches, this study employs restricted training, which requires a given rule base, to restrict the number of rules used and save apriori knowledge that was entered into the system before training. This is due to the fact that if a huge unrestricted rule base does not provide a good match to the given sample distribution, the training may not converge. Configuration of neural training is shown in Figure The definitions of the terms in this dialog are shown in Appendix I. Figure 6.10 Configuration of neural training (Source: FuzzyTech, 2001) ii) Training of fuzzification and defuzzification In a fuzzy system, membership function of these parameters found in the fuzzification (input variables) and defuzzification (output variables) can also be trained. The manner in which neuro-fuzzy model treats the parameters of the membership functions of variables is similar to the training of fuzzy rules. Figure 6.11 shows an example of the 161

17 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance variable after training. Figure 6.11 Training of output variable Summary of settings of neuro-fuzzy network modeling Fuzzyification Membership function (MBF) Standard membership function Fuzzy inference Input aggregator Min operator (linguistic AND) Result aggregator Max operator (maximum of the degree of support (DoS) value) Defuzzification Method of defuzzification Centre of Maximum (CoM) Neuro-fuzzy training Learning method Batch Random 162

18 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Winner neurons 1 Step Width (DoS) 0.1 Step Width (Term) 1% Stop criteria Max. Steps from 4000 to Max. Deviation from 50% to 1%, factor 0.75 Avg. Deviation 0.1% 6.4 Objectives of neuro-fuzzy network modeling of data centre energy performance Data centre energy performance involves a complex system of energy consuming components and elements. It comprises various energy consuming components and environmental and design variables. Neuro-fuzzy network combining the advantages of fuzzy systems, the explicit knowledge representation of knowledge, and the ability to cope with uncertainties with the learning power of neural network is a powerful technique for developing desired solution of integrating all these systems and variables into a whole system. The objectives of neuro-fuzzy network modeling in this study are as follows: a. To develop whole-system based models for data centre energy performance. b. To provide efficient and effective evaluation of data centre energy performance, and accurate estimation of energy-saving potential of data centres based on the models developed. 163

19 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance 6.5 Modeling methodology In this study, neuro-fuzzy network modeling consists of four steps. Figure 6.12 shows the systematic diagram of the modeling procedure. Parameter analysis Establish fuzzy logic model Neural network training Model verification Figure 6.12 Systematic diagram of modeling procedure Parameter analysis & establishment of fuzzy logic model Based on parameter analysis conducted in Chapter 5, variables that affect overall data centre energy performance are determined; linking those (input variables) to the key indicators (output variables) of data centre energy performance by fuzzy rule blocks, the structure of the model is determined. The results of parameter analysis are summarized as follows. The key indicators of total data centre energy performance are the total data centre energy consumption per unit gross floor area in term of kwh/m 2 /year and the energy consumption of IT equipment as a percentage of total data centre energy consumption. The significant energy consuming system of data centres are HVAC system and IT equipment. Physical variables gross floor area (m 2 ), IT footprint as a percentage of gross floor area, and total over-designing of data centre power demand plus energy consumptions of HVAC and IT equipment in terms of kwh/m 2 /year are set as the 164

20 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance significant variables of total data centre energy consumption. On the other hand, relative energy effectiveness of HVAC system which is denoted by the ratio of HVAC energy consumption to IT equipment energy consumption seems to be positively correlating to IT equipment energy consumption as a percentage of total data centre energy consumption. Based on the results of parameter analysis, model of total data centre energy performance is shown in Figure Figure 6.13 Model of overall data centre energy performance where, Grss_area is gross floor area of data centre (m 2 ) HVAC_energy is HVAC energy consumption per unit gross floor area (kwh/m 2 /year) IT_energy is IT equipment energy consumption per unit gross floor area (kwh/m 2 /year) ITfootprint is IT equipment footprint area as percentage of gross floor area (%) Total_overdesign is over-designing of total data centre power demand (-) Total_energy is total data centre energy consumption per unit gross floor area (kwh/m 2 /year) HVAC_Vs_IT is the ratio of HVAC energy consumption to IT equipment energy consumption (-) IT_Vs_Total is the IT equipment energy consumption as percentage of total data centre energy consumption (%) 165

21 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance As shown in section 5.2, affecting variable IT equipment energy consumption in terms of kwh/m 2 /year is affected by the over-designing of IT equipment power demand and IT equipment footprint area as percentage of gross floor area. These relationships are described in the model of IT equipment energy performance (Figure 6.14). Figure 6.14 Model of IT equipment energy performance where, IT_overdesign is the over-designing rate of IT equipment power demand Based on the results of parameter analysis in Section 5.4, variables such as energy consumptions of HVAC and the ratio of HVAC energy consumption to IT equipment energy consumption, which indicate energy performance of HVAC system, are also affected by other variables. Figure 6.15 depicts that HVAC energy consumption (kwh/m 2 /year) is affected by IT equipment energy consumption (kwh/m 2 /year), gross floor area (m 2 ), IT equipment footprint area as percentage of gross floor area and the over-designing of HVAC system. The ratio of HVAC energy consumption to IT equipment energy consumption is determined by the over-designing of HVAC system. 166

22 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance Figure 6.15 Model of HVAC system energy performance Moreover, as shown in Figure 5.12 and Figure 5.21, over-designing of HVAC system power demand and total data centre power demand are both affected by the over-designing of IT equipment power demand. Figure 6.16 Integrated model of total data centre energy performance 167

23 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance As stated earlier, neuro-fuzzy network is suitable for the purpose of modeling complex systems. Based on the models above, it is practical to integrate all the affecting variables and systems to a whole-system based model of total data centre energy performance (Figure 6.16). In this model, the preliminary input variables are gross floor area (m 2 ), over-designing of IT equipment power demand, and IT equipment footprint as a percentage of gross floor area. Rule block level 1 acts as the intermediate level collecting the information from input variables and transmit it to the next level defined as rule block level 2. On this step, the information that over-designing of IT equipment power demand determines the over-designing of HVAC cooling capacity and the over-designing of total data centre power demand is input to the model and passed to the next level of rule block. Rule block level 2 also acts as the intermediate level collecting the information from input variables and transmitting it to the next level defined as rule block level 3. Rule blocks 2and 4 (RB2 & 4) together illustrate the structure of HVAC system energy performance. Rule block 3 describes the significant variables that affect IT equipment energy consumption. Rule block level 3 is defined as the same as that in the previous model of total data centre energy performance (Figure 6.13). It receives the information transmitted from rule block level 2 and other independent significant variables including gross area, IT equipment footprint as percentage of gross floor area. Based on the information 168

24 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance received, the values of the output variables can be determined Neural network training Data obtained from investigations conducted in the data centres studied are then used to train the model. This training data consist of both the observed values of input variables and output variables. Based on the correlationship analysis conducted in Chapter four, data centre 6 seems often the disturbing point of the analysis. This can be explained by its significantly larger area than other facilities and the low occupancy rate of the floor area. For this reason, the data of data centre 6 is excluded from the training. However, this data centre will be used to as a new data to test the validity of the trained model and evaluate the energy saving potential. Appendix J shows the training data of the model Model verification After the training, the information of the data is learned and stored into the model, and the significance levels of the various fuzzy rules are determined. Those rules have minor influences on output variables that are removed from the rule blocks. The trained model can then be used to predict the energy performance of a data centre. Moreover, by changing one or more values of the input variables, the influences of that or those variables can be examined. 169

25 Chapter 6: Neuro-fuzzy Network Modeling of Data Centre Energy Performance However, sometime the neural training may not converge, which may lead to inaccurate prediction and analysis. This is due to many reasons. The main reason is the constraint of sample size. In this study, only 6 sets of data obtained from the selected data centres are available for the training and testing of the model. Moreover, there are significant variations among data centres in terms of energy consumption, power demand and systems efficiencies. These can lead to a poor convergence of the model developed. Improper model structure and unrestricted and great number of fuzzy rules can also result in improper model. In this case, further re-examination of model structure and more sample data are needed. Figure 6.17 Training result in 3-D plot Figure 6.17 depicts the trained model of total data centre energy performance in a 3-Dimensional plot. Relationships between HVAC energy consumption (vertical axis) and Gross area and Over-designing of IT equipment power demand (horizontal axes) are depicted graphically. 170