Energy Conversion and Management 50 (2009) 1411–1418

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Non-linear HVAC computations using least square support vector machines

Mahendra Kumar *, I.N. KarDepartment of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110 016, India

a r t i c l e i n f o

Article history:Received 30 July 2008Accepted 8 March 2009Available online 8 April 2009

Keywords:HVACThermal comfortPsychrometric chartPrediction modelLeast square support vector machines

0196-8904/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.enconman.2009.03.009

* Corresponding author. Tel.: +91 11 26591093; faxE-mail address: mahendray71@gmail.com (M. Kum

a b s t r a c t

This paper aims to demonstrate application of least square support vector machines (LS-SVM) to modeltwo complex heating, ventilating and air-conditioning (HVAC) relationships. The two applications consid-ered are the estimation of the predicted mean vote (PMV) for thermal comfort and the generation of psy-chrometric chart. LS-SVM has the potential for quick, exact representations and also possesses a structurethat facilitates hardware implementation. The results show very good agreement between function val-ues computed from conventional model and LS-SVM model in real time. The robustness of LS-SVM mod-els against input noises has also been analyzed.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The primary purpose of a HVAC system is to make occupantscomfortable in terms of thermal comfort. The indicator of humanthermal comfort is called the thermal sensation index. A numberof thermal sensation indices have been studied for design of HVACsystem. However, the most widely used thermal sensation index isthe predicted mean vote (PMV).

PMV is a function of six variables. Out of six variables two arehuman variables and remaining four are environmental variables.The human variables are clothing insulation and activity levelwhereas environmental variables are air temperature, relativehumidity, relative air velocity and mean radiant temperature.The values of PMV have a range from �3 to +3, which correspondsto the occupant’s feeling from cold to hot, while the null value ofPMV means neutral. The conventional PMV model given by [1] isa set of non-linear recursive relations and it requires iterativelycomputing the roots of the non-linear equations which may takelong computation time. Therefore, it has been suggested to use ta-bles for determining the thermal sensation index. Similar to PMV,thermodynamic properties of moist air are also very important foranalyzing HVAC system. These properties are also given by non-linear relations and psychrometric chart is used to determine theseproperties. In this case, dry bulb air temperature and relativehumidity are measured by sensor and other properties such asdew point temperature, humidity ratio and enthalpy can find frompsychrometric chart.

ll rights reserved.

: +91 11 26581606.ar).

It is always required to have real time knowledge of PMV index,humidity ratio, dew point temperature and enthalpy of air to de-sign an efficient air-conditioning controller. The conventionalmodels for finding these variables are computationally intensiveso these models are difficult to use in real time application. Thisnecessitates predictive models, which require less computationaleffort, robust against input noise and facilitate hardware realiza-tion. A robust predictive model can be designed by employingfunction approximation methods.

A review of literature reveals that there is extensive research inthe realm of function approximation using soft computing tech-niques [2–8]. A few researchers have contributed to the field offunction approximation for thermal sensation index and psychro-metric variables. Ref. [2] presented modeling of PMV index andpsychrometric variables using back propagation neural networks.In paper [2] out of six input variables two input variables are con-sidered as constant and corresponding to these constant inputvariables correction networks have been added in neural networkarchitecture. Atthajariyakul and Leepahakpreeda [3] have devel-oped feed forward neural network model to capture the relationsof the conventional PMV model of Fanger, quantitatively. The pa-per presented a feed forward neural network model as an explicitfunction of the relation of the PMV index to accessible inputvariables. Mittal and Zhang [4] have presented artificial neuralnetwork based psychrometric predictor. In this paper neuralnetwork technique has been used for computing psychrometricvariables. Literature shows that most of work in function approxi-mation for PMV and psychrometric variables has been done byusing neural networks. The sensitivity of these predictive modelsto noise and uncertainties has not been studied. The inputs tothe models have always some noises and uncertainties in practical

Nomenclature

T air temperatureg mechanical efficiencyfc1 clothing area factorhc convective heat transfer coefficientIcl thermal resistance from skin to outer surface of the

clothed bodyl relative air velocitytc1 mean temperature of outer surface of clothed bodyW3 humidity ratio of thermal spaceWs humidity ratio at saturationW air humidity ratio

T3 dry bulb temperature of thermal spacetmrt mean radiant temperatureM

ADumetabolic Rate

PWS saturation vapour pressuretdb dry bulb temperaturePW vapour pressureU relative humiditytdp dew point temperatureh air enthalpy

1412 M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418

applications. The output of the model in such cases will also getinfluenced by these noises and uncertainties and this is not desir-able for design purposes. The predictive models which are less sen-sitive to noises and uncertainties are always preferred. So therobustness of the models is also an important aspect to beanalyzed.

In the literature, LSSVM models are used to approximate non-linear relationship between input variables and output variableswith certain accuracy [6,7] and [9–12]. LSSVM, originated fromSVM (support vector machines) is a powerful methodology forsolving problems in non-linear classification, function estimationand density estimation. LSSVM is computationally less intensivethan traditional modeling techniques such as BPNN (back propaga-tion neural networks), MLR (multivariate linear regression) andPLS (partial least square regression) and also facilitates hardwareimplementation. The work for computing PMV index and psychro-metric variables using LSSVM has not been identified in open liter-ature. Therefore in the present study, modeling of PMV index andpsychrometric variables has been done by LSSVM. The proposedmodel reduces computational effort as compared to conventionalmodel and shown to be robust against noises. The accuracy ofLS-SVM model in predicting output is found to be excellent. Sothe model can replace conventional model for practical purposesin both applications.

The main contributions of this paper are summarized as below:

(i) LSSVM model has been developed for the prediction of PMVindex in wide range of human variables and environmentalvariables for a HVAC control system. The performance of thismodel has been compared with conventional model given by[1].

(ii) LSSVM model has also been developed to determine psy-chrometric variables such as humidity ratio, dew point tem-perature and enthalpy of air. The performance of this modelhas been compared with conventional model which isderived from standard formulae.

(iii) The robustness of the presented LSSVM models againstnoises and uncertainties has been analyzed by adding noisein testing data and the performance is compared with con-ventional models. The noise which has been added to testingdata resembles with practical measuring uncertainty in themeasurement of input variables.

2. Data acquisition and data preprocessing

The training data set for off-line training of predictive models isobtained from conventional models. The conventional model forcomputing PMV index is given by [1] and the conventional model

for computing psychrometric variables is given by standard formu-lae [13]. Both these models have been briefly discussed in follow-ing subsections.

2.1. Predicted mean vote

PMV is a thermal sensation index given by [1] and is interna-tionally standardized (ISO-1987). It is easy to use the PMV whencontrolling an air-conditioning system because PMV shows the hu-man thermal sensation with one value for all seasons. It is effectivefor all temperatures that are neither extremely high nor extremelylow. The PMV value is zero where thermal sensation is neutral, po-sitive where thermal sensation is warm or hot and negative wherethermal sensation is cool or cold. The PMV value depends upon hu-man factors as well as environmental factors. The environmentalfactors are dry bulb temperature of conditioned air, humidity ofair, mean radiant temperature and air velocity. The human factorsare activity level and thermal resistance of clothes. Thermal sensa-tion indicator and thermal variables which affect PMV are shownin Fig. 1.

The time dependent relationship between these variables andPMV value is given by following equations [1]:

PMV¼ð0:352e�0:042ðM=ADuÞ þ0:032Þ

� MADuð1�gÞ�0:35 43�0:061

MADuð1�gÞ�1204:82W3

� ��

�0:42M

ADuð1�gÞ�50

� ��0:0023

MADuð44�1204:82W3Þ

�0:0014M

ADuð34�T3Þ

�3:4�10�8fc1 ðtc1þ273Þ4�ðtmrtþ273Þ4h i

� fc1hcðtc1�T3Þ�ð1Þ

where tc1 is given by equation

tc1 ¼ 35:7� 0:032M

ADuð1� gÞ � 0:18Icl

� 3:4� 10�8fc1 ðtc1 þ 273Þ4 � ðtmrt þ 273Þ4h i

þ fc1hcðtc1 � T3Þh i

ð2Þand hc by

hc ¼2:05ðtc1 � T3Þ0:25 for 2:05 ðtc1 � T3Þ0:25

> 10:4ffiffiffiffilp

10:4ffiffiffiffilp for 2:05 ðtc1 � T3Þ0:25

< 10:4ffiffiffiffilp

(ð3Þ

The training data set is generated by using Eqs. (1)–(3). The valuesfor dry bulb temperature have been taken between 20 �C and 28 �Cin steps of 0.1 �C, relative humidity between 20% and 95% in steps of7.5%, relative air velocity 0.1–0.5 m/s in steps of 0.1 m/s, clothing

Fig. 1. PMV and thermal sensation.

M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418 1413

insulation 0–1.5 in steps of 0.25. The activity level has beenassumed 50 kcal/hrm2, corresponding to seated, quiet condition.The mean radiant temperature has been assumed equal to dry bulbtemperature. Total 6035 data sets were generated to train the LS-SVM network. The testing data sets are selected in such a way thatthese differ from training data set. The deviation of testing datafrom training data is maximum possible within the upper and lowerbounds of training data. The conventional model uses Eqs. (1)–(3)for computing PMV. The solution of these equations requires a lotof computational effort and time because these are non-linear andrecursive in nature. The hardware realization of this model is alsodifficult. Presently, due to above two reasons the conventional mod-el is difficult to use in real time application.

2.2. Psychrometric Chart

A psychrometric chart is a graphical representation of thermo-dynamic properties of moist air at constant pressure. The chartgraphically expresses how various properties of moist air relateto each other. The psychrometric chart depicts the non-linear rela-tionship between dry bulb temperature, relative humidity, dewpoint temperature, wet bulb temperature, humidity ratio and en-thalpy. By knowing three independent properties of moist air(one of which is pressure), the other properties can be determinedby using psychrometric chart. For generating the training data set,we have used the standard formulae given in [13]. The modelbased on standard formulae is known as standard model. Thishas been expressed as follows:

The saturation pressure over liquid water is given by For0 �C 6 tdb < 200 �C

lnðPwsÞ ¼ �5:8002� 103=T � 5:5163� 4:8640� 10�2T

þ 4:1765� 10�5T2 � 1:4452� 10�8T3 þ 6:5460 lnðTÞ ð4Þ

where T = 273.16 + tdb Humidity ratio at saturation, Vapour pres-sure (Pw), W, h

Fig. 2. Schematic diagram of psychrometric var

Ws ¼ 0:62198Pws=ðP � PwsÞPw ¼ UPws

W ¼ 0:62198Pw=ðP � PwÞ ð5Þh ¼ 1:006tdb þWð2501þ 1:805tdbÞ ð6Þ

Dew point temperature tdp

B ¼ lnðPwÞ

if tdp � 0�C;tdp ¼ 6:09þ 12:608Bþ 0:459B2 ð7Þif 0�C < tdp � 93�C;

tdp ¼ 6:54þ 14:526Bþ 0:7389B2 þ 0:09486B3 þ 0:4569P0:1984w ð8Þ

Knowing tdb in �C and U, the following parameters can be calculatedusing (4)–(8): tdp in �C, h in kJ kg�1, W in kg [water] kg�1[dry air] ata standard atmospheric pressure (P = 101325 Pa). The training dataset for LSSVM model is generated by using Eqs. (4)–(8). The valuesfor dry bulb temperature has been taken from 10 �C to 50 �C in stepsof 1 �C, relative humidity from 20% to 97.5% in steps of 1.25%. Total2583 training data sets were generated to train the LS-SVM net-work. The conventional model for computing psychrometric vari-ables uses Eqs. (4)–(8). The model has limitations as it requireslot of computation effort and time. The hardware realization of thismodel is also difficult.

Three computation networks have been shown in Fig. 2 forcomputing psychrometric variables. The pressure of moist air isalso required for these computations but it has not been men-tioned in Fig. 2 as pressure remains constant for a particularplace.

After obtaining training data set, the preprocessing of data setis carried out to eliminate inconsistency of the input data.Traditional preprocessing techniques include (i) linear transfor-mation (ii) statistical standardization. Here statistical standardi-zation preprocessing technique is used to normalize the inputdata set.

iable computations with constant pressure.

1414 M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418

3. Basics of least square support vector machine

In last few decades, neural networks has proven to be a power-ful soft computing tool in many areas like statistics, system andcontrol theory, signal processing, information theory and others.Despite many advantages, there are still remaining some weakpoints for these classical neural network approaches, like localminima solution and the issue of choosing the number of hiddenunits. These disadvantages can be overcome by means of a newclass of networks called support vector machines. SVM was devel-oped by [14] to solve the classification problem, but recently,applications of SVM have been successfully extended to regressionand density estimation problem. However, the fact that SVMs arebasically non-parametric techniques, their use in dynamical sys-tems and control is more complicated. That is why standard SVMmethods have been developed mainly for static problems like clas-sification, static non-linear function and density estimation. Toovercome above disadvantages, LS-SVM is developed and its usehas been extended to dynamical problems of recurrent neural net-works and optimal control [15]. In this paper LS-SVM has beenused for non-linear function estimation problem.

3.1. LS-SVM for non-linear function estimation

In this section we briefly introduce LS-SVM, which can be usedfor regression problems [9,14].

The LS-SVM model for function estimation has the followingrepresentation in the feature space

yðxÞ ¼ wTuðxÞ þ b ð9Þ

where x 2 Rn; y 2 R: u is the non-linear mapping between the in-put space to the output space. Given a training set fxk; ykg

Nk¼1, the

optimization problem can be formulated in the form

minw;b;e

Iðw; eÞ ¼ 12

wT wþ c12

XN

k¼1

e2k ð10Þ

subject to equality constraints

yk ¼ wTuðxkÞ þ bþ ek; k ¼ 1; . . . ;N

The Lagrangian comes in the form

Lðw; b; e;aÞ ¼ Iðw; eÞ �XN

k¼1

akfwTuðxkÞ þ bþ ek � ykg

where ak e R are Lagrange multipliers and are called support values.The conditions for optimality are given by

@L@w¼ 0! w ¼

XN

k¼1

akuðxkÞ;

@L@b¼ 0!

XN

k¼1

ak ¼ 0;

@L@ek¼ 0! ak ¼ cek; k ¼ 1; . . . N;

@L@ak¼ 0! wTuðxkÞ þ bþ ek � yk ¼ 0; k ¼ 1; . . . N

The above equation, after elimination of w and e can be written inmatrix form as:

0 IT

I #þ c�1I

" #b

a

� �¼

0y

� �ð11Þ

where

Y ¼ ½y1; . . . yN� I ¼ ½1; . . . 1� a ¼ ½a1; . . . aN�

and the kernel function can be given by

#kl ¼ uðxkÞTuðxlÞ ¼ Kðxk; xlÞ; k; l ¼ 1; . . . N ð12Þ

According to Mercer’s condition [15], there exists a mapping u andan expansion

Kðx; yÞ ¼X

i

uiðxÞuiðyÞ; x; y 2 Rn ð13Þ

if and only if, for any g(x) such thatR

g(x)2dx is finite, one hasZKðx; yÞgðxÞgðyÞdxdy � 0 ð14Þ

As a result, one can choose a kernel K(�,�) such that

Kðxk; xlÞ ¼ uðxkÞTuðxlÞ k; l ¼ 1; . . . N ð15Þ

The resulting LS-SVM model for function approximation becomes

yðxÞ ¼XN

k¼1

akKðx; xkÞ þ b ð16Þ

where a and b are the solution to the linear system (11), RBF kernelis given by:

Kðxk; xlÞ ¼ expf�kxk � xlk2 r2g:

Fig. 3 contains a graphical overview over the different steps inregression stage. The input pattern (for which prediction is to bemade) is mapped into feature space by a map U. Then dot productsare computed with the images of the training patterns under themap U. This corresponds to evaluating kernel functions k(xi,x). Fi-nally the dot products are added up using the weights ai. This plusthe constant term b yields the final prediction output. The processdescribed here is very similar to regression in a neural network,with the difference, that in the SV case the weights in the inputlayer are a subset of the training pattern.

3.2. Implementation of the LS-SVM

The algorithm for the final implementation can be describedthrough the following steps:

1. Consider the data set fxk; ykgNk¼1. The data set is preprocessed to

eliminate inconsistency of input.2. Instead of training the LS-SVM model with total input data set,

the proposed algorithm intends to train the model with themost relevant standard data. In this step, the amount of trainingdata set should be determined.

3. In this step appropriate training data set is extracted from thewhole standard training data set and the extracted data set isused to get predictive model using standard LS-SVM algorithm[10].

4. By solving Eq. (11), we can get the values of a and b. Then theestimated PMV or psychrometric variable value can be easilyobtained according to Eq. (16).

Fig. 3. Architecture of a regression machine constructed by the LS-SVM algorithm.

0 10 20 30 40 50 60 70 80 90 100-2

-1.5

-1

-0.5

0

0.5

1

1.5

1.25

0.75

0.25

-0.25

-0.75

-1.25

-1.75

Samples

PMV

inde

x

ActualPredicted

Fig. 4. Comparison of Fanger model and LSSVM model for PMV computation.

0 10 20 30 40 50 60 70 80 90 100-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Samples

Abso

lute

erro

r

Fig. 5. Absolute error of LS-SVM model in comparison to Fanger model.

M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418 1415

4. Results and discussion

The results obtained from LSSVM models and conventionalmodels for both HVAC computations have a very good correlation.The ability of LSSVM techniques to model the highly non-linearbehaviour in a manner suitable for hardware and control imple-mentations is clearly apparent. The results are presented in a com-parative manner. The performance of LSSVM model is comparedwith Fanger model for prediction of PMV. The performance ofLSSVM model is also compared with standard model for predictionof psychrometric variables. To describe the performance of theabove models, we chose the following statistical indices:

� Maximum absolute error = max jyi � �yij� Mean absolute error = 1

N

PNi¼1jyi � �yij

� Mean squared error =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1N�1

PNi¼1½yi � �yi�2

q� Correlation coefficient = covðy;�yÞ

ryr�y

where yi is actual output, �yi is predicted output, N is number oftesting samples, ry; r�y are standard deviations and cov iscovariance.

The performance of LSSVM model is tested for test input datasamples without any noise added to any of the inputs. The test in-put data is selected in such a way that it totally differs from thetraining data. The predictive model gives maximum error for thetest input data. The performance of the predictive model is testedagain for data samples with noise added to test inputs. The detailedresults for PMV index and psychrometric chart are discussedbelow.

4.1. In the absence of noise

The prediction accuracy is compared by computing meansquare error, mean absolute error, maximum absolute error andpercentage of data sets having relative error more than 1%and 5% for the test data. These performance indices are shown inTable 1.

It is clear from the Table 1 that prediction accuracy is excellentin case of PMV approximation and LS-SVM model is giving outputvery close to Fanger model. The PMV values using LSSVM and Fan-ger model are shown in Fig. 4 and absolute error between the twomodels is shown in Fig. 5. Fig. 4 shows that the PMV obtained fromLSSVM model is closely follow the PMV obtained from Fanger mod-el and these could not be differentiated visually in the figure. The

Table 1Performance of LS-SVM in predicting PMV, humidity ratio, dew point temperatureand enthalpy in absence of noise.

Performanceindex

PMV Humidity ratio Dew pointtemperature

Enthalpy

Correlationcoefficient

0.999991 0.9999989 0.999995 0.9999995

Mean squarederror

1.0158 � 10�5 2.3575 � 10�10 0.002125 0.00176

Mean absoluteerror

0.0021934 9.31 � 10�6 0.020788 0.034682

Maximumabsolute error

0.0097094 5.55 � 10�5 0.22822 0.118783

Percentage datasets havingrelative errormore than 1%

12 0 9 0

Percentage datasets havingrelative errormore than 5%

4 0 4 0

maximum absolute error is 0.0097 as shown in Fig. 5. This erroris very small and may be considered equal to zero for practicalapplications. Thus LS-SVM model is an exact replica of Fanger mod-el. This model can be used in place of Fanger model for all practicalpurposes.

The performance indices shown in Table 1 show that deviationof LS-SVM model from standard model is very close to zero. Forabsolute humidity and enthalpy, all outputs of the LSSVM modelhave relative error less than 1%. In case of dew point temperatureprediction there are nine outputs which have relative error morethan 1%. This shows that prediction accuracy of this predictivemodel is excellent for predicting humidity ratio, dew point temper-ature and enthalpy of air. Fig. 6 shows comparison between pre-dicted and standard values of humidity ratio. The differencebetween predicted and actual value of humidity ratio is very smalland it can not be identified visually in the figure. Figs.7–9 showabsolute error between predicted and standard values of humidityratio, dew point temperature and enthalpy, respectively. Themaximum absolute errors are 0.000155 kg[water] kg�1[dry air],0.4989 �C, 0.573799 kJ kg�1 for humidity ratio, dew point

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Samples

Abso

lute

hum

idity

PredictedActual

Fig. 6. Comparison of predicted and actual humidity ratio in kg/kg (dry air).

0 10 20 30 40 50 60 70 80 90 100-1

0

1

0.5

0.25

0.75

-0.75

-0.25

-0.5

x 10-4

Samples

Abso

lute

erro

r

Fig. 7. Absolute error of predicted model in case of humidity ratio in kg/kg (dry air).

0 10 20 30 40 50 60 70 80 90 100-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Sample

Abso

lute

erro

r

Fig. 8. Absolute error of predicted model in case of dew temperature in �C.

1416 M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418

temperature and enthalpy, respectively and these can be consid-ered equal to zero for practical purposes. Thus, LS-SVM modelcan replace standard model for all practical purposes.

4.2. In presence of noise

(a) Sources of noise: The six input parameters for computingPMV index are prone to uncertainties. This uncertainty incase of air temperature is due to malfunctioning of mea-surement system itself and other is due to spatial variationin temperature in thermal space to be conditioned. Theuncertainty due to measurement system can happen ineach environmental variable. The uncertainty in case ofrelative air velocity is due to non-uniform air distributionin space caused by obstruction due to curtains, chairsetc. The uncertainties in human variables are due to activ-ity level of occupant changes with activity and precisemeasurement of activity level is not possible. There is widerange in the value of clothing insulation and precise mea-

0 10 20 30 40 50 60 70 80 90 100-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Sample

Abso

lute

erro

r

Fig. 9. Absolute error of predicted point model in case of enthalpy in kJ kg�1.

0 10 20 30 40 50 60 70 80 90 100-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Sample no.

Noi

se M

ag d

eg C

Fig. 10. Magnitude of noise added to dry bulb air temperature.

Table 4Performance indices of LS-SVM model and standard model for air dew pointtemperature prediction in presence of noise.

Performance index Noise in single inputvariable

Noise in multi inputvariables

LS-SVMmodel

Standardmodel

LS-SVMmodel

Standardmodel

Correlation coefficient 0.999935 0.99994 0.999326 0.999320Mean squared error 0.0145511 0.013538 0.1514768 0.15460Mean absolute error 0.10607217 0.103009 0.29783 0.3003328Maximum absolute error 0.244310 0.1938 1.17208 1.14588Percentage data sets

having relative errormore than 5%

4 4 17 18

Table 5Performance indices of LS-SVM model and standard model for prediction of airenthalpy in presence of noise.

M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418 1417

surement of clothing insulation is also not possible. Simi-larly for computing psychrometric variables air tempera-ture and relative humidity are used as input parameters.These input parameters are also prone to uncertainties.These uncertainties in case of air temperature and relativehumidity are due to malfunctioning of measurement sys-tem itself.

(b) Interpretation of noise: In this case the prediction accuracyof LSSVM model in the presence of noise is analyzed intwo ways: first way is by adding noise in one input vari-able at a time and second way is adding noise in morethan one input variable simultaneously. The test data isadded with uniformly distributed noise U(�a,a). The mag-nitude of ‘a’ was selected based on measuring accuracy ofsensors. The measuring accuracy in case of temperatureand humidity sensors which are commercially available iswithin ±0.2 �C and ±2%, respectively. The noise added totest input resembles with actual measuring uncertaintyobserved in the measurement of input variable. The noiseadded in air temperature is shown in Fig. 10. The test datawas added with noise and then performance of LSSVMmodel in terms of accuracy compared with Fanger modeland standard model. The results are presented in Tables2–5. The comparisons of models shown in Tables 2–5 hasbeen done with respect to ideal case i.e. conventionalmodel in absence of noise. It is clear from the Table 2and Fig. 11 that the performance of LSSVM model is betterthan Fanger model.

The prediction accuracy of LS-SVM model and standard modelin computing air humidity ratio, dew point temperature and air en-

Table 2Performance indices of LS-SVM model and Fanger model for PMV prediction inpresence of noise.

Performance index Noise in single inputvariable

Noise in multi inputvariables

LS-SVMmodel

Fangermodel

LS-SVMmodel

Fangermodel

Correlation coefficient 0.99965 0.99966 0.99962 0.999372Mean squared error 0.00037 0.0008067 0.0003988 0.0010838Mean absolute error 0.014463 0.0245983 0.0155784 0.03022Maximum absolute error 0.079132 0.0566 0.0729386 0.06011Percentage data sets

having relative errormore than 5%

23 46 27 48

Table 3Performance indices of LS-SVM model and standard model for humidity ratioprediction in presence of noise.

Performance index Noise in single inputvariable

Noise in both input variables

LS-SVMmodel

Standardmodel

LS-SVMmodel

Standardmodel

Correlationcoefficient

0.999925 0.999926 0.99952 0.99953

Mean squared error 1.36 � 10�8 1.355 � 10�8 8.7784 � 10�8 8.56 � 10�8

Mean absolute error 8.92 � 10�5 9.0212 � 10�5 0.0002265 0.0002247Maximum absolute

error0.000392 0.000396 0.0008857 0.000878

Percentage datasets havingrelative errormore than 5%

0 0 8 7

thalpy in presence of noise is presented in Tables 3–5, respectively.This comparison has been done with respect to ideal case i.e. stan-dard model in absence of noise. It is clear from Tables 3 and 4 thatall six performance indices for LSSVM and standard model arenearly equal. The results, presented in Table 5, shows that LSSVMmodel is slightly better than standard model in terms of therobustness against noises.

The results presented in Table 1–5 are summarized as below:

Performance index Noise in single inputvariable

Noise in multi inputvariables

LS-SVMmodel

Standardmodel

model Standardmodel

Correlation coefficient 0.999958 0.99992 0.999735 0.999697Mean squared error 0. 093713 0.17311 0.56180 0.647086Mean absolute error 0.232496 0.34581 0.5738 0.61189Maximum absolute error 1.10705 1.20 2.2042 2.410Percentage data sets having

relative error more than 5%0 0 0 0

0 10 20 30 40 50 60 70 80 90 100-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Samples

Abso

lute

erro

r in

term

s of

PM

V

LSSVM modelFanger model

Fig. 11. Magnitude of error for LSSVM and Fanger model in case of muti-inputnoise.

1418 M. Kumar, I.N. Kar / Energy Conversion and Management 50 (2009) 1411–1418

1. The prediction accuracy of LSSVM model is found to be excel-lent in estimating PMV, specific humidity, dew point tempera-ture and enthalpy in absence of noise. The numerical valuesof linear correlation coefficients are found to be greater than0.99999, for all four predictions as shown in Table 1.

2. The LSSVM model is found to be more robust against single aswell as multi-input noises than Fanger model used for predict-ing PMV. The LSSVM model is also found to be more robustagainst single as well as multi-input noises than standardmodel used for predicting air enthalpy.

3. The LSSVM model is found to be as robust against single as wellas multi-input noises as the standard model used for predictingspecific humidity and dew point temperature.

5. Conclusions

Least square support vector machines based approach hasbeen proposed to predict the PMV index and the psychrometricvariables. These parameters are widely used in heating,ventilating and air-conditioning applications. The excellent accu-racy was obtained for prediction of these parameters usingLSSVM method. It is evident from Eqs. (1)–(8) that expressionsfor PMV index and psychometric variables are highly non-linearfunctions of its inputs and computationally intensive. UsingLSSVM approach this process becomes straight forward and isvery well suited for direct implementation using hardware chips.The computations are performed much faster with this approach.Implementation of LSSVM method using hardware chips makesthem attractive for real time application. The robustness aspectof a LSSVM model has been analyzed. In some cases LSSVMmodels found more robust than conventional models but in

other cases these models found as robust as the conventionalmodels.

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