A Method for Merging Similar Zones to Improve Intelligent Models for Real Estate Appraisal

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1 A Method for Merging Similar Zones to Improve Intelligent Models for Real Estate Appraisal Tadeusz Lasota 1, Edward Sawiłow 1, Bogdan Trawiński 2 Marta Roman 3, Paulina Marczuk 3, Patryk Popowicz 3 1 Wrocław University of Environmental and Life Sciences, Dept. of Spatial Management ul. Norwida 25/27, Wrocław, Poland 2 Wrocław University of Technology, Department of Information Systems Wybrzeże Wyspiańskiego 27, Wrocław, Poland 3 Wrocław University of Technology, Faculty of Computer Science and Management Wybrzeże Wyspiańskiego 27, Wrocław, Poland {tadeusz.lasota, edward.sawilow}@up.wroc.pl, bogdan.trawinski@pwr.edu.pl Abstract. A method for property valuation based on the concept of merging different areas of the city into uniform zones reflecting the characteristics of the real estate market was worked out. The foundations of the method were verified by experimental testing the accuracy of the models devised for the prediction of real estate prices built over the merged zones. The experiments were conducted using real-world data of sales transactions of residential premises completed in a Polish urban municipality. Two machine learning techniques implemented in the WEKA environment were employed to generate property valuation models. The comparative analysis of the methods was made with the nonparametric Friedman and Wilcoxon statistical tests. The study proved the usefulness of merging of similar areas which resulted in better reliability and accuracy of predicted prices. Keywords: real estate appraisal, predictive models, machine learning, linear regression, decision trees 1 Introduction The current system of real estate appraisal in Poland is not flawless and is based mainly on the analysis of real estate market which can be performed in a limited scope. Such an analysis is carried out using market transactions of similar properties located nearby the property being appraised, and the requirements as for the similarity are not unambiguous. The weakness of this system and binding standards consist in the difficulties in acquiring and assessing input data. The appraisers tend to find identical objects, however those are mostly sparse on the market or even lacking. To address this difficult problem the authors propose a method for merging different areas of an urban municipality into uniform zones reflecting the similar characteristics of the real estate market. The uniform zones may constitute the basis for elaborating more reliable and accurate models for real property valuation.

2 Real estate market modelling and predicting real property value have been an intensively developed area of research for many years. Numerous approaches to real estate appraisal ranged from statistical through operational research to computational intelligence techniques have been proposed and experimentally evaluated recently. Various methods can be found in rich literature on the topic including models built based on statistical multiple regression and neural networks [1], [2], [3], linear parametric programming [4], decision trees [5], rough set theory [6], fuzzy systems [7], and hybrid approaches [8]. For a decade we have been working out methods for generating data driven regression models to aid in real estate appraisal based on fuzzy systems and neural networks as both single models [9], [10] and multiple models built using various resampling techniques [11], [12], [13], [14], [15]. A relatively good accuracy provided evolving fuzzy models applied to cadastral data [16], [17]. We have also explored methods to predict from a data stream of real estate sales transactions based on ensembles of genetic fuzzy systems and neural networks [18], [19]. In this paper we propose a novel algorithm for property valuation based on the concept of merging different areas of an urban municipality into uniform zones reflecting the characteristics of the real estate market. The algorithm was evaluated by experimental testing the accuracy of the models for the prediction of real estate prices built over the merged zones. The experiments were conducted using real-world data of sales transactions of residential premises completed in a Polish urban municipality and linear regression and model pruned tree algorithms taken from WEKA [20]. 2 Method for merging similar zones of an urban municipality The core of the proposed method is an algorithm for merging similar zones to constitute firm grounds for real estate appraisal. The area of an urban municipality was partitioned into about 250 zones based on a self-governmental land-use plan as shown in Fig. 1. The idea of merging consists in finding zones in which the prices of premises change similarly in the course of time. Such zones are then merged into bigger areas encompassing greater number of sales transactions of similar nature which allow for constructing more reliable and accurate property valuation models. Fig. 1. Zones of an urban municipality based on a land-use plan considered in the paper

3 Fig. 2. Three periods with different trends of real estate price variability The considered time span should be divided into a number of periods according to the dynamics of real estate prices. In Fig. 2 three periods with different linear trends of price variability are illustrated. The consecutive steps of the algorithm are as follows. 1) Partition the land into a number of uniform zones i (i=1,2,,n) to consider. 2) Partition the considered time span into a number of periods j (j=1,2,,m) with a uniform trend of price changes. 3) For each intersection ij of a zone and period set the minimum number of sales transactions to obtain a credible linear trend function. 4) Remove outliers from each intersection ij of a zone and period. 5) Exclude zones which do not contain the minimum number of transactions. 6) For each intersection ij determine the linear trend function y=f ij (t)=kt+b, where t time, y price per square metre, k the slope, and b the intercept. b e 7) For each zone i compute the values of linear trend functions: y ii and y ii at the beginning b and end e of each period j as shown in Fig. 3. 8) Assign the weights a j to discriminate among individual periods in respect of their importance (age), e.g. the earlier the period the lower the weight. Then normalize the weights to sum to one according to Formula 1. w j = a j m j=1 a j (1) Fig. 3. Denotation of trend function values for two zones and three periods

4 9) Assign the weights c j to discriminate among individual periods in respect of their credibility to determine the trend functions, e.g. the smaller number of sales transactions within the period the lower the weight. Then normalize the weights to sum to one according to Formula 2 where n ij and n kj the number of transactions in the intersection of j-th period and i-th and k-th zones respectively. v j = c j m j=1 c j = mmm( n ii, n kk ) m mmm(n ii, n kk ) j=1 (2) 10) Determine similarity measures D ik between each pair of zones i and k applying Formulas 3, 4, 5, or 6. Each formula contains the division by 2 because two deviations are computed for each period, i.e. for the beginning and end. D ii = 1 2m D ii = 1 2 D ii = 1 2 D ii = 3 j=1 m j=1 m j=1 m j=1 y ii b y b kk + y e ii y e kk w j y b ii y b kk + y e ii y e kk v j y b ii y b kk + y e ii y e kk w jv j y b ii y b kk + y e ii y e kk 2 3 j=1 w j v j (3) (4) (5) (6) 11) Normalize the similarity measures D ik by dividing them by the average price per square metre in the respective zones i and k applying Formulas 7 and 8. In Formula 7 N i denotes the number of transactions in the zone i and P il stands for the price of individual transaction in this zone. In turn, in Formula 8 N k denotes the number of transactions in the zone k and P kl stands for the price of individual transaction in this zone. i d ii = 1 N i d k ii = 1 N k D ii N i l=1 P ii D ii N k l=1 P kk (7) (8) 12) Create a similarity matrix which contains the values of the normalized similarity i k measures d ii and d ii between each pair of zones i and k determined by Formulas 7 and 8. 13) Set the criteria for merging zones according to the Formulas 9 and 10, where δ denotes the maximum acceptable deviation.

5 i d ii < 0, δ > (9) d ii ii < 0, δ > (10) 14) Merge each pair of zones that satisfies the criteria (9) and (10) creating as a result bigger zones encompassing the transactions from both component zones. For each new zone remove outliers and then determine new linear trend functions and their values at the beginning and end of each period. 15) Repeat merging until no pair of zones i and k meets the criteria (9) and (10). As a result the algorithm produces a new partition of the area into smaller number of zones with the greater cardinality which should allow for constructing the more credible and in consequence more accurate predictive models for real estate appraisal. In order to create predictive models we need to update all prices for the last day of the considered timespan using trend functions. The procedure, which is very similar to our Delta method [18], is illustrated in Fig. 4. It is performed for each zone separately starting from the first period. Fig. 4. Updating the prices of a premises for the end of period 1 and i-th zone 16) At the time point t p for a given transaction x p compute the value of the trend function F i1 (t p ). Then, calculate the deviation of the price per square metre from the trend value ΔP(x p )=P(x p ) F i1 (t p ). Finally work out the updated price per square metre of the premises Ṗ e1 (x p ) by adding this deviation to the trend value in the end point of the first period t e1 using the formula Ṗ e1 (x p ) =ΔP(x p )+ F i1(t e1 ), where F i1 (t e1 ) is the trend function value in the i-th zone at the time point t e1. 17) Update the so obtained values of Ṗ e1 (x p ) of all premises for the end points of individual periods applying the same approach and the trend functions of successive periods. Finish the update when the last day of the considered timespan is reached. 18) Update analogously the prices of transactions from the second and next periods. As a result all the transaction prices are updated for the last day of the considered timespan. The revised prices together with the other attributes of the premises, such as usable area, age, number of storeys in a building, and distance from the city centre, constitute the basis for generating predictive models for real estate appraisal.

6 3 Experimental evaluation of the proposed method The goal of evaluating experiments was to prove the utility of merging similar zones. The experiments aimed to show that the predictive models for real estate appraisal built over the merged zones surpass in terms of accuracy the ones created over original component zones. The study was performed employing real-world data on sales transactions taken from a cadastral system and a public registry of real estate transactions. The dataset used in experiments was drawn from an unrefined dataset containing above records referring to residential premises transactions accomplished in one Polish urban municipality, i.e. a city with the population of and area of about 300 square kilometres within 16 years from 1998 to The final dataset after cleansing was confined to sales transaction data of residential premises (flats) where the land was leased on terms of perpetual usufruct and counted 31,553 samples. This dataset was then divided by the experts into four classes according to the year of a building construction as shown in Table 1. The properties encompassed by individual classes differ in construction technology and dynamics of relative market value. The differences in behaviour of property prices over time within individual classes are so big that the estimation of the values of properties should be carried out for individual classes separately. Table 1. Classes of real estate considered in the paper Class Property type Year of construction No. of trans. 1 Flats in buildings over land in perpetual usufruct before ,178 2 Flats in buildings over land in perpetual usufruct ,329 3 Flats in buildings over land in perpetual usufruct ,017 4 Flats in buildings over land in perpetual usufruct ,029 The algorithm for merging similar zones was run for individual classes of properties presented in Table 1 with the following parameters: three periods of accomplished transactions: ; ; , importance weights of individual periods: a 1 =0.6, a 2 =0.7, and a 3 =0.9, threshold value of the deviation to merge zones δ=0.2. Next, all transactional prices were updated for the last day of the considered timespan, i.e. for June 30, The update was performed within both merged zones and component ones using the algorithm described in the previous section. For further comparative analysis four datasets comprising merged zones were employed. They are shown in Table 2 where the numbers of datasets 1, 2, 3, and 4 correspond to the denotation of classes in Table 1. Due to removing outliers after merging the number of transactions in merged zones is usually less than the sum of transactions over component zones.

7 Dataset Merged zones Table 2. Datasets used in evaluation experiments Component zones. Trans. in merged zones Transactions in component zones Set , 122, 329, 117, 730, 167, 46, , 165, 375, 65, 1216, 224, 401, 3998, 58, 37, 194, 366 Set , 178, 276, 53, 104, 57, 42, 2700, 318, 41 Set , 94, 269, 480, 127, , 539 Set , 74, 202, 96, 373, 96, 36, 278, , , 359, 152, 394, 159, , 117, 268, 22, 83 In order to perform experiments the holdout method was employed. The transactions within each considered zone were randomly split into two parts in such way that 70% of transactions constituted the training set and remaining 30 % formed the test set. Two following types of predictive models were built in each zone using WEKA procedures over training sets [20]: LRM Linear Regression Model. The algorithm is a standard statistical approach to build a linear model which uses the least mean square method in order to adjust the parameters of the linear function. M5P Pruned Model Tree. The algorithm implements routines for generating M5 model trees. It is based on decision trees, however, instead of having values at tree's nodes, it contains a multivariate linear regression model at each node. The input space is divided into cells using training data and their outcomes, then a regression model is built in each cell as a leaf of the tree. Five following attributes of premises proposed by professional appraisers were applied to the LRM and M5P machine learning algorithms. As four input features: usable area of a flat (Area), age of a building construction (Age), number of storeys in a building (Storeys), the distance of the building from the city centre (Centre) were taken, in turn, price per square metre (Price) was the output variable. For comparing the accuracy of the models we used subsets of Set-1, Set-2, Set-3, and Set-4, namely the test sets of all merged zones. From each test subset 100 transactions were randomly drawn and the performance measure Mean Absolute Percentage Error (MAPE) was computed according to Formula 11, where y i a and y i p denote the actual and predicted values respectively and n stands for the number of transactions in the test set. n MMMM = 1 p a n y i y i a 100% y i i=1 (11) The drawing procedure was repeated 50 times and the final score was computed as the arithmetic mean of the results obtained in individual iterations.

8 4 Analysis of experimental results The performance of LRM and M5P models built over training datasets for the zones before and after merging is depicted in Figures 5 and 6. It is clearly seen that the models constructed over merged zones surpass the ones over component zones in terms of accuracy expressed in MAPE. Moreover, the M5P models reveal better accuracy than LRM ones over transaction datasets both before and after merging. The individual MAPE values ranging from 4 to 11 per cent were regarded by professional appraisers as a good result especially when you take into account the confined number of features which could be drawn from cadastral systems. Fig. 5. Average MAPE values for LRM and M5P before and after merging over 50 iterations Fig. 6. Comparison of MAPE values for LRM and M5P before and after merging The performance of LRM and M5P models for individual iterations for the zones before and after merging are illustrated in Figure 7. It is also visible that the majority of models constructed over merged zones outperform the ones built over component zones. However, in order to prove the significance of differences one should refer to statistical tests. Based on the output of individual iterations statistical tests of significance were made. We employed nonparametric tests, namely the Friedman test followed by the paired Wilcoxon test [21]. Average rank positions of models determined during Friedman test for the comparison of LRM before and after merging, M5P before and after merging, and LRM and M5P after merging are presented in Tables 3, 4, and 5, respectively. The ranks produced by the Friedman test mean the lower rank value the better model. In turn, the results of the paired Wilcoxon test indicate whether the differences are statistically significant.

9 Fig. 7. MAPE values for LRM and M5P before and after merging for individual 50 iterations

10 The zero hypothesis stated there were not significant differences in accuracy between a given pair of models. In Tables 3, 4, and 5 + denotes that the model in the left column performed significantly better than, significantly worse than, and ~ statistically equivalent to the one in the right column, respectively. The significance level considered for the null hypothesis rejection was 5%. The main outcome is as follows: LRM after merging showed significantly better performance than LRM before merging over all datasets but one; M5P after merging surpassed significantly M5P before merging over all datasets. In turn, M5P after merging revealed significantly lower MAPE than LRM after merging over all datasets but one. Table 3. Results of statistical tests for comparison LRM before and after merging zones Dataset Friedman Wilcoxon Result LRM after avg. rank LRM before avg. rank p-value p-value Set Set Set Set ~ Table 4. Results of statistical tests for comparison M5P before and after merging zones Dataset Friedman Wilcoxon Result M5P after avg. rank M5P before avg. rank p-value p-value Set Set Set Set Table 5. Results of statistical tests for comparison LRM and M5P after merging zones Dataset Friedman Wilcoxon Result LRM after avg. rank M5P after avg. rank p-value p-value Set Set ~ Set Set Conclusions and Future Work The problem of spatial object classification for the purposes of real estate appraisal was tackled in the paper. A novel algorithm for merging similar zones of the area of the city to obtain bigger homogeneous regions was devised. The similarity among zones was considered in respect of analogous changes of real property prices over

11 time. The measure of zone similarity based on the values of the linear trend functions within the selected time periods was also proposed. The proposed method was evaluated experimentally using real-world data of sales transactions of residential premises derived from a cadastral system and a public registry of real estate transactions. Machine learning procedures were acquired from the WEKA data mining system to build linear regression models (LRM) and pruned model trees (M5P) over merged and corresponding component zones. Mean absolute percentage error (MAPE) was employed as the performance measure and the statistical nonparametric Friedman and Wilcoxon tests were applied to examine the significance of the outcome. Experimental results proved the utility of the proposed algorithm. The models built over zones after merging revealed significantly better accuracy than the ones constructed on the basis of transactional data taken from the zones before merging. The results of our research constitute a step towards the development an intelligent system to support appraisers work providing them with the tool for accomplishing more reliable and accurate property valuations. Moreover, they may be utilized for preparing the maps of real property values for the purposes of mass taxation. It is planned to conduct further experiments to tune the parameters of the proposed algorithm using the other real-world datasets. Moreover, the other machine learning techniques will be used to construct real estate models such as neural networks, support vector regression, and genetic fuzzy systems. The comparison with the valuation methods routinely employed by professional appraisers will be also accomplished. References 1 Kontrimas, V., Verikas, A.: The mass appraisal of the real estate by computational intelligence. Applied Soft Computing 11:1, pp (2011) 2 Zurada, J., Levitan, A.S., Guan, J.: A Comparison of Regression and Artificial Intelligence Methods in a Mass Appraisal Context. Journal of Real Estate Res. 33:3, (2011) 3 Peterson, S., Flangan, A.B.: Neural Network Hedonic Pricing Models in Mass Real Estate Appraisal. Journal of Real Estate Research 31:2, (2009) 4 Narula, S.C., Wellington, J.F., Lewis, S.A.: Valuating residential real estate using parametric programming. European Journal of Operational Research 217, (2012) 5 Antipov, E.A., Pokryshevskaya, E.B.: Mass appraisal of residential apartments: An application of Random forest for valuation and a CART-based approach for model diagnostics. Expert Systems with Applications 39, (2012) 6 D Amato, M.: Comparing Rough Set Theory with Multiple Regression Analysis as Automated Valuation Methodologies. Int. Real Estate Review 10:2, (2007) 7 Kusan, H., Aytekin, O., Özdemir, I.: The use of fuzzy logic in predicting house selling price. Expert Systems with Applications 37:3, (2010) 8 Musa, A.G., Daramola, O., Owoloko, A., Olugbara, O.: A Neural-CBR System for Real Property Valuation. Journal of Emerging Trends in Computing and Information Sciences 4:8, (2013) 9 Król D., Lasota T., Nalepa W., Trawiński B.: Fuzzy system model to assist with real estate appraisals. In H.G. Okuno, A. Ali (Eds.): IEA/AIE-2007, LNAI 4570, pp , Springer, Heidelberg (2007)

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