# Discussion on the Choice of Decomposition Level for Wavelet Based Hydrological Time Series Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Study Area

^{2}, accounts for about 8% of the whole basin area, and the basin outlet is the Yingluoxia streamflow station.

#### 2.2. Precipitation and Streamflow Data

_{mean}, x

_{max}, x

_{min}, C

_{v}, and C

_{s}denote the mathematical mean, maximum, minimum, coefficient of variation and coefficient of skewness, respectively. The precipitation and streamflow series in the calibration period show similar scattered variations as those in the verification period (i.e., similar C

_{v}values of 2.72 and 2.58, 1.11 and 0.96), but the former have higher positive skewness (i.e., C

_{s}= 4.97 compared with 4.13, 3.55 compared to 2.65). The precipitation data fall within the range of 0–34.70 mm in the calibration period, while those in the verification period fall within the range of 0–22.00 mm, being smaller than the former. The same results can be found in the streamflow data: the streamflow data fall within the range of 5.01–788.00 m

^{3}/s in the calibration period, and those in the verification period are in the range of 4.08–504.00 m

^{3}/s. According to the statistical characters, it is thought that extreme streamflow values in the verification period can be accurately modeled as long as a proper model is built by using the calibration data sets.

#### 2.3. Discrete Wavelet Transform and Modeling Design

_{0}and b

_{0}are constants, integer j is the decomposition level, and k is the time translation factor; ψ

^{*}(t) is the complex conjugate. In practice, the dyadic DWT in Equation (2) is used commonly by assigning a

_{0}= 2 and b

_{0}= 1 [32]:

_{j}(t) of the original series under the level j can be reconstructed by Equation (4), and their sum is the original series:

_{2}(N), where N is the series length. When conducting a wavelet-based ANN model, it needs to determine the most suitable decomposition level from 1 to M. Along with the increase of decomposition level j, more sub-signals and detailed information of series at larger temporal scales would appear. More information may contribute to better performance of the model, but more input neutrals may reduce the computing efficiency and decrease the stability of the model. Therefore, it is important to select a suitable decomposition level for wavelet-neutral modeling.

_{i}

^{bs}and Y

_{i}

^{sim}denote the observation and simulated series, and Y

_{mean}

^{bs}and Y

_{mean}

^{sim}denote their mean values, respectively; n is the series length. The NSE coefficient is widely used for assessment of the model’s performance [33]. R is an index commonly used to describe the correlation of two series. AARE can measure the error between the observation and simulated data [34]. In general, high NSE and R values and small AARE values indicate good modeling performance.

## 3. Results

#### 3.1. Wavelet Decomposition of Precipitation

_{2}1827) is used here. By considering both the deterministic characteristics of the series and the mathematical properties of the wavelets, we used the method in Reference [35] and chose the “db3” mother wavelet to analyze the precipitation series in the verification and calibration periods. They are decomposed into 11 sub-signals by the discrete wavelet decomposition method. The sub-signals of the original series under “D” levels are reconstructed by detailed wavelet coefficients, and the sub-signal under the “A” level is reconstructed by approximate wavelet coefficients, and it is usually the mean or trend of the original series. The method of significance testing of DWT proposed by Sang [35] is used here to judge the sub-signals of series under each level belonging to true components or noise.

^{10}-day periodicity (D10). A10 represents the approximation component at the 10-level of decomposition. Lower detail levels have higher frequencies, which represent the rapidly changing component of the dataset, whereas the higher detail levels have lower frequencies, which represent the slowly changing component of the dataset. The approximation components (A10) represent the slowest changing component of the dataset (including the trend).

#### 3.2. Forecasting by Different Decomposition Levels

## 4. Discussion

_{10}(5479)) for the study. From Figure 5 we can find that both in the calibration and verification processes, the modeling results under level 3 are much worse than those under level 6, because the annual and inter-annual variability of precipitation series cannot be clearly identified when using decomposition level 3. Therefore, the third important understanding gained from the above results is that the choice of a suitable decomposition level should meet the need of the accurate identification of series’ characteristics and composition, which is the basis of wavelet-based modeling. To be specific, the choice of decomposition level should be based on the characteristics of the analyzed series, but should not consider the series length or other factors, and unreasonable decomposition of the original series would cause bad wavelet-aided modeling performance.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Cheng, C.T.; Xie, J.X.; Chau, K.W.; Layeghifard, M. A new indirect multi-step-ahead prediction model for a long-term hydrologic prediction. J. Hydrol.
**2008**, 361, 118–130. [Google Scholar] [CrossRef] - Tiwari, M.K.; Chatterjee, C. Uncertainty assessment and ensemble flood forecasting using bootstrap based artificial neural networks (banns). J. Hydrol.
**2010**, 382, 20–33. [Google Scholar] [CrossRef] - Alvisi, S.; Franchini, M. Fuzzy neural networks for water level and discharge forecasting with uncertainty. Environ. Model. Softw.
**2011**, 26, 523–537. [Google Scholar] [CrossRef] - Jain, A.; Kumar, A.M. Hybrid neural network models for hydrological time series forecasting. Appl. Soft. Comput.
**2007**, 7, 585–592. [Google Scholar] [CrossRef] - Nourani, V.; Alami, M.T.; Aminfar, M.H. A combined neural-wavelet model for prediction of ligvanchai watershed precipitation. Eng. Appl. Artif. Intel.
**2009**, 22, 466–472. [Google Scholar] [CrossRef] - Kisi, O. Wavelet regression model for short-term streamflow forecasting. J. Hydrol.
**2010**, 389, 344–353. [Google Scholar] [CrossRef] - Kişi, Ö. Wavelet regression model as an alternative to neural networks for monthly streamflow forecasting. Hydrol. Process.
**2009**, 23, 3583–3597. [Google Scholar] - Sang, Y.F. Improved wavelet modeling framework for hydrological time series forecasting. Water Resour. Manag.
**2013**, 27, 2807–2821. [Google Scholar] [CrossRef] - Nourani, V.; Baghanam, A.H.; Adamowski, J.; Kisi, O. Applications of hybrid wavelet–artificial intelligence models in hydrology: A review. J. Hydrol.
**2014**, 514, 358–377. [Google Scholar] [CrossRef] - Kwon, H.H.; Lall, U.; Khalil, A.F. Stochastic simulation model for nonstationary time series using an autoregressive wavelet decomposition: Applications to rainfall and temperature. Water Resour. Res.
**2007**, 43. [Google Scholar] [CrossRef] - Cannas, B.; Fanni, A.; See, L.; Sias, G. Data preprocessing for river flow forecasting using neural networks: Wavelet transforms and data partitioning. Phys. Chem. Earth.
**2006**, 31, 1164–1171. [Google Scholar] [CrossRef] - Rajurkar, M.; Kothyari, U.; Chaube, U. Modeling of the daily rainfall-runoff relationship with artificial neural network. J. Hydrol.
**2004**, 285, 96–113. [Google Scholar] [CrossRef] - Adamowski, J.; Sun, K. Development of a coupled wavelet transform and neural network method for flow forecasting of non-perennial rivers in semi-arid watersheds. J. Hydrol.
**2010**, 390, 85–91. [Google Scholar] [CrossRef] - Grossmann, A.; Morlet, J. Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Ana.
**1984**, 15, 723–736. [Google Scholar] [CrossRef] - Maheswaran, R.; Khosa, R. Wavelet–volterra coupled model for monthly stream flow forecasting. J. Hydrol.
**2012**, 450, 320–335. [Google Scholar] [CrossRef] - Torrence, C.; Compo, G.P. A practical guide to wavelet analysis. B. Am. Meteorol. Soc.
**1998**, 79, 61–78. [Google Scholar] [CrossRef] - Adamowski, J.F. River flow forecasting using wavelet and cross-wavelet transform models. Hydrol. Process.
**2008**, 22, 4877–4891. [Google Scholar] [CrossRef] - Kişi, Ö. Neural network and wavelet conjunction model for modelling monthly level fluctuations in Turkey. Hydrol. Process.
**2009**, 23, 2081–2092. [Google Scholar] - Nourani, V.; Mogaddam, A.A.; Nadiri, A.O. An ann-based model for spatiotemporal groundwater level forecasting. Hydrol. Process.
**2008**, 22, 5054–5066. [Google Scholar] [CrossRef] - Shoaib, M.; Shamseldin, A.Y.; Melville, B.W. Comparative study of different wavelet based neural network models for rainfall-runoff modeling. J. Hydrol.
**2014**, 515, 47–58. [Google Scholar] [CrossRef] - Tiwari, M.K.; Chatterjee, C. Development of an accurate and reliable hourly flood forecasting model using wavelet-bootstrap-ANN (WBANN) hybrid approach. J. Hydrol.
**2010**, 394, 458–470. [Google Scholar] [CrossRef] - Zhou, H.C.; Peng, Y.; Liang, G.H. The research of monthly discharge predictor-corrector model based on wavelet decomposition. Water Resour. Manag.
**2008**, 22, 217–227. [Google Scholar] [CrossRef] - Nourani, V.; Komasi, M.; Mano, A. A multivariate ann-wavelet approach for rainfall-runoff modeling. Water Resour. Manag.
**2009**, 23, 2877–2894. [Google Scholar] [CrossRef] - Aussem, A.; Campbell, J.; Murtagh, F. Wavelet-based feature extraction and decomposition strategies for financial forecasting. J. Comput. Intell. Finan.
**1998**, 6, 5–12. [Google Scholar] - Partal, T.; Kişi, Ö. Wavelet and neuro-fuzzy conjunction model for precipitation forecasting. J. Hydrol.
**2007**, 342, 199–212. [Google Scholar] [CrossRef] - Kisi, O.; Shiri, J. Precipitation forecasting using wavelet-genetic programming and wavelet-neuro-fuzzy conjunction models. Water Resour. Manag.
**2011**, 25, 3135–3152. [Google Scholar] [CrossRef] - Shoaib, M.; Shamseldin, A.Y.; Melville, B.W.; Khan, M.M. Hybrid wavelet neuro-fuzzy approach for rainfall-runoff modeling. J. Comput. Civil Eng.
**2014**, 30, 04014125. [Google Scholar] [CrossRef] - Shoaib, M.; Shamseldin, A.Y.; Melville, B.W.; Khan, M.M. Runoff forecasting using hybrid Wavelet Gene Expression Programming (WGEP) approach. J. Hydrol.
**2015**, 527, 326–344. [Google Scholar] [CrossRef] - Nourani, V.; Kisi, Ö.; Komasi, M. Two hybrid artificial intelligence approaches for modeling rainfall-runoff process. J. Hydrol.
**2011**, 402, 41–59. [Google Scholar] [CrossRef] - Li, Z.; Xu, Z.; Shao, Q.; Yang, J. Parameter estimation and uncertainty analysis of swat model in upper reaches of the heihe river basin. Hydrol. Process.
**2009**, 23, 2744–2753. [Google Scholar] [CrossRef] - Percival, D.B.; Walden, A.T. Wavelet Methods for Time Series Analysis; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Daubechies, I. Ten Lectures on Wavelets; Society for industrial and applied Mathematics: Philadelphia, PA, USA, 1992. [Google Scholar]
- Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Jain, A.; Srinivasulu, S. Development of effective and efficient rainfall-runoff models using integration of deterministic, real-coded genetic algorithms and artificial neural network techniques. Water Resour. Res.
**2004**, 40, W04302. [Google Scholar] [CrossRef] - Sang, Y.F. A practical guide to discrete wavelet decomposition of hydrologic time series. Water Resour. Manag.
**2012**, 26, 3345–3365. [Google Scholar] [CrossRef]

**Figure 1.**Location of the Heihe River Basin in China, and the hydrological station used in this study.

**Figure 2.**Correlation coefficients of the daily precipitation and streamflow series under different time delays.

**Figure 4.**Energy functions of the precipitation (

**a**); and streamflow (

**b**) series and the reference energy function with 95% confidence interval.

**Figure 5.**NSE values of the WA-ANN modeling results by two decomposition types under 10 decomposition levels. The result under the decomposition level 0 is obtained by using original series.

Series Type | Period | Statistical Characters | ||||
---|---|---|---|---|---|---|

x_{mean} | x_{max} | x_{min} | C_{v} | C_{s} | ||

Precipitation (mm) | calibration | 0.85 | 34.70 | 0 | 2.72 | 4.97 |

verification | 0.84 | 22.00 | 0 | 2.58 | 4.13 | |

Streamflow (m³/s) | calibration | 50.20 | 788.00 | 5.01 | 1.11 | 3.55 |

verification | 49.37 | 503.00 | 4.08 | 0.96 | 2.65 |

Period | Index | Mode Name * | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T-I-0 | T-I-1 | T-I-2 | T-I-3 | T-I-4 | T-I-5 | T-I-6 | T-I-7 | T-I-8 | T-I-9 | T-I-10 | ||

Calibration | NSE | 0.54 | 0.65 | 0.70 | 0.77 | 0.80 | 0.81 | 0.82 | 0.83 | 0.83 | 0.83 | 0.82 |

R | 0.74 | 0.81 | 0.84 | 0.88 | 0.89 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | |

AARE | 0.41 | 0.36 | 0.32 | 0.27 | 0.26 | 0.25 | 0.25 | 0.24 | 0.24 | 0.24 | 0.24 | |

Verification | NSE | 0.43 | 0.51 | 0.60 | 0.70 | 0.76 | 0.79 | 0.81 | 0.82 | 0.81 | 0.80 | 0.80 |

R | 0.67 | 0.73 | 0.78 | 0.84 | 0.87 | 0.89 | 0.91 | 0.91 | 0.91 | 0.90 | 0.90 | |

AARE | 0.43 | 0.40 | 0.35 | 0.30 | 0.27 | 0.25 | 0.24 | 0.23 | 0.24 | 0.25 | 0.25 |

Period | Index | Mode Name * | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T-II-0 | T-II-1 | T-II-2 | T-II-3 | T-II-4 | T-II-5 | T-II-6 | T-II-7 | T-II-8 | T-II-9 | T-II-10 | ||

Calibration | NSE | 0.54 | 0.65 | 0.69 | 0.75 | 0.78 | 0.80 | 0.79 | 0.73 | 0.62 | 0.57 | 0.56 |

R | 0.74 | 0.81 | 0.83 | 0.87 | 0.88 | 0.89 | 0.89 | 0.85 | 0.79 | 0.76 | 0.75 | |

AARE | 0.41 | 0.36 | 0.34 | 0.29 | 0.27 | 0.26 | 0.26 | 0.31 | 0.36 | 0.39 | 0.40 | |

Verification | NSE | 0.43 | 0.51 | 0.61 | 0.70 | 0.70 | 0.74 | 0.75 | 0.71 | 0.50 | 0.44 | 0.25 |

R | 0.67 | 0.73 | 0.78 | 0.85 | 0.84 | 0.86 | 0.87 | 0.84 | 0.72 | 0.67 | 0.58 | |

AARE | 0.43 | 0.40 | 0.35 | 0.29 | 0.31 | 0.28 | 0.29 | 0.32 | 0.39 | 0.45 | 0.40 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, M.; Sang, Y.-F.; Liu, C.; Wang, Z.
Discussion on the Choice of Decomposition Level for Wavelet Based Hydrological Time Series Modeling. *Water* **2016**, *8*, 197.
https://doi.org/10.3390/w8050197

**AMA Style**

Yang M, Sang Y-F, Liu C, Wang Z.
Discussion on the Choice of Decomposition Level for Wavelet Based Hydrological Time Series Modeling. *Water*. 2016; 8(5):197.
https://doi.org/10.3390/w8050197

**Chicago/Turabian Style**

Yang, Moyuan, Yan-Fang Sang, Changming Liu, and Zhonggen Wang.
2016. "Discussion on the Choice of Decomposition Level for Wavelet Based Hydrological Time Series Modeling" *Water* 8, no. 5: 197.
https://doi.org/10.3390/w8050197