# Combined Forecasting of Rainfall Based on Fuzzy Clustering and Cross Entropy

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## Abstract

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## 1. Introduction

## 2. Improved Fuzzy Clustering Model

#### 2.1. Research Data

#### 2.2. An Introduction of Ant Colony Algorithm

_{ij}is the distance between city i and city j. τ

_{ij}(t) is the amount of information between city i and city j at time t. We use it to simulate the actual ant anterin, set a total of m ants, the term p

_{ij}(t) represents the probably of the k-th ant being transferred between city i and city j at time t:

_{ij}indicates the degree of transfer expectation between city i and j. When a = 0, the algorithm is the traditional greedy algorithm; and when b = 0, it becomes a pure positive feedback heuristic algorithm. After n moments, the ants can finish all the cities and complete a cycle. In this case, the amount of information on each path is updated according to the following formula:

_{k}is the length of the path traveled by the ant k in this cycle. After several cycles, the calculation can be terminated according to the appropriate stop condition.

#### 2.3. Basic Principles of Fuzzy Clustering

_{k}(k = 1, 2, …, n) into m fuzzy clusters and obtains the clustering center of each cluster so that the objective function is minimized. The objective function is defined as:

_{ij}is the membership function, c

_{i}is the i-th clustering center, h is the fuzzy weight index. μ

_{ij}∈ (0, 1) and:

^{3}), the time complexity of the first step of the algorithm is reached O(n

^{4}logn).

#### 2.4. Improvement of Fuzzy Clustering by Ant Colony Algorithm

_{i}|X

_{j}= (x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{im})}, i = 1, 2, …, n} is a collection of data to be clustered, τ

_{ij}(t) is amount of information between X

_{i}and X

_{j}at time t. When all the ants have completed a path search, it is said the algorithm carried out a search cycle. In the t search period, the path selection probability can be expressed as:

_{s}|d

_{sj}≤ r

_{j}, s = 1, 2, …, N}, and the other parameters are consistent with the above.

_{ij}(t), then X

_{i}is merged into the X

_{j}field. Make C

_{j}= {X

_{i}|d

_{ij}< r

_{j}, i = 1, 2, …, k}, C

_{j}represents all the data sets that are merged into X

_{j}, and we find the cluster center:

_{ij}(t), and the general initial value of the fuzzy clustering membership matrix is obtained, and ${\overline{c}}_{j}$ is used as the initial of fuzzy clustering center.

_{ij}(t) (in this paper we set 4 cycles) and ${\overline{c}}_{j}$, then iterate according to Equations (8) and (9). When the optimization process slows down, we use the ant colony algorithm once or twice to optimize until the accuracy requirements are reached.

#### 2.5. Method Validation

_{ij}(t)). When calculating p

_{ij}(t), set ρ = 0.7, a = 1, b = 1, η = 1, τ

_{ij}(0) = 0. The results are shown in Figure 3.

_{ij}(t) improves the number of samples with the increase of the number. Figure 3b shows that when the error rate is large, the random number and fuzzy clustering calculation using p

_{ij}(t) are not very different, but when the error rate becomes smaller, the p

_{ij}(t) fuzzy clustering is not changed, and the fuzzy clustering using random number calculation time is rising rapidly. Therefore, the clustering method adopted in this paper is more scientific and effective.

## 3. Rainfall Forecasting Model Based on CE

#### 3.1. Combined Forecasting Model

_{t}, ω

_{ij}is the weight of the i-th model at time t, and ${\widehat{y}}_{it}$ is the predicted value of the i-th model at time t, then the problem of combined forecasting is described as follows:

#### 3.2. The CE Model

_{i}(x) of the single forecasting method multiplied by the corresponding weight. According to the central limit theorem, if a variable is the sum of many independent random factors, we can treat the variable as following a normal distribution, and thus the rainfall value at a certain time can be considered as satisfying a normal distribution. The minimum CE is used to determine the probability distribution of the different forecasting methods, so the combined probability distribution of the rainfall is obtained.

_{i}is the mean value and σ

_{i}is the variance.

_{ij}is the weight of the i-th single method.

_{ij}) to ensure that $-{{\displaystyle \int}}^{\text{}}{g}^{\ast}\left(x\right)\mathrm{ln}f\left(x;{\omega}_{it}\right)dx$ reaches the minimum value, which is equivalent to the maximum value problem:

_{{S(x)>γ}}is called the indicator function:

_{ij}), ω

_{0}is the initial weight, γ is the target estimation parameter, and L represents the estimated target value of a low probability event.

_{ij}and λ to zero, we can obtain:

_{it}= w

_{0}, set iteration number z = 1;

## 4. Results and Analysis

#### 4.1. Predictive Stability Comparison Results

#### 4.2. The Influence of Clustering Method on Prediction Results

- Scenario 1: Traditional c-means clustering method
- Scenario 2: We do not cluster historical data.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The Beijing–Tianjin–Hebei administrative divisions and change of monthly rainfall from 1960 to 2013. (

**a**) Beijing–Tianjin–Hebei administrative divisions; (

**b**) monthly rainfall from 1960 to 2013.

**Figure 3.**Clustering calculation time. (

**a**) calculation time when error rate is constant; (

**b**) calculation time when data amount is constant.

Month | 12, 1, 2 | 3, 4, 5 | 6, 7, 8 | 9, 10, 11 |
---|---|---|---|---|

value | 0.1 | 0.3 | 0.5 | 0.7 |

ARIMA | GM | RBF | CE | |||||
---|---|---|---|---|---|---|---|---|

MRPE | RMSE | MRPE | RMSE | MRPE | RMSE | MRPE | RMSE | |

2011 | 12.13% | 5.53% | 11.15% | 4.49% | 10.99% | 4.21% | 10.02% | 3.93% |

2012 | 9.44% | 4.99% | 13.15% | 4.97% | 8.65% | 5.01% | 9.45% | 4.32% |

2013 | 11.10% | 7.12% | 7.22% | 6.19% | 7.01% | 4.49% | 6.88% | 4.17% |

Average | 10.89% | 5.88% | 10.51% | 5.22% | 8.88% | 4.57% | 8.78% | 4.14% |

S1 | S2 | |||
---|---|---|---|---|

MRPE | RMSE | MRPE | RMSE | |

2011 | 10.04% | 3.89% | 13.01% | 7.12% |

2012 | 9.41% | 4.34% | 12.04% | 8.14% |

2013 | 6.92% | 4.17% | 10.29% | 7.71% |

Average | 8.79% | 4.19% | 11.78% | 7.66% |

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**MDPI and ACS Style**

Men, B.; Long, R.; Li, Y.; Liu, H.; Tian, W.; Wu, Z. Combined Forecasting of Rainfall Based on Fuzzy Clustering and Cross Entropy. *Entropy* **2017**, *19*, 694.
https://doi.org/10.3390/e19120694

**AMA Style**

Men B, Long R, Li Y, Liu H, Tian W, Wu Z. Combined Forecasting of Rainfall Based on Fuzzy Clustering and Cross Entropy. *Entropy*. 2017; 19(12):694.
https://doi.org/10.3390/e19120694

**Chicago/Turabian Style**

Men, Baohui, Rishang Long, Yangsong Li, Huanlong Liu, Wei Tian, and Zhijian Wu. 2017. "Combined Forecasting of Rainfall Based on Fuzzy Clustering and Cross Entropy" *Entropy* 19, no. 12: 694.
https://doi.org/10.3390/e19120694