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Fuzzy Regression Model Using Trapezoidal Fuzzy Numbers for Re-auction Data

Il Kyu Kim1, Woo-Joo Lee2, Jin Hee Yoon3, and Seung Hoe Choi4

1Department of Real Estate and Finance, Gwangju University, Gwangju, Korea, 2Department of Mathematics, Yonsei University, Seoul, Korea, 3School of Mathematics and Statistics, Sejong University, Seoul, Korea, 4School of Liberal Arts and Science, Korea Aerospace University, Goyang, Korea
Correspondence to: Seung Hoe Choi (shchoi@kau.ac.kr)
Received January 5, 2016; Revised March 22, 2016; Accepted March 24, 2016.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Re-auction happens when a bid winner defaults on the payment without making second in-line purchase declaration even after determining sales permission. This is a process of selling under the court’s authority. Re-auctioning contract price of real estate is largely influenced by the real estate business, real estate value, and the number of bidders. This paper is designed to establish a statistical model that deals with the number of bidders participating especially in apartment re-auctioning. For these, diverse factors are taken into consideration, including ratio of minimum sales value from the point of selling to re-auctioning, number of bidders at the time of selling, investment value of the real estate, and so forth. As an attempt to consider ambiguous and vague factors, this paper presents a comparatively vague concept of real estate and bidders as trapezoid fuzzy number. Two different methods based on the least squares estimation are applied to fuzzy regression model in this paper. The first method is the estimating method applying substitution after obtaining the estimators of regression coefficients, and the other method is to estimate directly from the estimating procedure without substitution. These methods are provided in application for re-auction data, and appropriate performance measure is also provided to compare the accuracies.

Keywords : Re-auction, Trapezoidal fuzzy number, Fuzzy regression model, Least squares estimation
1. Introduction

There are two types of real estate auction. The one is a compulsory auction which is requested by the creditor when the real estate is owned by the debtor. The other is a voluntary auction requested by a holder of a real right granted by way of security. Re-auction happens when a successful bidder has not paid the balance until payment due without second in-line purchase declaration even after determining sales permission. It is a process of selling under the court’s authority. Reasons for defaulting on the payment can involve legal problems or economic problems. Legal reasons may be caused by analyzing the real estate title, assessing market price, and granting loans. Especially when the real estate market is in decline, re-auction is likely done because of the default on the payment of purchasing the deal at a higher price than the market price rather than because of the flaw of real estate or mistake of the analysis of real estate title. Thus, a bidding pricing is very important in real estate auction [1?3].

However, many bidders often ruin the investment estimate the bidding price simply since they based on their past experiences, intuitions, and market price. Dealing the real estate auction does not occur under normal condition: it occurs under a special circumstance where a bidder assesses the bidding price [3, 4]. As a result, price of the winning bid decreases as the number of bidders increases, and vice versa. Thus, in such type of deals, estimating the number of bidders is extremely significant.

Study on the real estate auction has been conducted in various fields, which may range from examining problems of auctioning process, analyzing the legal title, to discovering different types of case studies. Nevertheless, there has not been much study neither on the bidding price nor on the degree of bidding which chiefly affects the bidding price. One of the major problems of establishing a mathematical model of the real estate auction is the problem of uncertainty [2, 4]. There are two types of uncertainties: stochastic uncertainty whose uncertainty can be naturally resolved as time passes or as the experiment proceeds, and fuzzy uncertainty whose uncertainty is caused by an unclear distinction between different groups.

It seems more realistic to literally if not qualitatively express the investment value of real estate, which is deemed to influence the real estate auction, rather than presenting them as real numbers with definitive values. Hence, Zadeh [5, 6] suggested fuzzy theory in order to express literal variables in mathematical form. More specifically, he established the fuzzy control theory to explain fuzzy uncertainty in terms of ambiguity and vagueness, and to establish a necessary system for handling information expressed in such ambiguous or vague manners. Fuzzy theory quantifies ambiguity and vagueness by applying the degree of fuzziness and fuzzy measure. It also handles inference, evaluation, decision making based on quantified data.

This paper considers the ratio of minimum sales value from the point of selling to the point of re-auctioning, number of bidders at the time of selling, and investment value of the real estate as discrimination variable in order to construct a statistical model of the number of bidders participating in re-auctioning. Since investment value of the real estate and the number of bidders are ambiguous and vague in this particular suggested model, trapezoid fuzzy number is used as a statistical variable. As well, this paper uses the least square method in order to estimate the regression model in accordance with the number of bidders, using the trapezoid fuzzy number.

Also, two different methods based on the least squares estimations are applied to fuzzy regression model. The first method is to find the estimated values applying substitution after obtaining the estimators of regression coefficients, and the second method is to find the estimated values directly from the estimating procedure without substitution. These methods are provided in application for re-auction data, and proper performance measure is also provided to compare the accuracies in Section 2.

2. Fuzzy Regression Model Using Trapezoidal Fuzzy Numbers

Fuzzy regression analysis was first introduced by Tanaka and Hayashi [7] and Tanaka et al. [8] in order to apply linguistic or vague data to regression analysis. Fuzzy regression analysis using triangular fuzzy numbers and trapezoidal fuzzy number have been studied in many works [9?15]. The theoretical studeis regarding fuzzy regression model have been investigated in [16?18].

A fuzzy number can be expressed differently due to subjective points of view while a crisp number is expressed uniquely. However, expression of a fuzzy number is based on the objective observation, therefore a fuzzy number includes subjective concept as well as objective concept. Triangular fuzzy numbers and trapezoidal fuzzy numbers have been widely used to express these two different kinds of concepts. To analyze re-auction data, we introduce a fuzzy regression model with trapezoidal fuzzy input and outputs as follows:

$Y(Xi)=A0?A1?Xi1???Ap?Xip?Ei,$

where Xij is the j-th observation of i-th explanatory variable, Ai is the fuzzy regression coefficient, and Y (Xi) is the response variable. ? and ? is the addition and multiplication of fuzzy numbers, respectively. For the arithmetic operations, see [19, 20]. The membership function MA(·) of a trapezoidal fuzzy number (TrFN) $A=(lae,lap,rap,rae)$

can be expressed as follows:

$μA(x)={0ifxrae,$

where $lae$ and $rae$ are two end points [13, 21], $lap$ and $rap$ and are two peak points (see Figure 1).

If $rae-rap=lap-lae$, the trapezoidal fuzzy number A is called a symmetric trapezoidal fuzzy number (symmetric TrFN). If the left endpoint $lae$ is a positive number, then A is called a positive TrFN. From practical point of view, note that the independent variables which are assigned with only positive or nonnegative fuzzy numbers, have been frequently used in several studies [13, 22?24]. And if $lap=rap=pa,A=(lae,pa,rae)$, is a trangular fuzzy number (TFN) [7, 10, 11, 25, 26].

The support of a fuzzy set is defined by S(A) = {xR | MA(x) > 0} = (lA(0), rA(0)). For any α in [0, 1], the α-level set is defined by A(α) = {xR | MA(x) ≥ α}, and α-level set can be expressed by A(α) = [lA(α), rA(α)], where $lA(α)=lae+(lap-lae)α$ and $rA(α)=rae+(rap-rae)α$. From the model (1) with positive TrFN as input and output, α-level set of Y (Xi) with trapezoidal fuzzy inputs Xi = [Xi1, Xi2, ···, Xip] can be expressed by Y (Xi)(α) = [lY(Xi)(α), rY(Xi)(α)], where

$lY(Xi)(a)=∑k=0plAk(α)·lXik(α)+lEi(α)$

and

$rY(Xi)(a)=∑k=0prAk(α)·rXik(α)+rEi(α).$

Zadeh [6] suggested the extension principle to define the membership functions of a function of fuzzy numbers. Operations between fuzzy numbers can be defined based on the extension principle. Further he suggested the resolution identity. A fuzzy number can be expressed by α-level set based on the resolution identity. Therefore, Y (Xi) can be estimated based on its α- level set Y (Xi)(α) after estimating the regression coefficients Ak(k = 0, ···, p). The fuzzy regression coefficient Ak can be estimated as follows:

In case the membership function of regression coefficient Ak is known, Ak can be estimated by estimating the parameters of membership function of Ak using specific estimated α-level set Ak(α).

However, if the membership function of Ak is unknown, it should be estimated applying non-parametric method to estimated alpha-level sets of the regression coefficients after appropriate number of alpha-level sets are properly calculated. In case the membership function of regression coefficient AK is unknown, Ak can be estimated by estimating finite number of α-level sets of Ak and applying proper estimation method for its membership function.

Next, the estimation procedure for the regression coefficients is proposed when the membership functions of those are known. In order to estimate the regression coefficients of (1), the regression model using trapezoidal fuzzy numbers for re-auction, 5 steps are proposed as follows: we suggest following 5 steps to find the regression model (1) using (3) and (4).

Step 1: Use the least squares method to find the intermediate estimators l?Ak(1) and r?Ak(1) of lAk(1) and rAk(1) by minimizing

$∑i=1n(lYi(1)-∑k=0plAk(1)·lXik(1))2=Min!$

and

$∑i=1n(rYi(1)-∑k=0prAk(1)·rXik(1))2=Min!,$

respectively.

Step 2: For some α* ∈ (0, 1), find the intermediate estimators l?Ak(α*) and r?Ak(α*), using the formula defined in step 1. That is,

$∑i=1n(lYi(α*)-∑k=0plAk(α*)·lXik(α*))2=Min!$

and

$∑i=1n(rYi(α*)-∑k=0prAk(α*)·rXik(α*))2=Min!,$

respectively. Then find the estimators l?Ak(α*) and r?Ak(α*) of lAk(α) and rAk(α) satisfying

$l^Ak(α*)=Min{l?Ak(α*),l^Ak(1)}$

and

$r^Ak(α*)=Max{r?Ak(α*),r^Ak(α*)}$

to be TrFNs.

Step 3: For arbitrary α*α ∈ (0, 1), find the intermediate estimators l?Ak(α) and r?Ak(α), using the formula defined in step 2. Then find the estimators l?Ak(α) and r?Ak(α) of lAk(α) and rAk(α) satisfying

$l^Ak(α)={Max{l?Ak(α*),Min{l?Ak(α),l^Ak(1)}}ifα*<αMin{l?Ak(α),l^Ak(α*)}ifα*≥α$

to be TrFNs. The {l?Ak(αj): j = 1, ···, s} and {r?Ak(αj): j = 1, ···, s} estimated in step 3 are used to estimate the membership function of the regression coefficients. Note that when the membership function is given as a nonlinear curve, more number of αjs are needed to estimate the regression coefficients.

Step 4: To estimate the membership function M?k(·) of ?k, we use α-level set ?k(αj) = [l?Ak(αj), r?Ak(αj)](j = 1, ···, s), where s is the number of α-level sets that we are going to estimate. For this, we apply the least squares method to {(r?Ak(αj), αj)|j = 1, ···, s} and {(l?Ak(αj), αj)|j = 1, ···, s}. We estimate the membership function M?k(·) of ?k satisfying

$MA^k(l^Ak(1))=MA^k(r^Ak(1))=1.$

Step 5: We find the estimated value $Y^i1(Xi)$ of Y (Xi) using ?k(k = 0, ···, p) obtained from above 4 steps and the fuzzy regression model (1) as follows:

$Y^i1(Xi)=A^0?A^1?Xi1???A^p?Xip.$

In ordinary regression, there is only one method to find the estimated value of responsible variable. That is, after finding the estimates of the regression coefficients we use the regression model by substitution. On the other hand, we consider two different methods in this paper to find the estimated value of Y (Xi) of fuzzy regression model. The first method to find $Y^i1(Xi)$ after estimating ?k and applying substitution to fuzzy regression model following above 5 steps. The second method is applying above steps directly to Y (Xi) not estimating coefficient of regression ?k, which will be denoted by $Y^i2(Xi)$. Note that there are some methods which estimate response variables directly without estimating the regression coefficients. It might be simple to estimate $Yi2(Xi)$ directly, but this method has a drawback, which makes it unable to estimate $Yi2(Xi)$ when a new value of independent variable is given. We compare $Y^i1(Xi)$ and $Y^i2(Xi)$ through re-auction data provided in next section.

3. Fuzzy Regression Model for Re-auction Data

To find the statistical model for real estate re-auction data, we investigated the potential variables which may affect the number of bidders. Consequently, we found out that affect the number of total bidders has to do with the ratio between the price of lowest bid, the total number of bids, and the perceived market value of the real estate. In order to find the statistical model for the number of bidders based on the study in advance, we’ve actually used 18 samples of specific area of Seoul in 2013. The number of bidders on re-auction is known to be dependent on the number of bidders who participated on the original auction prior to failure of bidding. According to the expert view, it is empirically known that the number of bidders of re-auction shows specific patterns when the number of bidders of prior auction was 13, 420, and over 20. Auction participants of targeted real estate are heavily influenced by the subtle changes in the market climate [2]. Therefore, the number of bidders should reflect facts both subjective and objective, thus should be expressed in terms of trapezoidal fuzzy numbers. In addition, it is also known that the price of lowest bid of re-auction gets 20% smaller than prior auction [3]. The number of bidders just prior to the re-auction serves as the crucial factors which decides outcome. However, the main reason for failure of bidding has to do with unreasonably high winning price, so the reasonable bidding price is really important.

The real estate value at re-auction is determined by factors such as the average lowest price of sales, location, market climate, similar sales in the neighbor, and various subjective opinions held by the auction participants. Therefore, the value of real estate at re-auction can be expressed in term of trapezoidal fuzzy numbers as follows (See Figure 2):

(8, 9, 10, 10)(very good), (7, 8, 8, 9)(good), (5, 6, 7, 8)(somewhat good), (4, 5, 5, 6)(average), (2, 3, 4, 5) (somewhat bad), (1, 2, 2, 3)(bad), (0, 0, 1, 2)(very bad).

The number of failure of bidding at re-auction can be expressed in terms of the ratio of the lowest bidding price due to failure of bidding, because it is known that the price of lowest bid of re-auction gets 20% smaller than prior auction. The fuzzy regression model for ratio of the lowest bidding price due to failure of bidding (xi1), the number of bidders of prior auction (xi2), investment value of the real estimate (Xi3), number of bid participants Yi is as follows:

$Yi=A0?A1?xi1?A2?xi2?A3?xi3?Ei$

where Yi, Xi3, and the regression coefficient Aj(j = 0, ···, 3) are TrFN, and xik(k = 1, 2) are crisp numbers (i = 1, ···, 18) (See Table 1).

The fuzzy regression model using 5 step method that we proposed in Section 2 for 18 samples of specific area of Seoul is as follows:

$Y^i1(Xi)=-5.34?(2.07,2.33,3.75,4.11)?xi1?0.03?xi2?(1.01,1.17,1.45,1.72)?Xi3.$

Here, the constant term A0 and the coefficient A2 of the number of bid participants of first auction (xi2) are calculated as crisp numbers after applying Min-Max operation in step 3 in Section 2. Here, the 0 spreads of ?0 and ?2, estimated from step 1 to step 5, explains that the number of bidders of prior auction can affect only the mode of number of bid participants.

Figure 3 shows the observed number of bidders and estimated number of bidders $Y^i1(Xi)$ obtained by estimating the coefficients Ak. Figure 4 shows the observed number of bidders and estimated number of bidders $Y^i2(Xi)$ which is obtained directly from the proposed steps.

In order to compare the efficiencies of the estimated fuzzy regression model, we calculate the distance between the observed value and estimated value to find the errors for accuracies. For this, we propose distance between two fuzzy numbers as follows [2, 13, 21, 27]:

$M=1n∑i=1nd(Yi,Y^i),$

where

$d(Yi,Y^i)=∫-∞∞?μYi(x)-μY^i(x)?dx∫-∞∞?μYi(x)?dx+hd(Yi(0),Y^i(0)).$

Here, Yi(0) and ?i(0) are the supports of Yi and ?i. And hd is the Hausdorff metric defined by hd(A, B) = infaAinfbB|a?b|.

Table 2 shows the results of estimated values and errors obtained from (4) based on Hasdorff metric formula. It is shown in Table 2 that total sum of the error between the observed value and the estimated value $Y^i2$ obtained directly is more than $Y^i1$ which is obtained after estimating the coefficients Ak.

The estimated fuzzy regression model shows that the higher the value of real estate is, the more the number of bidders of re-auction increase. Further the larger the number of failure of bidding of auction for some real estimate is, the more the number of bidders of re-auction increase. In addition, the less the number of bid participants of first auction is, the number of bidders of re-auction increase. This result doesn’t seem to coincide with the results of common re-auction. However, it seems that the increase of investment value based on the decrease of real estate market, and several times of defaults affected the results.

4. Conclusions

In order to find the fuzzy regression model for re-auction, we expressed the independent and dependent values in terms of trapezoidal fuzzy numbers. Eighteen data of re-auction which are collected from specific area of Seoul are used to find the fuzzy regression model. Several independent data such as ratio of the lowest bidding price due to failure of bidding, the number of bidders of prior auction, investment value of the real estimate are used to estimate the number of bid participants of re-auction. In this paper, two different least squares estimation methods are applied to fuzzy regression model. That is, estimating method applying substitution after obtaining the estimators of regression coefficients, and the other method is to estimate directly from the estimating procedure without substitution. These methods are provided in application for re-auction data, and proper performance measure is also provided to compare the accuracies. The results showed that the estimated values which are obtained directly performed better than the other method. This research shows that the number of defaults after first auction, the number of bid participants in first auction and the investment value of the real estate can affect the number of bid participants of re-auction. The number of defaults after first auction and the increase investment value can positively affect the number of bid participants of re-auction. However, the number of bid participants in first auction can negatively affect the number of bid participants of re-auction. Further research is needed to apply various methods to analyze re-auction data. And data collection covering wide scope is also needed to be considered in our next research.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Figures
Fig. 1.

Trapezoidal fuzzy number $A=(lae,lap,rap,rae)$.

Fig. 2.

Fuzzy numbers for the value of real estate.

Fig. 3.

The observed and estimated number of bid participants: $Y^i1(Xi)$.

Fig. 4.

The observed and estimated number of bid participants: $Y^i2(Xi)$.

TABLES

### Table 1

Fuzzy data for sales volumes

No.xi1xi2Xi3Yi
10.82(8,9,10,10)(14,16,18,20)
20.82(4,5,5,6)(2,4,6,8)
30.81(4,5,5,6)(0,1,2,3)
40.641(8,9,10,10)(10,13,16,19)
50.641(5,6,7,8)(0,1,2,3)
611(2,3,4,5)(1,2,3,4)
713(5,6,7,8)(1,3,6,8)
80.642(5,6,7,8)(0,1,3,4)
90.84(8,9,10,10)(2,4,6,8)
1017(4,5,5,6)(1,2,3,4)
110.82(8,9,10,10)(1,2,6,7)
120.81(5,6,7,8)(0,1,3,4)
130.641(7,8,8,9)(2,4,6,8)
14118(8,9,9,10)(6,8,10,12)
15110(8,9,9,10)(4,6,8,10)
160.641(8,9,9,10)(3,4,6,7)
170.641(2,3,4,5)(1,2,3,4)
1815(4,5,5,6)(2,3,4,5)

### Table 2

Estimated values $Y^i1$ and $Y^i2$ and errors

No.$Y^i1$Error$Y^i2$Error
1(4.31, 6.96, 12.12, 15.13)2.2182(5.10, 6.96, 10.13, 11.06)2.7347
2(0.27, 2.29, 4.86, 8.23)0.2380(1.07, 2.29, 2.86, 4.16)2.1868
3(0.31, 2.32, 4.89, 8.26)2.1442(1.11, 2.32, 2.89, 4.23)2.1380
4(4.01, 6.62, 11.56, 14.50)2.1820(4.81, 6.62, 9.56, 10.47)2.0685
5(0.99, 3.12, 7.20, 11.05)2.3592(1.78, 3.12, 5.20, 7.02)2.2911
6(0, 0.46, 4.19, 7.36)1.7725(0, 0.46, 2.19, 3.32)1.9045
7(1.66, 3.89, 8.48, 12.47)1.5775(2.45, 3.89, 6.48, 8.37)1.5775
8(0.95, 3.08, 7.16, 11.02)2.0267(1.75, 3.08, 5.16, 6.95)2.0265
9(4.23, 6.89, 12.06, 15.06)2.1795(5.03, 6.89, 10.06, 10.93)2.1708
10(0.50, 2.58, 5.44, 8.88)0.1570(1.30, 2.58, 3.43, 4.67)1.7917
11(4.31, 6.96, 12.12, 15.13)2.1411(5.10, 6.96, 10.13, 11.06)2.1271
12(1.32, 3.49, 7.80, 11.71)2.1381(2.11, 3.49, 5.80, 7.68)2.1296
13(3.00, 5.45, 8.65, 12.78)1.8626(3.80, 5.45, 6.65, 8.74)1.8626
14(4.13, 6.87, 10.87, 15.40)0.9088(4.93, 6.87, 8.86, 10.86)1.7841
15(4.42, 7.15, 11.15, 15.67)1.7869(5.22, 7.15, 9.14, 11.37)1.7869
16(4.01, 6.62, 10.11, 14.50)2.1705(4.81, 6.62, 8.11, 10.47)2.1605
17(0, 0, 2.84, 5.88)0.0064(0, 0, 0.84, 1.84)2.2442
18(0.57, 2.65, 5.51, 8.95)2.6700(1.37, 2.65, 3.50, 4.79)1.7477
M1.69662.0407

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Biographies

Il Kyu Kim obtained his Ph.D. degree in Mathematics majoring mathematica statistics from Yonsei University, Seoul, Korea in 1998. He is currently full professor of Gwanju University. His main research interests are mathematical prediction method using the soft computing and statistical prediction in real estate investment and theory of real estate auction.

Woo-Joo Lee obtained a Ph.D. in Mathematics majoring mathematical statistics from Yonsei University, Seoul, Korea. His main research interests are data mining, fuzzy time series and intelligent systems.

Jin Hee Yoon received Bachelor, Master and Ph.D. degree in Mathematics from Yonsei university. She is currently faculty of school of Mathematics and Statistics at Sejong University, Seoul, Korea. Her research interests are fuzzy regression analysis, fuzzy time series and F-transform. She is a board member of KIIS (Korean institute of Intelligent Systems) and has been working as an editor, guest editor and editorial board member of The Scientific world journal, Information, International journal of Applied Economics and International Journal of Fuzzy logic and Intelligent Systems etc. Also, she has been working as an organizer and committee member of several international conferences such as 2012, 2013 NIMS Hot topic workshop, 2012, 2013 ICIS, 2014, 2015 FUZZ IEEE, 2015 ISIS, 2016 SCIS&ISIS, 2016 ICFMsquare 2016, ICMF 2016 etc.

Seung Hoe Choi obtained his Ph.D. degree in Mathematical Statistics from Yonsei University Korea in 1994. Since 1996, he is currently full professor of Korea Aerospace University. His main research interests are mathematical prediction method using the soft computing and statistical prediction in sports like soccer, baseball and basketball.

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