# Predicting Multiple Functions of Sustainable Flood Retention Basins under Uncertainty via Multi-Instance Multi-Label Learning

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Data Acquisition

**Figure 1.**Map of 372 identified sustainable flood retention basins (SFRBs) in the wider Central Scotland area (UK).

ID | Variable and Unit | ID | Variable and Unit |
---|---|---|---|

1 | Engineered (%) | 21 | Impermeable Soil Proportion (%) |

2 | Dam Height (m) | 22 | Seasonal Influence (%) |

3 | Dam Length (m) | 23 | Site Elevation (m) |

4 | Outlet Arrangement and Operation (%) | 24 | Vegetation Cover (%) |

5 | Aquatic Animal Passage (%) | 25 | Algal Cover in Summer (%) |

6 | Land Animal Passage (%) | 26 | Relative Total Pollution (%) |

7 | Floodplain Elevation (m) | 27 | Mean Sediment Depth (cm) |

8 | Basin and Channel Connectivity (m) | 28 | Organic Sediment Proportion (%) |

9 | Wetness (%) | 29 | Flotsam Cover (%) |

10 | Proportion of Flow within Channel (%) | 30 | Catchment Size (km^{2}) |

11 | Mean Flooding Depth (m) | 31 | Urban Catchment Proportion (%) |

12 | Typical Wetness Duration (day/year) | 32 | Arable Catchment Proportion (%) |

13 | Estimated Flood Duration (day/year) | 33 | Pasture Catchment Proportion (%) |

14 | Basin Bed Gradient (%) | 34 | Viniculture Catchment Proportion (%) |

15 | Mean Basin Flood Velocity (cm/s) | 35 | Forest Catchment Proportion (%) |

16 | Wetted Perimeter (m) | 36 | Natural Catchment Proportion (%) |

17 | Maximum Flood Water Volume (m^{3}) | 37 | Groundwater Infiltration (%) |

18 | Flood Water Surface Area (m^{2}) | 38 | Mean Depth of the Basin (m) |

19 | Mean Annual Rainfall (mm) | 39 | Length of Basin (m) |

20 | Drainage (cm/day) | 40 | Width of Basin (m) |

**Figure 2.**Typical examples for six different types of sustainable flood retention basins. (

**A**): Hydraulic Flood Retention Basin; (

**B**): Traditional Flood Retention Basin; (

**C**): Sustainable Flood Retention Wetland; (

**D**): Aesthetic Flood Treatment Wetland; (

**E**): Integrated Flood Retention Wetland; (

**F**): Natural Flood Retention Wetland.

SFRB Type | Type Name | Dominant Function | Typical Examples |
---|---|---|---|

1 | Hydraulic Flood Retention Basin (HFRB) | Hydroelectricity generation and drinking water supply | Hydropower reservoir; highly engineered and large-scale drinking water reservoir (Figure 2A) |

2 | Traditional Flood Retention Basin (TFRB) | Flood control | Former drinking water reservoir; traditional flood retention basin (Figure 2B) |

3 | Sustainable Flood Retention Wetland (SFRW) | Retention and drainage | Sustainable drainage systems or best management practices such as some retention and detention basins (Figure 2C) |

4 | Aesthetic Flood Treatment Wetland (AFTW) | Integrated floodwater treatment | Some modern constructed treatment wetlands and integrated constructed wetlands (Figure 2D) |

5 | Integrated Flood Retention Wetland (IFRW) | Multiple recreation functions (e.g., fishing, water sports) | Some artificial water bodies within parks or near motorways; Reservoirs providing recreational activities (Figure 2E) |

6 | Natural Flood Retention Wetland (NFRW) | Nature conservation | Natural or semi-natural lakes and large ponds, potentially with restricted access (Figure 2F) |

#### 2.2. Evaluation of Uncertainty

#### 2.2.1. Evaluation of Input Uncertainty

_{i}be one sample of SFRB data, and x

_{ij}be the j-th variable of the sample x

_{i}

_{,}a set of values (K) following the Gaussian probability distribution function N (x, μ, σ) (Equation (1)) are generated to represent each variable x

_{ij}.

_{i}= {X

_{i}

^{1}, X

_{i}

^{2},…, X

_{i}

^{K}}. Hence, the variability or uncertainties of the variables of each SFRB are well-modeled by a set of values, which provides a potential way to capture the real and intrinsic properties of input data in an intuitive and effective fashion. Namely, the true value of each variable is well-estimated by a set of values instead of a single potential error (bias) value.

**Table 3.**The generated new data based on the uncertainty model (taking Glensherup Reservoir (Figure 3) as a representative example).

Variable | Engineered (%) | Dam Height (m) | Dam Length (m) | Outlet Operation (%) | Width of Basin (m) |
---|---|---|---|---|---|

X_{i} | 80.00 | 20.00 | 250.00 | 75.00 | 200.00 |

C.l. | 0.80 | 0.70 | 0.80 | 0.85 | 0.90 |

σ | 1.25 | 0.81 | 7.54 | 0.71 | 2.63 |

X_{i}^{1} | 80.02 | 19.34 | 250.34 | 74.50 | 200.24 |

X_{i}^{2} | 80.55 | 19.99 | 249.71 | 75.26 | 202.46 |

X_{i}^{3} | 79.82 | 20.26 | 250.50 | 75.05 | 200.29 |

X_{i}^{4} | 79.87 | 20.09 | 244.95 | 74.94 | 201.56 |

X_{i}^{5} | 81.23 | 20.32 | 248.35 | 75.30 | 199.40 |

X_{i}^{K} | 79.80 | 20.08 | 250.55 | 75.37 | 198.02 |

_{i}: Original data; and C.l.: Confidence level.

**Figure 4.**The histogram of the newly generated data for the variable “Engineered” (%) for Glensherup Reservoir based on the uncertainty model (μ = 80, σ = 1.25 and K = 100); Normal fit (μ' = 80.06 and σ' = 1.29); PDF, probability density function.

#### 2.2.2. Evaluation of Output Uncertainty

#### 2.2.2.1. Preliminaries

_{1}, l

_{1}), (x

_{2}, l

_{2}), ..., (x

_{m}, l

_{m})}, where ${x}_{i}\in x$ is an example and ${l}_{i}\in l$ is the corresponding label.

_{1}, l

_{1}), (x

_{2}, l

_{2}), ..., (x

_{m}, l

_{m})}, where ${X}_{i}=\left\{{x}_{i}^{1},{x}_{i}^{1},\dots ,{x}_{i}^{ni}\right\}$ is a set of instances (e.g., X

_{i}is the set of instances generated by the uncertainty model in this study), ni is the number of corresponding instances and ${l}_{i}\in l$ is its label.

_{1}, l

_{1}), (x

_{2}, l

_{2}), ..., (x

_{m}, l

_{m})}, where x

_{i}$\in $ x is an example, ${L}_{i}=\left\{{l}_{i}^{1},{l}_{i}^{2},\dots ,{l}_{i}^{li}\right\}$ is a set of labels (e.g., L

_{i}is the multiple types of one SFRB), l

_{i}is the number of labels of the example and ${l}_{i}\in l$ is the corresponding label.

_{1}, L

_{1}), (X

_{2}, L

_{2}), ..., (X

_{m}, L

_{m})}, where X

_{i}is represented by a set of instances $\left\{{x}_{i}^{1},{x}_{i}^{1},\dots ,{x}_{i}^{ni}\right\}$ and ${L}_{i}=\left\{{l}_{i}^{1},{l}_{i}^{2},\dots ,{l}_{i}^{li}\right\}$.${L}_{i}=\left\{{l}_{i}^{1},{l}_{i}^{2},\dots ,{l}_{i}^{li}\right\}$ is a set of labels of the example X

_{i}. This is a combinational problem of multi-instance learning and multi-label learning, which is called a multi-instance multi-label problem. To deal with such challenges, the multi-instance multi-label learning (MIML) was introduced. Zhou and Zhang [18] first formalized multi-instance multi-label learning by transferring it into a typical multi-instance learning or multi-label learning problem, which resulted in the algorithm MIML-SVM. In this study, each SFRB is represented by a set of instances modeled by the proposed uncertainty function, and it usually exhibits multiple functions simultaneously, which naturally fits the multi-instance multi-label learning scheme.

#### 2.2.2.2. Multi-Instance Multi-Label Support Vector Machine

_{1}, L

_{1}), (X

_{2}, L

_{2}), ..., (X

_{N}, L

_{N}), } be a data set, where X

_{i}={x

_{i}

^{1}, x

_{i}

^{2},…, x

_{i}

^{K}} is the i-th example consisting of K instances, and L

_{i}are the corresponding class labels. MIML-SVM learning [18] is involved in the following four phases:

- (1)
- Data modeling: Construct a data set Γ
_{MIML}consisting of MIML examples (X_{i}, L_{i}) according to the proposed uncertainty model. - (2)
- Clustering: Perform k-medoids clustering on the MIML examples to obtain k-medoids: M
_{k}. - (3)
- Transformation: Transform the MIML learning task into a multi-label learning task by calculating the distance between each MIML example and M
_{k}medoids, and a new data set (Γ_{ML}) is further generated. - (4)
- Multi-label learning: Learn the newly generated data set Γ
_{ML}based on the multi-label SVM classification scheme [42].

#### 2.3. Evaluation Metrics for Assessing the Performance of the MIML-SVM Algorithm

_{1}are applied to evaluate the performance of traditional supervised learning (i.e., SVM in this study).

_{i}, L

_{i}), i = 1, 2,…, m} be a multi-instance multi-label evaluation data set and L is the set of labels, where X

_{i}= {x

_{i}

^{1}, x

_{i}

^{2},…, x

_{i}

^{K}} is the i-th sample consisting of K number of instances, and L

_{i}is the corresponding class labels and ${L}_{i}\subseteq L$. Given P

_{i}= {p

_{1}, p

_{2},…, p

_{n}} is the set of labels predicted by the multi-instance multi-label learning algorithms, r

_{i}(p

_{j}) indicates the rank of a predicted label p

_{j}by the ranking method. Average Precision (Avg. Pre., Equation (5)) is used to evaluate the document ranking performance for query retrieval. When average precision equals one, it means that the ranking runs perfectly.

_{i}is the set of actual labels of a given instance X

_{i}; r

_{i}(p

_{j}) indicates the rank of a predicted label p

_{j}by the ranking method; p indicates the class label, where the rank is higher than that of class label p

_{j}.

_{j}) equals to one, if ${p}_{i}\notin {L}_{i}$, otherwise it equals to zero.

_{i}in L. p

_{a}and p

_{b}indicate the predicted class labels.

^{ml}, Equation (9)) evaluates how many times an instance-label pair is classified incorrectly [43].

## 3. Results and Discussion

^{2}. It provides drinking water to the city Glendevon, which indicates that the water body belongs to SFRB type 2. Furthermore, it allows the community to undertake recreational activities such as fishing and walking, which shows that the reservoir also belongs to SFRB type 5. The traditional classifier SVM predicted the reservoir to be SFRB type 2. After integrating uncertainty, the reservoir was predicted to belong to SFRB types 2 and 5 (Table 2).

**Table 4.**Classification results of sustainable flood retention basins by using a multi-instance multi-label learning algorithm-support vector machine (MIML-SVM) and SVM. (mean ± standard deviation).

Algorithm | K | Evaluation Criteria | ||||
---|---|---|---|---|---|---|

Ham.-L. ^{ml} | Ran.-Loss | One-Error | Coverage | Avg. Pre. | ||

MIML-SVM | 10 | 0.094 ± 0.029 | 0.051 ± 0.027 | 0.097 ± 0.057 | 0.768 ± 0.057 | 0.935 ± 0.031 |

30 | 0.092 ± 0.022 | 0.049 ± 0.020 | 0.082 ± 0.060 | 0.761 ± 0.149 | 0.935 ± 0.029 | |

50 | 0.091 ± 0.018 | 0.049 ± 0.018 | 0.087 ± 0.033 | 0.750 ± 0.120 | 0.935 ± 0.020 | |

70 | 0.093 ± 0.028 | 0.049 ± 0.024 | 0.087 ± 0.058 | 0.758 ± 0.191 | 0.934 ± 0.029 | |

100 | 0.089 ± 0.023 | 0.048 ± 0.015 | 0.087 ± 0.048 | 0.753 ± 0.119 | 0.936 ± 0.024 | |

Mean | 0.092 ± 0.024 | 0.049 ± 0.021 | 0.088 ± 0.051 | 0.758 ± 0.157 | 0.935 ± 0.027 | |

SVM | Accuracy | Precision | Recall | F_{1} | ||

0.847 ± 0.023 | 0.842 ± 0.025 | 0.847 ± 0.013 | 0.843 ± 0.024 |

^{ml}: Hamming-loss; Ran.-Loss: Ranking-Loss; Avg. Pre.: Average Precision.

**Figure 5.**Glensherup Reservoir (56°7'48'' N, 3°24'3'' W) is a typical example of a sustainable flood retention basin (SFRB) belonging to SFRB types 2 and 5.

**Figure 6.**Garnqueen Loch (55°53'24'' N, 4°3'5'' W) is a typical example of a sustainable flood retention basin (SFRB) belonging to SFRB types 3, 5 and 6.

## 4. Conclusions and Recommendations

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Yang, Q.; Boehm, C.; Scholz, M.; Plant, C.; Shao, J.
Predicting Multiple Functions of Sustainable Flood Retention Basins under Uncertainty via Multi-Instance Multi-Label Learning. *Water* **2015**, *7*, 1359-1377.
https://doi.org/10.3390/w7041359

**AMA Style**

Yang Q, Boehm C, Scholz M, Plant C, Shao J.
Predicting Multiple Functions of Sustainable Flood Retention Basins under Uncertainty via Multi-Instance Multi-Label Learning. *Water*. 2015; 7(4):1359-1377.
https://doi.org/10.3390/w7041359

**Chicago/Turabian Style**

Yang, Qinli, Christian Boehm, Miklas Scholz, Claudia Plant, and Junming Shao.
2015. "Predicting Multiple Functions of Sustainable Flood Retention Basins under Uncertainty via Multi-Instance Multi-Label Learning" *Water* 7, no. 4: 1359-1377.
https://doi.org/10.3390/w7041359