Boscovich Fuzzy Regression Line
Abstract
:1. Introduction
2. Materials and Methods
 (1)
 the sum of the positive and negative residuals (in the sense of y axis) shall be equal,
 (2)
 the sum of the best absolute values of all the residuals shall be as small as possible.
3. Fuzzy Set Preliminaries
 (a)
 ${\mu}_{\tilde{A}}\left(x\right)$ is normal, i.e., $\exists {x}_{0}\in \mathbb{R}$ with ${\mu}_{\tilde{A}}\left({x}_{0}\right)=1$,
 (b)
 ${\mu}_{\tilde{A}}\left(x\right)$ is fuzzy convex,
 (c)
 ${\mu}_{\tilde{A}}\left(x\right)$ is upper semicontinuous on $\mathbb{R}$,
 (d)
 ${\mu}_{\tilde{A}}\left(x\right)$ is compactly supported, i.e., $cl\left\{x\in \mathbb{R};{\mu}_{\tilde{A}}\left(x\right)>0\right\}$ is compact, where $cl\left(A\right)$ denotes the closure of the set A.
 (a)
 commutative, i.e., $\tilde{A}\oplus \tilde{B}=\tilde{B}\oplus \tilde{A}$,
 (b)
 associative, i.e., $\tilde{A}\oplus (\tilde{B}\oplus \tilde{C})=(\tilde{A}\oplus \tilde{B})\oplus \tilde{C}$.
 (a)
 for any $\lambda ,\nu \in \mathbb{R}$ with $\lambda \nu \ge 0$ and any $\tilde{A}\in {\mathbb{R}}_{T}$ that$$(\lambda +\nu )\xb7\tilde{A}=\lambda \xb7\tilde{A}\oplus \nu \xb7\tilde{A},$$
 (b)
 for any $\lambda \in \mathbb{R}$ and any $\tilde{A},\tilde{B}\in {\mathbb{R}}_{T}$ that$$\lambda \xb7(\tilde{A}\oplus \tilde{B})=\lambda \xb7\tilde{A}\oplus \lambda \xb7\tilde{B},$$
 (c)
 for any $\tilde{A}\in {\mathbb{R}}_{T}$ and $\lambda \in \mathbb{R}$ that$$\tilde{A}\lambda ={\left({m}_{\tilde{A}}\lambda ,{\alpha}_{\tilde{A}},{\beta}_{\tilde{A}}\right)}_{T},$$and$$\tilde{A}+\lambda ={\left({m}_{\tilde{A}}+\lambda ,{\alpha}_{\tilde{A}},{\beta}_{\tilde{A}}\right)}_{T}.$$
4. Boscovich Fuzzy Regression Line
Algorithm 1 Boscovich fuzzy regression line 
Ensure: The best estimates of ${\tilde{A}}_{0}$ and ${a}_{1}$

5. Numerical Examples
6. Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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PLR  OPRL  MOFLR  FLS  FLAR  RFR  BFRL  

$\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  $\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  $\mathit{\omega}=\mathbf{0.1}$  $\mathit{\omega}=\mathbf{0.5}$  $\mathit{\omega}=\mathbf{0.99}$  
SFN1  12.40  17.89  12.40  17.89  220.42  35.35  14.53  10.14  9.50    9.17 
SFN2  3.86  6.45  3.51  4.78  73.73  11.89  5.09  3.23  3.24  3.23  3.61 
NFN1                144.13  133.81  151.98  161.09 
NFN2                15.21  14.09  14.57  15.30 
NFN3                2.98  2.29  2.90  3.03 
o1  o2  o3  

Dataset  i  ${\mathit{y}}_{\mathit{i}}$  ${\underline{\mathit{\upsilon}}}_{\mathit{i}}$  ${\overline{\mathit{\upsilon}}}_{\mathit{i}}$  ${\mathit{y}}_{\mathit{i}}$  ${\underline{\mathit{\upsilon}}}_{\mathit{i}}$  ${\overline{\mathit{\upsilon}}}_{\mathit{i}}$  ${\mathit{y}}_{\mathit{i}}$  ${\underline{\mathit{\upsilon}}}_{\mathit{i}}$  ${\overline{\mathit{\upsilon}}}_{\mathit{i}}$ 
SFN1  3  $\mathbf{1}$  1.8  1.8  8  $\mathbf{35}$  $\mathbf{35}$  $\mathbf{1}$  $\mathbf{35}$  $\mathbf{35}$ 
SFN2  4  $\mathbf{4}$  0.4  0.4  2  $\mathbf{1.6}$  $\mathbf{1.6}$  $\mathbf{4}$  $\mathbf{1.6}$  $\mathbf{1.6}$ 
NFN1  7  $\mathbf{170}$  12  12  70.9  $\mathbf{65}$  $\mathbf{77}$  $\mathbf{170}$  $\mathbf{65}$  $\mathbf{77}$ 
NFN2  6  $\mathbf{3}$  1.5  1.7  22  $\mathbf{10}$  $\mathbf{0.01}$  $\mathbf{3}$  $\mathbf{10}$  $\mathbf{0.01}$ 
NFN3  1  $\mathbf{9.5}$  0.17  0.4  2.5  $\mathbf{3}$  $\mathbf{4}$  $\mathbf{9.5}$  $\mathbf{3}$  $\mathbf{4}$ 
PLR  OPRL  MOFLR  FLS  FLAR  RFR  BFRL  

$\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  $\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  $\mathit{\omega}=\mathbf{0.1}$  $\mathit{\omega}=\mathbf{0.5}$  $\mathit{\omega}=\mathbf{0.99}$  
SFN1o1  35.99  56.88  35.99  23.03 *  220.58  39.57  22.50  15.74  14.17    14.59 
SFN2o1  7.58  12.70  4.33 *  6.29 *  73.74  12.29  5.82  3.76  3.01  3.76  3.49 
SFN1o2  131.05  131.05  42.03 *  44.30 *  836.00  132.99  79.48  52.97  41.08    53.02 
SFN2o2  8.21  10.17    5.62 *  96.52  15.44  6.51  3.90  3.82  4.60  3.97 
SFN1o3  132.57  132.38  42.56 *  45.06 *  836.13  145.87  81.79  54.64  42.06    54.61 
SFN2o3  10.70  15.72  5.23 *  6.96 *  96.54  16.10  7.69  4.80  3.93  5.64  4.60 
FLS  FLAR  RFR  BFRL  

NFN1o1  180.45  146.88  166.43  189.74 
NFN2o1  19.23  15.34  20.14  19.14 
NFN3o1  5.30  2.29  5.32  5.03 
NFN1o2  207.69  183.84  192.53  214.43 
NFN2o2  15.74  14.89  17.04  16.54 
NFN3o2  5.86  4.94  6.57  6.15 
NFN1o3  259.93  205.94  393.57  259.24 
NFN2o3  25.98  19.47  28.62  25.64 
NFN3o3  11.63  5.58  11.69  10.60 
PLR  OPRL  

$\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  $\mathit{h}=\mathbf{0}$  $\mathit{h}=\mathbf{0.5}$  
${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  
SFN1  ${(3.85,3.85,3.85)}_{T}$  ${(2.10,0.00,0.00)}_{T}$  ${(4.15,5.57,5.57)}_{T}$  ${(1.97,0.00,0.00)}_{T}$  ${(3.85,3.85,3.85)}_{T}$  ${(2.10,0.00,0.00)}_{T}$  ${(4.15,5.57,5.57)}_{T}$  ${(1.97,0.00,0.00)}_{T}$ 
SFN2  ${(1.28,0.83,0.83)}_{T}$  ${(0.13,0.00,0.00)}_{T}$  ${(1.39,1.23,1.23)}_{T}$  ${(0.11,0.00,0.00)}_{T}$  ${(1.03,0.63,0.63)}_{T}$  ${(0.14,0.01,0.01)}_{T}$  ${(0.94,0.98,0.98)}_{T}$  ${(0.15,0.00,0.00)}_{T}$ 
SFN1o1  ${(1.68,6.02,6.02)}_{T}$  ${(1.51,0.59,0.59)}_{T}$  ${(2.27,9.32,9.32)}_{T}$  ${(1.33,1.27,1.27)}_{T}$  ${(1.68,6.02,6.02)}_{T}$  ${(1.51,0.59,0.59)}_{T}$  ${(4.15,5.57,5.57)}_{T}$  ${(1.20,0.00,0.00)}_{T}$ 
SFN2o1  ${(2.85,1.25,1.25)}_{T}$  ${(0.03,0.00,0.00)}_{T}$  ${(3.05,2.00,2.00)}_{T}$  ${(0.02,0.00,0.00)}_{T}$  ${(1.29,0.82,0.82)}_{T}$  ${(0.13,0.00,0.00)}_{T}$  ${(1.44,1.12,1.12)}_{T}$  ${(0.11,0.00,0.00)}_{T}$ 
SFN1o2  ${(5.53,26.47,26.47)}_{T}$  ${(1.32,2.84,2.84)}_{T}$  ${(5.52,25.25,25.25)}_{T}$  ${(1.33,3.25,3.25)}_{T}$  ${(3.85,3.85,3.85)}_{T}$  ${(2.10,0.00,0.00)}_{T}$  ${(4.15,5.57,5.57)}_{T}$  ${(1.98,0.00,0.00)}_{T}$ 
SFN2o2  ${(0.53,1.60,1.60)}_{T}$  ${(0.16,0.00,0.00)}_{T}$  ${(1.09,1.83,1.83)}_{T}$  ${(0.11,0.00,0.00)}_{T}$      ${(1.44,1.12,1.12)}_{T}$  ${(0.11,0.00,0.00)}_{T}$ 
SFN1o3  ${(6.68,26.19,26.19)}_{T}$  ${(1.89,2.94,2.94)}_{T}$  ${(4.13,24.07,24.07)}_{T}$  ${(1.04,3.64,3.64)}_{T}$  ${(3.85,3.85,3.85)}_{T}$  ${(2.10,0.00,0.00)}_{T}$  ${(4.15,5.57,5.57)}_{T}$  ${(1.97,0.00,0.00)}_{T}$ 
SFN2o3  ${(4.01,1.77,1.77)}_{T}$  ${(0.02,0.00,0.00)}_{T}$  ${(3.63,2.52,2.52)}_{T}$  ${(0.01,0.00,0.00)}_{T}$  ${(1.29,0.82,0.82)}_{T}$  ${(0.13,0.00,0.00)}_{T}$  ${(1.44,1.12,1.12)}_{T}$  ${(0.11,0.00,0.00)}_{T}$ 
MOFLR  

$\mathit{\omega}=\mathbf{0.1}$  $\mathit{\omega}=\mathbf{0.5}$  $\mathit{\omega}=\mathbf{0.99}$  
${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  
SFN1  ${(4.95,36.80,36.80)}_{T}$  ${(1.71,3.20,3.20)}_{T}$  ${(4.95,7.36,7.36)}_{T}$  ${(1.71,0.64,0.64)}_{T}$  ${(4.95,1.84,1.84)}_{T}$  ${(1.71,0.16,0.16)}_{T}$ 
SFN2  ${(1.38,2.95,2.95)}_{T}$  ${(0.12,0.50,0.50)}_{T}$  ${(1.38,0.59,0.59)}_{T}$  ${(0.12,0.10,0.10)}_{T}$  ${(1.37,0.30,0.30)}_{T}$  ${(0.12,0.05,0.05)}_{T}$ 
SFN1o1  ${(3.25,36.80,36.80)}_{T}$  ${(1.71,3.20,3.20)}_{T}$  ${(3.25,7.36,7.36)}_{T}$  ${(1.71,0.64,0.64)}_{T}$  ${(3.25,3.72,3.72)}_{T}$  ${(1.71,0.32,0.32)}_{T}$ 
SFN2o1  ${(2.38,2.95,2.95)}_{T}$  ${(0.06,0.50,0.50)}_{T}$  ${(2.38,0.59,0.59)}_{T}$  ${(0.06,0.10,0.10)}_{T}$  ${(2.38,0.30,0.30)}_{T}$  ${(0.07,0.05,0.05)}_{T}$ 
SFN1o2  ${(4.95,166.40,166.40)}_{T}$  ${(1.71,3.20,3.20)}_{T}$  ${(4.95,33.28,33.28)}_{T}$  ${(1.71,0.64,0.64)}_{T}$  ${(4.94,16.81,16.81)}_{T}$  ${(1.71,0.32,0.32)}_{T}$ 
SFN2o2  ${(1.38,14.95,14.95)}_{T}$  ${(0.12,\mathbf{0.17},\mathbf{0.17})}_{T}$  ${(1.38,2.99,2.99)}_{T}$  ${(0.12,\mathbf{0.03},\mathbf{0.03})}_{T}$  ${(1.35,1.51,1.51)}_{T}$  ${(0.12,\mathbf{0.02},\mathbf{0.02})}_{T}$ 
SFN1o3  ${(3.25,166.40,166.40)}_{T}$  ${(1.71,3.20,3.20)}_{T}$  ${(3.25,33.28,33.28)}_{T}$  ${(1.71,0.64,0.64)}_{T}$  ${(3.26,16.81,16.81)}_{T}$  ${(1.71,0.32,0.32)}_{T}$ 
SFN2o3  ${(2.38,14.95,14.95)}_{T}$  ${(0.06,\mathbf{0.17},\mathbf{0.17})}_{T}$  ${(2.38,2.99,2.99)}_{T}$  ${(0.06,\mathbf{0.03},\mathbf{0.03})}_{T}$  ${(2.38,1.51,1.51)}_{T}$  ${(0.07,\mathbf{0.02},\mathbf{0.02})}_{T}$ 
FLS  FLAR  RFR  BFRL  

${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\mathbf{a}}^{\mathbf{T}}$  b  g  d  h  ${\tilde{\mathit{A}}}_{0}$  ${\mathit{a}}_{\mathbf{1}}$  
SFN1  ${(4.95,1.84,1.84)}_{T}$  ${(1.71,0.16,0.16)}_{T}$  ${(5.61,1.80,1.80)}_{T}$  ${(1.48,0.20,0.20)}_{T}$            ${(5.70,2.32,2.32)}_{T}$  1.46 
SFN2  ${(1.38,1.48,1.48)}_{T}$  ${(0.12,0.03,0.03)}_{T}$  ${(1.41,0.10,0.10)}_{T}$  ${(0.12,0.03,0.03)}_{T}$  $\left[\begin{array}{cc}0.81& 0.16\end{array}\right]$  0.18  0.18  −0.03  −0.02  ${(1.20,0.49,0.49)}_{T}$  0.13 
SFN1o1  ${(3.25,1.84,1.84)}_{T}$  ${(1.71,0.16,0.16)}_{T}$  ${(4.14,1.80,1.80)}_{T}$  ${(1.77,0.20,0.20)}_{T}$            ${(2.44,2.32,2.32)}_{T}$  1.98 
SFN2o1  ${(2.38,0.15,0.15)}_{T}$  ${(0.06,0.03,0.03)}_{T}$  ${(2.00,0.10,0.10)}_{T}$  ${(0.08,0.03,0.03)}_{T}$  $\left[\begin{array}{cc}2.37& 0.06\end{array}\right]$  0.39  0.39  −0.77  −0.77  ${(1.90,0.49,0.49)}_{T}$  0.10 
SFN1o2  ${(4.95,8.32,8.32)}_{T}$  ${(1.71,0.16,0.16)}_{T}$  ${(5.60,1.80,1.80)}_{T}$  ${(1.48,0.20,0.20)}_{T}$            ${(5.70,8.80,8.80)}_{T}$  1.46 
SFN2o2  ${(1.38,0.75,0.75)}_{T}$  ${(0.12,\mathbf{0.01},\mathbf{0.01})}_{T}$  ${(1.42,0.21,0.21)}_{T}$  ${(0.12,0.02,0.02)}_{T}$  $\left[\begin{array}{cc}1.57& 0.10\end{array}\right]$  0.18  0.19  0.14  0.12  ${(1.20,0.64,0.64)}_{T}$  0.13 
SFN1o3  ${(3.25,8.32,8.32)}_{T}$  ${(1.71,0.16,0.16)}_{T}$  ${(4.05,1.80,1.80)}_{T}$  ${(1.79,0.20,0.20)}_{T}$            ${(2.44,8.80,8.80)}_{T}$  1.98 
SFN2o3  ${(2.38,0.75,0.75)}_{T}$  ${(0.06,\mathbf{0.01},\mathbf{0.01})}_{T}$  ${(2.00,0.21,0.21)}_{T}$  ${(0.08,0.02,0.02)}_{T}$  $\left[\begin{array}{cc}2.78& 0.03\end{array}\right]$  0.18  0.18  0.03  0.03  ${(1.90,0.64,0.64)}_{T}$  0.10 
FLS  FLAR  RFR  BFRL  

${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\tilde{\mathit{A}}}_{\mathbf{1}}$  ${\mathbf{a}}^{\mathbf{T}}$  b  g  d  h  ${\tilde{\mathit{A}}}_{\mathbf{0}}$  ${\mathit{a}}_{\mathbf{1}}$  
NFN1  ${(24.47,4.85,4.46)}_{T}$  ${(34.05,4.95,5.80)}_{T}$  ${(25.46,4.68,4.82)}_{T}$  ${(32.90,4.90,5.45)}_{T}$  $\left[\begin{array}{cc}21.66& 35.16\end{array}\right]$  0.16  0.17  0.67  0.39  ${(27.07,12.28,13.16)}_{T}$  32.32 
NFN2  ${(12.93,1.29,1.70)}_{T}$  ${(0.54,0.04,0.01)}_{T}$  ${(12.86,1.30,1.58)}_{T}$  ${(0.57,0.02,0.02)}_{T}$  $\left[\begin{array}{cc}12.65& 0.55\end{array}\right]$  0.09  0.09  0.13  0.13  ${(12.93,1.63,1.79)}_{T}$  0.54 
NFN3  ${(1.31,0.16,0.29)}_{T}$  ${(0.13,0.01,0.02)}_{T}$  ${(0.51,0.15,0.29)}_{T}$  ${(0.17,0.01,0.02)}_{T}$  $\left[\begin{array}{cc}1.31& 0.13\end{array}\right]$  0.06  0.15  0.08  0.10  ${(1.09,0.26,0.54)}_{T}$  0.14 
NFN1o1  ${(33.94,4.85,4.46)}_{T}$  ${(31.87,4.95,5.80)}_{T}$  ${(25.46,4.70,4.82)}_{T}$  ${(32.90,4.94,5.45)}_{T}$  $\left[\begin{array}{cc}35.02& 27.99\end{array}\right]$  0.16  0.18  0.17  0.02  ${(37.05,12.28,13.16)}_{T}$  29.79 
NFN2o1  ${(13.61,1.29,1.70)}_{T}$  ${(0.20,0.04,0.01)}_{T}$  ${(13.00,1.30,1.58)}_{T}$  ${(0.50,0.02,0.02)}_{T}$  $\left[\begin{array}{cc}13.25& 0.19\end{array}\right]$  0.10  0.12  0.06  0.02  ${(13.61,1.63,1.79)}_{T}$  0.20 
NFN3o1  ${(6.80,0.16,0.29)}_{T}$  ${(0.22,0.01,0.02)}_{T}$  ${(0.51,0.15,0.29)}_{T}$  ${(0.17,0.01,0.02)}_{T}$  $\left[\begin{array}{cc}8.62& 0.36\end{array}\right]$  0.01  0.05  0.18  0.26  ${(5.10,0.26,0.54)}_{T}$  −0.09 
NFN1o2  ${(24.47,9.92,10.67)}_{T}$  ${(34.05,3.78,4.36)}_{T}$  ${(25.46,4.55,4.82)}_{T}$  ${(32.90,5.03,5.45)}_{T}$  $\left[\begin{array}{cc}34.14& 29.79\end{array}\right]$  0.15  0.17  0.73  0.49  ${(27.07,15.59,17.22)}_{T}$  32.32 
NFN2o2  ${(12.93,0.99,1.76)}_{T}$  ${(0.54,0.19,\mathbf{0.02})}_{T}$  ${(12.86,0.95,1.58)}_{T}$  ${(0.57,0.13,0.02)}_{T}$  $\left[\begin{array}{cc}12.86& 0.47\end{array}\right]$  0.10  0.10  0.07  0.11  ${(12.93,2.69,1.58)}_{T}$  0.54 
NFN3o2  ${(1.31,2.38,3.11)}_{T}$  ${(0.13,\mathbf{0.13},\mathbf{0.16})}_{T}$  ${(0.51,0.20,0.41)}_{T}$  ${(0.17,0.01,0.01)}_{T}$  $\left[\begin{array}{cc}1.50& 0.11\end{array}\right]$  −0.01  0.11  0.68  0.71  ${(1.09,0.61,0.99)}_{T}$  0.14 
NFN1o3  ${(33.94,9.92,10.67)}_{T}$  ${(31.87,3.78,4.36)}_{T}$  ${(25.46,4.71,4.82)}_{T}$  ${(32.90,4.97,5.45)}_{T}$  $\left[\begin{array}{cc}53.26& 25.81\end{array}\right]$  0.19  0.23  3.44  3.25  ${(37.05,15.59,17.22)}_{T}$  29.79 
NFN2o3  ${(13.61,0.99,1.76)}_{T}$  ${(0.20,0.19,\mathbf{0.02})}_{T}$  ${(13.00,0.95,1.58)}_{T}$  ${(0.50,0.13,0.02)}_{T}$  $\left[\begin{array}{cc}13.69& 0.10\end{array}\right]$  0.21  0.11  −0.12  −0.03  ${(13.61,2.69,1.58)}_{T}$  0.20 
NFN3o3  ${(6.80,2.38,3.11)}_{T}$  ${(0.22,\mathbf{0.13},\mathbf{0.16})}_{T}$  ${(0.51,0.20,0.41)}_{T}$  ${(0.17,0.01,0.01)}_{T}$  $\left[\begin{array}{cc}8.44& 0.37\end{array}\right]$  0.34  0.42  −0.55  −0.44  ${(5.10,0.61,0.99)}_{T}$  −0.09 
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Škrabánek, P.; Marek, J.; Pozdílková, A. Boscovich Fuzzy Regression Line. Mathematics 2021, 9, 685. https://doi.org/10.3390/math9060685
Škrabánek P, Marek J, Pozdílková A. Boscovich Fuzzy Regression Line. Mathematics. 2021; 9(6):685. https://doi.org/10.3390/math9060685
Chicago/Turabian StyleŠkrabánek, Pavel, Jaroslav Marek, and Alena Pozdílková. 2021. "Boscovich Fuzzy Regression Line" Mathematics 9, no. 6: 685. https://doi.org/10.3390/math9060685