# New Product Short-Term Demands Forecasting with Boxplot-Based Fractional Grey Prediction Model
## 3 The Proposed Method  
This research aims to improve the prediction accuracy of FGM (1,1) by determining the coefficient \(\alpha\) sets, which affect the background values proposed by [PERSON] et al. (2013) The core technique in this paper is to employ the new fractional order grey model based on boxplots and to find the coefficient \(\alpha\) sets and will be named BP-FGM (1,1) hereafter. The most suitable background values would be located between \((x^{(1)}(k-1)\), \(x^{(1)}(k))\), shown as the area colored in grey in Figure 2, where the dotted line represents the background values of the traditional GM model when \(\alpha\) is set as a constant 0.5. Firstly, it is necessary to build the shapes of attribute sample distributions and the triangular membership functions (MF), shown in Figure 3, by adopting the suitable ranges defined by the box-and-whisker plots (called boxplots hereafter). As in Section 2.1, boxplots contains three sections and each section contains its upper bound (Q1) and lower bound (Q3). The appropriate value bounds [L, U] defined in the boxplots and called the inner fences, where L is the lower fence, defined as \(1.5\times\text{IQR}\) lower than Q1 and U is the upper fence defined as \(1.5\times\text{IQR}\) higher than Q3. Observations distributed outside the bounds can be recognized as outliers, while those distributed within the bounds [L, U] are recognized as reasonable observations. However, it is easy to misclassify a reasonable observation as an outlier when sample sizes are small. The estimated bounds are thus modified as  
\[\text{L}=\left\{ \begin{array}{ll}\text{Q1}-1.5\times\text{IQR}\,&\text{L}\leq min\\ min&\,&\text{L}>min\end{array} \right. \tag{5}\]  
\[\text{U}=\left\{ \begin{array}{ll}\text{Q3}+1.5\times\text{IQR}\,&\text{U}\geq max\\ max&\,&\text{U}<max\end{array} \right. \tag{6}\]  
where \(min\) and \(max\) denote the minimum and maximum values of the observations, respectively, and are presented in Figure 4.  
Figure 3: The reasonable ranges can be defined by the box-and-whisker plots. Figure 2: The possible area indicating where background values would be located. When the domain bounds of observations are obtained, the possible sample distribution can be demonstrated by drawing a triangular membership function (MF), using L, Q2, and U to represent it, as shown in Figure 5. The reason to take Q2 (or the median, Me) as the central location is that it is more sensitive than the average (or mean) when sample sizes are small. The MF is formulated as:  
\[\text{MF}(x)=\left\{ \begin{array}{ll}1&,x=\text{CL}\\ \frac{x-L}{U-L}&,x<\text{CL}\\ \frac{U-L}{U-\text{CL}}&,x>\text{CL}\end{array} \right. \tag{7}\]  
The procedure for building the BP-FGM (1,1) model are described in following steps, as shown in Figure 6. Let the original sequence be: \(\left\{x^{(0)}(1),x^{(0)}(2),\ldots,x^{(0)}(n)\right\}\). _Step 1_. Obtain coefficient \(\alpha\) set by MF process, which is  
\[\left\{MF\Big{(}x^{(0)}(1)\Big{)},MF(x^{(0)}(2)),\ldots,MF(x^{(0)}(n))\right\} \tag{8}\]  
where \(MF\Big{(}x^{(0)}(1)\Big{)}=1\)_when_\(i=1\). _Step 2_. Generate a new sequence through the accumulating generation operator (AGO):  
\[X^{(1)}=\left\{x^{(1)}(1),x^{(1)}(2),\ldots,x^{(1)}(n)\right\}\,;\,x^{(1)}(1)= x^{(0)}(1) \tag{9}\]  
\[x^{(r)}(\kappa)=\sum_{i=1}^{\kappa}\frac{\Gamma(r+k-i)}{\Gamma(k-i+1)\Gamma(r )}x^{(0)}(i),\,k=2,3,\ldots,n \tag{10}\]  
_Step 3_. Calculate the background values, \(z^{(r)}(\kappa)=\frac{1}{2}\Big{[}x^{(r)}(k)+x^{(r)}(k-1)\Big{]}\), which can be written as  
\[z^{(1)}(\kappa)=(1-\alpha)x^{(1)}(k-1)+\alpha x^{(1)}(k),\,\,\,\alpha\epsilon (0,1),\,\,\,k=2,3,\ldots,n \tag{11}\]  
_Step 4_. Calculate the whitening differential equation:  
\[x^{(r)}(\kappa)-x^{(r)}(\kappa-1)+az^{(r)}(\kappa)=b \tag{12}\]  
Figure 4: Using triangular MFs to form the shape of the attribute distribution. Figure 5: Representing the membership function. \[\frac{dx^{(r)}(t)}{dt}+ax^{(r)}(t)=b \tag{13}\]  
_Step 5_.