Comparative Study O Estimate House Price Usig Statistical Ad Neural Network Model Azme Bi Khamis, Nur Khalidah Khalilah Biti Kamarudi Abstract: This study was coducted to compare the performace betwee Multiple Liear Regressio (MLR) model ad Neural Network model o estimate house prices i New York. A sample of 1047 houses is radomly selected ad retrieved from the Math10 website. The factors i predictio house prices icludig livig area, umber of bedrooms, umber of bathrooms, lot size ad age of house. The methods used i this study are MLR ad Artificial Neural Network. It was foud that, the value of R 2 i Neural Network model is higher tha MLR model by 26.475%. The value of Mea Squared Error (MSE) i Neural Network model also lower compared to MLR model. Therefore, Neural Network model is prefered to be used as alterative model i estimatig house price compared to MLR model. Idex Terms: multiple liear regressio, artificial eural etworks, estimate, house price, mea square error, R 2, model performace 1. Itroductio Housig market is of great importat for the ecoomy activities. Housig costructio ad reovatio boost the ecoomy through a icrease i the aggregate expeditures, employmet ad volume of house sales. They also simulate the demad for relevat idustries such as household durables. The oscillatio of house prices affects the value of asset portfolio for most households for whom a house is the largest sigle asset [16]. A accurate predictio o the house price is importat to prospective homeowers, developers, ivestors, appraisers, tax assessors ad other real estate market participats, such as, mortgage leders ad isurers [6]. Traditioal house price predictio is based o cost ad sale price compariso lackig of a accepted stadard ad a certificatio process. Therefore, the availability of a house price predictio model helps fill up a importat iformatio gap ad improve the efficiecy of the real estate market [2]. Accordig to [17], the results show that the houses with more bedrooms ad bathrooms are priced higher. A relatively ew house is more expesive tha a old house ad a house with a garde is priced higher tha oe without a garde. Recet studies further justify the ecessity of housig price aalysis with a coclusio that housig sector plays a sigificat role i actig as a leadig idicator of the real sector of the ecoomy ad assets prices help forecast both iflatio ad output [5][[27][4]. May previous studies fid empirical evidece supportig the sigificat iterrelatios betwee house price ad various ecoomic variables, such as icome, iterest rates, costructio costs ad labor market variables [18][24][28]. Azme Khamis is Associate Professor i statistics at Uiversity Tu Hussei O Malaysia. E-mail: azme@uthm.edu.my Nur Khalidah Khalilah Biti Kamarudi is curretly pursuig masters degree program i statistics i Uiversity Tu Hussei O Malaysia E-mail: hw130037@siswa.uthm.edu.my Housig market is illiquid ad heterogeeous i both physical ad geographical perspectives, which makes forecastig house price a difficult task. Moreover, the subtle iteractios betwee house price ad other macroecoomic fudametals make the predictio further complicated. The chage i house prices ca either reflect a atioal pheomeo, such as the effect of moetary policy, or be attributed to local factors circumstaces that specific to each geographic market [8]. It ca either idicate the chages i the real sector variables, such as labor iput ad productio of goods, or be affected by the activities i the omial sector, i.e., fiacial market liberalizatio [10]. Apart from the above close lik to housig ivestmet, house prices have a strog lik with both icome ad iterest rates both via a stadard housig demad fuctio ad a housig supply fuctio. O the demad side, [19] propose a theoretical model of house price determiatio that is drive by chages i icome ad iterest rates. [9] studied idicate that i advaced ecoomies real house prices have fluctuated aroud a upward tred at least sice 1970 at euro market, geerally attributed i the literature to risig demad for housig space liked to icreasig per capita icome as well as a growig populatio o the demad side. [12] applied artificial eural etwork to evaluate the curret market situatio durig the world ecoomic crisis i 2008 ad predicted the future performace of property i order to help ivestors ad other market players i makig importat decisios. Although artificial eural etwork has bee limitedly used for valuatio or forecastig property price, studies carried out to compare the accuracy of liear regressio ad artificial eural etwork discovered that the latter has superiority compared to the former. [23] compared liear regressio ad artificial eural etwork i predictig housig value. [22] used artificial eural etwork ad compared its accuracy with that of liear regressio i predictig of housig price. [13] i their research also compared liear regressio ad artificial eural etwork i the mass appraisal cotext. [7] simulated a hypothesis i relatio to valuig real estate value i Madrid. Forecastig has some degree of ucertaity. However, a high degree of sophisticatio has bee developed recetly, with a rage of advaced quatitative ad qualitative procedure used by istitutioal ivestors i property forecastig, icludig judgemetal procedures, causal or ecoometric procedures, ad time series ad tred aalysis procedures [20]. This study aimed to compare betwee MLR model ad 126
Neural Network model to predict the house prices i New York. Secodary data from 1047 houses i New York is used i artificial eural etwork to predict the house price ad determie whether the predictio is good or ot. The secodary data was collected i year 2012. The data cosist of house price, livig area, umber of bedrooms, umber of bathrooms, lot size ad age of house [3][21]. The livig area, umber of bedrooms, umber of bathrooms, lot size ad age of house will be i iput layer while house price will be i output layer. There is a total of 1047 data poits i which 70% was used for traiig, 15% for validatio ad aother 15% for testig. All 1047 experimetal data sets are divided for traiig, validatio ad testig. Usig Neural Network Toolbox (tool) i MATLAB, differet etwork cofiguratio with differet umber of hidde euros is traied ad their performace is checked. There are 733 data sets are used for traiig, 157 data sets for validatio ad 157 data sets for testig. 2. Research Methodology 2.1 Multiple Liear Regressio Regressio is a fudametal operatio i statistics ad icludes techiques for modellig ad aalyzig several variables at a time. Regressio aalysis is used for explaiig the relatioship betwee a depedet variable, usually deoted by Y, ad a umber of idepedet variables, X 1, X 2,..., X p. The idepedet variables are also kow as predictor or explaatory variables. I most regressio aalyses, the variables are assumed to be cotiuous. I simple regressio, there is oly oe idepedet variable. However, most real world applicatios ivolve more tha oe variable which ifluece the outcome variable. The model for Multiple Liear Regressio ca be represeted as: E Y X = β 0 + β 1 X 1 + + β p X p where β 0 is called itercept ad β j are called slopes or regressio coefficiets. The differece betwee the predicted ad the actual value of Y is called the error (ε) or ca be writte as ε = Y Y. The, regressio equatio ca be express as: Y i = β 0 + β 1 X i,1 + + β p X i,p + ε i where Y i is the actual value ad ε i is the error for the i th observatio. We write X i,j for the j th predictor variable measured for the i th observatio. The mai assumptios for the errors ε i is that E(ε i ) = 0 ad var(ε i ) = ζ 2. Also the ε i are radomly distributed. The predicted value is also deoted by Y. The various errors are give as: SSE = Y i Y 2 i i=1 SSR = Y Y i 2 ; i=1 ; SST = Y i Y 2 i=1 SSE is the Sum of Squares of Error, SSR is the Sum of Squares of Regressio, ad SST is the Sum of Squares Total. R-square is the square of the correlatio betwee the respose values ad the predicted respose values. It is also called the square of the multiple correlatio coefficiets ad the coefficiet of multiple determiatios. The coefficiet of determiatio is the overall measure of the usefuless of a regressio. It is give as; R 2 = 1 - SSE SST The value of R 2 ca rage betwee 0 ad 1, a higher value idicates a better model. I terms of the sample, the estimate of the populatio total variace (SST) is deoted by Mea Sum of Squares Total (MST). MST is obtaied as SST/(-1) where is the sample size. Similarly, the estimated residual or error is called Mea Squared Error (MSE) ad is calculated as, MSE = SSE / (-p-1) where is the sample size, ad p is the umber of exploratory variables. A better estimate of the coefficiet of determiatio is made by the Adjusted-R squared statistic: SSE 2 ( p 1) R adj = 1 = 1 MSE SST MST ( 1) The F-test i oe way Aalysis of Variace (ANOVA) is also used as a statistic to fid the goodess of fit of the model. It is calculated as: F test = explaied variace uexplaied variace = SSR p SSE ( p 1) 2.2 Artificial Neural Network Artificial eural etwork (ANN) is a artificial itelligece model origially desiged to replicate the huma brai s learig process. ANN is distributed through a dese web of itercoectios. A eural etwork is formed by a series of euros or odes that are orgaized i layers. Neural etworks cosist of processig uits (artificial euros) ad coectios (weights) betwee those uits. The processig uits trasport icomig iformatio o their outgoig coectios to other uits. The iput iformatio is simulated with specific values stored i those weights that give these etworks the capacity to lear, memorize, ad create relatioships betwee data. Each euro i a layer is coected with each euro i the ext layer through a weighted coectio. The value of the weight w ij idicates the stregth of the coectio betwee the i th euro i a layer ad the j th euro i the ext oe. The structure of a eural etwork is formed by a iput layer, oe or more hidde layers, ad the output layer or ca be summarized as (iput, hidde ode, output). The umber of euros i a layer ad the umber of layers depeds strogly o the complexity of the problem studied. Therefore, the optimal etwork architecture must be determied. The geeral scheme of a typical three-layered ANN architecture is give i Fig. 1. The w ij is the weight of the coectio betwee the i th ad the j th ode. The euros i the iput layer receive the data ad trasfer them to euros i the first hidde layer through the weighted liks. Here, the data are mathematically processed ad the result is trasferred to the euros i the ext layer. Ultimately, the euros i the 127
last layer provide the etwork s output. The j th euro i a hidde layer processes the iput data, x i by: Step 1. Calculatig the weighted sum ad addig a bias term, δ j accordig to equatio 1: f(et) j = i=1 x i w ij + δ j for j = 1, 2,, (1) Step 2. Trasformig the f(et) j through a suitable mathematical trasfer fuctio or activatio fuctio, ad Step 3. Trasferrig the result to euros i the ext layer. Various trasfer fuctios are available [15][25]. Some of the most commoly used activatio fuctios are; (i) is liear fuctio, g(x) = x. It is obvious that the iput uits use the idetity fuctio. Sometimes a costat is multiplied by the et iput to form a liear fuctio; (ii) sigmoid fuctio, g x = 1 1+ e x. This fuctio is especially advatageous for use i eural etworks traied by back-propagatio, because it is easy to differetiate, ad thus ca dramatically reduce the computatio burde for traiig. It applies to applicatios whose desired output values are betwee 0 ad 1, ad Figure 2 (iii) hyperbolic taget fuctio, e g x) e x x e e x (. This fuctio has similar properties to x the sigmoid fuctio. It works well for applicatios that yield output values i the rage of -1 ad 1. Figure 1. Geeral structure of a eural etwork with iput layer, two hidde layers ad output layer The mathematical process through which the etwork achieves learig ca be pricipally igored by the fial user. I this way, the etwork ca be viewed as a black box that receives a vector with m iputs ad provides a vector with outputs. Here we will give oly a brief descriptio of the learig process; more details are provided for example i the review by [26]. The etwork lears from a series of examples that form the traiig data set. Traiig is formed by a vector X(ip) im = (x i1, x i2,., x im ) of iputs ad a vector Y(out) i = (y i1, y i2,..., y i ) of outputs. The objective of the traiig process is to approximate the fuctio f betwee the vectors X(ip) im ad the Y(out) i, Y(out) i = f(x(ip) im ). This is achieved by chagig iteratively the values of the coectio weights, w ij accordig to a suitable mathematical rule called the traiig algorithm. The values of the weights are chaged by usig the steepest descet method to miimize a suitable fuctio used as the traiig stoppig criteria. Oe of the fuctios most commoly used is the sum-of squared residuals give by equatio (2), E = 1 2 m i=1 j =1 (2) y ij y ij 2 where y ij ad y ij * are the actual ad etwork s j th output correspodig to the i th iput vector, respectively. The curret weight chage o a give layer is give by equatio (3): w ij = η de dw ij (3) where η is a positive costat called the learig rate. To achieve faster learig ad avoid local miima, a additioal term is used ad equatio (3) becomes: w ij k = η de dw ij + μ w ij k 1 (4) where μ is the mometum term ad w ij k 1 is the chage of the weight w ij from the (k-1) th learig cycle. The learig rate cotrols the weight update rate accordig to the ew weight chage ad the mometum acts as a stabilizer, beig aware of the previous weight chage. The fuctio give by equatio (2) is also used as the criterio to optimize the etwork architecture because it depeds o the umber of hidde layers ad the umber of euros therei. To fid the optimal architecture, the most commo approach is to plot the value of E i equatio (2) as a fuctio of the umber of odes i the hidde layer (q). Backpropagatio is the most commo learig algorithm for feed forward etwork. Backpropagatio simply the gradiet descet method to miimize the total squared error of output computed by the et. Traiig a etwork by backpropagatio ivolves three stages: the feed forward of the iput traiig patter, the backpropagatio of the associated error, ad the adjustmet of the weights [15]. The umber of hidde euros with lowest MSE will be choose as a optimum umber of hidde euros, ad model with higher R 2 ad lower MSE was cosidered to be a good model. 3. Data Aalysis ad Results 3.1 Regressio aalysis The relatioship betwee depedet variable with idepedet variables was performed usig Pearso correlatio. It was foud that the correlatio coefficiet idicated that house prices are positive ad strogly sigificace to livig area (0.776), bathroom (0.670), bedrooms (0.471) ad fireplace (0.460), but egative relatioship to age of house (-0.363). Meawhile, house price is ot sigificace to lot size. The value of F statistic is 380.696 ad p-value is 0.0000, meas that the model is suitable ad ca be fitted to the data. The coefficiet of determiatio R 2 = 0.646 ad Adj-R 2 = 0.645, it shows that 64.5% variace i house price ca be explaied by livig area, umber of bathrooms, umber of bedrooms, lot size ad age of house. The regressio equatio for the house price ca be writte as follow; 128
House price = 27467.001 + 67.211*Livig area -216.718*Age+ 16,402.244*Bathrooms +10,099.817*Fireplace 5167.260*Bedrooms. 3.2 Artificial Neural Network The umber of odes or euros i hidde layer are determied by trial ad error process. We starts our trial ad error with 2 odes ad the process is repeated util 15 odes. The researcher compares the MSE value ad R value for all umber of odes. The lowest MSE value with higher R value will be selected as optimum umber of odes i hidde layer. Based o Table 8, the lowest MSE value is 1.293E9 with 10 odes i hidde layer ad correlatio coefficiet is 0.9039. Hece, 10 odes are selected as optimum umber of odes i hidde layer. Table 8 : Optimum umber of odes i hidde layer Number of odes MSE R Number of odes MSE R 2 1.574 E9 0.8118 9 1.401 E9 0.8135 3 2.566 E9 0.8215 10 1.293 E9 0.9039 4 1.843 E9 0.8341 11 2.108 E9 0.8845 5 1.443 E9 0.8378 12 1.507 E9 0.8752 6 1.468 E9 0.8563 13 1.433 E9 0.8588 7 1.476 E9 0.8665 14 1.434 E9 0.8231 8 1.476 E9 0.8239 15 1.499 E9 0.8025 LA Iput Layer Hidde Layer BA BE LS HP Output Layer AH Figure 2 : Architecture of Artificial Neural Network with three layers Figure 4 : Performace plot 129
The architecture i Figure 2 shows that there are 5 iput layers, 1 hidde layer cosists of 10 odes ad 1 output layer. The iputs are livig area i square feet (LA), umber of bathrooms (BA), umber of bedrooms (BE), lot size i acre (LS) ad age of house i year (AH). The output is house price (HP).Figure 4 shows retraied performace (MSE) graph of eural etwork model, created durig its traiig. The traiig stopped after 23 epochs because the test error icreased. It is a useful diagostic tool to plot the traiig, validatio, ad test errors to check the progress of traiig. The result here is reasoable because the traiig set error ad the validatio set error have similar characteristics, ad it does't appear that ay sigificace over fittig has occurred. After iitial traiig of eural etwork model, it is retraied for 23 epochs ad performace MSE is obtaied 1.293E9 i traiig. 5. Refereces [1] M. H. Beale, M. T. Haga, & H. B. Demuth, Neural Network Toolbox User s Guide. The MathWorks, Ic., 2013. [2] C. A. Calhou, Property valuatio models ad house price idexes for The Provices of Thailad: 1992 2000. Housig Fiace Iteratioal, 17: 31 41, 2003. [3] Creative Research Systems. (.d.). Retrieved December 22, 2013, from Survey System: http://www.surveysystem.com/correlatio.htm., 2013. [4] S. Das, R. Gupta, & A. Kabudi, Could we have predicted the recet dowtur i the South Africa housig market? Joural of Housig Ecoomics 4:325-335, 2009. [5] M. Fori, M. Halli, M. Lippi, & L. Reichli, Do fiacial variables help forecastig iflatio ad real activity i the euro area? Joural of Moetary Ecoomics 6: 1243-1255, 2003. [6] J. Frew, & G. D. Jud, Estimatig the value of apartmet buildigs, The J. Real Estate Res., 25: 77 86, 2003). [7] J. Gallego, & Mora-Esperaza (2004). Artificial itelligece applied to real estate valuatio: A example for the appraisal of Madrid. Catastro: 255-265. Figure 5 : Regressio plot Based o Figure 5, the value for R is 0.0.9039 ad the value of R 2 is 0.817. This shows that 81.7% of total variatio i house price was explaied by livig area, umber of bathrooms, umber of bedrooms, lot size ad age of house. The value of R 2 ad MSE for MLR model is 0.644 ad 1.633E9. The value of R 2 ad MSE for Neural Network model is 0.817 ad 1.293E9. The R 2 value for Neural Network model is higher compared to MLR model. The value of MSE i Neural Network model is lower compared to MLR model. Therefore, Neural Network model is prefered to predict house price. 4. Coclusio The model s accuracy i predictig house price was measured by a umber of criteria. The value of R 2 ad MSE were compared to select preferred model. By usig ANN, the R 2 value was icrease about 26.475% higher tha MLR. It ca be coclude Neural Network model is prefered to predict house price compared to MLR model ad ca be used as a alterative way to estimate house price i future. [8] L. Gattii, & P. Hiebert, Forecastig ad assessig euro area house prices through the les of key fudametals, Workig Paper Series, No.124/ October, 2010, 2010 [9] N. Girouard,, M. Keedy, P. va de Noord, & C. Adr e, Recet house price developmets: The role of fudametals, OECD Ecoomics Departmet Workig Paper No. 475, 2006 [10] R. Gupta, S.M. Miller, & D.V. Wyk, Fiacial market liberalizatio, moetary policy, ad housig price dyamics. Workig paper No. 201009, Dept. of Eco., Uiversity of Pretoia. 2010. [11] H. Hossei, A.Khairil,, H. T. Huam,, K. Naser,, & R. Mohse, Artificial eural etworks: Applicatios i maagemet. World Applied Scieces Joural, 14 (7), 1008-1019, 2011. [12] A. Khalafallah, Neural etwork based model for predictig housig market performace. Tsighua Sciece ad Techology, 13(1): 325-328, 2008. [13] M. Khashei, & M. Bijari, A artificial eural etwork (p, d, q) model for time series forecastig, Expert Systems with Applicatios, 37: 479-489, 2010. 130
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