### 1. Introduction

### 2. Fuzzy Regression Model Using Trapezoidal Fuzzy Numbers

##### (1)

$$Y({X}_{i})={A}_{0}\xe2\u0160\u2022{A}_{1}\xe2\u0160\u2014{X}_{i1}\xe2\u0160\u2022\xe2\u2039\xaf\xe2\u0160\u2022{A}_{p}\xe2\u0160\u2014{X}_{ip}\xe2\u0160\u2022{E}_{i},$$*X*

*is the*

_{ij}*j*-th observation of

*i*-th explanatory variable,

*A*

*is the fuzzy regression coefficient, and*

_{i}*Y*(

*X*

*) is the response variable. âŠ• and âŠ— is the addition and multiplication of fuzzy numbers, respectively. For the arithmetic operations, see [19, 20]. The membership function*

_{i}*M*

*(Â·) of a trapezoidal fuzzy number (TrFN) $A=({l}_{a}^{e},{l}_{a}^{p},{r}_{a}^{p},{r}_{a}^{e})$*

_{A}##### (2)

$${\mathrm{\xce\xbc}}_{A}(x)=\{\begin{array}{lll}\hfill 0\hfill & \text{if}\hfill & x<{l}_{a}^{e}\hfill \\ \hfill \frac{x-{l}_{a}^{e}}{{l}_{a}^{p}-{l}_{a}^{e}}\hfill & \text{if}\hfill & {l}_{a}^{e}<x<{l}_{a}^{p}\hfill \\ \hfill 1\hfill & \text{if}\hfill & {l}_{a}^{p}<x<{r}_{a}^{p}\hfill \\ \hfill \frac{{r}_{a}^{e}-x}{{r}_{a}^{e}-{r}_{a}^{p}}\hfill & \text{if}\hfill & {r}_{a}^{e}<x<{r}_{a}^{p}\hfill \\ \hfill 0\hfill & \text{if}\hfill & x>{r}_{a}^{e},\hfill \end{array}$$*A*is called a symmetric trapezoidal fuzzy number (symmetric TrFN). If the left endpoint ${l}_{a}^{e}$ 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, 22â€“24]. And if ${l}_{a}^{p}={r}_{a}^{p}={p}_{a},A=({l}_{a}^{e},{p}_{a},{r}_{a}^{e})$, is a trangular fuzzy number (TFN) [7, 10, 11, 25, 26].

*S*(

*A*) = {

*x*âˆˆ

*R*|

*M*

*(*

_{A}*x*) > 0} = (

*l*

*(0),*

_{A}*r*

*(0)). For any*

_{A}*Î±*in [0, 1], the

*Î±*-level set is defined by

*A*(

*Î±*) = {

*x*âˆˆ

*R*|

*M*

*(*

_{A}*x*) â‰¥

*Î±*}, and

*Î±*-level set can be expressed by

*A*(

*Î±*) = [

*l*

*(*

_{A}*Î±*),

*r*

*(*

_{A}*Î±*)], where ${l}_{A}(\mathrm{\xce\pm})={l}_{a}^{e}+({l}_{a}^{p}-{l}_{a}^{e})\mathrm{\xce\pm}$ and ${r}_{A}(\mathrm{\xce\pm})={r}_{a}^{e}+({r}_{a}^{p}-{r}_{a}^{e})\mathrm{\xce\pm}$. From the model (1) with positive TrFN as input and output,

*Î±*-level set of

*Y*(

*X*

*) with trapezoidal fuzzy inputs*

_{i}*X*

*= [*

_{i}*X*

_{i}_{1},

*X*

_{i}_{2}, Â·Â·Â·,

*X*

*] can be expressed by*

_{ip}*Y*(

*X*

*)(*

_{i}*Î±*) = [

*l*

_{Y(Xi)}(

*Î±*),

*r*

_{Y(Xi)}(

*Î±*)], where

##### (3)

$${l}_{Y({X}_{i})}(a)=\underset{k=0}{\overset{p}{\xe2\u02c6\u2018}}{l}_{{A}_{k}}(\mathrm{\xce\pm})\xc2\xb7{l}_{{X}_{ik}}(\mathrm{\xce\pm})+{l}_{{E}_{i}}(\mathrm{\xce\pm})$$##### (4)

$${r}_{Y({X}_{i})}(a)=\underset{k=0}{\overset{p}{\xe2\u02c6\u2018}}{r}_{{A}_{k}}(\mathrm{\xce\pm})\xc2\xb7{r}_{{X}_{ik}}(\mathrm{\xce\pm})+{r}_{{E}_{i}}(\mathrm{\xce\pm}).$$*Î±*-level set based on the resolution identity. Therefore,

*Y*(

*X*

*) can be estimated based on its*

_{i}*Î±*- level set

*Y*(

*X*

*)(*

_{i}*Î±*) after estimating the regression coefficients

*A*

*(*

_{k}*k*= 0, Â·Â·Â·,

*p*). The fuzzy regression coefficient

*A*

*can be estimated as follows:*

_{k}*A*

*is known,*

_{k}*A*

*can be estimated by estimating the parameters of membership function of*

_{k}*A*

*using specific estimated*

_{k}*Î±*-level set

*A*

*(*

_{k}*Î±*).

*A*

*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*

_{k}*A*

*is unknown,*

_{K}*A*

*can be estimated by estimating finite number of*

_{k}*Î±*-level sets of

*A*

*and applying proper estimation method for its membership function.*

_{k}*lÌ‚*

_{Ak}(1) and

*rÌ‚*

_{Ak}(1) of

*l*

_{Ak}(1) and

*r*

_{Ak}(1) by minimizing

*Î±*

^{*}âˆˆ (0, 1), find the intermediate estimators

*lÌ„*

_{Ak}(

*Î±*

^{*}) and

*rÌ„*

_{Ak}(

*Î±*

^{*}), using the formula defined in step 1. That is,

*lÌ‚*

_{Ak}(

*Î±*

^{*}) and

*rÌ‚*

_{Ak}(

*Î±*

^{*}) of

*l*

_{Ak}(

*Î±*) and

*r*

_{Ak}(

*Î±*) satisfying

*Î±*

^{*}â‰

*Î±*âˆˆ (0, 1), find the intermediate estimators

*lÌ„*

_{Ak}(

*Î±*) and

*rÌ„*

_{Ak}(

*Î±*), using the formula defined in step 2. Then find the estimators

*lÌ‚*

_{Ak}(

*Î±*) and

*rÌ‚*

_{Ak}(

*Î±*) of

*l*

_{Ak}(

*Î±*) and

*r*

_{Ak}(

*Î±*) satisfying

*lÌ‚*

_{Ak}(

*Î±*

*):*

_{j}*j*= 1, Â·Â·Â·,

*s*} and {

*rÌ‚*

_{Ak}(

*Î±*

*):*

_{j}*j*= 1, Â·Â·Â·,

*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

*Î±*

*s are needed to estimate the regression coefficients.*

_{j}*M*

_{Ã‚k}(Â·) of

*Ã‚*

*, we use*

_{k}*Î±*-level set

*Ã‚*

*(*

_{k}*Î±*

*) = [*

_{j}*lÌ‚*

_{Ak}(

*Î±*

*),*

_{j}*rÌ‚*

_{Ak}(

*Î±*

*)](*

_{j}*j*= 1, Â·Â·Â·,

*s*), where

*s*is the number of

*Î±*-level sets that we are going to estimate. For this, we apply the least squares method to {(

*rÌ‚*

_{Ak}(

*Î±*

*),*

_{j}*Î±*

*)|*

_{j}*j*= 1, Â·Â·Â·,

*s*} and {(

*lÌ‚*

_{Ak}(

*Î±*

*),*

_{j}*Î±*

*)|*

_{j}*j*= 1, Â·Â·Â·,

*s*}. We estimate the membership function

*M*

_{Ã‚k}(Â·) of

*Ã‚*

*satisfying*

_{k}*Y*(

*X*

*) using*

_{i}*Ã‚*

*(*

_{k}*k*= 0, Â·Â·Â·,

*p*) obtained from above 4 steps and the fuzzy regression model (1) as follows:

*Y*(

*X*

*) of fuzzy regression model. The first method to find ${\widehat{Y}}_{i}^{1}({X}_{i})$ after estimating*

_{i}*Ã‚*

*and applying substitution to fuzzy regression model following above 5 steps. The second method is applying above steps directly to*

_{k}*Y*(

*X*

*) not estimating coefficient of regression*

_{i}*Ã‚*

*, which will be denoted by ${\widehat{Y}}_{i}^{2}({X}_{i})$. Note that there are some methods which estimate response variables directly without estimating the regression coefficients. It might be simple to estimate ${Y}_{i}^{2}({X}_{i})$ directly, but this method has a drawback, which makes it unable to estimate ${Y}_{i}^{2}({X}_{i})$ when a new value of independent variable is given. We compare ${\widehat{Y}}_{i}^{1}({X}_{i})$ and ${\widehat{Y}}_{i}^{2}({X}_{i})$ through re-auction data provided in next section.*

_{k}### 3. Fuzzy Regression Model for Re-auction Data

*x*

_{i}_{1}), the number of bidders of prior auction (

*x*

_{i}_{2}), investment value of the real estimate (

*X*

_{i}_{3}), number of bid participants

*Y*

*is as follows:*

_{i}*Y*

*,*

_{i}*X*

_{i}_{3}, and the regression coefficient

*A*

*(*

_{j}*j*= 0, Â·Â·Â·, 3) are TrFN, and

*x*

*(*

_{ik}*k*= 1, 2) are crisp numbers (

*i*= 1, Â·Â·Â·, 18) (See Table 1).

*A*

_{0}and the coefficient

*A*

_{2}of the number of bid participants of first auction (

*x*

_{i}_{2}) are calculated as crisp numbers after applying Min-Max operation in step 3 in Section 2. Here, the 0 spreads of

*Ã‚*

_{0}and

*Ã‚*

_{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.

*A*

*. Figure 4 shows the observed number of bidders and estimated number of bidders ${\widehat{Y}}_{i}^{2}({X}_{i})$ which is obtained directly from the proposed steps.*

_{k}*Y*

*(0) and*

_{i}*Å¶*

*(0) are the supports of*

_{i}*Y*

*and*

_{i}*Å¶*

*. And*

_{i}*h*

*is the Hausdorff metric defined by*

_{d}*h*

*(*

_{d}*A*,

*B*) =

*inf*

_{a}_{âˆˆ}

_{A}*inf*

_{b}_{âˆˆ}

*|*

_{B}*a*â€“

*b*|.

*A*

*.*

_{k}