Electric energy consumption of domestic sector in India

Rev. Téc. Ing. Univ. Zulia. Vol. 37, Nº 3, 41 - 50, 2014

Electric energy consumption of domestic sector in India S.Saravanan1, S.Kannan2, C.Thangaraj3 [email protected], Department of EEE, Kalasalingam University, Tamil Nadu, India 2 [email protected], Department of EEE, Ramco Institute of Technology, Tamil Nadu, India. 3 Former Vice Chancellor, Anna University of Technology Chennai, Chennai, India 1

Abstract The objective of this study is to describe the development of an electric energy consumption model which is able to predict the future electricity consumption (EC) of the domestic sector (DS) in India. Computational intelligence (CI) techniques such as Artificial Neural Network (ANN), Least Square Support Vector Machine (LS-SVM) and Adaptive Neuro Fuzzy Inference System (ANFIS)are used. Population, Per capita Gross Domestic Product (GDP) and electricity production are used as the input variables. The EC-DS is the predicted output variable. A 31-year data set is used to train the network and 7-year data set is used to test the network. Mean Absolute Percentage Error (MAPE), Mean Bias Error (MBE) and Root Mean Square Error (RMSE)are used as performance evaluation criteria. Results obtained using ANN, LS-SVM and ANFIS methods are compared with each other. The performance of ANFIS is found to be more efficient than ANN and LS-SVM. The future projections are carried out for the period between 2015and 2025. Keywords: ANFIS, ANN, electricity production, India, LS-SVM, per capita GDP, population

1. INTRODUCTION Electric energy plays a vital role in our day-to-day life activity because it is needed in almost all type of human activities, such as domestic, industrial production, agriculture and other activities. The limited availability of energy resources and production related to environmental issues, warrant effective and efficient methodologies of production and uses. Electric energy is one of the important development parameters in any developing country. In India, growing human population, large scale industries, continuing pressure for better living standards and higher economic growth rate require more electric energy. In parallel, the consumption of electricity has also been increasing. EC is an important component for technical, social and economic development of all countries.

Fig. 1 Sector wise electricity consumption (EC) in the year 2011 The electricity consumption (EC) data was collected from energy statistics of India released for the year 2013 by Central Statistics Office, Government of India.India’s net EC has increased at an average rate of 7.3 % annually from 2000 to 2012. The EC was 43.724 Tera Watt hour (TWh) in 1970 and has increased to 612.645TWh in 2010. The sector wise EC for the year 2011 is shown in Fig. 1. In India, EC of domestic sector (DS) is the second biggest consumer of the electric energy. In the year 2011, 21.56% of electricity is consumed in DS of India. Therefore prediction of EC-DS is of prime importance of the future energy strategies, such as identification and analysis of energy issues and development. Fig. 2 indicates that the EC-DShas increased approximately 3 times between the year 1995 and 2010. It has increased from 3840 Giga Watt hour (GWh) in 1970 to 146080 GWh in 2010 with an annual average growth rate of 9.87 %.

41


Residential EC (GWh)

140000 120000 100000 80000 60000 40000 20000 0 1971

1976

1981

1986

1991

1996

2001

2006

2011

Year Fig. 2 India’s EC-DSbetween 1971 and 2010 years EC-DS is usually dependent on many factors such as population, per capita GDP, electricity price, installed capacity, gross electricity generation, total subscribership, and household income. Generally the EC-DS is directly related to population of the country. The population increase leads to corresponding increase in EC. Per capita Gross Domestic Product (GDP) is a measure of economic activities of all countries. The aim of this study is to provide a prediction model for EC-DS using population, per capita GDP and electricity production of India.

S. No. 1.

Table 1 Literature survey for forecasting electric energy demand/consumption and inputs Techniques Name of the author Output variables Input variables used PSO and GA M.H. Amjadi et al. Electricity GDP, population, number of demand customers and average price electricity Residential electricity demand Residential and Industrial sectors electric energy demand Energy consumption

Household total final consumption expenditure, real energy prices and energy demand trend Installed capacity, Gross electricity generation, population total subscribership

Energy demand

GDP, population, import and export

KadirKavaklioglu et al.

Electricity consumption

population, Gross National Product, imports and exports

SVR

KadirKavaklioglu

Electricity consumption

Population, Gross National Product, imports and exports.

8.

GA

HalimCeylan et al

Energy demand

Energy demand, population, import, export

9.

ABC and PSO

Mustafa Servet et al

Electricity energy demand


10.

PSO and ACO

Mustafa Servet et al

Energy demand


11.

PSO and ACO

AlperUnler

Energy demand


12.

ACO

M. Duran Toksari

Energy demand


2.

Structural time series

ZaferDilaver

3.

ANN, Linear Regression, Nonlinear regression

Mehmet Bilgili et al

4.

ANN

Murat Kankal et al

5.

ANN

Zong et al

6.

ANN

7.

Lester C Hunt

Gross Domestic Product (GDP), population, import, export amounts and employment

42


In recent years, the prediction of electricity/energy demand/consumptionis a subject of widespread current interest among the academicians and policy makers. Since 1980, much research was conducted on the application of computational intelligence (CI) techniques such as Artificial neural network (ANN) (Mehmet Bilgili et al; 2012, Murat Kankal et al; 2011, Geem, Zong Woo et al; 2009), Support Vector Regression (SVR) (KadirKavaklioglu; 2011), Genetic Algorithm (GA) (Amjadi et al; 2010, Ceylan et al; 2004), Particle Swarm Optimization (PSO) (Amjadi et al; 2010, Mustafa ServetKıran; 2012, UnlerAlper, 2008), Artificial Bee Colony (ABC) (Mustafa ServetKıran et al; 2012) and Ant Colony Optimization (ACO) (Mustafa ServetKıran; 2012, UnlerAlper; 2008, Duran Toksari; 2007, Kermanshahi; 2002). Some methods frequently used as a forecasting tooland summarized in Table 1. 2. THE PROPOSED ALGORITHMS In this study, the estimation of EC-DS based on dominant input variables was modeled using ANN, ANFIS and LS-SVM. Thesealgorithms begin with the selection of anEC-DS model and the related input data. Input variables include population, per capita GDP and electricity production. The collected data is partly used for training the network; remaining data are used to examine the accuracy of the techniques. The steps of the analysis are as follows: i) Selection the input variables and finalize the output variable ii) Normalize the input/ output variable(s) between 0 to 1 iii) Divide the available data into training and testing sets iv) Using the raining data different experimentation was conducted to select the parameters in order to minimize the error. v) Based on least MAPE, the following parameters has been selected a. For ANN, select the number of hidden layer neurons and learning algorithm. b. For ANFIS, select the number of membership functions and their types c. For LS-SVM, select the kernel function and optimization parameters vi) After training, evaluate the model using testing data vii) Display the minimum error with their parameters of each techniques The ANN, ANFIS and LS-SVM are discussed in details in Appendix. 3. SELECTION OF INPUT VARIABLES Generally in any model, the important point is the selection of the input variables to provide the best modeling result and the prediction of the output variable. Due to this reason, the input variables are selected based on the correlation coefficients. These correlation coefficients which measure the relationship between any one of the selected input variable and predicted output variable. Such correlation coefficient between the ECDS and the input variables used in problem are given in Table 2. Table 2 Correlation coefficient between the independent variables and EC Variables Population Per capita GDP Electricity production

Correlation coefficient (R2) 0.8877 0.8810 0.9031

The available 38-year data have been divided between training and testing, training data set are used to make the network parameters of the model and testing data set are used to assess the performance of the model. In this study, the available 38-year data are partly considered for training the model (31-year data) and remaining used for checking, i.e., to verify the exactness of the model (7-year data) after training. The data are scaled between 0 and 1 to avoid numerical overflows due to very small/large weights. 4. PERFORMANCE CRITERIA FOR ESTIMATION OF ERROR The best model in terms of accuracy is selected based on the mean absolute percentage error (MAPE), which is computed by (1). MAPE =

1 𝑛

(𝐴𝑖 −𝑃 𝑖 ) 𝑛 𝑖=1 𝐴𝑖

× 100

(1)

In addition, Root Mean Square Error (RMSE), multiple fraction of variance (R2) error and Mean Bias Error(MBE) are also considered as follows: RMSE =

1 𝑛

𝑛 𝑖=1 (𝐴𝑖

− 𝑃𝑖 )2

(2) 43


R2 = 1 – MBE =

1 𝑛

𝑛 2 𝑖=1 (𝐴𝑖 −𝑃 𝑖 ) 𝑛 (𝐴 )2 𝑖=1 𝑖

(3)

𝑛 𝑖=0

(4)

𝑃𝑖 − 𝐴𝑖

wherePi, Ai are the predicted and actual values, and ‘n’ denoted the total number of data in the testing set. MAPE and RMSE measure the residual errors, which gives the difference between the predicted and actual values. MBE indicates if the predicted data are over/under estimated. 5. RESULTS AND DISCUSSION In this section, the result obtained by computational intelligence (CI) techniques such as ANN, ANFIS and LS-SVMare compared with each other. 5.1 Results obtained using ANN The type of the learning algorithm and the number of neurons in the hidden layer associated with each input, decide the accuracy of an ANN model. In order to attain an effective training for ANN algorithm, several experiments were conducted.For finding the best results, in this work trial and error method is used to determine the learning algorithm and number of neurons in the hidden layer. Three different types of learning algorithms are considered such as Scaled conjugate gradient (‘trainscg’), Conjugate gradient back propagation with Fletcher-Reeves updates (‘traincgf’) and Levenberg-Marquardt back propagation (‘trainlm’), and the numbers of neurons in the hidden layers are varied from 2 to 10. The training process continues until the desired error between the actual and predicted output is reached. The predicted EC_DSS results are given in Table 3. Table 3 MAPE with respect to number of hidden neurons Hidden neurons ‘trainscg’

0.8466

0.7003

0.8884

0.8437

0.7747

0.8377

0.8618

0.7978

0.7052

‘traincgf’

1.4081

1.3727

3.0432

0.7167

1.5819

2.5306

1.1782

0.9695

1.0801

‘trainlm’

0.8028

0.7156

0.7657

2.2289

0.7399

0.7348

0.7977

0.7412

0.7172

2

3

4

5

6

7

8

9

10

Based on the results given in Table 3, it obvious that the best architecture of ANN consist of three layers such as, one input layer (no transfer function) with three neurons; one hidden layer with log-sigmoid (‘logsig’) transfer function having three neurons; one output layer with linear (‘purelin’) transfer function having one neuron. The Scaled conjugate gradient (‘trainscg’) method is used in the present study. It updates the weight and bias values according to Scaled conjugate gradient method. The simulation results of ANN model with minimum MAPE and the comparison between actual and predicted electricity consumption (GWh) results are given in Table 5. Table 4 Test Results with different number MFs and their types gaussmf gauss2mf gbellmf pimf trimf NMF* MAPE NMF* MAPE NMF* MAPE NMF* MAPE NMF* MAPE 4,4,4 1.0241 3,4,3 0.6743 2,4,3 1.0009 2,4,3 1.2287 3,4,2 1.2951 4,4,3 1.0428 3,2,3 0.8704 3,4,3 1.0401 2,3,3 1.2696 2,2,2 1.3296 2,4,2 1.1451 3,4,2 0.9766 3,2,3 1.0563 2,4,2 1.3348 3,4,3 1.4641 3,3,3 1.2577 2,4,4 1.0034 3,3,2 1.1001 3,4,2 1.3745 2,3,2 1.5253 2,4,4 1.2593 2,4,3 1.0046 4,4,3 1.1094 3,3,3 1.5111 4,4,2 1.8561 NMF* Number of membership function

trapmf NMF* MAPE 3,2,2 1.0678 4,4,3 1.369 2,4,3 1.4205 2,3,2 1.4553 3,4,3 1.4587

psigmf NMF* MAPE 3,4,4 0.9077 3,4,3 1.0013 4,3,3 1.0307 3,3,3 1.0368 2,4,2 1.1941

5.2 Results obtained using ANFIS The type of the fuzzy membership functions (MFs) and number of MFs associated with each input decide the accuracy of ANFIS model. In order to realize effective training of the ANFIS algorithm, several experiments were conducted. For finding the best results, in this work trial and error method is followed for deciding the optimal number of MFs and the types of MFs. Eight different types of MFs are considered such as Gaussian curve built-in MF (‘gaussmf’),Gaussian combination MF (‘gauss2mf’), generalized bell MF (‘gbellmf’), Πshaped built-in MF (‘pimf’), triangular MF (‘trimf’), Trapezoidal-shaped built-in MF (‘trapmf’), Built-in MF composed of product of two sigmoidal shaped MF (‘psigmf’) and Built-in MF composed of difference between two sigmoidal MF (‘dsigmf’) and the number of MFs are varied between 2, 2, 2 and 4, 4, 4. The results are given in Table 4. 44


The training process continues until the MAPE reaches the minimum value between the actual and the predicted output.Built-in MF composed of product of two sigmoidal shaped MF (‘psigmf’) and Built-in MF composed of difference between two sigmoidal MF (‘dsigmf’) produce the same result. Hence, Built-in MF composed of difference between two sigmoidal MF (‘dsigmf’) results is not displayed in the Table 4. Of all the combinations, Table 4 shows the best five minimum results in terms of MAPE. Those MFs having higher MAPE are neglected. From Table 4, it is evident that the ‘gauss2’ MF with number of MFs 3, 4, 3 is found to have a better performance in terms of MAPE (0.6743). The simulation results for ANFIS model with minimum MAPE and comparison between actual and predicted electricity consumption (GWh) results are given in Table 5. 5.3 Results obtained using LS-SVM The type of the kernel function (KF) and the optimization of two control parameters such as Gaussian radial basis kernel (γ) and the regularization constant (σ) decide the model evaluation of LS-SVM model. In order to achieve effective training of the LS-SVM algorithm, several experiments were conducted. For finding the best results, in our work, trial and error method is followed for deciding the optimal number of control parameters and the types of KF. Three different types of KF are considered such as linear, polynomial, Gauss radial basis function (RBF). The control parameter γ is varied from 10 to 70 with a step size of 1 and the control parameter σ is varied between 0.01 and 0.9 with a step size of 0.01. The Gauss RBF KF with design values of γ (14) and σ (0.8) was found to give the best performance in term of MAPE (0.8265) value.The simulation results for LS-SVM model with minimum MAPE and comparison between actual and predicted electricity consumption (GWh) results are given in Table 5. 5.4 Comparison ofCI techniques results

Year 1994 1997 1998 2000 2001 2003 2008

Table 5 EC-DS Actual Vs. Predicted EC-DS (GWh) Predicted Actual ANN ANFIS 43344 43205.94 43913.02 55267 55270.09 55650.82 60346 60204.01 60308.85 70520 70983.88 70476.87 75629 74456.91 75122.79 83355 84016.14 82558.17 120918 119295.29 119751.40 MAPE 0.7003 0.6743

LS-SVM 43217.04 55609.31 60405.14 70679.14 74957.00 84448.72 118077.77 0.8265

From the results in Table 5, the actual value of EC-DS in the year 1994 was 43344 GWh and in the year 2008 the same was 120918GWh. The EC-DScalculated using ANN is 43205.94 GWhfor the year 1994, and 119295.29 GWh for the year 2008. The EC-DScalculated using ANFIS was43913.02 GWh for the year 1994, and 119751.40GWh for the year2008. The EC-DScalculated using LS-SVM is 43217.04GWh for the year 1994, and 118077.77GWh for the year 2008. The results for the remaining years are also given in Table 5. It is difficult to make comparison between the accuracy of the models selected, but the numerical value shows that the ANFIS model gives the best results compared to ANN and LS-SVM as shown in Table 5. Next to the ANFIS, the ANN produces better results than LS-SVM. Table 6 Comparison of ANN, ANFIS and LS-SVM error results MAPE RMSE R2 MBE ANN

0.7003

819.28

0.99988

278.11

ANFIS

0.6743

624.10

0.99993

228.15

LS-SVM

0.8265

1187.85

0.99976

283.56

According to the obtained results, the deviations in percentage of predicted results in the testing years are in the range of -1.32% to 2.35% for all the CItechniques: ANN, LS-SVM and ANFIS, as shown in Figure 3. For our analysis, testing data are selected randomly from the available 38 years data. 45


In the ANN result, the error deviation is approximately +1.55% in positive side and -0.79% in the negative side. In the LS-SVM result, the percentage of error deviation is +2.35% in positive side and -1.32 % in the negative side. Based on the ANFIS results, it is evident that the percentage of error deviation is approximately 0.96% in positive side and -1.32% negative side.When compared to ANN and LS-SVM, ANFIS produces better results in term of deviation in percentage, almost closer to the actual results, for most of the time. 2.5

Deviation (%)

2.0 ANN

1.5

ANFIS

LS-SVM

1.0 0.5 0.0 -0.5 1

2

3

4

5

6

7

-1.0 -1.5

Intervel Fig. 3 Comparison of ANN, ANFIS and LS-SVM deviation (%)

In addition, error results of ANN, ANFIS and LS-SVM are compared with each other with respect to four different categories of error and isshown in Table 6. Based on the above results the ANN model has the error value in MAPE (0.7003), RMSE (819.28), MBE (278.11) and R 2 (0.99988); the LS-SVM model has the error value in MAPE (0.8265), RMSE (1187.85), MBE (283.56) and R2 (0.99976); ANFIS model has less error value in MAPE(0.6743), RMSE (624.10), MBE (228.15)and R2(0.99993) value is very close to 1. Based on the comparison of error results, it can be easily seen that the ANFIS outputs have less error value than the ANN and LS-SVM model. Next to ANFIS, ANN has less errors in predicted outputs. Table 7 ANFIS parameters of predicted input variables with minimum MAPE Population 5 gauss2mf 0.0301

No of MFs Type MAPE

Per capita GDP 2 gauss2mf 0.8902

Electricity production 2 gauss2mf 1.1868

Table 8 Predicted input variables Year

Population(Millions)

Per capita GDP(MINR)

ElectricityProduction(TWh)

2015

1308.645

585984.691

1275.33003

2016

1325.468

618258.391

1343.974156

2017

1342.290

650790.391

1413.561326

2018

1359.113

683509.243

1483.875721

2019

1375.935

716361.066

1554.741822

2020

1392.758

749309.372

1626.041669

2021

1409.581

782326.252

1697.677433

2022

1426.403

815385.477

1769.530726

2023

1443.226

848471.102

1841.526076

2024

1460.048

881573.425

1913.615533

2025

1476.871

914686.459

1985.768223 46


6. FUTURE ESTIMATION The future estimation of EC-DS for the year (2015 to 2025) using ANN and ANFIS model (with same model parameters) was made. To forecast the EC-DS, input variables (population, per capita GDP and electricity production) should be analyzed and their trends for the future need are to be predicted. The prediction has been made based on the historical data from the year 1975 using ANFIS technique. Table 7 shows that the number of MF and their types with minimum MAPE error. The forecasted results of input variables based on historical data are given in Table 8. Population is given in Millions, Per capita GDP in Million Indian Rupees (MINR) andElectricityProduction in Tera Watt hour (TWh). Based on the predicted input variables (Table 8), the EC-DS predicted for the period from 2015 to 2025 using ANN and ANIFS is given in Table 9. Table 9 predicted EC-DS estimation for India Year

ANN

ANFIS

(GWh) 2015

186688.2

184134.1

2016

196635.7

192703.6

2017

206419.3

201321.3

2018

215978.5

209975.9

2019

225263.9

218657.5

2020

234240.1

227359.9

2021

242880.1

236077.6

2022

251161.0

244804.9

2023

259070.6

253538.7

2024

266604.3

262276.3

2025

273763.5

271016.8

7. CONCLUSIONS Accurate predictions of EC are quite important for effective enactment of electricity energy policies and vital when demand grows faster. The aim of this study is to support the planners with alternative better forecasting methods. In this paper, EC-DS of India was modelled as a function of input variables (population, per capita GDP and electricity production) using ANN, ANFIS and LS-SVM. The models performed effectively in the statistical test (R2) conducted, indicating the significance of selected input variables in the prediction of EC-DS. The proposed ANFIS model estimate the EC-DS better than the ANN and LS-SVM models in terms of MAPE, MBE and RMSE. The ANFIS algorithm is suitable for handling complex and uncertain data because it comprises the features of both ANN and Fuzzy system. Next to ANFIS result, ANN gives better results than LSSVM methods.The results of the presented study are helpful to give a new direction to the energy planning studies by policy designers, manufacturer of power system components and independent power producers among others. In 2025, the EC-DS of India may reach between 271016 GWh and 273763 GWh. In future, forecasting of EC-DS can also be investigated with GA-ANN, PSO-ANN and other hybrid techniques. ACKNOWLEDGMENT The authors gratefully acknowledge the Management of Kalasalingam University, Krishnankoil, Tamil Nadu, India for their constant support and encouragement extended during this research. They are also thankful to the Department of Science and Technology, New Delhi, for its support through the project: SR/S4/MS/364/06.

47


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APPENDIX 8. TECHNIQUES USED ANN, ANFIS, LSSVM are used to predict the EC-DSfor India. 8.1 ANN ANNs are computational modeling tools that have emerged in 1980 and found extensive applications in the field of data mining, pattern recognition, classification and forecasting. ANN is capable of developing a mathematical model to deal with the unknown relationship between input and output data set (Mehmet Bilgili et al; 2012). Most used ANN type for forecasting is multi-layer perceptron (MLP). It is proved that the MLP with error back propagation algorithm is an appropriate tool for long term load forecasting (Kermanshahi et al; 2002)also one hidden layer is enough to approximate any function (Sousa et al; 2007), if presented with enough hidden nodes. The MLP consists of three layers such as input layer, output layer and one or more hidden layers containing nodes which help to capture the nonlinearity in the data (Saravanan et al; 2012). One of the most important parameters was the number of hidden neurons; it determines the efficiency of ANN. The number of hidden neurons depends mostly on the specific application of interest and estimated by ‘‘trial and error” (Mehmet Bilgili et al; 2012). The features of ANN are learning ability, generalization and parallel processing. These features would help to solve complex problems precisely and flexibly. 8.2 ANFIS ANFIS, as a hybrid intelligent system that enhances the ability to automatically learn and adapt, is being used by researchers in various engineering systems. ANFIS is a combination of ANN and Fuzzy Inference System (FIS) which combines the learning capabilities of neural network and reasoning capabilities of Fuzzy logic. ANN provides effective learning from scratch by adjusting the interconnections between layers and fast computations. Fuzzy Inference System (FIS) is a popular computing framework based on the concept of fuzzy set theory. It allows thinking and reasoning capability for the Fuzzy logic. ANFIS is a multilayer feed-forward network which is applied to map an input space to an output space using a combination of neural network learning algorithms and fuzzy reasoning (Khoshnevisan; 2014). The structure of ANFIS model is shown in Figure 4.The FIS applies a fractional non-linear relationship to map its input space to the output space by a number of fuzzy IF-THEN rules (RoohollahNoori; 2013). The learning usually applies to the membership function (MF) of the IF-THEN rules of the fuzzy systems (GholamrezaKarimi; 2013). FIS type selection is one of the main steps for ANFIS development. FIS advantage is that it does not require knowledge of the underlying physical process as a precondition for its application (DaliborPetkovic; 2013). ANFIS has the capability of fast and accurate learning, extension of capacity, excellent explanation facilities in the form of semantically meaningful fuzzy rules and the ability to adopt both data and existing expert knowledge (Heidar; 2013).

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Layer 1 (Fuzzification)

Layer 2 (Rules)

A1 I1

R1

Layer 3 (Normalization)

w1

N

Layer 4 (Defuzzification)

W’1

f1

Layer 5 (Output)

W’1f1

A2 Output B1

R2

I2

N

W’2

W2

f2

W’2f2

A2 Fig. 4 Structure of ANFIS model 8.3 LS-SVM The SVM, was proposed by Vapnik and his co-workers in 1995, is a powerful machine learning method based on the statistical learning theory and the structural risk minimization principles (Chong Wu et al; 2006, SamuiPijush et al; 2011). SVM deals the convex optimization problems by solving the Quadratic programming problem. The size of matrix involved in the Quadratic programming problem is directly proportional to the number of training points. Hence, to reduce the complexity of optimization process, a modified version, called LS-SVM is used. It is an extension of standard SVM. It changes inequality constraint into an equality constraint, thus transform theQuadratic programming problems intolinear equations in order to minimize the complexity nature optimization problem(Fuwei; 2009). The optimization problem can be solved by introducing the Lagrange function and Karush–Kuhn–Tucker conditions. The final resulting LS-SVM model for function estimation will be given in (5). 𝑁

𝑎 − 𝑎 ∗ 𝐾 𝑏, 𝑏𝑖 + 𝑐, 0 ≤ 𝑎, 𝑎∗ ≤ ⋎ (5)

𝑦 𝑥 = 𝑖=1

where a and a* are the Lagrange multipliers, K(b, bi) is the kernel function, c is the bias value and ⋎ is the regularization parameter, determining the trade-off between the fitting error minimization and smoothness of the estimated function (Ghaedi; 2014).

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