# Support Vector Regression for Rainfall-Runoff Modeling in Urban Drainage: A Comparison with the EPA’s Storm Water Management Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Support Vector Regression

_{1}, y

_{1}), (x

_{2}, y

_{2}),…, (x

_{l}, y

_{l})} ⊂ X × R, in SVR the goal is to find a function f(x) that has at most ε deviation from the actually obtained targets y

_{i}for all the training data and, at the same time, is as flat as possible. The errors that are smaller than ε are not of concern, while the errors greater than ε are unacceptable. In the case of linear functions in the form

^{2}need to be minimized. Therefore, this problem can be seen as a convex optimization problem in the form

_{i}, y

_{i}) with ε precision. In cases where this is not possible, or a specified error can be tolerated, slack variables ζ

_{i}, ζ

_{i}

^{*}can be introduced (Figure 2) in the otherwise infeasible constraints of the optimization problem, which can be stated according to the formulation:

_{i}.

_{i}, y

_{i}) with corresponding α

_{i}

^{*}= C are positioned outside the ε-insensitive tube around f. Secondly, α

_{i}α

_{i}

^{*}= 0, that is a set of dual variables α

_{i}, α

_{i}

^{*}, which are both simultaneously nonzero, does not exist. Finally, from α

_{i}

^{*}∈ (0, C) results ζ

_{i}

^{*}= 0, and the second factor in Equation (11) has to vanish. Therefore, b can be evaluated by the following:

_{i}∈ (0, C) and α

_{i}

^{*}∈ (0, C).

_{i}by a map Φ: X→F, where F is some feature space [22]. The Support Vector algorithm only depends on dot products between the various patterns, so it is sufficient to know and use a kernel $k({x}_{i},{x}_{j})=\langle \Phi ({x}_{i}),\Phi ({x}_{j})\rangle $ instead of Φ(∙) explicitly. This allows us to rewrite the Support Vector algorithm as follows:

_{i}, x

_{j}) corresponding to a dot product in some feature space F have to satisfy the Mercer’s condition [22].

#### 2.2. SWMM

_{u}, Conductivity K

_{s}and Initial Deficit). In order to simplify the calibration process, the width value for each sub-basin has been manually evaluated, in accordance with the suggestions of Rossman [25]. The constraints on the parameters were selected based on their physical meaning.

#### 2.3. Case Studies

## 3. Results and Discussion

^{2}. The latter is defined as

_{pi}is the predicted discharge for data point i, Q

_{mi}is the measured discharge for data point i, and Q

_{a}is the averaged value of the measured discharges.

^{3}/s.

_{max}< 0.2. For t/t

_{max}> 0.2 rainfall intensity is reduced, except for a short time interval around t/t

_{max}= 0.5. None of the events that have been considered in the calibration process has similar characteristics. Being a non-physically based method, SVR may lead to unexpected results in some cases.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Butler, D.; Davies, J. Urban Drainage, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Coombes, P.J. Transitioning Drainage into Urban Water Cycle Management. In Proceedings of the WSUD2015 Conference, Engineers Australia, Sydney, Australia, 2015.
- Beven, K.J. Rainfall-Runoff Modelling: The Primer, 2nd ed.; Wiley-Blackwell: Oxford, UK, 2012. [Google Scholar]
- Coombes, P.J.; Babister, M.; McAlister, T. Is the Science and Data underpinning the Rational Method Robust for use in Evolving Urban Catchments. In Proceedings of the 36th Hydrology and Water Resources Symposium, Engineers Australia, Hobart, Australia, 2015.
- ASCE Task Committee on Application of Artificial Neural Networks in Hydrology. Artificial Neural Networks in Hydrology. I: Preliminary Concepts. J. Hydrol. Eng.
**2000**, 5, 115–123. - ASCE Task Committee on Application of Artificial Neural Networks in Hydrology Artificial Neural Networks in Hydrology. II: Hydrologic Applications. J. Hydrol. Eng.
**2000**, 5, 124–137. - Cristianini, N.; Shawe-Taylor, J. An Introduction to Support Vector Machines; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Vapnik, V. The Nature of Statistical Learning Theory; Springer: New York, NY, USA, 1995. [Google Scholar]
- Cortes, C.; Vapnik, V. Support vector networks. Mach. Learn.
**1995**, 20, 273–297. [Google Scholar] [CrossRef] - Boser, B.E.; Guyon, I.M.; Vapnik, V.N. A training algorithm for optimal margin classifiers. In Proceedings of the Annual Conference on Computational Learning Theory, Pittsburgh, PA, USA; ACM Press: New York, NY, USA, 1992; pp. 144–152. [Google Scholar]
- Vapnik, V.; Golowich, S.; Smola, A. Support vector method for function approximation, regression estimation, and signal processing. In Advances in Neural Information Processing Systems; Mozer, M.C., Jordan, M.I., Petsche, T., Eds.; MIT Press: Cambridge, MA, USA, 1997; pp. 281–287. [Google Scholar]
- Dibike, Y.B.; Velickov, S.; Solomatine, D.; Abbott, M.B. Model induction with sup-port vector machines: Introduction and application. J. Comput. Civil Eng.
**2001**, 15, 208–216. [Google Scholar] [CrossRef] - Bray, M.; Han, D. Identification of support vector machines for runoff modelling. J. Hydroinf.
**2004**, 6, 265–280. [Google Scholar] - Tripathi, S.H.; Srinivas, V.V.; Nanjundiah, R.S. Downscaling of precipitation for climate change scenarios: A support vector machine approach. J. Hydrol.
**2006**, 330, 621–640. [Google Scholar] [CrossRef] - Chen, H.; Guo, J.; Wei, X. Downscaling GCMs using the smooth support vector machine method to predict daily precipitation in the Hanjiang basin. Adv. Atmos. Sci.
**2010**, 27, 274–284. [Google Scholar] [CrossRef] - Garcìa Nieto, P.J.; Martinez Torres, J.; Araùjo Fernàndez, M.; Ordònez Galàn, C. Support vector machines and neural networks used to evaluate paper manufactured using Eucalyptus globulus. Appl. Math. Model.
**2012**, 36, 6137–6145. [Google Scholar] [CrossRef] - Antonanzas, J.; Urraca, R.; Martinez-de-Pison, F.J.; Antonanzas-Torres, F. Solar irradiation mapping with exogenous data from support vector regression machines estimations. Energy Convers. Manag.
**2015**, 100, 380–390. [Google Scholar] [CrossRef] - Yu, P.S.; Chen, S.T.; Chang, I.F. Support vector regression for real-time flood stage forecasting. J. Hydrol.
**2006**, 328, 704–716. [Google Scholar] [CrossRef] - Hosseini, S.M.; Mahjouri, N. Integrating Support Vector Regression and a geomorphologic Artificial Neural Network for daily rainfall-runoff modeling. Appl. Soft Comput.
**2016**, 38, 329–345. [Google Scholar] [CrossRef] - Raghavendra, S.N.; Deka, P.C. Support vector machine applications in the field of hydrology: A review. Appl. Soft Comput.
**2014**, 19, 372–386. [Google Scholar] [CrossRef] - Chang, C.C.; Lin, C.J. Training ν-support vector classifiers: Theory and algorithms. Neural Comput.
**2001**, 13, 2119–2147. [Google Scholar] [CrossRef] [PubMed] - Smola, A.J.; Scholkopf, B. A tutorial on support vector regression. Stat. Comput.
**2004**, 14, 199–222. [Google Scholar] [CrossRef] - Karush, W. Minima of functions of several variables with inequalities as side constraints. Master’s Thesis, Department of Mathematics, University of Chicago, Chicago, IL, USA, 1939. [Google Scholar]
- Kuhn, H.W.; Tucker, A.W. Nonlinear programming. In Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probabilistics; University of California Press: Oakland, CA, USA, 1951; pp. 481–492. [Google Scholar]
- Rossman, L.A. Storm Water Management Model User’s Manual, Version 5.0; U.S. Environmental Protection Agency: Cincinnati, OH, USA, 2004.
- Box, M.J. A new method of constrained optimization and comparison with other methods. Comput. J.
**1965**, 8, 42–52. [Google Scholar] [CrossRef] - Barco, J.; Wong, K.M.; Stenstrom, M.K. Automatic Calibration of the U.S. EPA SWMM Model for a Large Urban Catchment. J. Hydraul. Eng.
**2008**, 134, 466–474. [Google Scholar] [CrossRef] - Calomino, F.; Paoletti, A. Le Misure di Pioggia e di Portata Nei Bacini Sperimentali Urbani in Italia; CSDU: Milan, Italy, 1994. [Google Scholar]

**Figure 2.**Example of nonlinear Support Vector regression. Errors do not matter as long as they are less than ε, while the deviations are penalized in a linear fashion, as shown in the top graph, where the “Loss” is the penalty for larger than ε deviations.

**Figure 4.**Simulation of the testing event 718 (Merate basin), (

**a**) rainfall; (

**b**) storm hydrograph; (

**c**) predicted versus observed discharge (SVM); (

**d**) predicted versus observed discharge (SWMM).

**Figure 5.**Simulation of the testing event 730 (Merate basin). (

**a**) Rainfall; (

**b**) storm hydrograph; (

**c**) predicted versus observed discharge (SVM); (

**d**) predicted versus observed discharge (SWMM).

**Figure 6.**Simulation of the testing event 402 (Cascina Scala basin), (

**a**) rainfall; (

**b**) storm hydrograph; (

**c**) predicted versus observed discharge (SVM); (

**d**) predicted versus observed discharge (SWMM).

**Figure 7.**Simulation of the testing event 430 (Cascina Scala basin), (

**a**) rainfall; (

**b**) storm hydrograph; (

**c**) predicted versus observed discharge (SVM); (

**d**) predicted versus observed discharge (SWMM).

Basin: | MERATE | |||||

n IMP | n PERV | IMP D.S. | PERV D.S. | S_{u} | K_{s} | i. m. d. |

(mm) | (mm) | (mm) | (mm/h) | (m/m) | ||

0.13 | 0.45 | 1.1 | 5.5 | 120 | 5 | 0.2 |

Basin: | CASCINA SCALA | |||||

n IMP | n PERV | IMP D.S. | PERV D.S. | S_{u} | K_{s} | i. m. d. |

(mm) | (mm) | (mm) | (mm/h) | (m/m) | ||

0.014 | 0.4 | 1.25 | 2.46 | 150 | 7.4 | 0.15 |

Basin | Event [28] | Total Rainfall | Q_{max} (m^{3}/s) | t_{peak} | Total Runoff (mm) | RMSE (m^{3}/s) | R^{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(mm) | Obs. | SVR | SWMM | Obs. | SVR | SWMM | Obs. | SVR | SWMM | SVR | SWMM | SVR | SWMM | ||

Merate | 718 | 13.80 | 0.614 | 0.548 | 0.653 | 76' | 74' | 74' | 4.23 | 4.51 | 4.73 | 0.0187 | 0.0282 | 0.984 | 0.963 |

730 | 18.40 | 0.263 | 0.230 | 0.301 | 334' | 338' | 332' | 7.77 | 7.89 | 8.18 | 0.0115 | 0.0183 | 0.971 | 0.926 | |

Cascina Scala | 402 | 13.55 | 0.140 | 0.136 | 0.135 | 94' | 98' | 94' | 7.49 | 7.70 | 7.47 | 0.0076 | 0.0911 | 0.969 | 0.964 |

418 | 25.96 | 0.319 | 0.294 | 0.381 | 38' | 40' | 40' | 10.78 | 10.75 | 11.6 | 0.0097 | 0.0298 | 0.986 | 0.868 | |

419 | 7.80 | 0.270 | 0.236 | 0.312 | 20' | 22' | 18' | 2.43 | 2.67 | 2.90 | 0.0128 | 0.0283 | 0.964 | 0.823 | |

430 | 5.99 | 0.071 | 0.066 | 0.070 | 54' | 62' | 50' | 1.26 | 1.91 | 1.94 | 0.0079 | 0.0105 | 0.751 | 0.567 |

Basin | Event | Rainfall | Q_{max} Error (%) | Runoff Error (%) | ||
---|---|---|---|---|---|---|

(mm) | SVR | SWMM | SVR | SWMM | ||

Merate | 718 | 13.80 | −10.7 | 6.4 | 6.6 | 11.8 |

730 | 18.40 | −12.5 | 14.4 | 1.5 | 5.2 | |

Cascina Scala | 402 | 13.55 | −3.3 | −3.7 | 2.8 | −0.3 |

418 | 25.96 | −7.8 | 19.4 | −0.3 | 7.9 | |

419 | 7.80 | −12.8 | 15.4 | 9.9 | 19.6 | |

430 | 5.99 | −6.2 | −1.7 | 51.6 | 54.0 |

© 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 by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Granata, F.; Gargano, R.; De Marinis, G.
Support Vector Regression for Rainfall-Runoff Modeling in Urban Drainage: A Comparison with the EPA’s Storm Water Management Model. *Water* **2016**, *8*, 69.
https://doi.org/10.3390/w8030069

**AMA Style**

Granata F, Gargano R, De Marinis G.
Support Vector Regression for Rainfall-Runoff Modeling in Urban Drainage: A Comparison with the EPA’s Storm Water Management Model. *Water*. 2016; 8(3):69.
https://doi.org/10.3390/w8030069

**Chicago/Turabian Style**

Granata, Francesco, Rudy Gargano, and Giovanni De Marinis.
2016. "Support Vector Regression for Rainfall-Runoff Modeling in Urban Drainage: A Comparison with the EPA’s Storm Water Management Model" *Water* 8, no. 3: 69.
https://doi.org/10.3390/w8030069