By introducing new variables \(x\) and \(\mathrm\) as

$$\beginx=}_\overline_}},\end$$

(A1)

let us define an approximated line graph (polyline) in Fig. 3b as

$$\beginy=\left\_\left(x-_\right)+_, \left(x<_\right)\\ _\left(x-_\right)+_, \left(x\ge _\right),\end\right.\end$$

(A3)

where \(_\), \(_\), \(_\), and \(_\) are the coefficients of the approximated polyline that should be estimated. For the measured values \(\_,_\}\) (\(_<_\)) and \(\_,_\}\) (\(_\ge _\)), the weighted mean squared error \(\alpha (_,_,_;\,\,_)\) between the measured values and polyline in Eq. (A3) can be obtained as follows:

$$\begin\alpha \left(_,_,_;\,\,_\right)=__\cdot _-\left(_\left(_-_\right)+_\right)\right|}^\\ +__\cdot _-\left(_\left(_-_\right)+_\right)\right|}^, \left(_<_,_\ge _\right)\end$$

(A4)

where \(w\) is the weight defined by the variance of \(EX\), \(^\), as

$$\beginw=\frac^},\end$$

(A5)

where the standard deviation \(\sigma\) of \(EX\) is presented in Table 1 and Fig. 3b. By fixing the coefficient \(_\) to an arbitrary value and defining \(\) as

$$\begin^}}\left(_\right)=x-_,\end$$

(A6)

the condition minimizing \(\alpha \left(_,_,_;\,\,_\right)\) is given by

$$\begin\frac\frac_,\,\,_,\,\,_;\,\,_\,\right)}_}=____^}}\left(_\right)}^-____^}}\left(_\right)+____^}}\left(_\right)=0,\end$$

(A7)

$$\begin\frac\frac_,\,\,_,\,\,_;\,\,_\,\right)}_}=____^}}_\right)}^-____^}}\left(_\right)+____^}}\left(_\right)=0,\end$$

(A8)

$$\begin\frac\frac_,\,_,\,_;\,_\,\right)}_}=_\left(__+__\right)-\left(___+___\right)\\ +____^}}\left(_\right)+____^}}\left(_\right)=0.\end$$

(A9)

By substituting \(_\) into Eq. (A7), and \(_\) in Eq. (A8) into Eq. (A9), \(\widehat_}(_)\), which minimizes \(\alpha \left(_,_,_;\,\,_\right)\) with fixed \(_\), is estimated by

$$\begin\widehat_}\left(_\right)=\frac___+___-\left(___+___\right)__}_}^_+_^_-\left(__+__\right)__},\end$$

(A10)

where

$$\begin_=___^}}\left(_\right), _=___^}}\left(_\right),\\ _=___^}}\left(_\right)}^, _=___^}}\left(_\right)}^,\\ _=____^}}\left(_\right),_=____^}}\left(_\right).\end$$

(A11)

Subsequently, \(\widehat_}\left(_\right)\) and \(\widehat_}\left(_\right)\) minimizing \(\alpha \left(_, _, _;\, _\right)\) with a fixed \(_\) are estimated by substituting Eq. (A10) into Eq. (A7) and (A8), respectively.

$$\begin\widehat_}\left(_\right)=\frac_-_\widehat_}\left(_\right)}_},\end$$

(A12)

$$\begin\widehat_}\left(_\right)=\frac_-_\widehat_}\left(_\right)}_}.\end$$

(A13)

Using \(\widehat_}\left(_\right)\), \(\widehat_}\left(_\right)\), and \(\widehat_}\left(_\right)\), \(\alpha \left(\widehat_}\left(_\right),\widehat_}\left(_\right),\widehat_}\left(_\right);\,_\right)\) is calculated for each of the various values \(\left\_\right\}\). Thus, the coefficient \(\widehat_}\) that minimizes \(\alpha \left(_,_,_;\,\,_\right)\) in Eq. (A4) can be obtained as follows:

$$\begin\widehat_}=\mathrm\underset_}}\alpha \left(\widehat_}\left(_\right),\widehat_}\left(_\right),\widehat_}\left(_\right);\,\,_\right).\end$$

(A14)

Finally, the optimum values of \(_\), \(_\), and \(_\) for the minimum condition can be determined by \(\widehat_}\left(\widehat_}\right)\), \(\widehat_}\left(\widehat_}\right)\), and \(\widehat_}\left(\widehat_}\right)\), respectively.