# A Refined Approach for Forecasting Based on Neutrosophic Time Series

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## Abstract

**:**

## 1. Introduction

## 2. Some Basic Definitions of Neutrosophic Set and Neutrosophic Time Series

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Neutrosophic Time Series Forecasting Algorithm

#### 3.1. The Proposed Method of Forecasting Based on First-Order NTS Data

**Step 1:**By depending on the range of the existing data set, determine the universe of discourse U as follows:

**Step 2:**Create a partition of the universe of discourse, to $m$ triangular neutrosophic numbers as follows:

- -
- Decide the suitable length ($Le$) of available time series data:
- ○
- Among the value ${D}_{v-1}$, ${D}_{v}$, calculate all absolute differences and take the average of these differences.
- ○
- Consider half the average as the initial length.
- ○
- According to the obtained result, use the base mapping table [42] to determine the base for the length of intervals.
- ○
- Round the result to determine the appropriate length of neutrosophic numbers.
- ○
- For example: if we have these time series data $30,50,80,120,100,70$, then the absolute differences will be $20,30,40,20,30$, and the average of these values $=28$. Then, half of the average will be 14 and this is the initial value of length. By using the base mapping table [42], the base for length $=10$ because 14 locates in the range $\left[11-100\right]$ and by rounding the length $14$ by the base ten, the result will equal $10$. Here, the appropriate length of neutrosophic numbers equals $10$.

- -
- Compute the number of triangular neutrosophic numbers $\left(m\right)$ as follows:$$m=\frac{{D}_{l}+{D}_{2}-{D}_{s}+{D}_{1}}{le}$$

**Step 3:**According to the numbers of triangular neutrosophic numbers on the universe of discourse and determined length ($le$), begin to construct the triangular neutrosophic numbers. The triangular neutrosophic numbers are ${\tilde{N}}_{1},{\tilde{N}}_{2},\dots ,{\tilde{N}}_{m}.$

**Step 4:**Make a neutrosophication process of the existing data:

**Step 5:**Construct the neutrosophic logical relationships (NLRs) as follows:

**Step 6:**Based on the NLR, begin to establish the neutrosophic logical relationship groups (NLRGs).

**Step 7:**Calculate the forecasted values as follows:

- -
- If NLRG of ${\tilde{N}}_{j}$ is empty (i.e., ${\tilde{N}}_{j}\to \varnothing ,$ or ${\tilde{N}}_{j}\to {\tilde{N}}_{j}$), then the forecasted value is the middle value of ${\tilde{N}}_{j}.$
- -
- If NLRG of ${\tilde{N}}_{j}$ is one-to-one (i.e., ${\tilde{N}}_{j}\to {\tilde{N}}_{k}),$ then the forecasted value is the middle value of ${\tilde{N}}_{k}.$
- -
- If NLRG of ${\tilde{N}}_{j}$ is one-to-many (i.e., ${\tilde{N}}_{j}\to {\tilde{N}}_{k1},{\tilde{N}}_{k2},\dots ,{\tilde{N}}_{kn}$), then the forecasted value is the average of the middle values of ${\tilde{N}}_{k1},{\tilde{N}}_{k2},\dots ,{\tilde{N}}_{kn}.$

**Step 8:**Use the following equations to calculate the forecasting error:

#### 3.2. The Proposed Method of Forecasting Based on High-Order NTS Data

- -
- All steps from 1 to 4 are the same as previously, but in step 5 we begin to construct the neutrosophic logical relationships (NLRs) of the $n$th order NTS, where $n\ge 2$.
- -
- Based on the NLR of the $n$th order, NTS begin to establish the neutrosophic logical relationship groups (NLRGs).
- -
- Calculate the forecasted values as follows:
- ○
- Rule 1: If the neutrosophication values of $dat{a}_{i}$ is ${\tilde{N}}_{l}$ and it is not caused by any other neutrosophication values and, by looking at the NLRG of this value, you cannot find the values which it depends on (i.e., $\ne \to {\tilde{N}}_{l}$), then the forecasted value in this case will equal—(i.e., leave it empty). The $\ne $ symbol means no value.
- ○
- Rule 2: If the neutrosophication value of $dat{a}_{i}$ is ${\tilde{N}}_{l}$ and it is caused by ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}$ (i.e., ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}\to {\tilde{N}}_{l}$), then look at the NLRG of ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}$, and
- If ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}\to \varnothing $, then the forecasted value at this year is the average of the middle value of ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}.$
- If ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}\to {\tilde{N}}_{j}$, then the forecasted value at this year is the middle value of ${\tilde{N}}_{j}.$
- If ${\tilde{N}}_{in}$, ${\tilde{N}}_{i\left(n-1\right)}$, …, ${\tilde{N}}_{ik}\to {\tilde{N}}_{j},{\tilde{N}}_{j1}$, ${\tilde{N}}_{j2}$, then the forecasted value at this year is the average of the middle value of ${\tilde{N}}_{j},{\tilde{N}}_{j1},{\tilde{N}}_{j2}.$

## 4. Numerical Examples

#### 4.1. Numerical Example 1

**Step 1:**Let the two proper positive numbers ${D}_{1}$ and ${D}_{2}$ be $5$ and $13$, determined by the expert. By selecting the largest and the smallest observation from all available data which are presented in Table 1, then ${D}_{l}$ $=19,337$ and ${D}_{s}=13,055$, respectively. Consequently, the universe of discourse U = $\left[\mathrm{13,055}-5,\mathrm{19,337}+13\left]=\right[\mathrm{13,050},\mathrm{19,350}\right].$

**Step 2:**Create a partition of the universe of discourse, to $m$ triangular neutrosophic numbers, as follows:

- -
- Determine the suitable length ($Le$) of available time series data:
- ○
- From Table 1, the average of absolute differences $=510.3.$
- ○
- The initial length $=\frac{510.3}{2}=255.15.$
- ○
- By using the base mapping table [42], the base for length of intervals $=100$, since it is located in the range $\left[\mathrm{101,1000}\right].$
- ○
- By rounding $255.15$ with regard to base $100$, then the appropriate length of neutrosophic numbers = 300.

- -
- Compute the number of triangular neutrosophic numbers $\left(m\right)$ as follows:$$m=\frac{19350-13050}{300}=21.$$

**Step 3:**According to the number of triangular neutrosophic numbers on the universe of discourse and determined length ($le$), begin to construct the triangular neutrosophic numbers as follows:

**Step 4:**Make a neutrosophication of the available time series data:

**Step 5:**Construct the neutrosophic logical relationships (NLRs) as in Table 2:

**Step 6:**Based on NLR, begin to establish the neutrosophic logical relationship groups (NLRGs) as in Table 3.

**Step 7:**Calculate the forecasted values as in Table 4:

- -
- Look at the neutrosophication value of 13055 in year 1971 which is ${\tilde{N}}_{1}$ as it appears in Table 1.
- -

- -
- Look at the neutrosophication value of 13,563 in year 1972 which is ${\tilde{N}}_{2}$ as it appears in Table 1, and because ${\tilde{N}}_{2}$ is caused by ${\tilde{N}}_{1}$ (i.e.,${\tilde{N}}_{1}\to {\tilde{N}}_{2}$), then
- -
- Go to Table 3, and look at the NLRG which starts with ${\tilde{N}}_{1}$, and we noted that it is ${\tilde{N}}_{1}\to {\tilde{N}}_{2}.$ Then the forecasted value of 13,563 is the middle value of ${\tilde{N}}_{2}$.

- -
- Look at the neutrosophication value of 18,876 in year 1992 which is ${\tilde{N}}_{19}$ as it appears in Table 1. Since ${\tilde{N}}_{19}$ is caused by ${\tilde{N}}_{21},$ then
- -
- Go to Table 3, and look at the NLRG which starts with ${\tilde{N}}_{21}$ (i.e.$,{\tilde{N}}_{21}\to {\tilde{N}}_{19},{\tilde{N}}_{21}\to {\tilde{N}}_{21}$). Then the forecasted value of 18876 is the average of the middle values of ${\tilde{N}}_{19}$, ${\tilde{N}}_{21}$, and it will equal 19,050.

#### 4.2. Numerical Example 2

## 5. Conclusion and Future Directions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Years | Actual Enrollments | $\mathbf{Neutrosophication}\text{}\mathbf{Values}\text{}\mathbf{of}\text{}\mathbf{Enrollments}\text{}\tilde{\mathit{N}}$ |
---|---|---|

1971 | 13,055 | ${\tilde{N}}_{1}$ |

1972 | 13,563 | ${\tilde{N}}_{2}$ |

1973 | 13,867 | ${\tilde{N}}_{3}$ |

1974 | 14,696 | ${\tilde{N}}_{6}$ |

1975 | 15,460 | ${\tilde{N}}_{8}$ |

1976 | 15,311 | ${\tilde{N}}_{8}$ |

1977 | 15,603 | ${\tilde{N}}_{8}$ |

1978 | 15,861 | ${\tilde{N}}_{9}$ |

1979 | 16,807 | ${\tilde{N}}_{13}$ |

1980 | 16,919 | ${\tilde{N}}_{13}$ |

1981 | 16,388 | ${\tilde{N}}_{11}$ |

1982 | 15,433 | ${\tilde{N}}_{8}$ |

1983 | 15,497 | ${\tilde{N}}_{8}$ |

1984 | 15,145 | ${\tilde{N}}_{7}$ |

1985 | 15,163 | ${\tilde{N}}_{7}$ |

1986 | 15,984 | ${\tilde{N}}_{10}$ |

1987 | 16,859 | ${\tilde{N}}_{13}$ |

1988 | 18,150 | ${\tilde{N}}_{17}$ |

1989 | 18,970 | ${\tilde{N}}_{20}$ |

1990 | 19,328 | ${\tilde{N}}_{21}$ |

1991 | 19,337 | ${\tilde{N}}_{21}$ |

1992 | 18,876 | ${\tilde{N}}_{19}$ |

${\tilde{N}}_{1}\to {\tilde{N}}_{2}$ | ${\tilde{N}}_{2}\to {\tilde{N}}_{3}$ | ${\tilde{N}}_{3}\to {\tilde{N}}_{6}$ | ${\tilde{N}}_{6}\to {\tilde{N}}_{8}$ | ${\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8}\to {\tilde{N}}_{9}$ | ${\tilde{N}}_{9}\to {\tilde{N}}_{13}$ | ${\tilde{N}}_{13}\to {\tilde{N}}_{13}$ | ${\tilde{N}}_{13}\to {\tilde{N}}_{11}$ | ${\tilde{N}}_{11}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8}\to {\tilde{N}}_{7}$ | ${\tilde{N}}_{7}\to {\tilde{N}}_{7}$ | ${\tilde{N}}_{7}\to {\tilde{N}}_{10}$ | ${\tilde{N}}_{10}\to {\tilde{N}}_{13}$ | ${\tilde{N}}_{13}\to {\tilde{N}}_{17}$ |

${\tilde{N}}_{17}\to {\tilde{N}}_{20}$ | ${\tilde{N}}_{20}\to {\tilde{N}}_{21}$ | ${\tilde{N}}_{21}\to {\tilde{N}}_{21}$ | ${\tilde{N}}_{21}\to {\tilde{N}}_{19}$ |

${\tilde{N}}_{1}\to {\tilde{N}}_{2}$ | ||

${\tilde{N}}_{2}\to {\tilde{N}}_{3}$ | ||

${\tilde{N}}_{3}\to {\tilde{N}}_{6}$ | ||

${\tilde{N}}_{6}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{7}\to {\tilde{N}}_{7}$ | ${\tilde{N}}_{7}\to {\tilde{N}}_{10}$ | |

${\tilde{N}}_{8}\to {\tilde{N}}_{7}$ | ${\tilde{N}}_{8}\to {\tilde{N}}_{8}$ | ${\tilde{N}}_{8}\to {\tilde{N}}_{9}$ |

${\tilde{N}}_{9}\to {\tilde{N}}_{13}$ | ||

${\tilde{N}}_{10}\to {\tilde{N}}_{13}$ | ||

${\tilde{N}}_{11}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{13}\to {\tilde{N}}_{11}$ | ${\tilde{N}}_{13}\to {\tilde{N}}_{13}$ | ${\tilde{N}}_{13}\to {\tilde{N}}_{17}$ |

${\tilde{N}}_{17}\to {\tilde{N}}_{20}$ | ||

${\tilde{N}}_{20}\to {\tilde{N}}_{21}$ | ||

${\tilde{N}}_{21}\to {\tilde{N}}_{19}$ | ${\tilde{N}}_{21}\to {\tilde{N}}_{21}$ |

Years | Actual Enrollments | Forecasted Values of Enrollments |
---|---|---|

1971 | 13,055 | $-$ |

1972 | 13,563 | $\mathrm{13,650}$ |

1973 | 13,867 | 13,950 |

1974 | 14,696 | $\mathrm{14,850}$ |

1975 | 15,460 | $\mathrm{15,450}$ |

1976 | 15,311 | $\mathrm{15,450}$ |

1977 | 15,603 | $\mathrm{15,450}$ |

1978 | 15,861 | $\mathrm{15,450}$ |

1979 | 16,807 | $\mathrm{16,950}$ |

1980 | 16,919 | $\mathrm{17,150}$ |

1981 | 16,388 | $\mathrm{17,150}$ |

1982 | 15,433 | 15,450 |

1983 | 15,497 | $\mathrm{15,450}$ |

1984 | 15,145 | $\mathrm{15,450}$ |

1985 | 15,163 | $\mathrm{15,600}$ |

1986 | 15,984 | $\mathrm{15,600}$ |

1987 | 16,859 | $\mathrm{16,950}$ |

1988 | 18,150 | $\mathrm{17,150}$ |

1989 | 18,970 | $\mathrm{19,050}$ |

1990 | 19,328 | $\mathrm{19,350}$ |

1991 | 19,337 | $\mathrm{19,050}$ |

1992 | 18,876 | $\mathrm{19,050}$ |

Years | Actual Values | Forecasted Values | ||||||
---|---|---|---|---|---|---|---|---|

Proposed | [43] | [44] | [45] | [46] | [14] | [17] | ||

1971 | 13,055 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ |

1972 | 13,563 | $13,650$ | 14,242.0 | 14,025 | 13,250 | 14,031.35 | 14,586 | 13,693 |

1973 | 13,867 | 13,950 | 14,242.0 | 14,568 | 13,750 | 14,795.36 | 14,586 | 13,693 |

1974 | 14,696 | $14,850$ | 14,242.0 | 14,568 | 13,750 | 14,795.36 | 15,363 | 14,867 |

1975 | 15,460 | $15,450$ | 15,774.3 | 15,654 | 14,500 | 14,795.36 | 15,363 | 15,287 |

1976 | 15,311 | $15,450$ | 15,774.3 | 15,654 | 15,375 | 16,406.57 | 15,442 | 15,376 |

1977 | 15,603 | $15,450$ | 15,774.3 | 15,654 | 15,375 | 16,406.57 | 15,442 | 15,376 |

1978 | 15,861 | $15,450$ | 15,774.3 | 15,654 | 15,625 | 16,406.57 | 15,442 | 15,376 |

1979 | 16,807 | $16,950$ | 16,146.5 | 16,197 | 15,875 | 16,406.57 | 15,442 | 16,523 |

1980 | 16,919 | $17,150$ | 16,988.3 | 17,283 | 16,833 | 17,315.29 | 17,064 | 16,606 |

1981 | 16,388 | $17,150$ | 16,988.3 | 17,283 | 16,833 | 17,315.29 | 17,064 | 17,519 |

1982 | 15,433 | $15,450$ | 16,146.5 | 16,197 | 16,500 | 17,315.29 | 15,438 | 16,606 |

1983 | 15,497 | $15,450$ | 15,474.3 | 15,654 | 15,500 | 16,406.57 | 15,442 | 15,376 |

1984 | 15,145 | $15,450$ | 15,474.3 | 15,654 | 15,500 | 16,406.57 | 15,442 | 15,376 |

1985 | 15,163 | $15,600$ | 15,474.3 | 15,654 | 15,125 | 16,406.57 | 15,363 | 15,287 |

1986 | 15,984 | $15,600$ | 15,474.3 | 15,654 | 15,125 | 16,406.57 | 15,363 | 15,287 |

1987 | 16,859 | $16,950$ | 16,146.5 | 15,654 | 16,833 | 16,406.57 | 15,438 | 16,523 |

1988 | 18,150 | $17,150$ | 16,988.3 | 16,197 | 16,667 | 17,315.29 | 17,064 | 17,519 |

1989 | 18,970 | $19,050$ | 19,144.0 | 17,283 | 18,125 | 19,132.79 | 19,356 | 19,500 |

1990 | 19,328 | $19,350$ | 19,144.0 | 18,369 | 18,750 | 19,132.79 | 19,356 | 19,000 |

1991 | 19,337 | $19,050$ | 19,144.0 | 19,454 | 19,500 | 19,132.79 | 19,356 | 19,500 |

1992 | 18,876 | $19,050$ | 19,144.0 | 19,454 | 19,500 | 19,132.79 | 19,356 | 19,500 |

Tool | Proposed | [43] | [44] | [45] | [46] | [14] | [17] |
---|---|---|---|---|---|---|---|

RMSE | 342.68 | 478.45 | 781.47 | 646.67 | 805.17 | 642.68 | 493.56 |

AFE (%) | 1.44 | 2.39 | 3.61 | 2.98 | 4.28 | 2.96 | 2.33 |

${\tilde{N}}_{1},{\tilde{N}}_{2}\to {\tilde{N}}_{3}$ |

${\tilde{N}}_{2},{\tilde{N}}_{3}\to {\tilde{N}}_{6}$ |

${\tilde{N}}_{3},{\tilde{N}}_{6}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{6},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{9}$ |

${\tilde{N}}_{8},{\tilde{N}}_{9}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{9},{\tilde{N}}_{13}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{13},{\tilde{N}}_{13}\to {\tilde{N}}_{11}$ ${\tilde{N}}_{13},{\tilde{N}}_{11}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{11},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{7}$ |

${\tilde{N}}_{8},{\tilde{N}}_{7}\to {\tilde{N}}_{7}$ |

${\tilde{N}}_{7},{\tilde{N}}_{7}\to {\tilde{N}}_{10}$ |

${\tilde{N}}_{7},{\tilde{N}}_{10}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{10},{\tilde{N}}_{13}\to {\tilde{N}}_{17}$ |

${\tilde{N}}_{13},{\tilde{N}}_{17}\to {\tilde{N}}_{20}$ |

${\tilde{N}}_{17},{\tilde{N}}_{20}\to {\tilde{N}}_{21}$ |

${\tilde{N}}_{20},{\tilde{N}}_{21}\to {\tilde{N}}_{21}$ |

${\tilde{N}}_{21},{\tilde{N}}_{21}\to {\tilde{N}}_{19}$ |

${\tilde{N}}_{1},{\tilde{N}}_{2}\to {\tilde{N}}_{3}$ | ||

${\tilde{N}}_{2},{\tilde{N}}_{3}\to {\tilde{N}}_{6}$ | ||

${\tilde{N}}_{3},{\tilde{N}}_{6}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{6},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ | ${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{9}$ | ${\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{7}$ |

${\tilde{N}}_{8},{\tilde{N}}_{9}\to {\tilde{N}}_{13}$ | ||

${\tilde{N}}_{9},{\tilde{N}}_{13}\to {\tilde{N}}_{13}$ | ||

${\tilde{N}}_{13},{\tilde{N}}_{13}\to {\tilde{N}}_{11}$ | ||

${\tilde{N}}_{13},{\tilde{N}}_{11}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{11},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ | ||

${\tilde{N}}_{8},{\tilde{N}}_{7}\to {\tilde{N}}_{7}$ | ||

${\tilde{N}}_{7},{\tilde{N}}_{7}\to {\tilde{N}}_{10}$ | ||

${\tilde{N}}_{7},{\tilde{N}}_{10}\to {\tilde{N}}_{13}$ | ||

${\tilde{N}}_{10},{\tilde{N}}_{13}\to {\tilde{N}}_{17}$ | ||

${\tilde{N}}_{13},{\tilde{N}}_{17}\to {\tilde{N}}_{20}$ | ||

${\tilde{N}}_{17},{\tilde{N}}_{20}\to {\tilde{N}}_{21}$ | ||

${\tilde{N}}_{20},{\tilde{N}}_{21}\to {\tilde{N}}_{21}$ | ||

${\tilde{N}}_{21},{\tilde{N}}_{21}\to {\tilde{N}}_{19}$ |

**Table 9.**Actual and forecasted values of enrollments based on the second order of the proposed method vs. the Gautam and Singh [47] method.

Years | Actual Enrollments | Second-Order Forecasted Values of the Proposed Method | Forecasted Values in [47] |
---|---|---|---|

1971 | 13,055 | $-$ | $-$ |

1972 | 13,563 | $-$ | $-$ |

1973 | 13,867 | 13,950 | 13,800 |

1974 | 14,696 | $14,850$ | 14,400 |

1975 | 15,460 | $15,450$ | 15,300 |

1976 | 15,311 | $15,450$ | 15,300 |

1977 | 15,603 | $15,450$ | 15,600 |

1978 | 15,861 | $15,450$ | 15,600 |

1979 | 16,807 | $16,950$ | 16,800 |

1980 | 16,919 | $16,950$ | 16,800 |

1981 | 16,388 | $16,350$ | 16,200 |

1982 | 15,433 | $15,450$ | 15,300 |

1983 | 15,497 | $15,450$ | 15,300 |

1984 | 15,145 | 15,450 | 15,000 |

1985 | 15,163 | $15,150$ | 15,000 |

1986 | 15,984 | $16,050$ | 15,900 |

1987 | 16,859 | $16,950$ | 16,800 |

1988 | 18,150 | $18,150$ | 18,000 |

1989 | 18,970 | $19,050$ | 18,900 |

1990 | 19,328 | $19,350$ | 19,200 |

1991 | 19,337 | $19,350$ | 19,200 |

1992 | 18,876 | $18,750$ | 18,600 |

**Table 10.**Error measures of the proposed method and the Gautam and Singh method [47].

Tool | Proposed | [47] |
---|---|---|

MSE | 19,823.4 | 24,443.4 |

AFE (%) | 0.60 | 0.81 |

${\tilde{N}}_{1},{\tilde{N}}_{2},{\tilde{N}}_{3}\to {\tilde{N}}_{6}$ |

${\tilde{N}}_{2},{\tilde{N}}_{3},{\tilde{N}}_{6}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{3},{\tilde{N}}_{6},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{6},{\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{9}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8},{\tilde{N}}_{9}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{8},{\tilde{N}}_{9},{\tilde{N}}_{13}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{9},{\tilde{N}}_{13},{\tilde{N}}_{13}\to {\tilde{N}}_{11}$ |

${\tilde{N}}_{13},{\tilde{N}}_{13},{\tilde{N}}_{11}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{13},{\tilde{N}}_{11},{\tilde{N}}_{8}\to {\tilde{N}}_{8}$ |

${\tilde{N}}_{11},{\tilde{N}}_{8},{\tilde{N}}_{8}\to {\tilde{N}}_{7}$ |

${\tilde{N}}_{8},{\tilde{N}}_{8},{\tilde{N}}_{7}\to {\tilde{N}}_{7}$ |

${\tilde{N}}_{8},{\tilde{N}}_{7},{\tilde{N}}_{7}\to {\tilde{N}}_{10}$ |

${\tilde{N}}_{7},{\tilde{N}}_{7},{\tilde{N}}_{10}\to {\tilde{N}}_{13}$ |

${\tilde{N}}_{7},{\tilde{N}}_{10},{\tilde{N}}_{13}\to {\tilde{N}}_{17}$ |

${\tilde{N}}_{10},{\tilde{N}}_{13},{\tilde{N}}_{17}\to {\tilde{N}}_{20}$ |

${\tilde{N}}_{13},{\tilde{N}}_{17},{\tilde{N}}_{20}\to {\tilde{N}}_{21}$ |

${\tilde{N}}_{17},{\tilde{N}}_{20},{\tilde{N}}_{21}\to {\tilde{N}}_{21}$ |

${\tilde{N}}_{20},{\tilde{N}}_{21},{\tilde{N}}_{21}\to {\tilde{N}}_{19}$ |

Years | Actual Enrollments | Third-Order Forecasted Values of the Proposed Method | Forecasted Values in [47] | Forecasted Values in [8] | Forecasted Values in [9] |
---|---|---|---|---|---|

1971 | 13,055 | $-$ | $-$ | $-$ | $-$ |

1972 | 13,563 | $-$ | $-$ | $-$ | $-$ |

1973 | 13,867 | $-$ | $-$ | $-$ | $-$ |

1974 | 14,696 | 14,850 | 14,400 | 14,500 | 14,750 |

1975 | 15,460 | 15,450 | 15,300 | 15,500 | 15,750 |

1976 | 15,311 | 15,450 | 15,300 | 15,500 | 15,500 |

1977 | 15,603 | 15,450 | 15,600 | 15,500 | 15,500 |

1978 | 15,861 | 15,750 | 15,600 | 15,500 | 15,500 |

1979 | 16,807 | 16,950 | 16,800 | 16,500 | 16,500 |

1980 | 16,919 | 16,950 | 16,800 | 16,500 | 16,500 |

1981 | 16,388 | 16,350 | 16,200 | 16,500 | 16,500 |

1982 | 15,433 | 15,450 | 15,300 | 15,500 | 15,500 |

1983 | 15,497 | 15,450 | 15,300 | 15,500 | 15,500 |

1984 | 15,145 | 15,150 | 15,000 | 15,500 | 15,250 |

1985 | 15,163 | 15,150 | 15,000 | 15,500 | 15,500 |

1986 | 15,984 | 16,050 | 15,900 | 15,500 | 15,500 |

1987 | 16,859 | 16,950 | 16,800 | 16,500 | 16,500 |

1988 | 18,150 | 18,150 | 18,000 | 18,500 | 18,500 |

1989 | 18,970 | 19,050 | 18,900 | 18,500 | 18,500 |

1990 | 19,328 | 19,350 | 19,200 | 19,500 | 19,500 |

1991 | 19,337 | 19,350 | 19,200 | 19,500 | 19,500 |

1992 | 18,876 | 18,750 | 18,600 | 18,500 | 18,750 |

Dates | Actual Values | Forecasted Values of the Proposed Method |
---|---|---|

01/11/2004 | 5656.17 | $-$ |

02/11/2004 | 5759.61 | 5760.17 |

03/11/2004 | 5862.85 | 5813.5 |

04/11/2004 | 5860.73 | 5900.17 |

05/11/2004 | 5931.31 | 5900.17 |

08/11/2004 | 5937.46 | 5903.02 |

09/11/2004 | 5945.2 | 5903.02 |

10/11/2004 | 5948.49 | 5940.17 |

11/11/2004 | 5874.52 | 5940.17 |

12/11/2004 | 5917.16 | 5903.02 |

15/11/2004 | 5906.69 | 5903.02 |

16/11/2004 | 5910.85 | 5903.02 |

17/11/2004 | 6028.68 | 5940.17 |

18/11/2004 | 6049.49 | 5940.17 |

19/11/2004 | 6026.55 | 5940.17 |

22/11/2004 | 5838.42 | 5830.17 |

23/11/2004 | 5851.1 | 5830.17 |

24/11/2004 | 5911.31 | 5903.02 |

25/11/2004 | 5855.24 | 5830.17 |

26/11/2004 | 5778.65 | 5813.5 |

29/11/2004 | 5785.26 | 5813.5 |

30/11/2004 | 5844.76 | 5860.17 |

1/12/2004 | 5798.62 | 5830.17 |

02/12/2004 | 5867.95 | 5860.17 |

03/12/2004 | 5893.27 | 5900.17 |

06/12/2004 | 5919.17 | 5900.17 |

07/12/2004 | 5925.28 | 5903.02 |

08/12/2004 | 5892.51 | 5903.02 |

09/12/2004 | 5913.97 | 5900.17 |

10/12/2004 | 5911.63 | 5903.02 |

13/12/2004 | 5878.89 | 5903.02 |

14/12/2004 | 5909.65 | 5900.17 |

15/12/2004 | 6002.58 | 5903.02 |

16/12/2004 | 6019.23 | 6040.17 |

17/12/2004 | 6009.32 | 6040.17 |

20/12/2004 | 5985.94 | 6040.17 |

21/12/2004 | 5987.85 | 6040.17 |

22/12/2004 | 6001.52 | 6040.17 |

23/12/2004 | 5997.67 | 6040.17 |

24/12/2004 | 6019.42 | 6040.17 |

27/12/2004 | 5985.94 | 6040.17 |

28/12/2004 | 6000.57 | 6040.17 |

29/12/2004 | 6088.49 | 6040.17 |

30/12/2004 | 6100.86 | 6080.17 |

31/12/2004 | 6139.69 | 6080.17 |

Tool | Proposed |
---|---|

RMSE | 42.05 |

AFE (%) | 0.005 |

**Table 16.**Error measures of the proposed method and other existing methods which solved the TAIEX2004 example.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdel-Basset, M.; Chang, V.; Mohamed, M.; Smarandache, F.
A Refined Approach for Forecasting Based on Neutrosophic Time Series. *Symmetry* **2019**, *11*, 457.
https://doi.org/10.3390/sym11040457

**AMA Style**

Abdel-Basset M, Chang V, Mohamed M, Smarandache F.
A Refined Approach for Forecasting Based on Neutrosophic Time Series. *Symmetry*. 2019; 11(4):457.
https://doi.org/10.3390/sym11040457

**Chicago/Turabian Style**

Abdel-Basset, Mohamed, Victor Chang, Mai Mohamed, and Florentin Smarandache.
2019. "A Refined Approach for Forecasting Based on Neutrosophic Time Series" *Symmetry* 11, no. 4: 457.
https://doi.org/10.3390/sym11040457