# Spatiotemporal Modes of Short Time Rainstorms Based on High-Dimensional Data: A Case Study of the Urban Area of Beijing, China

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Process

^{2}. Fourteen meteorological monitoring stations have continuously recorded the rainfall data of the Beijing urban area in recent years. The continuous monitoring data of 14 rainfall stations were selected from the database, which contains data from 2004 to 2016, at intervals of 5 min. The 14 rainfall stations were evenly distributed in urban areas, as shown in Figure 1.

## 3. Methodology

#### 3.1. Flowchart for Extraction of Rainfall Temporal and Spatial Modes

- (1)
- The rainstorm events were digitized and structured. High-dimensional arrays were established from temporal and spatial dimension perspectives.
- (2)
- Principal component analysis was used to map high-dimensional array to low latitude array.
- (3)
- Dynamic clustering was used to categorize samples to typical modes for describing the temporal and spatial distribution of rainstorms.
- (4)
- With the inverse calculation of principal component analysis, the low dimensional array was reduced to a high-dimensional array to express the spatiotemporal process of each rainstorm modes.

#### 3.2. Construction of High-Dimensional Array for Rainstorms

#### 3.3. Dimensionality Reduction of High Dimensional Array

- (1)
- The transformation matrix ${X}_{n\times m}$ was obtained by centralization of the matrix U,
- (2)
- Calculation of covariance matrix ${\sigma}_{m\times m}$ of ${X}_{n\times m}$,
- (3)
- Calculating the eigenvalues and eigenvectors of the variance matrix of M, M = ${\sigma}_{m\times m}$,
- (4)
- The dimension k, which can retain more than 90% information of original data, was obtained. K eigenvectors constitute the transformation matrix ${V}_{m\times k}$ as column vectors,
- (5)
- Descending dimensions by Equation (3).

#### 3.4. Clustering and Feature Selection

- (1)
- r initial cluster centers are set up: ${Z}_{1}\left(p\right),{Z}_{2}\left(p\right),\dots \dots {Z}_{r}\left(p\right)$, where p is the number of iterations.
- (2)
- Calculating the distance from samples x $\left(x\in X\right)$ to each cluster center, if ${D}_{x}\left(j\right)=min\left\{{D}_{x}\left(i\right)\right\}i=1,2,\cdots $r, then $x\in {S}_{j}$, where ${S}_{j}$ represents cluster j with the center of ${Z}_{j}$.
- (3)
- The new center of each cluster is calculated. The new center of ${Z}_{j}$ is calculated by Equation (4).$${Z}_{j}\left(p+1\right)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{x}_{i},j=1,2,\cdots r$$$${J}_{j}={\left[{\displaystyle \sum}_{x\in {S}_{j}\left(k\right)}\u23a2\u23a2x-{z}_{j}\left(k+1\right)\u23a5{\u23a5}^{2}\right]}^{\frac{1}{2}}$$
- (4)
- If ${Z}_{j}\left(p+1\right)\ne {Z}_{j}\left(p\right)$, j = 1,2,…r, then go to step (2); if ${Z}_{j}\left(p+1\right)={Z}_{j}\left(p\right)$, j = 1,2,…r, the calculation is over.

#### 3.5. Reconstruction of Rainstorms

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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Instantaneous Rainfall (mm) Every 5 min | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Date | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | Total |

2006-6-27 23:00 | 4.2 | 7.4 | 22.2 | 8.2 | 8.6 | 8.6 | 10.3 | 10.8 | 12.1 | 11.1 | 7.7 | 5 | 116.2 |

2006-6-30 22:45 | 3.8 | 3.3 | 3.8 | 4 | 3.4 | 3 | 7 | 19.7 | 18.9 | 18.4 | 5.4 | 8.9 | 99.6 |

2006-7-12 4:25 | 3 | 3 | 3 | 3 | 3.5 | 4.1 | 13.6 | 19.3 | 14.2 | 22.2 | 20.6 | 13.1 | 122.6 |

2006-7-12 19:05 | 4.5 | 4.5 | 7.3 | 20.3 | 24.2 | 16.3 | 11 | 13.2 | 14.8 | 11.6 | 9.6 | 8.6 | 145.9 |

2006-7-13 22:50 | 3.6 | 3.3 | 3.3 | 4.2 | 9.9 | 11.5 | 16 | 11.4 | 7.3 | 7.8 | 6.3 | 4.3 | 88.9 |

2006-7-31 8:50 | 10.6 | 7.1 | 3.3 | 9.7 | 12.7 | 16.7 | 20.7 | 28.7 | 18.2 | 17.2 | 15.7 | 11.7 | 172.3 |

2007-6-27 13:05 | 8.4 | 12.4 | 16.4 | 15.9 | 23.9 | 28.9 | 24.9 | 20.4 | 18.4 | 13.9 | 11.1 | 9.6 | 204.2 |

2008-6-23 14:55 | 4.8 | 3.5 | 5.8 | 4 | 9.2 | 11.7 | 16.3 | 9.2 | 10.3 | 7.7 | 10.7 | 8.5 | 101.7 |

2008-7-18 9:10 | 8.4 | 6.8 | 8.8 | 7.8 | 4.8 | 6.8 | 5.8 | 2.8 | 4.8 | 3.8 | 3.8 | 7.1 | 71.5 |

2009-7-22 17:15 | 6.5 | 11.5 | 16 | 16.2 | 8.5 | 20.4 | 24.9 | 19.9 | 17.9 | 12.9 | 5 | 6.5 | 166.2 |

2009-7-23 15:50 | 2 | 1 | 13.5 | 18.7 | 16.5 | 28 | 23 | 19.4 | 16.6 | 4.6 | 3.6 | 4.6 | 151.5 |

2009-8-7 16:20 | 2.1 | 3.4 | 2.7 | 6.6 | 9.3 | 8.6 | 9 | 9.6 | 6 | 3.5 | 5 | 2.2 | 68 |

2011-7-1 7:05 | 4 | 5.3 | 4.2 | 8.6 | 4.8 | 12.2 | 16.5 | 19.1 | 19.9 | 19.1 | 19 | 14.5 | 147.2 |

2013-7-31 19:20 | 5.1 | 0.6 | 5.1 | 1.1 | 13.6 | 0.1 | 5.6 | 0.6 | 3.6 | 1.1 | 11.1 | 7.8 | 55.4 |

2014-6-10 14:05 | 4.9 | 6.9 | 4.6 | 7.5 | 16.5 | 26.9 | 52.5 | 18.8 | 39.3 | 25.2 | 16.5 | 21.2 | 240.8 |

2014-6-15 18:55 | 8.5 | 2.3 | 6.2 | 9.6 | 14.2 | 13.1 | 8.7 | 8.4 | 8.6 | 9.1 | 11.2 | 7.3 | 107.2 |

2014-7-16 18:55 | 16.3 | 8.8 | 22 | 6.3 | 5.8 | 8.5 | 7.3 | 15.1 | 15.6 | 25 | 26.9 | 20 | 177.6 |

2014-8-23 22:30 | 0.3 | 1.8 | 7.9 | 14.3 | 16.8 | 12.7 | 18.3 | 22.8 | 12.3 | 7.8 | 8.4 | 4.8 | 128.2 |

2014-8-30 21:50 | 4.9 | 9.4 | 7.9 | 14.6 | 7.7 | 13.4 | 8.4 | 10.9 | 12.8 | 7.7 | 11.2 | 13.9 | 122.8 |

2015-8-23 14:10 | 2.5 | 4.5 | 5.7 | 9.5 | 15.6 | 7.4 | 3.7 | 2 | 5.3 | 9.9 | 4.2 | 5.1 | 75.4 |

2016-8-6 22:05 | 7.5 | 7 | 11 | 7 | 5.5 | 5.1 | 6 | 7.1 | 5.5 | 5.5 | 6.2 | 17.3 | 90.7 |

2016-9-7 18:25 | 5 | 3.7 | 1.3 | 21.3 | 26 | 32.5 | 38.1 | 22.8 | 6.8 | 18.3 | 9.5 | 5.3 | 190.6 |

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**MDPI and ACS Style**

Liu, W.; Chen, S.; Tian, F.
Spatiotemporal Modes of Short Time Rainstorms Based on High-Dimensional Data: A Case Study of the Urban Area of Beijing, China. *Water* **2021**, *13*, 3597.
https://doi.org/10.3390/w13243597

**AMA Style**

Liu W, Chen S, Tian F.
Spatiotemporal Modes of Short Time Rainstorms Based on High-Dimensional Data: A Case Study of the Urban Area of Beijing, China. *Water*. 2021; 13(24):3597.
https://doi.org/10.3390/w13243597

**Chicago/Turabian Style**

Liu, Wei, Sheng Chen, and Fuchang Tian.
2021. "Spatiotemporal Modes of Short Time Rainstorms Based on High-Dimensional Data: A Case Study of the Urban Area of Beijing, China" *Water* 13, no. 24: 3597.
https://doi.org/10.3390/w13243597