# A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Set on ${\mathbb{R}}^{n}$

- (1)
- There exists ${x}_{0}\in {\mathbb{R}}^{n}$ such that $\tilde{u}({x}_{0})=1$;
- (2)
- The $\alpha $-cut of $\tilde{u}$, ${\tilde{u}}_{\alpha}:=\{x\in {\mathbb{R}}^{n}:\tilde{u}(x)\ge \alpha \}$, $\alpha \in (0,1]$, is a convex and compact set of ${\mathbb{R}}^{n}$;
- (3)
- ${\tilde{u}}_{0}:=cl\{x\in {\mathbb{R}}^{n}:\tilde{u}(x)>0\}$, the support of $\tilde{u}$, is compact.

**Remark**

**1.**

- (1)
- $\tilde{u}{-}_{h}\tilde{u}=\{0\};$
- (2)
- $(\tilde{u}+\tilde{v}){-}_{h}\tilde{v}=\tilde{u};$
- (3)
- $\tilde{u}=\tilde{v}$ if and only if $\tilde{u}{-}_{h}\tilde{v}=\tilde{v}{-}_{h}\tilde{u}=\{0\}$;
- (4)
- ${S}_{{\tilde{u}}_{\alpha}{-}_{h}{\tilde{v}}_{\alpha}}={S}_{{\tilde{u}}_{\alpha}}-{S}_{{\tilde{v}}_{\alpha}}$.

#### 2.2. Fuzzy Random Variables (FRVs)

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Lemma**

**1.**

- (1)
- $Var(\tilde{u})=0;$
- (2)
- $Var(a\tilde{X}+b\tilde{Y})={a}^{2}Var(\tilde{X})+{b}^{2}Var(\tilde{Y})+2abCov(\tilde{X},\tilde{Y}),ab\ge 0,a,b\in \mathbb{R};$
- (3)
- $Var(a\xi )={\parallel a\parallel}^{2}Var\xi ,a\in {\mathbb{R}}^{n},r.v.\xi \ge 0;$
- (4)
- $Cov((a\tilde{X})+(b\tilde{Y}),c\tilde{Z})=acCov(\tilde{X},\tilde{Z})+bcCov(\tilde{Y},\tilde{Z}),ac\ge 0,bc\ge 0,a,b,c\in \mathbb{R};$
- (5)
- $Cov((a\tilde{X})+\tilde{u},b\tilde{Y}+\tilde{v})=abCov(\tilde{X},\tilde{Y}),ab\ge 0,a,b\in \mathbb{R},\tilde{u},\tilde{v}\in F({\mathbb{R}}^{n}).$

**Definition**

**1.**

**Lemma**

**2.**

- (1)
- if $\tilde{X}$ and $\tilde{Y}$ are independent, then $Cov(\tilde{X},\tilde{Y})=0$;
- (2)
- $|R(\tilde{X},\tilde{Y})|\le 1$;
- (3)
- $R(\tilde{X},\tilde{Y})=1$ if and only if $\tilde{Y}+(\lambda E\tilde{X})=E\tilde{Y}+(\lambda \tilde{X})$, a.e., $R(\tilde{X},\tilde{Y})=-1$ if and only if $\tilde{Y}+(\lambda \tilde{X})=E\tilde{Y}+(\lambda E\tilde{X})$, a.e., where $\lambda =\sqrt{Var\tilde{Y}/Var\tilde{X}},Var(\tilde{X})>0,Var(\tilde{Y})>0.$

**Theorem**

**1.**

- (1)
- ${D}_{2}({\tilde{X}}_{n},\tilde{X})\to 0(n\to \infty )$;
- (2)
- $\{{\tilde{X}}_{n}\}$ is a Cauchy sequence, i.e., ${\mathrm{lim}}_{m,l\to \infty}{D}_{2}({\tilde{X}}_{m},{\tilde{X}}_{l})=0$;
- (3)
- The series $\{\parallel {\tilde{X}}_{n}{\parallel}^{2},n\ge 1\}$ of random variables is uniformly integrable and ${\delta}_{2}({\tilde{X}}_{n},\tilde{X}))\to 0(n\to \infty )$ in probability.

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**6.**

## 3. A Fuzzy Set Valued ARMA Model Based on a Standardized Process

**Definition**

**2.**

**Example**

**1.**

**Definition**

**3.**

**Example**

**2.**

**Definition**

**4.**

**Definition**

**5.**

**Example**

**3.**

**Lemma**

**3.**

- (1)
- If the AR(1) model is causal, then an estimator of the parameter $\theta $ can be $\widehat{\theta}=\frac{\widehat{C}(1)}{\widehat{C}(0)}$, where$$\widehat{C}(1):=\frac{1}{m}\sum _{t=1}^{m-1}n{\int}_{0}^{1}{\int}_{{\mathbb{S}}^{n-1}}({S}_{{({\tilde{x}}_{t+1})}_{\alpha}}(x)-{S}_{{\overline{\tilde{x}}}_{\alpha}}(x))({S}_{{({\tilde{x}}_{t})}_{\alpha}}(x)-{S}_{{\overline{\tilde{x}}}_{\alpha}}(x))\mu (dx)d\alpha ,$$$$\widehat{C}(0):=\frac{1}{m}\sum _{t=1}^{m}n{\int}_{0}^{1}{\int}_{{\mathbb{S}}^{n-1}}{({S}_{{({\tilde{x}}_{t})}_{\alpha}}(x)-{S}_{{\overline{\tilde{x}}}_{\alpha}}(x))}^{2}\mu (dx)d\alpha ,$$
- (2)
- If the AR(1) with fuzzy data is not causal, then we may employ the least square method proposed by [20] to estimate the parameter $\theta $.

**Theorem**

**3.**

**Proof.**

**Remark**

**7.**

## 4. An Empirical Analysis of the ARMA($\mathbf{p},\mathbf{q}$) Models with Fuzzy Data

**Step 1**

**Step 2**

**Step 3**

**Step 4**

**Step 5**

**Remark**

**8.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The curves of the close value, low value, and high value for monthly Hang Seng Index (HSI). (https://www.hsi.com.hk/eng).

Year | Month | Data | Year | Month | Data |
---|---|---|---|---|---|

2009 | 1 | (13.278, 0.839, 2.485) | 2012 | 1 | (20.39, 2.07, 0.2) |

2 | (12.811, 0.177, 1.165) | 2 | (21.68, 1.411, 0.08) | ||

3 | (13.576, 2.23, 0.681) | 3 | (20.555, 0.181, 1.086) | ||

4 | (15.52, 2.188, 0.457) | 4 | (21.094, 1.059, 0.011) | ||

5 | (18.171, 2.316, 0.056) | 5 | (18.629, 0.251, 2.756) | ||

6 | (18.378, 1.002, 0.784) | 6 | (19.441, 1.385, 0.138) | ||

7 | (20.573, 3.387, 0.139) | 7 | (19.796, 1.086, 0.073) | ||

8 | (19.724, 0.132, 1.473) | 8 | (19.482, 0.032, 0.818) | ||

9 | (20.956, 1.529, 0.975) | 9 | (20.84, 1.764, 0.055) | ||

10 | (21.752, 1.447, 0.868) | 10 | (21.641, 0.874, 0.206) | ||

11 | (21.821, 0.819, 1.278) | 11 | (22.03, 0.932, 0.119) | ||

12 | (21.872, 0.939, 0.722) | 12 | (22.657, 0.969, 0.061) | ||

2010 | 1 | (20.121, 0.205, 2.551) | 2013 | 1 | (23.729, 0.869, 0.187) |

2 | (20.268, 1.185, 0.172) | 2 | (23.02, 0.575, 0.924) | ||

3 | (21.239, 0.664, 0.212) | 3 | (22.299, 0.323, 0.963) | ||

4 | (21.108, 0.345, 1.281) | 4 | (22.737, 1.314, 0.125) | ||

5 | (19.765, 0.974, 1.247) | 5 | (22.392, 0.102, 1.12) | ||

6 | (20.128, 0.917, 0.829) | 6 | (20.803, 1.377, 1.761) | ||

7 | (21.029, 1.251, 0.09) | 7 | (21.883, 1.764, 0.187) | ||

8 | (20.536. 0.164, 1.27) | 8 | (21.731, 0.266, 0.964) | ||

9 | (22.358, 1.828, 0.081) | 9 | (22.859, 0.911, 0.695) | ||

10 | (23.096, 0.592, 0.77) | 10 | (23.206, 0.566, 0.328) | ||

11 | (23.007, 0.224, 1.981) | 11 | (23.881, 1.418, 0.133) | ||

12 | (23.035, 0.653, 0.577) | 12 | (23.306, 0.593, 0.805) | ||

2011 | 1 | (23.447, 0.39, 0.987) | |||

2 | (23.338, 0.892, 0.644) | ||||

3 | (23.527, 1.404, 0.407) | ||||

4 | (23.72, 0.252, 0.748) | ||||

5 | (23.684, 1.165, 0.24) | ||||

6 | (22.398, 0.89, 1.308) | ||||

7 | (22.44, 0.829, 0.395) | ||||

8 | (20.534, 1.666, 2.274) | ||||

9 | (17.592, 0.593, 3.382) | ||||

10 | (19.864, 3.694, 0.409) | ||||

11 | (17.989, 0.376, 2.184) | ||||

12 | (18.434, 0.613, 0.6) |

i | ${\tilde{\mathit{w}}}_{\mathit{i}}$ | i | ${\tilde{\mathit{w}}}_{\mathit{i}}$ | i | ${\tilde{\mathit{w}}}_{\mathit{i}}$ |
---|---|---|---|---|---|

1 | (1.80482, 0.01, 0.01) | 21 | (1.30572, 0.001, 0.0001) | 41 | (−1.3595, 0.00008, 0.0001) |

2 | (−0.07992, 0.007, 0.008) | 22 | (1.42513, 0.0003, 0.0002) | 42 | (−2.33134, 0.001, 0.00012) |

3 | (0.39658, 0.01, 0.002) | 23 | (−0.4158, 0.0002, 0.0001) | 43 | (−0.40969, 0.00012, 0.0006) |

4 | (−1.08332, 0.0015, 0.001) | 24 | (1.61438, 0.0003, 0.001) | 44 | (0.6542, 0.0003, 0.0001) |

5 | (2.23829, 0.01, 0.001) | 25 | (−1.05773, 0.001, 0.00002) | 45 | (0.39926, 0.00003, 0.00001) |

6 | (−0.62423, 0.001, 0.001) | 26 | (−0.94833, 0.0001, 0.001) | 46 | (−0.46931, 0.00002, 0.0006 ) |

7 | (0.51366, 0.002, 0.001) | 27 | (0.95365, 0.0003, 0.001) | 47 | (0.86633, 0.0003, 0.00001) |

8 | (−0.08661, 0.0002, 0.0013) | 28 | (0.39198, 0.0002, 0.0001) | 48 | (−0.92372, 0.0002, 0.00008) |

9 | (−0.59418, 0.0002, 0.001) | 29 | (−0.07614, 0.00102, 0.0001) | 49 | (1.27746, 0.0001, 0.00002) |

10 | (0.03189, 0.002, 0.0012) | 30 | (1.22056, 0.0017, 0.00018) | 50 | (−1.4526, 0.0001, 0.001) |

11 | (−0.7378, 0.00021, 0.0013) | 31 | (−0.63084, 0.00016,0.00018) | 51 | (0.34892, 0.0002, 0.0001) |

12 | (−0.25014, 0.01, 0.0003) | 32 | (−0.63576, 0.001, 0.0001) | 52 | (−0.05535, 0.00012, 0.0001) |

13 | (0.685, 0.0013, 0.00011) | 33 | (−0.34, 0.001, 0.00008) | 53 | (−1.228, 0.0008, 0.0001) |

14 | (−0.80416, 0.0013, 0.0003) | 34 | (0.07628, 0.0001, 0.0002) | 54 | (0.14502, 0.0001, 0.00006) |

15 | (−0.74428, 0.0011, 0.0003) | 35 | (0.95536, 0.000016, 0.00011) | 55 | (−0.8395, 0.0001, 0.00032) |

16 | (−0.7955, 0.0002, 0.0001) | 36 | (−1.2167, 0.0001, 0.00011) | 56 | (−0.09626, 0.00009, 0.0006) |

17 | (0.34071, 0.001, 0.0001) | 37 | (1.18449, 0.0006, 0.0003) | 57 | (−0.85758, 0.0001, 0.00002) |

18 | (−0.30051, 0.001, 0.00017) | 38 | (−0.34369, 0.0002, 0.0003) | 58 | (0.76497, 0.00002, 0.001) |

19 | (−1.34985, 0.00031, 0.0005) | 39 | (1.09024, 0.0001, 0.00006) | 59 | (0.04501, 0.000016, 0.00001) |

20 | (0.4327, 0.0001, 0.0002) | 40 | (−0.13531, 0.0002, 0.0001) | 60 | (1.92838, 0.00008, 0.0002) |

i | Real Linguistic Monthly HSI | Predicted Linguistic Monthly HSI |
---|---|---|

61 | (22.035, 0.289, 1.434) | (23.274, 0.381, 0.236) |

62 | (22.836, 1.639, 0.15) | (23.662, 0.313, 0.368) |

63 | (22.151, 1.014, 0.688) | (23.182, 0.121, 0.502) |

64 | (22.133, 0.037, 1.091) | (23.311, 0.248, 0.308) |

65 | (23.081, 1.401, 0.128) | (23.402, 0.313, 0.470) |

66 | (23.19, 0.388, 0.207) | (23.431, 0.402, 0.487) |

67 | (24.756, 1.63, 0.156) | (24.217, 0.418, 0.501) |

68 | (24.742, 0.552, 0.492) | (24.406, 0.419, 0.537) |

69 | (22.932, 0.077, 2.43) | (24.100, 0.428, 0.558) |

70 | (23.998, 1.433, 0.048) | (23.579, 0.432, 0.563) |

**Table 4.**A comparison of the predicted close values obtained by the fuzzy set-valued autoregressive moving average (ARMA)(1,1) with the real close values in the monthly HSI.

i | Real Close Values | Predicted Close Values | Absolute Error | Relative Error |
---|---|---|---|---|

61 | 22.035 | 23.274 | 1.239 | 5.62% |

62 | 22.836 | 23.462 | 0.626 | 2.74% |

63 | 22.151 | 23.182 | 1.031 | 4.654% |

64 | 22.133 | 23.210 | 1.077 | 4.869% |

65 | 23.081 | 23.302 | 0.221 | 0.957% |

66 | 23.190 | 23.413 | 0.223 | 0.95% |

67 | 24.756 | 24.217 | 0.539 | 2.17% |

68 | 24.742 | 24.406 | 0.336 | 1.356% |

69 | 22.932 | 24.100 | 1.168 | 5.09% |

70 | 23.998 | 23.579 | 0.419 | 1.746% |

**Table 5.**A comparison of the predicted close values obtained by the classical AR(1) with the real close values in the monthly HSI.

i | Real Close Values | Predicted Close Values | Absolute Error | Relative Error |
---|---|---|---|---|

61 | 22.035 | 23.663 | 1.628 | 7.38% |

62 | 22.836 | 23.458 | 0.622 | 2.723% |

63 | 22.151 | 23.264 | 1.113 | 5.02% |

64 | 22.133 | 23.081 | 0.948 | 4.283% |

65 | 23.081 | 22.908 | 0.173 | 0.75% |

66 | 23.19 | 22.746 | 0.444 | 1.91% |

67 | 24.756 | 22.592 | 2.164 | 8.74% |

68 | 24.742 | 22.448 | 2.294 | 9.27% |

69 | 22.932 | 22.311 | 0.621 | 2.7% |

70 | 23.998 | 22.183 | 1.815 | 7.56% |

**Table 6.**A comparison of the predicted close values obtained by the classical AR(2) with the real close value in the monthly HSI.

i | Real Close Values | Predicted Close Values | Absolute Error | Relative Error |
---|---|---|---|---|

61 | 22.035 | 23.658 | 1.623 | 7.36% |

62 | 22.836 | 23.447 | 0.611 | 2.67% |

63 | 22.151 | 23.247 | 1.096 | 4.94% |

64 | 22.133 | 23.06 | 0.927 | 4.19% |

65 | 23.081 | 22.883 | 0.198 | 0.86% |

66 | 23.19 | 22.717 | 0.473 | 2.04% |

67 | 24.756 | 22.561 | 2.195 | 8.87% |

68 | 24.742 | 22.414 | 2.328 | 9.41% |

69 | 22.932 | 22.275 | 0.657 | 2.86% |

70 | 23.998 | 22.145 | 1.835 | 7.65% |

**Table 7.**A comparison of the predicted close values obtained by the classical AR(3) with the real close value in the monthly HSI.

i | Real Close Values | Predicted Close Values | Absolute Error | Relative Error |
---|---|---|---|---|

61 | 22.035 | 23.688 | 1.653 | 7.5% |

62 | 22.836 | 23.382 | 0.546 | 2.4% |

63 | 22.151 | 23.065 | 0.914 | 4.1% |

64 | 22.133 | 22.756 | 0.623 | 2.81% |

65 | 23.081 | 22.473 | 0.608 | 2.63% |

66 | 23.19 | 22.219 | 0.971 | 4.19% |

67 | 24.756 | 21.993 | 2.763 | 11.16% |

68 | 24.742 | 21.794 | 2.948 | 11.91% |

69 | 22.932 | 21.62 | 1.312 | 5.72% |

70 | 23.998 | 21.467 | 2.531 | 10.5% |

**Table 8.**A comparison of the predicted close values obtained by the classical ARMA(1,1) with the real close value in the monthly HSI.

i | Real Close Values | Predicted Close Values | Absolute Error | Relative Error |
---|---|---|---|---|

61 | 22.035 | 23.66 | 1.625 | 7.37% |

62 | 22.836 | 23.45 | 0.614 | 2.69% |

63 | 22.151 | 23.252 | 1.101 | 4.97% |

64 | 22.133 | 23.066 | 0.923 | 4.21% |

65 | 23.081 | 22.89 | 0.191 | 0.83% |

66 | 23.19 | 22.725 | 0.465 | 2% |

67 | 24.756 | 22.57 | 2.186 | 8.83% |

68 | 24.742 | 22.423 | 2.319 | 9.37% |

69 | 22.932 | 22.286 | 0.646 | 2.82% |

70 | 23.998 | 22.156 | 1.842 | 7.68% |

**Table 9.**A comparison of the proposed fuzzy set valued ARMA(1,1) with the classical AR(1), AR(2), AR(3), and ARMA(1,1) in the prediction errors for the close value of the monthly HSI.

i | fuzzy ARMA(1,1) | AR(1) | AR(2) | AR(3) | ARMA(1,1) |
---|---|---|---|---|---|

abs.err., rel.err. | abs.err., rel.err. | abs.err., rel.err. | abs.err., rel.err. | abs.err., rel.err. | |

61 | 1.239, 5.62% | 1.628, 7.38% | 1.623, 7.36% | 1.653, 7.5% | 1.625, 7.37% |

62 | 0.626, 2.74% | 0.622, 2.723% | 0.611, 2.67% | 0.546, 2.4% | 0.614, 2.69% |

63 | 1.031, 4.654% | 1.113, 5.02% | 1.096, 4.94% | 0.914, 4.1% | 1.101, 4.97% |

64 | 1.077, 4.869% | 0.948, 4.283% | 0.927, 4.19% | 0.623, 2.81% | 0.923, 4.21% |

65 | 0.221, 0.957% | 0.173, 0.75% | 0.198, 0.86% | 0.608, 2.63% | 0.191, 0.83% |

66 | 0.223, 0.95% | 0.444, 1.91% | 0.473, 2.04% | 0.971, 4.19% | 0.465, 2% |

67 | 0.539, 2.17% | 2.164, 8.74% | 2.195, 8.87% | 2.763, 11.16% | 2.186, 8.83% |

68 | 0.336, 1.356% | 2.294, 9.27% | 2.328, 9.41% | 2.948, 11.91% | 2.319, 9.37% |

69 | 1.168, 5.09% | 0.621, 2.7% | 0.657, 2.86% | 1.312, 5.72% | 0.646, 2.82% |

70 | 0.449, 1.746% | 1.815, 7.56% | 1.835, 7.65% | 2.531, 10.5% | 1.842, 7.68% |

ave. | 0.691, 3.02% | 1.182, 5.033% | 1.194, 5.085% | 1.487, 6.292% | 1.191, 5.076% |

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**MDPI and ACS Style**

Wang, D.; Zhang, L.
A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications. *Symmetry* **2018**, *10*, 324.
https://doi.org/10.3390/sym10080324

**AMA Style**

Wang D, Zhang L.
A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications. *Symmetry*. 2018; 10(8):324.
https://doi.org/10.3390/sym10080324

**Chicago/Turabian Style**

Wang, Dabuxilatu, and Liang Zhang.
2018. "A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications" *Symmetry* 10, no. 8: 324.
https://doi.org/10.3390/sym10080324