# An Innovative Hourly Water Demand Forecasting Preprocessing Framework with Local Outlier Correction and Adaptive Decomposition Techniques

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## Abstract

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## 1. Introduction

- The earlier models usually neglect the importance of outlier processing. As such, they cannot improve the accuracy of the hourly forecasting models. In this study, in order to improve the accuracy of water demand forecasting models, a global Isolation Forest model and a local Isolation Forest model have been compared to detect and correct outliers.
- In this study, an effective signal decomposition technique of CEEMDAN has been introduced to decompose a complex original signal into sub-signals, which makes water demand forecasting easier.
- A promising deep learning method of Gated Recurrent Unit (GRU) has been introduced and compared with the conventional ANN and Support Vector Regression (SVR) to explore the potential of the deep learning method for hourly water demand forecasting.
- To the best of our knowledge, this study is the first to integrate two preprocessing methods (i.e., signal decomposition and outlier detection and correction methods) for water demand forecasting.

## 2. Methods

#### 2.1. Isolation Forest

#### 2.2. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

#### 2.2.1. Empirical Mode Decomposition

- the number of extreme points and the zero crossings are equal or differ at the most by one; and
- the mean value of the upper and lower envelopes is zero at any point.

- Generate a new sequence $S\left(t\right)$ by calculating the mean value of the upper envelope and the lower envelope of the original signal $O\left(t\right)$.
- Subtracting $S\left(t\right)$ from $O\left(t\right)$ gives an IMF candidate $C\left(t\right)$, where $C\left(t\right)=O\left(t\right)-S\left(t\right)$.
- Set $C\left(t\right)$ as the original signal, repeat the above steps, and obtain the first IMF signal $im{f}_{1}$ when $C\left(t\right)$ satisfies the conditions of an IMF.
- Set ${O}_{1}\left(t\right)$ as the original signal, where ${O}_{1}\left(t\right)=O\left(t\right)-im{f}_{1}$. Repeat Steps (1) to (3) until the residue $R\left(t\right)$ is a monotonic function or satisfies the predefined stopping criterion, where $R\left(t\right)=O\left(t\right)-{\sum}_{i}^{I}im{f}_{i}$. Then, end the decomposition process, and all the IMFs and the residue of the original signal are obtained.

#### 2.2.2. Ensemble Empirical Mode Decomposition

- Add the Gaussian white noise ${w}^{k}\left(t\right)\text{}\left(k=1,2,\dots ,K\right)$ to the original signal $O\left(t\right)$ and determine ${O}^{k}\left(t\right)$, where ${O}^{k}\left(t\right)=O\left(t\right)+{w}^{k}\left(t\right)$.
- Decompose the improved original signal ${O}^{k}\left(t\right)$ by an EMD algorithm and determine all the$\text{}im{f}_{i}^{k}$ ($\text{}im{f}_{i}^{k}$ is the $i$th IMF of the EMD decomposition when adding the white noise at the $k$th time).
- Repeat Steps (1) and (2) for $K$ times. It is worth noting that the added Gaussian white noise is different each time.
- The true $\overline{im{f}_{i}}$ is the sequence constructed by the mean value of the $\text{}im{f}_{i}^{k}\left(k=1,2,\dots ,K\right)$:$$\overline{im{f}_{i}}=\frac{1}{K}{{\displaystyle \sum}}_{k=1}^{K}\text{}im{f}_{i}^{k}\left(i=1,2,\cdots I,k=1,2,\cdots ,K\right).$$

#### 2.2.3. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

- In CEEMDAN, the way to determine the first decomposition mode $\overline{im{f}_{1}}$ is the same as the EEMD. Decompose the signal ${O}^{k}\left(t\right)=O\left(t\right)+{\epsilon}_{0}{w}^{k}\left(t\right)$ for the $k$th computation of CEEMDAN, where ${\epsilon}_{0}$ is the signal-to-noise ratio. Calculate the first IMF as follows:$$\overline{im{f}_{1}}=\frac{1}{K}{{\displaystyle \sum}}_{k=1}^{K}\text{}im{f}_{1}^{k}.$$The first residue ${r}_{1}\left(t\right)$ is:$${r}_{1}\left(t\right)=O\left(t\right)-\overline{im{f}_{1}}.$$
- Determine the second decomposition mode $\overline{im{f}_{2}}$ and the residue ${r}_{2}\left(t\right)$ as follows:$$\overline{im{f}_{2}}=\frac{1}{K}{\sum}_{k=1}^{K}{E}_{1}\left({r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({w}^{k}\left(t\right)\right)\right),$$$${r}_{2}\left(t\right)={r}_{1}\left(t\right)-\overline{im{f}_{2}},$$
- Similarly, for $i$ = 3,4,$\cdots $, I, the $\overline{im{f}_{i}}$ and ${r}_{i}\left(t\right)$ can be calculated as follows:$$\overline{im{f}_{i}}=\frac{1}{K}{\sum}_{k=1}^{K}{E}_{1}\left({r}_{i-1}\left(t\right)+{\epsilon}_{i-1}{E}_{i-1}\left({w}^{k}\left(t\right)\right)\right),$$$${r}_{i}\left(t\right)={r}_{i-1}\left(t\right)-\overline{im{f}_{i}}.$$
- Repeat Step (3) until the residue $R\left(t\right)$ is a monotonic function or satisfies the predefined stopping criterion. The final residue can be calculated as follows:$$R\left(t\right)=O\left(t\right)-{\sum}_{i}^{I}\overline{im{f}_{i}}.$$Therefore, the original signal can be described as follows:$$O\left(t\right)={\sum}_{i}^{I}\overline{im{f}_{i}}+R\left(t\right).$$

#### 2.3. Artificial Neural Network

#### 2.4. Gated Recurrent Unit

#### 2.5. Support Vector Regression

#### 2.6. Research Flowchart

- Through the outlier detection and correction process, the original water demand data are cleaned (i.e., Step 1 in Figure 1). In this study, we have compared the proposed local outlier detection and correction method, and a global outlier detection and correction method. In the local outlier detection and correction method, water demand data are assumed to follow a specific distribution for each hour. First, the water demand data are classified hourly into 24 clusters. Then, 24 Isolation Forest models are built for each cluster to detect the outliers. The outliers are then corrected by the mean value of that hour instead of the global mean value. In the global method, an Isolation Forest model is built based on all the data of water demand, and the outliers are corrected by the global mean value.
- After outlier correction, the nonstationary and nonlinear time series signal is decomposed adaptively by CEEMDAN and turned into simpler sub-signals (i.e., Step 2 in Figure 1). It is easier for these sub-signals to extract the features.
- The GRU, SVR, and ANN models are built based on each sub-series to explore the efficiency of the proposed preprocessing framework on different forecasting models (i.e., Step 3 in Figure 1). The forecasting result of a water demand time series is the sum of the forecasting results of all the sub-signals.

## 3. Case Study

#### 3.1. Data Description

#### 3.2. Model Development

#### 3.3. Performance Evaluation Indices

## 4. Results and Discussion

#### 4.1. Outlier Detection and Correction

#### 4.2. Time Series Signal Decomposition

#### 4.3. Overall Performance

## 5. Conclusions

- The proposed preprocessing method can greatly improve the accuracy of hourly water demand forecasting models. The RMSE of the SVR, ANN, and GRU models has reduced by 57.5%, 27.8%, and 30.0%, respectively.
- The local outlier detection and correction method not only effectively identifies global outlier and outlier clusters that are overlapped with other normal data, but also reduce misidentification of normal samples.
- The CEEMDAN model is able to decompose nonstationary and nonlinear water demand time series into sub-signals with an obvious main power spectral density peak, which makes it easier to capture the signal characteristics for prediction.
- Despite the higher computational load, the GRU-based models always perform better than the ANN and SVR-based models. The IF-CEEMDAN-GRU model is the most accurate model among the twelve models examined in this study. The prediction by the SVR model without preprocessing is poor, but the IF-CEEMDAN-SVR model can achieve an accuracy close to that of the IF-CEEMDAN-GRU model with lower computational load. That is, the proposed method can also exert great potential on some of the conventional models.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The flowchart of the proposed method including outlier detection and correction, time series signal decomposition, and basic forecasting models.

**Figure 2.**The parametric heatmap for (

**a**) batch size and epoch, and (

**b**) the numbers of units in the first layer and the second layer of the Gated Recurrent Unit (GRU) models.

**Figure 3.**Scatter and density contour map of (

**a**) the observed water demand; (

**b**) the results of global outlier detection and correction; (

**c**) the results of local outlier detection and correction. Blue scatters represent the hourly water demand data in different hours. The right color bar shows the density contour map for water demand data. Red lines near the y-axis are the projections of water demand data.

**Figure 4.**Original signals, their corresponding Intrinsic Mode Functions (IMFs), and residue results of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) for (

**a**) initial time series, and (

**b**) time series processed by Isolation Forest.

**Figure 5.**Spectra of (

**a**) original signal, (

**c**) IMF 1–3, and (

**e**) IMF 4–8 of initial time series and spectra of (

**b**) original signal, (

**d**) IMF 1–3, and (

**f**) IMF 4–8 of time series processed by Isolation Forest.

**Figure 6.**Water demand forecasting results of: (

**a**) GRU, Artificial Neural Network (ANN), and Support Vector Regression (SVR); (

**b**) Isolation Forest (IF)-GRU, IF-ANN, and IF-SVR; (

**c**) CEEMDAN-GRU, CEEMDAN-ANN, and CEEMDAN-SVR; and (

**d**) IF-CEEMDAN-GRU, IF-CEEMDAN-ANN, and IF-CEEMDAN-SVR.

Model | MAE | RMSE | ${\mathit{R}}^{2}$ | r |
---|---|---|---|---|

SVR | 4.60996 | 5.58961 | 0.50343 | 0.75491 |

IF-SVR | 2.23167 | 2.95628 | 0.861099 | 0.928563 |

CEEMDAN-SVR | 2.60908 | 3.56389 | 0.798133 | 0.904639 |

IF-CEEMDAN-SVR | 1.79512 | 2.37339 | 0.910473 | 0.955239 |

ANN | 2.59636 | 3.60472 | 0.793481 | 0.89321 |

IF-ANN | 2.07612 | 3.17677 | 0.839606 | 0.917693 |

CEEMDAN-ANN | 2.99397 | 3.99805 | 0.745954 | 0.878257 |

IF-CEEMDAN-ANN | 1.9694 | 2.6012 | 0.892461 | 0.945378 |

GRU | 2.27949 | 3.32249 | 0.824554 | 0.910801 |

IF-GRU | 1.98076 | 3.26109 | 0.830978 | 0.916113 |

CEEMDAN-GRU | 1.67429 | 2.35978 | 0.911497 | 0.957639 |

IF-CEEMDAN-GRU | 1.52313 | 2.32435 | 0.914134 | 0.956688 |

Model | CPU Time (s) | Model | CPU Time (s) |
---|---|---|---|

SVR | 0.008 | CEEMDAN-SVR | 13.579 |

IF-SVR | 3.584 | IF-CEEMDAN-SVR | 17.134 |

ANN | 0.075 | CEEMDAN-ANN | 14.641 |

IF-ANN | 3.668 | IF-CEEMDAN-ANN | 18.529 |

GRU | 192.128 | CEEMDAN-GRU | 1934.022 |

IF-GRU | 194.508 | IF-CEEMDAN-GRU | 1996.464 |

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**MDPI and ACS Style**

Hu, S.; Gao, J.; Zhong, D.; Deng, L.; Ou, C.; Xin, P.
An Innovative Hourly Water Demand Forecasting Preprocessing Framework with Local Outlier Correction and Adaptive Decomposition Techniques. *Water* **2021**, *13*, 582.
https://doi.org/10.3390/w13050582

**AMA Style**

Hu S, Gao J, Zhong D, Deng L, Ou C, Xin P.
An Innovative Hourly Water Demand Forecasting Preprocessing Framework with Local Outlier Correction and Adaptive Decomposition Techniques. *Water*. 2021; 13(5):582.
https://doi.org/10.3390/w13050582

**Chicago/Turabian Style**

Hu, Shiyuan, Jinliang Gao, Dan Zhong, Liqun Deng, Chenhao Ou, and Ping Xin.
2021. "An Innovative Hourly Water Demand Forecasting Preprocessing Framework with Local Outlier Correction and Adaptive Decomposition Techniques" *Water* 13, no. 5: 582.
https://doi.org/10.3390/w13050582