1. Assessing Hydrological Model Performance Using Stochastic Simulation
Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
September 8, 2010

2. INTRODUCTION
Very often, in hydrology, the problems are not clearly understood for a meaningful analysis using physically-based methods.
Rainfall-runoff modeling
Empirical models – regression, ANN
Conceptual models – Nash LR
Physical models – kinematic wave
September 8, 2010

3. Regardless of which types of models are used, almost all models need to be calibrated using historical data.
Model calibration encounters a range of uncertainties which stem from different sources including
data uncertainty,
parameter uncertainty, and
model structure uncertainty.
September 8, 2010

4. The uncertainties involved in model calibration inevitably propagate to the model outputs.
Performance of a hydrological model must be evaluated concerning the uncertainties in the model outputs.
Uncertainties in model performance evaluation.
September 8, 2010

5. ASCE Task Committee, 1993
“Although there have been a multitude of watershed and hydrologic models developed in the past several decades, there do not appear to be commonly accepted standards for evaluating the reliability of these models. There is a great need to define the criteria for evaluation of watershed models clearly so that potential users have a basis with which they can select the model best suited to their needs”.
Unfortunately, almost two decades have passed and the above scientific quest remains valid.
September 8, 2010

6. SOME NATURES OF FLOOD FLOW FORECASTING
Incomplete knowledge of the hydrological process under investigation.
Uncertainties in model parameters and model structure when historical data are used for model calibration.
It is often impossible to observe the process with adequate density and spatial resolution.
Due to our inability to observe and model the spatiotemporal variations of hydrological variables, stochastic models are sought after for flow forecasting.
September 8, 2010

7. A unique and important feature of the flow at watershed outlet is its persistence, particularly for the cases of large watersheds.
Even though the model input (rainfall) may exhibit significant spatial and temporal variations, flow at the outlet is generally more persistent in time.
September 8, 2010

8. Illustration of persistence in flood flow series
A measure of persistence is defined as the cumulative impulse response (CIR).
September 8, 2010

9. The flow series have significantly higher persistence than the rainfall series.
We have analyzed flow data at other locations including Hamburg, Iowa of the United States, and found similar high persistence in flow data series.
September 8, 2010

10. The Problem of Lagged Forecast
September 8, 2010

11. September 8, 2010

12. CRITERIA FOR MODEL PERFORMANCE EVALUATION
Relative error (RE)
Mean absolute error (MAE)
Correlation coefficient (r)
Root-mean-squared error (RMSE)
Normalized Root-mean-squared error (NRMSE)
September 8, 2010

13. Coefficient of efficiency (CE) (Nash and Sutcliffe, 1970)
Coefficient of persistence (CP) (Kitanidis and Bras, 1980)
Error in peak flow (or stage) in percentages or absolute value (Ep)
September 8, 2010

14. September 8, 2010

15. Coefficient of Efficiency (CE)
The coefficient of efficiency evaluates the model performance with reference to the mean of the observed data.
Its value can vary from 1, when there is a perfect fit, to A negative CE value indicates that the model predictions are worse than predictions using a constant equal to the average of the observed data.
September 8, 2010

16. Model performance rating using CE (Moriasi et al., 2007)
Moriasi et al. (2007) emphasized that the above performance rating are for a monthly time step. If the evaluation time step decreases (for example, daily or hourly time step), a less strict performance rating should be adopted.
September 8, 2010

17. Coefficient of Persistency (CP)
It focuses on the relationship of the performance of the model under consideration and the performance of the naïve (or persistent) model which assumes a steady state over the forecast lead time.
A small positive value of CP may imply occurrence of lagged prediction, whereas a negative CP value indicates that performance of the considered model is inferior to the naïve model.
September 8, 2010

18. An example of river stage forcating
Model forecasting
CE=
ANN model
observation
September 8, 2010

19. Model forecasting CE=0.68466 CP= -0.3314 Naive forecasting CE=0.76315
ANN model
observation
Naïve model
September 8, 2010

20. Bench Coefficient
Seibert (2001) addressed the importance of choosing an appropriate benchmark series with which the predicted series of the considered model is compared.
September 8, 2010

21. The bench coefficient provides a general form for measures of goodness-of-fit based on benchmark comparisons.
CE and CP are bench coefficients with respect to benchmark series of the constant mean series and the naïve-forecast series, respectively.
September 8, 2010

22. The bottom line, however, is what should the appropriate benchmark series be for the kind of application (flood forecasting) under consideration.
We propose to use the AR(1) or AR(2) model as the benchmark for flood forecasting model performance evaluation.
A CE-CP coupled MPE criterion.
September 8, 2010

23. Demonstration of parameter and model uncertainties
September 8, 2010

24. Parameter uncertainties without model structure uncertainty
September 8, 2010

25. Parameter uncertainties without model structure uncertainty
September 8, 2010

26. Parameter uncertainties without model structure uncertainty
September 8, 2010

27. Parameter uncertainties with model structure uncertainty
September 8, 2010

28. Uncertainties in model performance RMSE
September 8, 2010

29. Uncertainties in model performance RMSE
September 8, 2010

30. Uncertainties in model performance CE
September 8, 2010

31. Uncertainties in model performance CE
September 8, 2010

32. Uncertainties in model performance CP
September 8, 2010

33. Uncertainties in model performance CP
September 8, 2010

34. It appears that the model specification error does not affect the parameter uncertainties. However, the bias in parameter estimation of AR(1) modeling will result in a poorer forecasting performance and higher uncertainties in MPE criteria.
September 8, 2010

35. ASYMPTOTIC RELATIONSHIP BETWEEN CE AND CP
Given a sample series { }, CE and CP respectively represent measures of model performance by choosing the constant mean series and the naïve forecast series as benchmark series.
The sample series is associated with a lag-1 autocorrelation coefficient
September 8, 2010

36. [A]
September 8, 2010

37. Given a data series with a specific lag-1 autocorrelation coefficient, we can choose various models for one-step lead time forecasting of the given data series.
Equation [A] indicates that, although the forecasting performance of these models may differ significantly, their corresponding (CE, CP) pairs will all fall on a specific line determined by .
September 8, 2010

38. Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.
September 8, 2010

39. The asymptotic CE-CP relationship can be used to determine whether a specific CE value, for example CE=0.55, can be considered as having acceptable accuracy.
The CE-based model performance rating recommended by Moriasi et al. (2007) does not take into account the autocorrelation structure of the data series under investigation, and thus may result in misleading recommendations.
September 8, 2010

40. Consider a data series with significant persistence or high lag-1 autocorrelation coefficient, say 0.8. Suppose that a forecasting model yields a CE value of 0.55 (see point C). With this CE value, performance of the model is considered satisfactory according to the performance rating recommended by Moriasi et al. (2007).
However, it corresponds to a negative value of CP (-0.125), indicating that the model performs even poorer than the naïve forecasting, and thus should not be recommended.
September 8, 2010

41. Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.
September 8, 2010

42. 1= 0.843 CE=0.686 at CP=0 1= 0.822 CE=0.644 at CP=0 1= 0.908
September 8, 2010

43. For these three events, the very simple naïve forecasting yields CE values of 0.686, 0.644, and respectively, which are nearly in the range of good to vary good according to the rating of Moriasi et al. (2007).
September 8, 2010

44. In the literature we have found that many flow forecasting applications resulted in CE values varying between 0.65 and With presence of high persistence in flow data series, it is likely that not all these models performed better than naïve forecasting.
September 8, 2010

45. Another point that worth cautioning in using CE for model performance evaluation is whether it should be applied to individual events or a constructed continuous series of several events.
Variation of CE values of individual events enables us to assess the uncertainties in model performance. Whereas some studies constructed an artifact of continuous series of several events, and a single CE value was calculated from the multiple-event continuous series.
September 8, 2010

46. CE value based on such an artifactual series cannot be considered as a measure of overall model performance with respect to all events.
This is due to that fact that the denominator in CE calculation is significant larger for the artifactual series than that of any individual event series, and thus the CE value of the artifactual series will be higher than the CE value of any individual event.
September 8, 2010

47. For example, the CE value by naïve forecasting for an artifactual flow series of the three events in Figure 1 is which is significant higher than the naïve-forecasting CE value of any individual event.
September 8, 2010

48. 1= 0.843 CE=0.686 at CP=0 1= 0.822 CE=0.644 at CP=0 1= 0.908
September 8, 2010

49. A nearly perfect forecasting model
CE=
CE=
CE=
CE=
CE=
CE=
CE=
CE=
CE=
September 8, 2010

50. A CE-CP COUPLED MPE CRITERION
Are we satisfied with using the constant mean series or naïve forecasting as benchmark?
Considering the high persistence nature in flow data series, we argue that performance of the autoregressive model AR(p) should be considered as a benchmark comparison for performance of other flow forecasting models.
September 8, 2010

51. From our previous experience in flood flow analysis and forecasting, we propose to use AR(1) or AR(2) model for benchmark comparison.
September 8, 2010

52. The asymptotic relationship between CE and CP indicates that when different forecasting models are applied to a given data series (with a specific value of 1, say *), the resultant (CE, CP) pairs will all fall on a line determined by Eq. [A] with 1= * .
September 8, 2010

53. In other words, points on the asymptotic line determined by 1= 
In other words, points on the asymptotic line determined by 1= * represent forecasting performance of different models which are applied to the given data series.
Using the AR(1) or AR(2) model as the benchmark, we need to know which point on the asymptotic line corresponds to the AR(1) or AR(2) model.
September 8, 2010

54. CE-CP relationships for AR(1) model
[B]
September 8, 2010

55. CE-CP relationships for AR(1) and AR(2) models
September 8, 2010

56. Example of event-1 September 8, 2010 1=0.843 AR(2) model
Data AR(2) modeling
AR(1) model
Data AR(1) modeling
September 8, 2010

57. Assessing uncertainties in (CE, CP) using modeled-based bootstrap resampling
September 8, 2010

58. Assessing uncertainties in MPE by bootstrap resampling (Event-1)
September 8, 2010

59. Assessing uncertainties in MPE by bootstrap resampling (Event-1)
September 8, 2010

60. Conclusions
Performance of a flow forecasting model needs to be evaluated by taking into account the uncertainties in model performance.
AR(2) model should be considered as the benchmark.
Bootstrap resampling can be helpful in evaluating the uncertainties in model performance.
September 8, 2010

61. As a final remark, we like to reiterate a remark made by Seibert (2001) a decade ago:
“Obviously there is the risk of discouraging results when a model does not outperform some simpler way to obtain a runoff series. But if we truly wish to assess the worth of models, we must take such risks. Ignorance is no defense.”
September 8, 2010

62. Thank you for your attention. Your comments are most welcome!
September 8, 2010

63. What exactly does ensemble mean?
“Ensemble” used in weather forecasting
Ensemble Prediction System (EPS)
Ensemble Streamflow Prediction (ESP)
Perturbation instead of stochastic variation
“Ensemble” in statistics
A collection of all possible outcomes of a random experiment.
September 8, 2010

64. In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1878, an ensemble (also statistical ensemble or thermodynamic ensemble) is an idealization consisting of a large number of mental copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in.
September 8, 2010

65. In both cases, the fact of including stochastic physics in the model gives rise to higher forecast scores values than using only an ensemble based on random perturbations to the initial conditions.
September 8, 2010