Post on 07-Jun-2019
Research ArticleAn Incremental Radial Basis Function Network Based onInformation Granules and Its Application
Myung-Won Lee and Keun-Chang Kwak
Department of Control and Instrumentation Engineering Chosun University 375 Seosuk-dong Dong-guGwangju 501-759 Republic of Korea
Correspondence should be addressed to Keun-Chang Kwak kwakchosunackr
Received 29 June 2016 Accepted 22 August 2016
Academic Editor Toshihisa Tanaka
Copyright copy 2016 M-W Lee and K-C Kwak This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper is concernedwith the design of an Incremental Radial Basis FunctionNetwork (IRBFN) by combining Linear Regression(LR) and local RBFN for the prediction of heating load and cooling load in residential buildings Here the proposed IRBFN isdesigned by building a collection of information granules through Context-based Fuzzy C-Means (CFCM) clustering algorithmthat is guided by the distribution of error of the linear part of the LRmodel After adopting a construct of a LR as globalmodel refineit through local RBFN that captures remaining and more localized nonlinearities of the system to be considered The experimentsare performed on the estimation of energy performance of 768 diverse residential buildingsThe experimental results revealed thatthe proposed IRBFN showed good performance in comparison to LR the standard RBFN RBFN with information granules andLinguistic Model (LM)
1 Introduction
During the past few decades we have witnessed a rapidgrowth in the number and variety of applications of fuzzylogic neural networks and evolutionary computing as aframework of computational intelligence [1ndash4]We especiallyshall concentrate on incremental construction of RadialBasis Function Network (RBFN) with the aid of informationgranules In general we design with the simplest linearmodels and then refine such linear models by incorporat-ing additional nonlinear model in system modeling Thecommonly used method becomes Linear Regression (LR)model [5] If LR model appears to be insufficient to predictfurther refinements are implemented This concept is astrong factor motivating the development of the incrementalmodels The effectiveness and superiority of this model havebeen demonstrated in the previous work The incrementalmodel introduced by Pedrycz and Kwak [6] represented anonlinear and complex characteristic more effectively thanconventional models There are several advantages of thisapproach First a commonly used framework of LR has beenusedThe nonlinear behavior of the system could be confined
to some limited regions of the input space and by adding onlya few patches in these regions becomes practically relevantand conceptually justifiable Furthermore it has establisheda comprehensive design platform offering a complete set-by-step procedure of the construction of the incrementalmodel [7] The clustering technique used in the design ofincremental model is based on Context-based Fuzzy C-Means (CFCM) clustering algorithm [8]This clustering algo-rithm generates information granules in the form of fuzzysets and estimate clusters by preserving the homogeneityof the clustered data points associated with the input andoutput variables In contrast to the context-free clusteringmethods [9ndash12] context-based fuzzy clustering is performedwith the use of the contexts produced in output space Theeffectiveness of the CFCM clustering has been successfullydemonstrated in the previous works [13ndash16]
In this paper we design the variant model with the fun-damental idea of incremental model For this purpose wedesign an Incremental Radial Basis Function Network(IRBFN) by incorporating LR and local RBFN for accuratequantitative prediction of energy performance of residentialbuildings We adopt a design of a LR as global model and
Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 3207627 6 pageshttpdxdoiorg10115520163207627
2 Computational Intelligence and Neuroscience
refine it through local RBFN that captures remaining andmore localized nonlinearities of the system with the aid ofinformation granulation Here the learning methods of localRBFN are performed by LSE and Back-Propagation (BP)This research on the topic of energy performance of buildingshas been recently raising concerns about energy waste Thecomputation of the heating and cooling load to performthe efficient building design is required to determine thespecifications of the heating and cooling equipment neededto maintain comfortable indoor air conditions [17 18] Theexperiments are performed on the estimation of energyperformance of 768 diverse residential buildings
This paper is organized in the following fashion InSection 2 the procedure steps of CFCM clustering methodsare described The entire design concept and process ofIRBFN are proposed in Section 3 The experimental resultsare performed and discussed in Section 4 Concluding com-ments are covered in Section 5
2 Context-Based Fuzzy C-Means Clustering
TheCFCMclustering introduced by Pedrycz [8] estimates thecluster centers preserving homogeneity with the use of fuzzygranulation First the contexts are produced from the outputvariable used in the modeling problem Next the cluster cen-ters are estimated by FCM clustering from input data pointsincluded in each context By forming fuzzy clusters in inputand output spaces CFCM clustering converts numerical datainto semantically meaningful information granules
In what follows we briefly describe the essence of CFCMclustering [8] In a batch-mode operation this clusteringdetermines the cluster centers and themembershipmatrix bythe following steps
Step 1 Select the number of contexts 119901 and cluster center 119888 ineach context respectively
Step 2 Produce the contexts in output space These contextswere generated through a series of triangular membershipfunctions equally spaced along the domain of an output vari-able However wemay encounter a data scarcity problem dueto small data included in some linguistic context Thus thisproblembrings about the difficulty to obtain clusters from theCFCMclusteringTherefore we use probabilistic distributionof output variable to produce the flexible linguistic contexts[13]
Step 3 Once the contexts have been formed the clustering isdirected by the provided fuzzy set of each context
Step 4 Initialize the membership matrix with random valuesbetween 0 and 1
Step 5 Compute 119888 fuzzy cluster centers using (1)Here fuzzification factor is generally used as fixed value119898 = 2
k119894=sum119873
119896=1119906119898
119894119896x119896
sum119873
119896=1119906119898119894119896
(1)
Local RBFN
Linear regression
Local RBFN
Figure 1 Combination of LR and local RBFN in the design ofincremental RBFN (o data points)
Step 6 Compute the objective function according to
119869 =
119888
sum
119894=1
119873
sum
119896=1
119906119898
1198941198961198892
119894119896 (2)
where 119889119894119896
is the Euclidean distance between 1198941015840th clustercenter and 1198961015840th data point The minimization of objectivefunction is obtained by iteratively updating the values of themembership matrix and cluster centers Stop if it is below acertain tolerance value
Step 7 Compute a new membership matrix using (3) Go toStep 5
119906119905119894119896=
119908119905119896
sum119888
119895=1(1003817100381710038171003817x119896 minus k1198941003817100381710038171003817 10038171003817100381710038171003817x119896minus k119895
10038171003817100381710038171003817)2(119898minus1) (3)
where 119906119905119894119896
represents the element of the membership matrixinduced by the 119894th cluster and 119896th data in the 119905th context119908
119905119896
denotes a membership value of the 119896th data point includedby the 119905th context
3 Incremental Radial Basis FunctionNetworks (IRBFN)
In this Section we focus on two essential phases of theproposed IRBFN (Incremental RBFN) as underlying princi-ple First we design a standard LR which could be treatedas a preliminary construct capturing the linear part of thedata Next the local RBFN is designed to eliminate errorsproduced by the regression part of the model Figure 1 showsthe example of nonlinear relationships and their modelingthrough a combination of LRmodel of a global character anda collection of local RBFN As shown in Figure 1 the LinearRegression exhibits a good match except for two local areasThese remaining regions are predicted by local RBFN withthe use of information granules through CFCM clusteringalgorithm Figure 2 shows the architecture and overall flowof processing realized in the design of the proposed IRBFN
The principle of the IRBFN is explained in the followingsteps
Step 1 Design of Linear Regression (LR) model in the input-output space 119911 = L(x b) with b denoting a vector of the
Computational Intelligence and Neuroscience 3
Linear regressionz
Y
x1
x2
Ew
1
w2
w3
w4
c1
c2
c3
c4
R1
R2
R3
R4
osum
sum
RBFN based onCFCM clustering
Figure 2 Main design process of incremental RBFN
regression hyperplane of the linear model b = [a 1198860]119879 thus
we obtain the predicted output by using LR as a global model[6] on the basis of the original data set a collection of input-error pairs is formed (x
119896 119890119896)
Step 2 Construction of the collection of contexts in the errorof the regressionmodel (119864
1 1198642 119864
119901) here 119901 is the number
of contexts the distribution of these fuzzy sets is obtainedthrough statistical method [7 8] mentioned in Section 2the contexts are characterized by triangular membershipfunctions with a 05 overlap between neighboring fuzzy sets
Step 3 CFCMclustering completed in the input-output spacefrom the contexts produced in the error space the obtainedcluster centers are used as the centers of receptive fields in thedesign of local RBFN as shown in Figure 1 for 119901 contexts and119888 clusters for each context the number of nodes in hiddenlayer is 119888 times 119901
Step 4 Calculation of output in the design of local RBFN thefinal output of RBFN is the weighted sum of the output valueassociated with each receptive field as follows
119900 =
119888times119901
sum
119894=1
119908119894119888119894 (4)
The receptive field functions are fixed and then the weightsof the output layer are directly estimated by LSE (Least SquareEstimate) and BP (Back-Propagation) These methods areknown as the most representative techniques frequently usedin conjunctionwith RBFN [2] In order to learn and adapt thearchitecture of RBFN to cope with changing environmentswe need BP learning if we use the steepest descent methodto tune the centers of radial basis function and the outputweights in the design of RBFN Otherwise we can directlyobtain the output weights as one-pass estimation usingLSE
Table 1 Optimal values of the fuzzification coefficient for selectednumber of contexts (p) and clusters (c)
p c3 4 5 6
3 26 21 22 234 22 21 22 25 19 19 19 26 17 18 18 2
Step 5 Calculation of final output of the proposed IRBFNthe granular result of the IRBFN is combined with the outputof the linear part
119884 = 119911 + 119864 (5)
In order to evaluate the overall performance we use standardroot mean square error (RMSE) defined as follows
RMSE = radic 1119873
119873
sum
119896=1
(119884119896minus 119910119896)2 (6)
4 Experimental Results
In the experiments we report on the design and performanceof the proposedmodels to assess the heating load and coolingload requirements of building as a function of buildingparameters All experiments were completed in the 10-foldcross-validation mode with a typical 60ndash40 split betweenthe training and testing data subsets We perform energyanalysis using 12 different building shapes simulated inEcotect [17 18]
These 12 building forms were generated by taking theelementary cube (35 times 35 times 35) where each building form iscomposed of 18 elementsThematerials used for each elemen-tary cube are the same for all building forms The selectionwas made by the newest and most common materials in thebuilding construction industry and by the lowest 119880-value[17] The buildings differ with respect to the glazing areathe glazing area distribution orientation overall height roofarea wall area surface area and relative compactness Thedata set comprises 768 samples and 8 features The attributesto be predicted in terms of the preceding 8 input attributes aretwo real valued responses (heating load and cooling load)
We obtained the experimental results with the twoessential parameters (119901 119888) controlling the granularity of theconstruct in the input and output spaceThe numerical rangeof the fuzzification factor (119898) used in the experiments isbetween 15 and 30 with the incremental step of 01 Table 1listed the optimal values of the fuzzification factor by theincrease of the number of contexts and clusters Figure 3shows the variation of the RMSE caused by the fuzzificationfactor in the case of 119901 = 119888 = 6 for heating load predictionHere the optimal values of the parameters are such that thetesting error becomes minimal
In the conventional method [14] the contexts wereproduced through triangular membership functions equally
4 Computational Intelligence and Neuroscience
1 15 2 25 3 35 421
22
23
24
25
26
27
28
29
3
31
Fuzzification factor (m)
RMSE
RMSE (training)RMSE (testing)
Figure 3 RMSE variation by the values of the fuzzification coeffi-cient (119901 = 119888 = 6)
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 100
20
40
60
80
100
120
140
160
Error
Num
ber o
f cou
nts
Figure 4 Histogram in the error space
spaced along the domain of an output variable However wemay encounter a data scarcity problem due to small amountsof data included in some contextThus we use a probabilisticdistribution of the output variable to obtain flexible contextsFor this the contexts in the error space are producedbased on a histogram shown in Figure 4 Probability DensityFunction (PDF) and Conditional Density Function (CDF)[13] Figure 5 shows the contexts (119901 = 6) generated inthe error space of LR model as one example among 10-fold cross-validation mode Figure 6 shows the predictionperformance based on local RBFN for the error of LR Asshown in Figure 6 the result clearly shows that the localRBFN with the use of information granulation has goodprediction capability Figures 7 and 8 show the performance
minus6 minus4 minus2 0 2 4 6 80
02
04
06
08
1
Error of LR
Deg
ree o
f mem
bers
hip
Figure 5 Contexts generated in the error space
0 50 100 150 200 250 300minus20
minus15
minus10
minus5
0
5
10
15
20
Number of testing data
Erro
r
Error of LRPredicted error of local RBFN
Figure 6 Prediction by local RBFN for the error of LR
0 50 100 150 200 250 3005
10
15
20
25
30
35
40
45
Number of testing data
Hea
ting
load
Desired outputIRBFN output
Figure 7 Prediction performance for heating load
Computational Intelligence and Neuroscience 5
0 50 100 150 200 250 30010
15
20
25
30
35
40
45
50
Number of testing data
Coo
ling
load
Desired outputIRBFN output
Figure 8 Prediction performance for cooling load
Table 2 Comparison results of RMSE for the prediction of heatingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 2936 2911MLP 36 2883 2890RBFN 36 3707 5199RBFN (CFCM) [14] 36 2767 3106LM 36lowast 4084 4388Proposed incrementalmethod
IRBFN LSE 36 2284 2826IRBFN BP 36 2353 2730
Table 3 Comparison results of RMSE for the prediction of coolingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 3180 3208MLP 36 3176 3226RBFN 36 3601 4812RBFN (CFCM) [14] 36 2866 3388LM 36lowast 3866 4296Proposed incrementalmethod
IRBFN LSE 36 2462 3102IRBFN BP 36 2555 3089
of IRBFN based on LSE for the prediction of heating loadand cooling load respectively Here the number of epochs is1000 and learning rate is 001 respectively As shown in thesefigures the proposed IRBFN showed good generalizationcapability for testing data set respectively Tables 2 and 3listed the comparison results of RMSE for the prediction
of heating and cooling load respectively As listed in thesetables the experimental results revealed that the proposedIRBFN showed good performance in comparison to LRMLP(Multilayer Perceptron) the conventional RBFN RBFN withCFCM clustering and LM (Linguistic Model)
5 Conclusions
We developed the incremental RBFN by combining LRand local RBFN for the prediction of heating load andcooling load of residential buildings It was found from theresult that the proposed IRBFN has good approximationand generalization capabilities with the aid of informationgranulation These results lead us to the conclusion that theproposed IRBFN combined by LR and local RBFN showeda good performance in comparison to the previous worksFor further research we shall design this model to optimizethe number of contexts and clusters per context based onevolutionary algorithm
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by research funds from ChosunUniversity 2012
References
[1] D Simon Evolutionary Optimization Algorithms JohnWiley ampSons 2013
[2] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy andSoft Computing A Computational Approach to Learning andMachine Intelligence Prentice Hall New York NY USA 1997
[3] S Sumathi and S Panneerselvam Computational IntelligenceParadigmsTheory andApplicationsUsingMATLAB CRCPressNew York NY USA 2010
[4] T P Trappenberg Fundamentals of Computational Neuro-science OxfordUniversity Press Oxford UK 2nd edition 2010
[5] G A F Seber Linear Regression Analysis Wiley Series in Prob-ability and Mathematical Statistics John Wiley amp Sons NewYork NY USA 1977
[6] W Pedrycz and K-C Kwak ldquoThe development of incrementalmodelsrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp507ndash518 2007
[7] W Pedrycz and F Gomide Fuzzy Systems Engineering TowardHuman-Centric Computing Wiley-Interscience 2007
[8] W Pedrycz ldquoConditional fuzzy C-meansrdquo Pattern RecognitionLetters vol 17 no 6 pp 625ndash631 1996
[9] L Hu and K C Chan ldquoFuzzy clustering in a complex networkbased on content relevance and link structuresrdquo IEEE Transac-tions on Fuzzy Systems vol 24 no 2 pp 456ndash470 2016
[10] P Fazendeiro and J V De Oliveira ldquoObserver-biased fuzzyclusteringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 1pp 85ndash97 2015
[11] T C Glenn A Zare and P D Gader ldquoBayesian fuzzy clus-teringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1545ndash1561 2015
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
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2 Computational Intelligence and Neuroscience
refine it through local RBFN that captures remaining andmore localized nonlinearities of the system with the aid ofinformation granulation Here the learning methods of localRBFN are performed by LSE and Back-Propagation (BP)This research on the topic of energy performance of buildingshas been recently raising concerns about energy waste Thecomputation of the heating and cooling load to performthe efficient building design is required to determine thespecifications of the heating and cooling equipment neededto maintain comfortable indoor air conditions [17 18] Theexperiments are performed on the estimation of energyperformance of 768 diverse residential buildings
This paper is organized in the following fashion InSection 2 the procedure steps of CFCM clustering methodsare described The entire design concept and process ofIRBFN are proposed in Section 3 The experimental resultsare performed and discussed in Section 4 Concluding com-ments are covered in Section 5
2 Context-Based Fuzzy C-Means Clustering
TheCFCMclustering introduced by Pedrycz [8] estimates thecluster centers preserving homogeneity with the use of fuzzygranulation First the contexts are produced from the outputvariable used in the modeling problem Next the cluster cen-ters are estimated by FCM clustering from input data pointsincluded in each context By forming fuzzy clusters in inputand output spaces CFCM clustering converts numerical datainto semantically meaningful information granules
In what follows we briefly describe the essence of CFCMclustering [8] In a batch-mode operation this clusteringdetermines the cluster centers and themembershipmatrix bythe following steps
Step 1 Select the number of contexts 119901 and cluster center 119888 ineach context respectively
Step 2 Produce the contexts in output space These contextswere generated through a series of triangular membershipfunctions equally spaced along the domain of an output vari-able However wemay encounter a data scarcity problem dueto small data included in some linguistic context Thus thisproblembrings about the difficulty to obtain clusters from theCFCMclusteringTherefore we use probabilistic distributionof output variable to produce the flexible linguistic contexts[13]
Step 3 Once the contexts have been formed the clustering isdirected by the provided fuzzy set of each context
Step 4 Initialize the membership matrix with random valuesbetween 0 and 1
Step 5 Compute 119888 fuzzy cluster centers using (1)Here fuzzification factor is generally used as fixed value119898 = 2
k119894=sum119873
119896=1119906119898
119894119896x119896
sum119873
119896=1119906119898119894119896
(1)
Local RBFN
Linear regression
Local RBFN
Figure 1 Combination of LR and local RBFN in the design ofincremental RBFN (o data points)
Step 6 Compute the objective function according to
119869 =
119888
sum
119894=1
119873
sum
119896=1
119906119898
1198941198961198892
119894119896 (2)
where 119889119894119896
is the Euclidean distance between 1198941015840th clustercenter and 1198961015840th data point The minimization of objectivefunction is obtained by iteratively updating the values of themembership matrix and cluster centers Stop if it is below acertain tolerance value
Step 7 Compute a new membership matrix using (3) Go toStep 5
119906119905119894119896=
119908119905119896
sum119888
119895=1(1003817100381710038171003817x119896 minus k1198941003817100381710038171003817 10038171003817100381710038171003817x119896minus k119895
10038171003817100381710038171003817)2(119898minus1) (3)
where 119906119905119894119896
represents the element of the membership matrixinduced by the 119894th cluster and 119896th data in the 119905th context119908
119905119896
denotes a membership value of the 119896th data point includedby the 119905th context
3 Incremental Radial Basis FunctionNetworks (IRBFN)
In this Section we focus on two essential phases of theproposed IRBFN (Incremental RBFN) as underlying princi-ple First we design a standard LR which could be treatedas a preliminary construct capturing the linear part of thedata Next the local RBFN is designed to eliminate errorsproduced by the regression part of the model Figure 1 showsthe example of nonlinear relationships and their modelingthrough a combination of LRmodel of a global character anda collection of local RBFN As shown in Figure 1 the LinearRegression exhibits a good match except for two local areasThese remaining regions are predicted by local RBFN withthe use of information granules through CFCM clusteringalgorithm Figure 2 shows the architecture and overall flowof processing realized in the design of the proposed IRBFN
The principle of the IRBFN is explained in the followingsteps
Step 1 Design of Linear Regression (LR) model in the input-output space 119911 = L(x b) with b denoting a vector of the
Computational Intelligence and Neuroscience 3
Linear regressionz
Y
x1
x2
Ew
1
w2
w3
w4
c1
c2
c3
c4
R1
R2
R3
R4
osum
sum
RBFN based onCFCM clustering
Figure 2 Main design process of incremental RBFN
regression hyperplane of the linear model b = [a 1198860]119879 thus
we obtain the predicted output by using LR as a global model[6] on the basis of the original data set a collection of input-error pairs is formed (x
119896 119890119896)
Step 2 Construction of the collection of contexts in the errorof the regressionmodel (119864
1 1198642 119864
119901) here 119901 is the number
of contexts the distribution of these fuzzy sets is obtainedthrough statistical method [7 8] mentioned in Section 2the contexts are characterized by triangular membershipfunctions with a 05 overlap between neighboring fuzzy sets
Step 3 CFCMclustering completed in the input-output spacefrom the contexts produced in the error space the obtainedcluster centers are used as the centers of receptive fields in thedesign of local RBFN as shown in Figure 1 for 119901 contexts and119888 clusters for each context the number of nodes in hiddenlayer is 119888 times 119901
Step 4 Calculation of output in the design of local RBFN thefinal output of RBFN is the weighted sum of the output valueassociated with each receptive field as follows
119900 =
119888times119901
sum
119894=1
119908119894119888119894 (4)
The receptive field functions are fixed and then the weightsof the output layer are directly estimated by LSE (Least SquareEstimate) and BP (Back-Propagation) These methods areknown as the most representative techniques frequently usedin conjunctionwith RBFN [2] In order to learn and adapt thearchitecture of RBFN to cope with changing environmentswe need BP learning if we use the steepest descent methodto tune the centers of radial basis function and the outputweights in the design of RBFN Otherwise we can directlyobtain the output weights as one-pass estimation usingLSE
Table 1 Optimal values of the fuzzification coefficient for selectednumber of contexts (p) and clusters (c)
p c3 4 5 6
3 26 21 22 234 22 21 22 25 19 19 19 26 17 18 18 2
Step 5 Calculation of final output of the proposed IRBFNthe granular result of the IRBFN is combined with the outputof the linear part
119884 = 119911 + 119864 (5)
In order to evaluate the overall performance we use standardroot mean square error (RMSE) defined as follows
RMSE = radic 1119873
119873
sum
119896=1
(119884119896minus 119910119896)2 (6)
4 Experimental Results
In the experiments we report on the design and performanceof the proposedmodels to assess the heating load and coolingload requirements of building as a function of buildingparameters All experiments were completed in the 10-foldcross-validation mode with a typical 60ndash40 split betweenthe training and testing data subsets We perform energyanalysis using 12 different building shapes simulated inEcotect [17 18]
These 12 building forms were generated by taking theelementary cube (35 times 35 times 35) where each building form iscomposed of 18 elementsThematerials used for each elemen-tary cube are the same for all building forms The selectionwas made by the newest and most common materials in thebuilding construction industry and by the lowest 119880-value[17] The buildings differ with respect to the glazing areathe glazing area distribution orientation overall height roofarea wall area surface area and relative compactness Thedata set comprises 768 samples and 8 features The attributesto be predicted in terms of the preceding 8 input attributes aretwo real valued responses (heating load and cooling load)
We obtained the experimental results with the twoessential parameters (119901 119888) controlling the granularity of theconstruct in the input and output spaceThe numerical rangeof the fuzzification factor (119898) used in the experiments isbetween 15 and 30 with the incremental step of 01 Table 1listed the optimal values of the fuzzification factor by theincrease of the number of contexts and clusters Figure 3shows the variation of the RMSE caused by the fuzzificationfactor in the case of 119901 = 119888 = 6 for heating load predictionHere the optimal values of the parameters are such that thetesting error becomes minimal
In the conventional method [14] the contexts wereproduced through triangular membership functions equally
4 Computational Intelligence and Neuroscience
1 15 2 25 3 35 421
22
23
24
25
26
27
28
29
3
31
Fuzzification factor (m)
RMSE
RMSE (training)RMSE (testing)
Figure 3 RMSE variation by the values of the fuzzification coeffi-cient (119901 = 119888 = 6)
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 100
20
40
60
80
100
120
140
160
Error
Num
ber o
f cou
nts
Figure 4 Histogram in the error space
spaced along the domain of an output variable However wemay encounter a data scarcity problem due to small amountsof data included in some contextThus we use a probabilisticdistribution of the output variable to obtain flexible contextsFor this the contexts in the error space are producedbased on a histogram shown in Figure 4 Probability DensityFunction (PDF) and Conditional Density Function (CDF)[13] Figure 5 shows the contexts (119901 = 6) generated inthe error space of LR model as one example among 10-fold cross-validation mode Figure 6 shows the predictionperformance based on local RBFN for the error of LR Asshown in Figure 6 the result clearly shows that the localRBFN with the use of information granulation has goodprediction capability Figures 7 and 8 show the performance
minus6 minus4 minus2 0 2 4 6 80
02
04
06
08
1
Error of LR
Deg
ree o
f mem
bers
hip
Figure 5 Contexts generated in the error space
0 50 100 150 200 250 300minus20
minus15
minus10
minus5
0
5
10
15
20
Number of testing data
Erro
r
Error of LRPredicted error of local RBFN
Figure 6 Prediction by local RBFN for the error of LR
0 50 100 150 200 250 3005
10
15
20
25
30
35
40
45
Number of testing data
Hea
ting
load
Desired outputIRBFN output
Figure 7 Prediction performance for heating load
Computational Intelligence and Neuroscience 5
0 50 100 150 200 250 30010
15
20
25
30
35
40
45
50
Number of testing data
Coo
ling
load
Desired outputIRBFN output
Figure 8 Prediction performance for cooling load
Table 2 Comparison results of RMSE for the prediction of heatingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 2936 2911MLP 36 2883 2890RBFN 36 3707 5199RBFN (CFCM) [14] 36 2767 3106LM 36lowast 4084 4388Proposed incrementalmethod
IRBFN LSE 36 2284 2826IRBFN BP 36 2353 2730
Table 3 Comparison results of RMSE for the prediction of coolingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 3180 3208MLP 36 3176 3226RBFN 36 3601 4812RBFN (CFCM) [14] 36 2866 3388LM 36lowast 3866 4296Proposed incrementalmethod
IRBFN LSE 36 2462 3102IRBFN BP 36 2555 3089
of IRBFN based on LSE for the prediction of heating loadand cooling load respectively Here the number of epochs is1000 and learning rate is 001 respectively As shown in thesefigures the proposed IRBFN showed good generalizationcapability for testing data set respectively Tables 2 and 3listed the comparison results of RMSE for the prediction
of heating and cooling load respectively As listed in thesetables the experimental results revealed that the proposedIRBFN showed good performance in comparison to LRMLP(Multilayer Perceptron) the conventional RBFN RBFN withCFCM clustering and LM (Linguistic Model)
5 Conclusions
We developed the incremental RBFN by combining LRand local RBFN for the prediction of heating load andcooling load of residential buildings It was found from theresult that the proposed IRBFN has good approximationand generalization capabilities with the aid of informationgranulation These results lead us to the conclusion that theproposed IRBFN combined by LR and local RBFN showeda good performance in comparison to the previous worksFor further research we shall design this model to optimizethe number of contexts and clusters per context based onevolutionary algorithm
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by research funds from ChosunUniversity 2012
References
[1] D Simon Evolutionary Optimization Algorithms JohnWiley ampSons 2013
[2] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy andSoft Computing A Computational Approach to Learning andMachine Intelligence Prentice Hall New York NY USA 1997
[3] S Sumathi and S Panneerselvam Computational IntelligenceParadigmsTheory andApplicationsUsingMATLAB CRCPressNew York NY USA 2010
[4] T P Trappenberg Fundamentals of Computational Neuro-science OxfordUniversity Press Oxford UK 2nd edition 2010
[5] G A F Seber Linear Regression Analysis Wiley Series in Prob-ability and Mathematical Statistics John Wiley amp Sons NewYork NY USA 1977
[6] W Pedrycz and K-C Kwak ldquoThe development of incrementalmodelsrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp507ndash518 2007
[7] W Pedrycz and F Gomide Fuzzy Systems Engineering TowardHuman-Centric Computing Wiley-Interscience 2007
[8] W Pedrycz ldquoConditional fuzzy C-meansrdquo Pattern RecognitionLetters vol 17 no 6 pp 625ndash631 1996
[9] L Hu and K C Chan ldquoFuzzy clustering in a complex networkbased on content relevance and link structuresrdquo IEEE Transac-tions on Fuzzy Systems vol 24 no 2 pp 456ndash470 2016
[10] P Fazendeiro and J V De Oliveira ldquoObserver-biased fuzzyclusteringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 1pp 85ndash97 2015
[11] T C Glenn A Zare and P D Gader ldquoBayesian fuzzy clus-teringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1545ndash1561 2015
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience 3
Linear regressionz
Y
x1
x2
Ew
1
w2
w3
w4
c1
c2
c3
c4
R1
R2
R3
R4
osum
sum
RBFN based onCFCM clustering
Figure 2 Main design process of incremental RBFN
regression hyperplane of the linear model b = [a 1198860]119879 thus
we obtain the predicted output by using LR as a global model[6] on the basis of the original data set a collection of input-error pairs is formed (x
119896 119890119896)
Step 2 Construction of the collection of contexts in the errorof the regressionmodel (119864
1 1198642 119864
119901) here 119901 is the number
of contexts the distribution of these fuzzy sets is obtainedthrough statistical method [7 8] mentioned in Section 2the contexts are characterized by triangular membershipfunctions with a 05 overlap between neighboring fuzzy sets
Step 3 CFCMclustering completed in the input-output spacefrom the contexts produced in the error space the obtainedcluster centers are used as the centers of receptive fields in thedesign of local RBFN as shown in Figure 1 for 119901 contexts and119888 clusters for each context the number of nodes in hiddenlayer is 119888 times 119901
Step 4 Calculation of output in the design of local RBFN thefinal output of RBFN is the weighted sum of the output valueassociated with each receptive field as follows
119900 =
119888times119901
sum
119894=1
119908119894119888119894 (4)
The receptive field functions are fixed and then the weightsof the output layer are directly estimated by LSE (Least SquareEstimate) and BP (Back-Propagation) These methods areknown as the most representative techniques frequently usedin conjunctionwith RBFN [2] In order to learn and adapt thearchitecture of RBFN to cope with changing environmentswe need BP learning if we use the steepest descent methodto tune the centers of radial basis function and the outputweights in the design of RBFN Otherwise we can directlyobtain the output weights as one-pass estimation usingLSE
Table 1 Optimal values of the fuzzification coefficient for selectednumber of contexts (p) and clusters (c)
p c3 4 5 6
3 26 21 22 234 22 21 22 25 19 19 19 26 17 18 18 2
Step 5 Calculation of final output of the proposed IRBFNthe granular result of the IRBFN is combined with the outputof the linear part
119884 = 119911 + 119864 (5)
In order to evaluate the overall performance we use standardroot mean square error (RMSE) defined as follows
RMSE = radic 1119873
119873
sum
119896=1
(119884119896minus 119910119896)2 (6)
4 Experimental Results
In the experiments we report on the design and performanceof the proposedmodels to assess the heating load and coolingload requirements of building as a function of buildingparameters All experiments were completed in the 10-foldcross-validation mode with a typical 60ndash40 split betweenthe training and testing data subsets We perform energyanalysis using 12 different building shapes simulated inEcotect [17 18]
These 12 building forms were generated by taking theelementary cube (35 times 35 times 35) where each building form iscomposed of 18 elementsThematerials used for each elemen-tary cube are the same for all building forms The selectionwas made by the newest and most common materials in thebuilding construction industry and by the lowest 119880-value[17] The buildings differ with respect to the glazing areathe glazing area distribution orientation overall height roofarea wall area surface area and relative compactness Thedata set comprises 768 samples and 8 features The attributesto be predicted in terms of the preceding 8 input attributes aretwo real valued responses (heating load and cooling load)
We obtained the experimental results with the twoessential parameters (119901 119888) controlling the granularity of theconstruct in the input and output spaceThe numerical rangeof the fuzzification factor (119898) used in the experiments isbetween 15 and 30 with the incremental step of 01 Table 1listed the optimal values of the fuzzification factor by theincrease of the number of contexts and clusters Figure 3shows the variation of the RMSE caused by the fuzzificationfactor in the case of 119901 = 119888 = 6 for heating load predictionHere the optimal values of the parameters are such that thetesting error becomes minimal
In the conventional method [14] the contexts wereproduced through triangular membership functions equally
4 Computational Intelligence and Neuroscience
1 15 2 25 3 35 421
22
23
24
25
26
27
28
29
3
31
Fuzzification factor (m)
RMSE
RMSE (training)RMSE (testing)
Figure 3 RMSE variation by the values of the fuzzification coeffi-cient (119901 = 119888 = 6)
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 100
20
40
60
80
100
120
140
160
Error
Num
ber o
f cou
nts
Figure 4 Histogram in the error space
spaced along the domain of an output variable However wemay encounter a data scarcity problem due to small amountsof data included in some contextThus we use a probabilisticdistribution of the output variable to obtain flexible contextsFor this the contexts in the error space are producedbased on a histogram shown in Figure 4 Probability DensityFunction (PDF) and Conditional Density Function (CDF)[13] Figure 5 shows the contexts (119901 = 6) generated inthe error space of LR model as one example among 10-fold cross-validation mode Figure 6 shows the predictionperformance based on local RBFN for the error of LR Asshown in Figure 6 the result clearly shows that the localRBFN with the use of information granulation has goodprediction capability Figures 7 and 8 show the performance
minus6 minus4 minus2 0 2 4 6 80
02
04
06
08
1
Error of LR
Deg
ree o
f mem
bers
hip
Figure 5 Contexts generated in the error space
0 50 100 150 200 250 300minus20
minus15
minus10
minus5
0
5
10
15
20
Number of testing data
Erro
r
Error of LRPredicted error of local RBFN
Figure 6 Prediction by local RBFN for the error of LR
0 50 100 150 200 250 3005
10
15
20
25
30
35
40
45
Number of testing data
Hea
ting
load
Desired outputIRBFN output
Figure 7 Prediction performance for heating load
Computational Intelligence and Neuroscience 5
0 50 100 150 200 250 30010
15
20
25
30
35
40
45
50
Number of testing data
Coo
ling
load
Desired outputIRBFN output
Figure 8 Prediction performance for cooling load
Table 2 Comparison results of RMSE for the prediction of heatingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 2936 2911MLP 36 2883 2890RBFN 36 3707 5199RBFN (CFCM) [14] 36 2767 3106LM 36lowast 4084 4388Proposed incrementalmethod
IRBFN LSE 36 2284 2826IRBFN BP 36 2353 2730
Table 3 Comparison results of RMSE for the prediction of coolingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 3180 3208MLP 36 3176 3226RBFN 36 3601 4812RBFN (CFCM) [14] 36 2866 3388LM 36lowast 3866 4296Proposed incrementalmethod
IRBFN LSE 36 2462 3102IRBFN BP 36 2555 3089
of IRBFN based on LSE for the prediction of heating loadand cooling load respectively Here the number of epochs is1000 and learning rate is 001 respectively As shown in thesefigures the proposed IRBFN showed good generalizationcapability for testing data set respectively Tables 2 and 3listed the comparison results of RMSE for the prediction
of heating and cooling load respectively As listed in thesetables the experimental results revealed that the proposedIRBFN showed good performance in comparison to LRMLP(Multilayer Perceptron) the conventional RBFN RBFN withCFCM clustering and LM (Linguistic Model)
5 Conclusions
We developed the incremental RBFN by combining LRand local RBFN for the prediction of heating load andcooling load of residential buildings It was found from theresult that the proposed IRBFN has good approximationand generalization capabilities with the aid of informationgranulation These results lead us to the conclusion that theproposed IRBFN combined by LR and local RBFN showeda good performance in comparison to the previous worksFor further research we shall design this model to optimizethe number of contexts and clusters per context based onevolutionary algorithm
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by research funds from ChosunUniversity 2012
References
[1] D Simon Evolutionary Optimization Algorithms JohnWiley ampSons 2013
[2] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy andSoft Computing A Computational Approach to Learning andMachine Intelligence Prentice Hall New York NY USA 1997
[3] S Sumathi and S Panneerselvam Computational IntelligenceParadigmsTheory andApplicationsUsingMATLAB CRCPressNew York NY USA 2010
[4] T P Trappenberg Fundamentals of Computational Neuro-science OxfordUniversity Press Oxford UK 2nd edition 2010
[5] G A F Seber Linear Regression Analysis Wiley Series in Prob-ability and Mathematical Statistics John Wiley amp Sons NewYork NY USA 1977
[6] W Pedrycz and K-C Kwak ldquoThe development of incrementalmodelsrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp507ndash518 2007
[7] W Pedrycz and F Gomide Fuzzy Systems Engineering TowardHuman-Centric Computing Wiley-Interscience 2007
[8] W Pedrycz ldquoConditional fuzzy C-meansrdquo Pattern RecognitionLetters vol 17 no 6 pp 625ndash631 1996
[9] L Hu and K C Chan ldquoFuzzy clustering in a complex networkbased on content relevance and link structuresrdquo IEEE Transac-tions on Fuzzy Systems vol 24 no 2 pp 456ndash470 2016
[10] P Fazendeiro and J V De Oliveira ldquoObserver-biased fuzzyclusteringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 1pp 85ndash97 2015
[11] T C Glenn A Zare and P D Gader ldquoBayesian fuzzy clus-teringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1545ndash1561 2015
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
4 Computational Intelligence and Neuroscience
1 15 2 25 3 35 421
22
23
24
25
26
27
28
29
3
31
Fuzzification factor (m)
RMSE
RMSE (training)RMSE (testing)
Figure 3 RMSE variation by the values of the fuzzification coeffi-cient (119901 = 119888 = 6)
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 100
20
40
60
80
100
120
140
160
Error
Num
ber o
f cou
nts
Figure 4 Histogram in the error space
spaced along the domain of an output variable However wemay encounter a data scarcity problem due to small amountsof data included in some contextThus we use a probabilisticdistribution of the output variable to obtain flexible contextsFor this the contexts in the error space are producedbased on a histogram shown in Figure 4 Probability DensityFunction (PDF) and Conditional Density Function (CDF)[13] Figure 5 shows the contexts (119901 = 6) generated inthe error space of LR model as one example among 10-fold cross-validation mode Figure 6 shows the predictionperformance based on local RBFN for the error of LR Asshown in Figure 6 the result clearly shows that the localRBFN with the use of information granulation has goodprediction capability Figures 7 and 8 show the performance
minus6 minus4 minus2 0 2 4 6 80
02
04
06
08
1
Error of LR
Deg
ree o
f mem
bers
hip
Figure 5 Contexts generated in the error space
0 50 100 150 200 250 300minus20
minus15
minus10
minus5
0
5
10
15
20
Number of testing data
Erro
r
Error of LRPredicted error of local RBFN
Figure 6 Prediction by local RBFN for the error of LR
0 50 100 150 200 250 3005
10
15
20
25
30
35
40
45
Number of testing data
Hea
ting
load
Desired outputIRBFN output
Figure 7 Prediction performance for heating load
Computational Intelligence and Neuroscience 5
0 50 100 150 200 250 30010
15
20
25
30
35
40
45
50
Number of testing data
Coo
ling
load
Desired outputIRBFN output
Figure 8 Prediction performance for cooling load
Table 2 Comparison results of RMSE for the prediction of heatingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 2936 2911MLP 36 2883 2890RBFN 36 3707 5199RBFN (CFCM) [14] 36 2767 3106LM 36lowast 4084 4388Proposed incrementalmethod
IRBFN LSE 36 2284 2826IRBFN BP 36 2353 2730
Table 3 Comparison results of RMSE for the prediction of coolingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 3180 3208MLP 36 3176 3226RBFN 36 3601 4812RBFN (CFCM) [14] 36 2866 3388LM 36lowast 3866 4296Proposed incrementalmethod
IRBFN LSE 36 2462 3102IRBFN BP 36 2555 3089
of IRBFN based on LSE for the prediction of heating loadand cooling load respectively Here the number of epochs is1000 and learning rate is 001 respectively As shown in thesefigures the proposed IRBFN showed good generalizationcapability for testing data set respectively Tables 2 and 3listed the comparison results of RMSE for the prediction
of heating and cooling load respectively As listed in thesetables the experimental results revealed that the proposedIRBFN showed good performance in comparison to LRMLP(Multilayer Perceptron) the conventional RBFN RBFN withCFCM clustering and LM (Linguistic Model)
5 Conclusions
We developed the incremental RBFN by combining LRand local RBFN for the prediction of heating load andcooling load of residential buildings It was found from theresult that the proposed IRBFN has good approximationand generalization capabilities with the aid of informationgranulation These results lead us to the conclusion that theproposed IRBFN combined by LR and local RBFN showeda good performance in comparison to the previous worksFor further research we shall design this model to optimizethe number of contexts and clusters per context based onevolutionary algorithm
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by research funds from ChosunUniversity 2012
References
[1] D Simon Evolutionary Optimization Algorithms JohnWiley ampSons 2013
[2] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy andSoft Computing A Computational Approach to Learning andMachine Intelligence Prentice Hall New York NY USA 1997
[3] S Sumathi and S Panneerselvam Computational IntelligenceParadigmsTheory andApplicationsUsingMATLAB CRCPressNew York NY USA 2010
[4] T P Trappenberg Fundamentals of Computational Neuro-science OxfordUniversity Press Oxford UK 2nd edition 2010
[5] G A F Seber Linear Regression Analysis Wiley Series in Prob-ability and Mathematical Statistics John Wiley amp Sons NewYork NY USA 1977
[6] W Pedrycz and K-C Kwak ldquoThe development of incrementalmodelsrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp507ndash518 2007
[7] W Pedrycz and F Gomide Fuzzy Systems Engineering TowardHuman-Centric Computing Wiley-Interscience 2007
[8] W Pedrycz ldquoConditional fuzzy C-meansrdquo Pattern RecognitionLetters vol 17 no 6 pp 625ndash631 1996
[9] L Hu and K C Chan ldquoFuzzy clustering in a complex networkbased on content relevance and link structuresrdquo IEEE Transac-tions on Fuzzy Systems vol 24 no 2 pp 456ndash470 2016
[10] P Fazendeiro and J V De Oliveira ldquoObserver-biased fuzzyclusteringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 1pp 85ndash97 2015
[11] T C Glenn A Zare and P D Gader ldquoBayesian fuzzy clus-teringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1545ndash1561 2015
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience 5
0 50 100 150 200 250 30010
15
20
25
30
35
40
45
50
Number of testing data
Coo
ling
load
Desired outputIRBFN output
Figure 8 Prediction performance for cooling load
Table 2 Comparison results of RMSE for the prediction of heatingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 2936 2911MLP 36 2883 2890RBFN 36 3707 5199RBFN (CFCM) [14] 36 2767 3106LM 36lowast 4084 4388Proposed incrementalmethod
IRBFN LSE 36 2284 2826IRBFN BP 36 2353 2730
Table 3 Comparison results of RMSE for the prediction of coolingload (mean value of 10-fold cross-validation) (lowast number of rules)
Methods Number ofnodes
RMSE(training)
RMSE(testing)
LR mdash 3180 3208MLP 36 3176 3226RBFN 36 3601 4812RBFN (CFCM) [14] 36 2866 3388LM 36lowast 3866 4296Proposed incrementalmethod
IRBFN LSE 36 2462 3102IRBFN BP 36 2555 3089
of IRBFN based on LSE for the prediction of heating loadand cooling load respectively Here the number of epochs is1000 and learning rate is 001 respectively As shown in thesefigures the proposed IRBFN showed good generalizationcapability for testing data set respectively Tables 2 and 3listed the comparison results of RMSE for the prediction
of heating and cooling load respectively As listed in thesetables the experimental results revealed that the proposedIRBFN showed good performance in comparison to LRMLP(Multilayer Perceptron) the conventional RBFN RBFN withCFCM clustering and LM (Linguistic Model)
5 Conclusions
We developed the incremental RBFN by combining LRand local RBFN for the prediction of heating load andcooling load of residential buildings It was found from theresult that the proposed IRBFN has good approximationand generalization capabilities with the aid of informationgranulation These results lead us to the conclusion that theproposed IRBFN combined by LR and local RBFN showeda good performance in comparison to the previous worksFor further research we shall design this model to optimizethe number of contexts and clusters per context based onevolutionary algorithm
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by research funds from ChosunUniversity 2012
References
[1] D Simon Evolutionary Optimization Algorithms JohnWiley ampSons 2013
[2] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy andSoft Computing A Computational Approach to Learning andMachine Intelligence Prentice Hall New York NY USA 1997
[3] S Sumathi and S Panneerselvam Computational IntelligenceParadigmsTheory andApplicationsUsingMATLAB CRCPressNew York NY USA 2010
[4] T P Trappenberg Fundamentals of Computational Neuro-science OxfordUniversity Press Oxford UK 2nd edition 2010
[5] G A F Seber Linear Regression Analysis Wiley Series in Prob-ability and Mathematical Statistics John Wiley amp Sons NewYork NY USA 1977
[6] W Pedrycz and K-C Kwak ldquoThe development of incrementalmodelsrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp507ndash518 2007
[7] W Pedrycz and F Gomide Fuzzy Systems Engineering TowardHuman-Centric Computing Wiley-Interscience 2007
[8] W Pedrycz ldquoConditional fuzzy C-meansrdquo Pattern RecognitionLetters vol 17 no 6 pp 625ndash631 1996
[9] L Hu and K C Chan ldquoFuzzy clustering in a complex networkbased on content relevance and link structuresrdquo IEEE Transac-tions on Fuzzy Systems vol 24 no 2 pp 456ndash470 2016
[10] P Fazendeiro and J V De Oliveira ldquoObserver-biased fuzzyclusteringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 1pp 85ndash97 2015
[11] T C Glenn A Zare and P D Gader ldquoBayesian fuzzy clus-teringrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1545ndash1561 2015
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 Computational Intelligence and Neuroscience
[12] A Proietti L Liparulo and M Panella ldquo2D hierarchical fuzzyclustering using kernel-basedmembership functionsrdquo Electron-ics Letters vol 52 no 3 pp 193ndash195 2016
[13] S-S Kim and K-C Kwak ldquoDevelopment of quantum-basedadaptive neuro-fuzzy networksrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 40 no 1 pp 91ndash100 2010
[14] W Pedrycz and A V Vasilakos ldquoLinguistic models and lin-guistic modelingrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 29 no 6 pp 745ndash757 1999
[15] K-C Kwak ldquoA design of genetically optimized linguistic mod-elsrdquo IEICE Transactions on Information and Systems vol E95-Dno 12 pp 3117ndash3120 2012
[16] W Pedrycz ldquoConditional fuzzy clustering in the design of radialbasis function neural networksrdquo IEEE Transactions on NeuralNetworks vol 9 no 4 pp 601ndash612 1998
[17] L Perez-Lombard J Ortiz and C Pout ldquoA review on buildingsenergy consumption informationrdquo Energy and Buildings vol40 no 3 pp 394ndash398 2008
[18] A Tsanas and A Xifara ldquoAccurate quantitative estimation ofenergy performance of residential buildings using statisticalmachine learning toolsrdquo Energy and Buildings vol 49 pp 560ndash567 2012
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014