The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/1753-8270.htm Forecasting house prices in Iran using GMDH Forecasting house prices Behrooz Nazemi and Mohsen Rafiean Faculty of Art and Architecture, Yazd University, Yazd, Iran Abstract Purpose – An accurate predictive model for forecasting urban housing price in Isfahan can be useful for Received 29 May 2020 Revised 4 July 2020 Accepted 13 July 2020 sellers and owners to take more appropriate actions about housing supplying. Also, it can help urban housing planners and policymakers in managing of the housing market and preventing an urban housing crisis in Isfahan. The purpose of this paper is forecasting housing price in Isfahan city of Iran until 2022 using group method of data handling (GMDH). Design/methodology/approach – This paper presents an accurate predictive model by applying the GMDH algorithm by using GMDH-Shell software for forecasting housing price in municipal boroughs of Isfahan city till the second half of 2022 based on creating time series and existing data. Alongside housing price, some other affecting factors have been also considered to control the forecasting process and make it more accurate. Furthermore, this research shows the housing price changes of boroughs on map using ArcMap. Findings – Based on forecasting results, the housing price will increase at all boroughs of Isfahan till second half of the year 2022. Amongst them, Borough 15 will have the highest percentage of the price increasing (28.27%) to year 2022 and Borough 6 will have the lowest percentage of the price increasing (8.34%) to the year 2022. About ranking of the boroughs in terms of housing price, Borough number 6 and 3 will keep their current position at the top and Borough number 15 will stay at the bottom. Research limitations/implications – In this research, just few factors have been selected alongside housing price to control the forecasting process owing to limitation of reliable data availability about affecting factors. Originality/value – The most remarkable point of this paper is reaching to a mathematical formula that can accurately forecast housing price in Isfahan city which has been rarely investigated in former studies, especially in simplified form. The technique used in this paper to forecast housing price in Isfahan city of Iran can be useful for other cities too. Keywords Forecasting, GIS, Algorithm, Housing price, Analyzing, GMDH Paper type Research paper 1. Introduction Nowadays, housing is not considered just as a roof over one’s head, and it has a crucial role in the urban economy and sustainable development (Golubchikov and Badyina, 2012). As an irreplaceable, consumable, marketable, producible and durable commodity (Ruonavaara, 2018), price of housing is determined based on supply and demand in the market (Rahadi et al., 2015). Forecasting the future trend of housing price is very necessary for all decisionmaker and takers to balance the housing market and react to the changes more appropriate (Liu and Liu, 2019). In today’s financial markets, including the housing market, analyzing existing data can be very helpful for predicting price movements (Tsantekidis et al., 2017). Choosing the suitable forecasting method depends significantly on data availability, kind of data, case study, the context of the research, financial and human resources and expected accuracy (Hyndman and Athanasopoulos, 2018). In previous studies different kinds of methods have been used to forecast housing price, some of the most important of them are auto regressive integrated moving average, random forest, fuzzy logic, support vector International Journal of Housing Markets and Analysis © Emerald Publishing Limited 1753-8270 DOI 10.1108/IJHMA-05-2020-0067 IJHMA machine, Bayesian approach, geographical detector technique, satellite imagery, spatial hedonic models, co-regionalized models, panel data-based approach, factor analysis, hedonic model and artificial neural networks (ANNs) (Li et al., 2017; Khalafallah, 2008; Morano et al., 2019; Stamou et al., 2017; Rossini, 2000; Morano et al., 2015; Wei and Cao, 2017; Bork and Møller, 2015; Kouwenberg and Zwinkels, 2014; Gu et al., 2011; Azadeh et al., 2012; Park and Bae, 2015; Ahn et al., 2012; Hadavandi et al., 2011; Waltl, 2016). In recent years, machine learning and ANN methods have been improved considerably in order to forecast housing price because of their advantages over traditional methods (Xu and Li, 2020; Dong, 2020). For example, Rui Liu and Lu Liu in the year 2019 predicted housing price based on long short-term memory incorporating modified genetic algorithm in their article that has shown more acceptable results over traditional techniques (Liu and Liu, 2019). Most of the previous studies about forecasting housing price using machine learning and artificial neural network methods have not presented a simplified and obvious predictive mathematical formula. So, one of the main goals of this article is using group method of data handling or group method of data handling (GMDH)-type artificial neural network for making an accurate predictive model in order to forecast housing price in different municipal boroughs of Isfahan (which is divided by the municipality of Isfahan) [1] and representing the relevant mathematical formula for each borough. Another main goal is showing the housing price changes of boroughs on map utilizing one of the most important geographic information system (GIS) software that known as ArcMap. Isfahan is one of the most important main cities in Iran located in 32°38’ 30” N latitude and 51°38’ 40” E longitude, about 340 km south of Tehran and the capital of Isfahan Province. The population of Isfahan city has been increased rapidly and considerably, mostly because of immigration, such as that based on last national census statistics it has grown more than 435% (more than 4 times) in comparison with 50 years ago (Assari et al., 2017) [1] [2]. Facility inequalities in regional scale, owning large and maternal industries, tourism and other important economics determinants have caused by knowing Isfahan as an attractive destination in terms of immigration (Assari et al., 2017). The certain consequence of this population growing has been increasing demand for housing in Isfahan city. Owing to appropriate response to the increasing demand, an accurate predictive model for forecasting urban housing price in Isfahan can be useful for developers, sellers, and owners to make more precise decisions and take more appropriate actions about housing supplying. Also, it can help urban housing planners and policy makers in managing of the housing market, price changes and prevent an urban housing crisis in Isfahan city of Iran. 2. Material and methods 2.1 Data In this research, data of housing price in different boroughs of Isfahan city have been collected from 1995 to 2019 for every six months. In order to increase accuracy and controlling process of the forecasting, 5 other factors have been selected alongside housing price (Rahadi et al., 2015). The value of these factors varies from one borough to another borough. The selected factors are average rent of 1 m2 residential unit (RP), average price of 1 m2 land (LP), population (P), average total costs of a household (CH) and average income of a household (IH). Relevant data to the research variables have been collected from Iran national statistics bureau (www.amar.org.ir), central bank of Iran (www.cbi.ir), World Bank (www.worldbank.org) and Isfahan municipality (www.isfahan.ir). 2.2 Group method of data handling algorithm The group method of data handling (GMDH) algorithm is very useful for modeling work like mathematics and computer ones (Onwubolu, 2014). This algorithm has been introduced by a scientist named Ivakhnenko as a multivariate analysis approach to solving complicated system problems (Stepashko et al., 2017). The basis of the GMDH approach is to create an analytical function in a quadratic node-based transfer network (Moroz, 2016). The GMDH network makes a connection between input and output data, that is indicated to the Volterra function series or the Kolmogorov–Gabor polynomial function (Yang et al., 2018): y ¼ a0 þ m X i¼1 ai xi þ m X m X aij xi xj þ i¼1 j¼1 XXX aijk xi xj xk þ . . . (1) Such mathematical expression can be produced by a set of partial quadratic polynomials, including only two variables (neurons) as follows (Kordnaeij et al., 2015): y ¼ Gðxi ; xj Þ ¼ a0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 x2i þ a5 x2j (2) By this means, in a network of related neurons the partial quadratic description is recursively used to create the general mathematical relation between giving inputs and output in Equation (Kordnaeij et al., 2015). The coefficients ai in equation (B) is classified by regression methods to minimize the difference between the actual output y and the assigned one y for each pair of input parameters xi, x j (Shaghaghi et al., 2017). In fact, a hierarchy of polynomials is built using the quadratic form produced in equation (A) whose constants are achieved by least-squares logic, then the constants of every quadratic function Gi are obtained to optimally fit the output throughout the whole set of output–input data pairs, in the form of Ahmadi et al. (2015): M X E¼ ðyi i M Gi Þ 2 ! min (3) Basically, all the probabilities of two independent variables out of the total n input variables are taken in order to create the regression in the GMDH algorithm, that best fits the dependent observations (yi, i = 1, 2, . . ., M) in terms of least squares logic (Ahmadi et al., 2019). For every row of M data triples, the following matrix equation can be readily achieved as Aa = Y, where a is the vector of unknown coefficients of the quadratic polynomial in the equation (B) (Shahsavar et al., 2019). a ¼ fa0 ; a1 ; a2 ; a3 ; a4 ; a5 g (4) Here Y is the vector of the output’s value of observation, as shown in equation (5) (Shahsavar et al., 2019): Y ¼ fy1 ; y2 ; y3 ; . . . ; yM gT (5) The least squares technique from multiple regression analysis leads to the solution of the normal equations in the form of Ghanadzadeh et al. (2014): Forecasting house prices IJHMA 0 1 B B1 B A ¼ B. B .. @ 1 x1p x1q x2p x2q .. .. . . xMp xMq x1p x1q x21p x2p x2q x22p .. .. . . xMp xMq x2Mp 1 a ¼ ð AT AÞ AT Y x21q x22q .. . x2Mq 1 C C C C C A (6) (7) which assignes the vector of the best coefficients of the quadratic equation (B) for the entire set of data triples, this procedure is repeated for each neuron of the next hidden layer based on the connectivity topology of the network (Varahrami, 2012). In the training of each layer, the neuron candidates are represented according to all the feasible combinations of input variables, then these neuron candidates are screened automatically regarding their ability to estimate the target variable and only those neurons having good predication powers (meets the criterion) are fed forward for the training of the next layer, and the rest is ignored (Jia et al., 2018). Special behavior properties of GMDH make it a perfect option to use in fields such as knowledge exploration, data mining, predicting, different types of modelling, optimization and pattern finding (Onwubolu, 2014; Moroz, 2016). In order to forecast housing price in Isfahan of Iran in this research, GMDH Shell software that is an advanced analytics suite that can represent complicated time series predicting, regression, classification, clustering and curve fitting tools based on GMDH algorithm was used [3]. Some of the most important characteristics of GMDH Shell software are assigning model structure automatically, preventing overfitting and performing acceptable with very small data sets and being quick even with more than 1,000 input variables. Some of the greatest global corporations like NASA, Xcel Energy, Cork Supply, Fike, Marketstar and Honeywell are customer of GMDH Shell software3. 2.3 Parameters setting After inserting data from Excel software to GMDH Shell software in data explorer section, the corresponding parameters have been set in the solver section of the software where the values of considered parameters have a significant influence on the performance of GMDH Shell to make a predictive model. The parameters are set based on trial and error method to obtaining the best result as follows: Reorder observations parameter is utilized to obtain uniform statistical properties of training and testing samples and to make them equally informative, this technique works perfectly for almost all problem types including time series models. This parameter has been set on pseudorandom function that is a deterministic function of a key and an input that is indistinguishable from a truly random function of the input. Validation strategy parameter is utilized to choose a method for model validation and sorting out. In this case the training/testing strategy, that splits the dataset into two parts and utilizes the training part to find model coefficients and implements the testing part to compare and choose a set of the best models, has been assigned as validation strategy. In order to obtain best results, train/test ratio differs from one borough to another one. For the Validation criterion parameter, that defines the model selection criterion, the Root Mean Square Error (RMSE) balance that is focused on the difference between training and testing sample RMSE, has been selected. The neuron function parameter sets the type of the internal function of neurons. The neurons are active, and each neuron can drop some of the function terms to increase overall predictive power of the model. The selected function for this parameter is: Forecasting house prices a þ xi þ xi :xj þ . . . The Max. Number of layers parameter values that sets the upper limit for the number of network layers created by the algorithm is 10. The selected value of the Initial layer width parameter that defines how many neurons are added to the set of inputs at each new layer is 50. 2.4 Geographic Information System GIS is a system that created to analyze, checking, manage and show spatial or geographic data [4]. Generally, GIS uses two kinds of data: spatial data and attribute data (Tomaszewski, 2014). Spatial data describe the absolute and relative locations of spatial (geographic) entities, while attributing data indicate to characteristics of spatial entries (Ehrgott et al., 2010). In this research Arc Map software as one of the most important parts of Esri’s ArcGIS suite of geospatial processing programs that is useful to display, modify, produce and analyze geospatial data [5] is regarded to show forecast results on the map. ArcMap makes user able to discover data within a set of data, symbolize properties and produce maps [6]. 3. Results In this article, only model details of Borough 1 is shown as an example. For other boroughs, just final forecasting results are presented on relevant maps using the same technique. The obtained results for Borough number 1 are shown in Table 1, Figure 1 and Table 2. About Table 2, errors have been calculated based on differences of existing data and predictions (till 2019) (Figure 2). The first 27 of existing data were used to recognize the pattern of created time series by the model. To forecast housing price till 2022, the next 6 observations after observation number 54 were considered. Predictions have been achieved based on below equations (according to GMDH algorithm). 0 HP ½tŠ ¼ þ0:940701*a þ 0:0615838* b 2:33448 ID Actual value Predictions ID Actual value Predictions ID Actual value Predictions 28 29 30 31 32 33 34 35 36 37 38 360.09 409.75 440.08 475.02 528.13 523.08 552.16 548.9 587.19 599.63 650.89 385.19 398.01 403.05 440.81 492.21 538.27 566.01 599.23 584.69 591.81 651.19 39 40 41 42 43 44 45 46 47 48 49 712.70 875.03 1000.02 1080.1 1076.3 1115.25 1104.12 1180.1 1245.01 1280.02 1440.3 769.94 866.81 958.37 997.66 1079.56 1126.92 1195.48 1179.14 1263.67 1291.17 1476.04 50 51 52 53 54 55 56 57 58 59 60 1550.81 1786.79 1900.91 2214.23 2350.11 – – – – – – 1531.67 1765.63 1877.47 2204.67 2351.28 2338.13 2504.01 2459.51 2659.31 2852.97 2798.66 Table 1. Actual values and predictions IJHMA Figure 1. Forecasting housing price for Borough 1 Index Table 2. Obtained errors Figure 2. Residuals Mean absolute percentage error (MAPE) Root Mean Square Error (RMSE) Mean absolute error (MAE) Residual sum Value 3.26% 4.98 24.84 0.77 a¼ b ¼ þ0:0792739*RP ½t 0:0254963*v þ 1:02278*Xþ2:77809 6Š*P ½t X¼ d ¼ 6Š 5919:2*RP ½t 6Š Forecasting house prices 0:0315856*P ½t 6Š þ 3076:71 0:491843*ˆ þ 1:4904*d þ1:47333 0:000407005*LP ½t 6Š* w þ 1:24173* w 68:9823 w ¼ þ0:0000697197*s *l þ 0:231829*s þ 0:608187*l þ 70:452 l ¼ þ0:00000169618*CH ½t 6Š* m þ 0:0117867*CH ½t m ¼ þ0:00445097*CH ½t ˆ¼ 6Š 0:000375373*LP ½t 6Š 2:93543* m þ 543:806 0:0157098*P ½t 6Š þ 1092:93 6Š*u þ 1:22365*u 64:5554 u ¼ þ0:0000843966* c *s þ 0:548268* c þ 0:255702*s þ 87:6753 s ¼ þ740:362*RP ½t c ¼ þ0:00000000287695*CH ½t 0:00223708*IH ½t v ¼ þ0:000504971*LP ½t 6Š 1:77642*LP ½t 6Š*IH ½t 6Š þ 199:15 6Š þ 0:00475612*CH ½t 6Š 6Š þ 178:369 6Š*P ½t 6Š 37:7863*LP ½t 6Š 0:0582863*P ½t 6Š þ 5199:12 In above equations: 0 HP ½tŠ = Housing price prediction [t] = Number of observation * = Multiply sign For example, to forecasting housing price about observation number 60 (2022 year): t = 60; and t 6 = 54 Value of considered factors for observation number 54 are as follows: (Table 3). When these values have been put in sub equations: Factor P LP RP CH IH Value 80986 1100 4.413 764925.67 830675.47 Table 3. Value of considered factors a ¼ 2798:877 IJHMA b ¼ 2729:48 Finally, the value of prediction for observation number 60 was achieved using a and b in main equation. The final predicted value for observation number 60 is 2798.66 $. Results of housing price prediction for all boroughs of Isfahan city have been presented in Table 4. In Figure 3, housing price in the year 2019 has been shown for each borough of Isfahan city that assigned by numbers (1, 2, 3. . ., 15) at relevant borders. Each borough has its own color on a map based on average housing price in there. The ranges of housing price on the map (Figure 3) have been selected based on geometrical interval classification method of Arc map software. The rate of housing price changes (per cent) based on forecasted values to the year 2022 for Isfahan boroughs that obtained by the used model, have been presented in Figure 4. 4. Discussion To avoid over-increasing the pages of the article, the process of housing price forecasting using considered technique has been presented only for Borough 1 of Isfahan city as an example. To measure the accuracy of housing price forecasting in Borough 1 of Isfahan city in comparison to existing data, interpretation of MAPE, RMSE and MAE is essential (Liu and Liu, 2019; Kordnaeij et al., 2015; Ahmadi et al., 2019; Ganti et al., 2018; Chen et al., 2017; Alfiyatin et al., 2017). To interpret of MAPE value, generally less than 10% can be acceptable (Borzemski and Wojtkiewicz, 2011). According to the obtained results, MAPE value of this work for the Borough number one of Isfahan is 3.26%. As shown in table 2, RMSE value is 4.98 in this research. Regarding scale of used data, RMSE value can be approved (Balouchi et al., 2015; Zhang, 2015). Considering used data range (maximum and minimum values), obtained MAE value, that is 24.84, can be confirmed as an acceptable value (Roy et al., 2016; Kim et al., 2017). For future housing price forecasting (from 2019 to Name of borough Table 4. Results of housing price forecasting for all boroughs Borough 1 Borough 2 Borough 3 Borough 4 Borough 5 Borough 6 Borough 7 Borough 8 Borough 9 Borough 10 Borough 11 Borough 12 Borough 13 Borough 14 Borough 15 Housing price in first half Housing price in second half Forecasted housing price to second of year 1995 of year 2019 half of year 2022 17.08 ($) 9.52 ($) 24.87 ($) 12.32 ($) 18.64 ($) 31.02 ($) 8.56 ($) 20.69 ($) 10.52 ($) 8.12 ($) 6.51 ($) 7.25 ($) 15.53 ($) 6.05 ($) 5.29 ($) 2350.11 ($) 1658.21 ($) 3700.23 ($) 2001.11 ($) 2763.25 ($) 4707.79 ($) 1502.38 ($) 3102.52 ($) 1608.33 ($) 1459.63 ($) 1096.78 ($) 1203.33 ($) 2640.74 ($) 999.32 ($) 850.85 ($) 2798.66 ($) 1923.52 ($) 4150.36 ($) 2464.01 ($) 3051.33 ($) 5100.58 ($) 1821.11 ($) 3450.22 ($) 1901.87 ($) 1702.65 ($) 1370.59 ($) 1450.81 ($) 3011.38 ($) 1250.61 ($) 1091.47 ($) Forecasting house prices Figure 3. Housing price in Isfahan city 2019 2022) that does not exist actual relevant data about them to compare, accuracy and validation of model results can be evaluated according to the following arguments:  Accuracy of the predictive model is acceptable for existing data owing to the obtained results and its ability for recognizing the general trends correctly.  In this work a multivariate method has been used for forecasting, other variables alongside housing price have been utilized to decrease the errors and provide more reasonable results.  In this article the targeted year is 2022, so it is a short-term forecasting. Generally, the lower selected period to forecasting will increase the accuracy.  Housing price is a stock variable, it cannot increase or decrease far more than the current value. In other words, housing price needs time to change considerably. Given that the results, the forecasted prices to 2022 are reasonable in comparison to current values. About other boroughs of Isfahan city, housing price has been forecasted based on the same technique, arguments and interpretations. Of course, every borough has its own forecasting equations that obtained by the model. IJHMA Figure 4. The rate of housing price changes in Isfahan city 2022 Owing to the obtained results, accuracy of housing price forecasting in Isfahan city housing market using GMDH-type artificial neural network that has been conducted in this work, is acceptable. Also, the relevant mathematical equations that produced by the model to forecast housing price in Isfahan city have been presented in the results section and simplified to be used easier by users. Finally, forecasting results for each borough of Isfahan in 2022 have been shown on the relevant maps that created by ArcMap GIS software to better analyze the price changes. So totally, the main goals of this article have been achieved. Based on forecasting results, the housing price will increase at all boroughs of Isfahan in the second half of 2022. Amongst them, Borough 15 will have the highest percentage of the price increasing (28.27%) to year 2022 and Borough 6 will have the lowest percentage of the price increasing (8.34%) to the year 2022. About ranking of the boroughs in terms of housing price, Borough number 6 and 3 will keep their current position at the top until the second half of 2022 because of predicting results. Accordingly, Borough number 15 of Isfahan will stay at the bottom of the housing price ranking. 5. Conclusion Humans have always been very interested in forecasting the future, predicting the prices in the housing market has not been exempted from this rule. Knowing about future housing price changes let the planners and policy makers be ready for quick reaction and handle the situation of the housing market. Also, using accurate forecasting results can help urban housing programs to be more feasible. Furthermore, recognizing the future trends of housing price changes can help owners, developers and consumers take and make better decisions. So, this article presents an accurate predictive model based on GMDH approach utilizing GMDH-Shell software for forecasting housing price in a case study of Isfahan city based on created time series and existing data. One of the most remarkable points of this work is reaching to a mathematical formula that accurately can forecast housing price in Isfahan city. This rarely has been presented in former studies, especially in simplified form. Achieving the formula and simplifying that, can help users to use it easily for analyzing and describing the status of the housing market and the changes in Isfahan city of Iran. Another most important point is using ArcMap GIS to providing quality and legible maps in this article. Descriptive maps can show the housing price changes more objective and visible over the time; this is useful for users to analyze the changes more precise with regard to the relevant locations. It should be said that the used technique in this research is not only for Isfahan city and can be used for other cities. As a limitation, in this research just few factors have been selected alongside housing price to control the forecasting process because of limitation of reliable data availability about affecting factors. For the future studies about the title of this article, using more related indices and periods to improve the accuracy of model is recommended. Also, using a more powerful computer in terms of processing abilities can improve the accuracy of results, in this research all calculations conducted on a personal computer with an Intel Core i7-3630QM @2.4 GHz CPU, 8 GB RAM and Windows 10 (64 bit) operational system. 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