Non-stationary frequency analysis

Figure 6.3-4: Akaike Information Criteria (AIC) based on the maximum likelihood of the fitting volume series to 7 distributions

and volume series. It means that it is reasonable to apply the non-stationary flood frequency analysis for both series.

6.3.2.2 Non-stationary models

For the application of non-stationary frequency analysis it is assumed that the trend is only caused by the time covariate. As outlined in chapter 6.3.1, LN3 and GEV3 will be analyzed for the non-stationary case of the peak series, and N2 and LN3 for the volume series. These distributions are briefly described in the Appendix A.

Based on the trend analysis and the selection of the original (parent) candidate distribution, the following non-stationary models for flood peak were proposed. Candidate ―parent‖ models and their parameters for the peak series are:

- Log-normal distribution (LN3): 𝑘 (shape), 𝛼 (scale), 𝜉 (location) - Generalized extreme value (GEV3): 𝑘 (shape), 𝜎 (scale),  (location)

Only trend in location and trend in scale (variance) are examined in the non-stationary analysis for this series. Furthermore, trend in location is assumed to be linear. Trend in scale follows an exponential behavior to ensure that the scale parameter is always positive. Table 6.3-3 illustrates the ―child‖ models constructed for examining the non-stationarity in the peak flow series.

Table 6.3-3: Non-stationary models for peak series

Parameters Log-normal based GEV based

q_M10 q_M11 q_M20 q_M21

Shape 𝑘 𝑘 𝑘 𝑘

Scale 𝛼 > 0 α t = exp(𝛼₀+ 𝛼₁𝑡) 𝜎 > 0 𝜎 t = exp(𝜎₀+ 𝜎₁𝑡)

Location 𝜉 𝜉(𝑡) = 𝜉₀+ 𝜉₁𝑡 𝜇 𝜇 = 𝜇₀+ 𝜇₁𝑡

The q_M10 and q_M20 are stationary models identical to the distributions used in 6.3.1. They are listed here as reference for the evaluation of the non-stationary models q_M11 and q_M21, respectively.

Candidate ―parent‖ models for the volume series are:

- Normal distribution (N2): 𝜎 (scale),  (location)

- Log-normal distribution (LN3): 𝑘(shape), 𝛼 (scale), 𝜉 (location)

Table 6.3-4 illustrates the ―child‖ models (but more general models) constructed for examining the non-stationarity in the volume data.

Table 6.3-4: Non-stationary models for volume series

Parameters Log-normal based Normal based

v_M10 v_M11 v_M20 v_M21

Shape 𝑘 𝑘 − −

Scale 𝛼 > 0 𝛼 t = exp(𝛼₀+ 𝛼₁𝑡) 𝜎 > 0 𝜎 t = exp(𝜎₀+ 𝜎₁𝑡)

Location 𝜉 𝜉(𝑡) = 𝜉₀+ 𝜉₁𝑡 𝜇 𝜇 = 𝜇₀+ 𝜇₁𝑡

The v_M10 and v_M20 are again stationary models identical to the distributions in chapter 6.3.1 and are used for reference in this section. It is necessary to note here that the time varying scale function should be selected with care, especially for the case of the normal distribution, because the scale parameter has a very large influence on the flood probabilities and the exponential function is not upper-bounded. In some cases, the logistic function may be more appropriate. In this study, to be consistent in the choice of the function to represent the time-varying scale, exponential function is used in all cases.

6.3.2.3 Parameter estimation

The parameters of the non-stationary models were estimated by the maximum likelihood method (Coles, 2001), with some modifications from its original form (Fisher, 1932). Time was used in the log-likelihood form as a covariate. The parameter sets estimated for the stationary cases were utilized as starting points. The optimization algorithm is the Shuffled Complex Evolution (SCE-UE) (Duan et al., 1993). For q_M11, q_21, v_M11 there are 5 parameters to be estimated for each model. For v_M21 there are 4 parameters to be estimated.

A criterion for choosing the model:

Coles (2001) described the deviance statistic for determining the suitable models when covariates are taken into account in the non-stationary case. Given M0, M1 are the two models considered and M0 is a sub-model of M1, the deviance statistic is defined as:

𝐷 = 2 × {𝐿₁ 𝑀₁ − 𝐿₀(𝑀₀)} (6.2)

Where 𝐿₀(𝑀₀), 𝐿₁ 𝑀₁ are the maximized log-likelihoods under models 𝑀₀, 𝑀₁ respectively.

Large values of 𝐷 imply that model M1 explains substantially more of the variation in the data than M0. Small values of 𝐷 suggest that the increase in model size does not bring worth-while improvements in the model‘s capacity to explain the data. A Chi-square based test is used to help

defining how large 𝐷 should be before the model M1 is preferred to model M0 (based on the asymptotic distribution of the deviance function).

Fitting results

Table 6.3-5: Summary of the fitted parameters for the proposed stationary (S) and non-stationary (NS) models

Location parameter Scale parameter Shape parameter

S NS S NS S NS

𝜉(𝜇) 𝜉₀(𝜇₀) 𝜉₁(𝜇₁) 𝛼 𝛼₀ 𝛼₁ 𝑘 𝑘₀

Peak

GEV3 45675 50565 -113 7118 8.571 0.0049 -0.1950 -0.155

LN3 48293 52290 -93.1 7291 8.667 0.0029 -0.0646 -0.1014

Volume N 4177927 4210009 -199 638713 13.08 0.0059 - -

LN3 4167936 4449238 -6528 638323 12.92 0.0081 -0.0308 -0.06

Figure 6.3-5: Akaike Information Criteria (AIC) based on the maximum likelihood of the fitting the non-stationary series to the proposed models: (left) - peak series, (right) - volume series; green denotes the stationary case, yellow is for the non-stationary case.

Peak series

Figure 6.3-5shows that both distributions are equally well suited for describing the time-varying peak series. The deviance statistic values are 10.850 and 12.076 for LN3 and GEV, respectively.

In both cases, the non-stationary models are superior to the stationary models at the significance level of 10%. The non-stationary GEV3 model can describe well the time-trend series. It is an upper-bound distribution in this case. As mentioned in Section 6.2, the maximum unit discharge

2005a). However, it is difficult to prove the physical limit of the runoff here. Therefore, in flood frequency analysis, upper-bounded distributions should be handled with care. The LN3 is hence chosen.

Volume series

Figure 6.3-5 shows that the LN3 distribution describes better the time-varying behavior of the volume series than the normal distribution. The deviance statistic is 3.794 and 9.352 for N2 and LN3, respectively. Applying the Chi square test leads to the conclusion that there is no improvement if applying v_M21 but there is a significant improvement in fitting the volume data to v_M22. Hence, LN3 is selected to model the volume series.

Figure 6.3-6 illustrates the time-varying probability density function which has been fitted to the peak and volume series using the non-stationary Log-Normal 3 parameter distribution.

Figure 6.3-6: estimated PDF of peak (left) and volume (right) for different year according to the non-stationary LN3

6.3.2.4 Standardization of the non-stationary series

In the non-stationary case, the homogeneity assumption in the distribution of the observations does not hold. Therefore, the model-testing procedure has to be modified. The idea is that the

―time-dependent‖ series is transformed to a ―time-independent‖ series. Coles (2001) suggested a way to standardize the original data which can be applied to homogenous or non-homogenous series. Log-Normal distribution was suited to fit to the series in both stationary and non-stationary analysis. Thus the standardization procedure for this distribution is introduced below.

When random variables 𝑋_𝑡 follow a LN3 distribution with three parameters 𝑘(𝑡) (shape), 𝛼(t) (scale), 𝜉(𝑡) (location) (see Appendix A), we say:

𝑋_𝑡 ~ 𝐿𝑁3(𝑘(𝑡), 𝛼 𝑡 ,𝜉 𝑡 ) (6.3)

Standardized variables 𝑍_𝑡 are then defined by:

𝑍_𝑡 =

−𝑘(𝑡)⁻¹log 1 −𝑘(𝑡) 𝑋_𝑡 − 𝜉(𝑡)

𝛼(𝑡) , 𝑘(𝑡) ≠ 0 𝑋_𝑡 − 𝜉(𝑡)

𝛼(𝑡) , 𝑘(𝑡) = 0

(6.4)

The standardized variables 𝑍_𝑡, hence, follow standard Normal distribution. Therefore, the probability and quantile plots of the observed 𝑧_𝑡 can be made. The above standardization form is also applied for the stationary case. Figure 6.3-7 shows the probability plots and quantile plots in both cases – stationary and non-stationary. In general, it illustrates good fits of the models to the series, although the fit in the tails of the peak series is not at high satisfaction. Furthermore, Figure 6.3-8 shows that the standardization does not change the rank of the data, while it changes the rank of the data in the non-stationary case. This implies a change in the dependence structure of the data (rank correlation) leading to a possible change in the bivariate model which will be applied later.

Figure 6.3-7: Residual diagnostic plots: probability plots (first column) and quantile plots (second columns) for the peak series (first line) and the volume series (second line) for stationary and non-stationary analysis

Figure 6.3-8: Standardized peak (first line) and volume series (second line) for the stationary case (left column) and the non-stationary case (right column); the label in vertical axis indicates the standardized value for the stationary case (first column) and for the non-stationary case (second column)

6.3.2.5 Time-trend in peak and volume series

Figure 6.3-9 illustrates the trend found in both location and scale parameter. Both series show a negative trend in the location parameter and a positive trend in the scale parameter. These trends will be used for extrapolating the flood hazard to ―a near‖ future (see Section 6.6).

Figure 6.3-9: Time-varying parameters of non-stationary models for flood peak (first line) and volume (second line); blue for the time series, green: location parameter; red: scale parameter