# Comparison of Statistical and Machine Learning Models for Pipe Failure Modeling in Water Distribution Networks

^{*}

## Abstract

**:**

^{2}between 0.695 and 0.927), but the Poisson Regression was the most suitable for predicting failures in pipes with lower failure rates. Regarding Machine Learning models, Bayes and ANNs exhibited low performance in the prediction of pipe failure condition. The GBT approach had the best performing classifier.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Methodology

#### 2.1.1. Statistical Models

_{i}, as shown in Equation (1) [24].

^{2}) and the root mean square error (RMSE), defined as follows.

#### 2.1.2. Machine Learning Models

_{i}(i = 1, 2, …, n) independent variables and an observed data Y, the Bayes formula can be written as shown in Equation (10) [38].

_{i}is the number of input variables, and N

_{c}is the number of classes. An automated approach is adopted to choose the best ANN configuration considering one or two hidden layers with eight neurons. Several activation functions are tested (i.e., exponential, logistic, sigmoid, and hyperbolic tangent). For each model, the selected values of the parameters are presented in Appendix A.

#### 2.2. Case Study

## 3. Results and Discussion

#### 3.1. Statistical Models

^{2}and RMSE. These results confirmed that the Poisson Regression’s ability for generalization (i.e., the model’s ability to adapt properly to a new range of inputs) is better than that of the other techniques. The advantage of the Poisson Regression is to recognize the non-negative nature of the predicted variable. The application of this model is suitable for predicting failures in pipes with lower failure rates, such as pipes with large diameters and small lengths.

_{i}) for a given observation period (T) is calculated by considering the pipe length (L

_{i}), the total class length (L

_{group}), and the failures predicted for the group (FR

_{group}) [5].

#### 3.2. Machine Learning Models

^{2}of 0.452 for asbestos-cement pipes and 0.724 for PVC pipes. For the individual pipe analysis, results gave only 4% correct classifications for asbestos-cement pipe failure and 15% for PVC pipe failure. These results and other findings in previous studies underline the need for each WDN to develop its own failure model [1,62]. All the networks have substantive differences, and the effect of specific variables in the models is dependent on the WDN characteristics.

## 4. Conclusions

^{2}between 0.695 and 0.927 and RMSE between 45 and 22 for the test sample), but the application of Poisson Regression was the most suitable for predicting failures in pipes with lower failure rates. Regarding ML models, all methods but the ANNs presented acceptable performance. The GBT approach presented the best performing classifier (ACU of 0.998 and 0.990 for the test sample of asbestos-cement and PVC pipes, respectively). The GBT model has a greater ability to accurately predict pipe failure when an imbalanced database is used. Furthermore, the assumptions and trade-offs of the GBT model are more transparent than in other artificial intelligence techniques.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameter | Value | |
---|---|---|

Asbestos-Cement Pipes | PVC Pipes | |

Number of trees | 300 | 300 |

Maximal depth | 5 | 4 |

Learning rate | 0.3 | 0.1 |

Parameter | Value | |
---|---|---|

Asbestos-Cement Pipes | PVC Pipes | |

Gamma | 5.0 | 10.0 |

C | 10.0 | 30.0 |

Epsilon | 0.001 | 0.001 |

Parameter | Value | |
---|---|---|

Asbestos-Cement Pipes | PVC Pipes | |

Input layers | 10 | 10 |

Hidden layers | 2 | 1 |

Hidden layer neurons | 8 | 8 |

Training cycles | 2000 | 2000 |

Learning rate | 0.2 | 0.2 |

Activation function of hidden layers | Sigmoid | Sigmoid |

Activation function of the output layer | Sigmoid | Sigmoid |

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**Figure 3.**Relationship between: (

**a**) Pipe diameter and failures normalized by the number of pipes of each diameter; (

**b**) Pipe age and failures normalized by the number of pipes of each age range; (

**c**) Pipe length and failures normalized by the number of pipes of each length range; (

**d**) Pipe material and failures normalized by the total of pipes of each material.

**Figure 5.**Average observations and predictions of failure rate based on: (

**a**) Asbestos-cement pipe diameter; (

**b**) Asbestos-cement pipe age; (

**c**) PVC pipe diameter; (

**d**) PVC pipe age.

Variable | Name | Type | Description |
---|---|---|---|

Physical | Diameter | Numerical | Pipe diameter in mm |

Age | Numerical | Pipe age in years | |

Length | Numerical | Pipe length in m | |

Environmental | Moisture content | Nominal | Soil moisture content (continually wet, generally moist and generally dry) |

Soil contraction and expansion potential | Nominal | Soil contraction and expansion potential (very low, low, moderate, and high) | |

Precipitation | Numerical | Precipitation in m | |

Operational | Land use | Nominal | Land use (residential, commercial, industrial, and institutional) |

Valves | Numerical | Number of valves on the pipe | |

Hydrants | Numerical | Number of hydrants connected to the pipe | |

Previous failures | Numerical | Number of previous failures recorded on the pipe |

Predicted Condition | ||||
---|---|---|---|---|

Yes | No | Recall | ||

Actual condition | Yes | True positive (TP) | False negative (FN) | TP/P |

No | False positive (FP) | True negative (TN) | TN/N | |

Precision | TP/(TP + FP) | TN/(TN + FN) | ||

Total positive | Total negative |

Variable | Asbestos-Cement | PVC | ||||||
---|---|---|---|---|---|---|---|---|

Linear Regression | Poisson Regression | Linear Regression | Poisson Regression | |||||

$\mathit{\beta}$ | p-Value | $\mathit{\beta}$ | p-Value | $\mathit{\beta}$ | p-Value | $\mathit{\beta}$ | p-Value | |

Diameter (mm) | −0.457 | 0.000 | −0.074 | 0.000 | −0.401 | 0.000 | −0.009 | 0.000 |

Length (km) | 2.707 | 0.000 | 0.034 | 0.000 | 0.919 | 0.000 | 0.002 | 0.000 |

Age (years) | 0.162 | 0.000 | −0.001 | 0.008 | 0.679 | 0.000 | −0.001 | 0.000 |

Intercept | n/a | n/a | 4.466 | 0.001 | n/a | n/a | 5.810 | 0.000 |

Material | Equation |
---|---|

Asbestos-cement | $\mathrm{FR}=\text{}{0.202\text{}L}^{1.5}{\text{}/\text{}\mathrm{DA}}^{2}$ |

PVC | $\mathrm{FR}=\text{}{0.00795\text{}\mathrm{LA}}^{0.5}{/\mathrm{D}}^{0.5}$ |

Performance Metric | Dataset | Linear Regression | Poisson Regression | EPR |
---|---|---|---|---|

R^{2} | Train data | 0.693 | 0.923 | 0.877 |

Test data | 0.695 | 0.927 | 0.885 | |

RMSE | Train data | 45.31 | 22.87 | 31.12 |

Test data | 44.93 | 22.09 | 31.10 |

Model | Accuracy | F-measure | ||
---|---|---|---|---|

Asbestos-Cement | PVC | Asbestos-Cement | PVC | |

Bayes | 94.83% | 93.69% | 25.66% | 6.26% |

GBT | 99.52% | 99.79% | 72.00% | 46.43% |

SVM | 99.47% | 99.83% | 66.67% | 52.18% |

ANN | 98.99% | 99.61% | 42.10% | 7.40% |

Bayes | Predicted | GBT | Predicted | ||||||
---|---|---|---|---|---|---|---|---|---|

Yes | No | Recall(%) | Yes | No | Recall(%) | ||||

Actual | Yes | 39 | 6 | Actual | Yes | 27 | 14 | ||

No | 220 | 4106 | 86.67 | No | 7 | 4323 | 65.85 | ||

Precision (%) | 15.06 | 99.85 | 94.91 | Precision (%) | 79.41 | 99.68 | 99.84 | ||

SVM | Predicted | ANN | Predicted | ||||||

Yes | No | Recall(%) | Yes | No | Recall(%) | ||||

Actual | Yes | 23 | 18 | 56.10 | Actual | Yes | 16 | 25 | 39.02 |

No | 5 | 4325 | 99.88 | No | 19 | 4311 | 99.56 | ||

Precision (%) | 82.14 | 99.59 | Precision (%) | 45.71 | 99.42 |

Bayes | Predicted | GBT | Predicted | ||||||
---|---|---|---|---|---|---|---|---|---|

Yes | No | Recall(%) | Yes | No | Recall(%) | ||||

Actual | Yes | 27 | 1 | 96.69 | Actual | Yes | 13 | 15 | 46.43 |

No | 807 | 11977 | 96.43 | No | 15 | 12769 | 99.88 | ||

Precision (%) | 3.24 | 99.99 | Precision (%) | 46.43 | 99.88 | ||||

SVM | Predicted | ANN | Predicted | ||||||

Yes | No | Recall(%) | Yes | No | Recall(%) | ||||

Actual | Yes | 2 | 26 | 7.14 | Actual | Yes | 12 | 16 | 42.86 |

No | 24 | 12760 | 99.81 | No | 6 | 12778 | 99.95 | ||

Precision (%) | 7.69 | 99.80 | Precision (%) | 66.67 | 99.87 |

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**MDPI and ACS Style**

Giraldo-González, M.M.; Rodríguez, J.P.
Comparison of Statistical and Machine Learning Models for Pipe Failure Modeling in Water Distribution Networks. *Water* **2020**, *12*, 1153.
https://doi.org/10.3390/w12041153

**AMA Style**

Giraldo-González MM, Rodríguez JP.
Comparison of Statistical and Machine Learning Models for Pipe Failure Modeling in Water Distribution Networks. *Water*. 2020; 12(4):1153.
https://doi.org/10.3390/w12041153

**Chicago/Turabian Style**

Giraldo-González, Mónica Marcela, and Juan Pablo Rodríguez.
2020. "Comparison of Statistical and Machine Learning Models for Pipe Failure Modeling in Water Distribution Networks" *Water* 12, no. 4: 1153.
https://doi.org/10.3390/w12041153