# Comparison between Quantile Regression Technique and Generalised Additive Model for Regional Flood Frequency Analysis: A Case Study for Victoria, Australia

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

^{2}(mean: 317.5 km

^{2}and median: 270.5 km

^{2}). Figure 1 shows the distributions of the selected catchments. The annual maximum flood record length ranges 26–62 years (mean 38 years); with 77% of the stations having record lengths of 34–42 years.

_{6,2}); mean annual rainfall (rain); mean annual evapotranspiration (evap); stream density (sden); mainstream slope (S1085); and fraction forest cover (forest). The catchment characteristics data used in this study are summarised in Table 1.

## 3. Methodology

#### 3.1. Quantile Regression Technique (QRT)

_{T}of return period T years) and the catchment characteristics (A

_{1}, A

_{2},…, A

_{n}) is the power-form function in the form [29]:

_{T})), β is the vector of regression coefficients (β =${\mathsf{\alpha}}_{0},$ ${\mathsf{\alpha}}_{1,}$…, ${\mathsf{\alpha}}_{n}$), X is the matrix of the physiographic characteristics or the explanatory variables (X =$\mathrm{log}\left({\mathrm{A}}_{1}\right)$) and e is the matrix of the error (e =$\mathrm{log}\left({\mathsf{\epsilon}}_{0}\right)$). However, if the model error is additive (i.e., Equation (4)), it is not possible to linearise the power-form model by a logarithmic transformation and the model coefficients needs to be estimated by some nonlinear optimisation method.

#### 3.2. GAM

_{T}), E(Y) denotes the expected value, and g(E(Y)) denotes the link function that links the expected value to the predictor variables x

_{1},…, x

_{p}. The terms ${s}_{1}\left({x}_{1}\right)$,…, ${s}_{P}\left({x}_{P}\right)$ denote smooth, nonparametric functions.

#### 3.3. Cluster Analysis

#### 3.4. Validation

## 4. Results

^{2}), p-value and standard error of estimate (SEE). R

^{2}values range from 0.69 to 0.53, respectively for Q

_{2}to Q

_{100}. The R

^{2}values are found to be particularly small for higher ARIs, indicating that the variance explained by catchment characteristics becomes smaller, resulting in higher model error variance of prediction for higher ARI flood estimation. All the R

^{2}values are modest to reasonable, indicating that the prediction equations are generally well-fitting. The SEE ranges from 0.22 to 0.32 for Q

_{2}to Q

_{100}. SEE is found to be lowest in Q

_{2}and highest in Q

_{100}. The predictor variables selected in the final model with the p-statistics value of ≤0.05 are shown in Table 2. From Table 2, the area and I

_{6,2}appear to be the most important variables for estimating Q for log-log linear model. These two variables are common with all the prediction equations. The next most important predictor variable is found as rain which appears in every prediction equation except for Q

_{2}and Q

_{5}. Only for Q

_{2}

_{,}sden is selected whereas rain is absent as predictor variable. For Q

_{5}, both rain and sden are selected as predictor variable. Overall, the prediction equations show consistency in selection of independent variables except for Q

_{2}and Q

_{5}. Table 2 shows the variables of the developed prediction equations.

_{pred}/Q

_{obs}ratio and the median relative error (RE). Figure 2 depicts boxplots of RE values for the log-log linear model for the combined group’s selected test catchments. Figure 2 shows that the median RE values (represented by the black line within a box) match very well with the 0:0 line for ARIs of 5 and 20 years and are relatively good for ARIs of 10 and 50 years. Some underestimation is observed for ARI of 2 years. The underestimation is remarkable for an ARI of 100 years. The ARI of 2 years has the lowest spread in the RE band, which is represented by the total spread of the box. The RE band for ten years ARI is very similar to the RE band for two years ARI. The RE band for 100 years ARI is more than twice than the RE bands for 2 and 10 years. These results show that in terms of RE, 10 years ARI achieve the best results, followed by 2 years ARI. Higher ARI flood quantiles are associated with a higher degree of spread in the RE, which could indicate a higher standard deviation of the estimate. This matches the findings of Haddad and Rahman [31] and Rahman et al. [32]. This is generally the case in RFFA as essentially, we are making predictions beyond the limits of the original data space.

_{pred}/Q

_{obs}ratio values of the selected 114 catchments for the log-log linear model. It is found that the median Q

_{pred}/Q

_{obs}ratio values (represented by the thick black line within a box) are located closer to 1:1 line (the horizontal line in Figure 3), for ARIs of 2, 5,10, 20 and 50 years (the best agreement is for ARI of 20 years). However, for ARI of 100 years, the median Q

_{pred}/Q

_{obs}ratio value is located a short distance below the 1:1 line, and for ARI of 2 years, the median Q

_{pred}/Q

_{obs}ratio value is located a short distance above the 1:1 line. In terms of the spread of the Q

_{pred}/Q

_{obs}ratio values, ARI of 2 years exhibits the lowest spread followed by ARI of 10 years. Furthermore, the spreads of the Q

_{pred}/Q

_{obs}ratio values for 50 and 100 years are very similar, which are remarkably larger than 2 and 10 years.

^{2}), generalised cross-validation (GCV) statistic and p-statistics. The R

^{2}values range from 0.69 to 0.44; particularly, smaller R

^{2}values are found for the higher ARIs indicating a weaker model. The R

^{2}values for lower ARIs seem to be quite reasonable (0.62–0.69). The GCV values vary from 501 to 82, 994 for Q

_{2}to Q

_{100}. The lowest value of GCV is found for Q

_{2}and the highest one is found for Q

_{100}. This indicates that the cross-validation error increases with increasing ARIs.

_{6,2}and evap appear to be the most important variables for estimating flood quantiles using GAM, as these three variables are common in all the prediction equations. The next most important predictor variable is rain, which appears in all the prediction models except for Q

_{2}. Another predictor variable, which is found statistically significant in Q

_{2}, Q

_{5}and Q

_{10}is sden. Overall, Q

_{20}, Q

_{50}and Q

_{100}models show a consistency in the selection of predictor variables (with area, I

_{6,2}and evap). The selected predictor variables are shown in Table 3.

_{pred}/Q

_{obs}ratio values associated with the GAM models for the combined group for the six ARIs. It is found that the median Q

_{pred}/Q

_{obs}ratio values are located very close to 1:1 line, for ARIs of 5 and 10 years, showing the best agreement for ARI of 10 years. However, for all the ARIs, the median Q

_{pred}/Q

_{obs}ratio values are located within a short distance above the 1:1 line except for ARI of 100 years. For this ARI, there is a noticeable overestimation by the GAM model. These results indicate a slight to noticeable overestimation of the predicted flood quantiles for all the ARIs. In terms of the spread of the Q

_{pred}/Q

_{obs}ratio values, ARI of 2 years exhibits the lowest spread, whereas 10 and 20 years of ARIs show similar spread. Furthermore, the spreads of the Q

_{pred}/Q

_{obs}ratio values for 50 and 100 years of ARIs are very similar, which are remarkably larger than 2, 5 and 10 years of ARIs.

^{2}values of the ten different RFFA models. From Table 4, R

^{2}values from GAM models are found to be higher than those from the respective log-log linear models for smaller ARIs. It has also been revealed that GAM models based on clustering groups produce better results, for example, models for smaller ARIs produce higher R

^{2}values. For example, the R

^{2}values of Q

_{2}, Q

_{5}, and Q

_{10}for GAM models in the combined group are 0.83, 0.73, and 0.70, respectively, which are 10%, 8%, and 4% higher than the respective log-log linear models. GAM models, on the other hand, have lower R

^{2}values than respective log-log linear models for higher ARIs (e.g., 0.67, 0.58 and 0.51, which are 1%, 10% and 17% lower than respective log-log linear model). Furthermore, the GAM models of clustering groups produce better results for Q

_{2}with a maximum value of 0.90. Overall, the log-log linear models give better performance for higher ARIs (i.e., 20, 50 and 100 years) and GAM models show better performance for smaller ARIs (i.e., 2, 5 and 10 years).

_{pred}/Q

_{obs}) values are summarised for 5 log-log linear models and 5 GAM models. The median ratio values are important as these are an effective indicator of overestimation or underestimation (i.e., a measure of bias) of the prediction model. The highest Q

_{pred}/Q

_{obs}ratio is 1.16, which is found for the log-log linear model for clustering group A1 for ARI of 50 years, and the lowest median Q

_{pred}/Q

_{obs}ratio is 0.83, which is found for GAM model for clustering group A2 data of 10 years of ARI.

_{pred}/Q

_{obs}ratio values range from 0.90 to 1.09. The smallest and highest median ratio values are found for 100 years of ARI for the log-log linear model of the clustering group B2 and log-log linear model of the clustering group B1, respectively. The overall smallest median Q

_{pred}/Q

_{obs}ratio values for the log-log linear models are found as 0.96, which is for the clustering group B2 and the highest median Q

_{pred}/Q

_{obs}ratio value for log-log linear model is found for the clustering group B1, which is 1.01. The overall median ratio values range from 0.96 to 1.01, which indicate a very small percentage of difference between different groups of the log-log linear models. Most of the median Q

_{pred}/Q

_{obs}ratio values obtained from the log-log linear model are in the range of 0.95 to 0.99, which indicate a slight underestimation in the prediction of flood quantiles. The best result is obtained for 20 and 5 years ARIs for the combined group, with the median ratio value of 1.00. In summary, log-log linear model-based RFFA techniques show a very reasonable and consistent median Q

_{pred}/Q

_{obs}ratio value.

_{pred}/Q

_{obs}ratio values range from 0.83 to 1.16. The smallest and highest median Q

_{pred}/Q

_{obs}ratio values are found for ARIs of 10 years for the clustering group A2 and 50 years ARI for the clustering group A1, respectively. The overall smallest median Q

_{pred}/Q

_{obs}ratio value for GAM is found for clustering group A2 with median Q

_{pred}/Q

_{obs}ratio of 0.97. The overall highest median Q

_{pred}/Q

_{obs}ratio value is found for combined group with median ratio of 1.08. The overall median Q

_{pred}/Q

_{obs}ratio value ranges from 0.98 to 1.08, which indicates that GAM tends to make an overestimation. Moreover, the overall median Q

_{pred}/Q

_{obs}ratio values for the GAM models are higher compared with respective log-log linear models. Most of the median Q

_{pred}/Q

_{obs}ratio values are found above 1.00 for the GAM models, which indicates again an overestimation. Lower values of median Q

_{pred}/Q

_{obs}ratio values for GAM are mostly found for the clustering group A2 that range from 0.83 to 1.14, which are comparatively lower than median Q

_{pred}/Q

_{obs}ratio values of the log-log linear models of the clustering group A2. For clustering group A2, median Q

_{pred}/Q

_{obs}ratio values are lower for the GAM than the log-log linear models for higher ARIs i.e., for 10, 20 and 50 years. However, in most cases, the median Q

_{pred}/Q

_{obs}ratio values of GAM are greater than the respective log-log linear models. Overall, median Q

_{pred}/Q

_{obs}ratio values indicate that the log-log linear models produce better predictions than GAM in the higher ARIs.

## 5. Discussion

## 6. Conclusions

^{2}, median RE and median Q

_{pred}/Q

_{obs}ratio values, it is found that log-log linear models from clustering group A1 outperform the respective GAM models. However, for smaller ARIs (i.e., 2, 5, and 10 years), GAM based RFFA models perform almost similar or better than the log-log linear models. This is as expected, since for smaller floods (i.e., for smaller ARIs), catchments generally tend to behave more non-linearly, i.e., a higher loss value. For higher ARIs (e.g., 50 and 100 years), catchments behave more linearly, hence log-log linear regression models are expected to perform better, which is confirmed in this study. There are predictor variables, which were previously found by [33,34] to be insignificant in RFFA, but are found statistically significant for the GAM models developed here. For example, evap is found statistically significant for most of the GAM models as opposed to previous RFFA studies in Australia. It is found that area, I

_{6,2}and rain are the most significant predictor variables for the log-log linear models. For the GAM models, the most important predictor variables are area, I

_{6,2}, rain and evap.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Boxplots of relative error (RE) values for log-log linear model of combined group (* represents outliers).

**Figure 3.**Boxplots of Q

_{pred}/Q

_{obs}ratio values for log-log linear model of combined group (* represents outliers).

**Figure 5.**Boxplots of Q

_{pred}/Q

_{obs}ratio values for GAM model of combined group (* represents outliers).

Variable | Unit | Notation | Min | Mean | Max | SD |
---|---|---|---|---|---|---|

Catchment area | km^{2} | area | 3 | 317.54 | 997 | 244.65 |

Catchment shape factor | - | SF | 0.281 | 0.79 | 1.4341 | 0.22 |

Mainstream slope | m/km | S1085 | 0.8 | 13.38 | 69.9 | 12.30 |

Stream density | km/km^{2} | sden | 0.52 | 1.53 | 4.25 | 0.53 |

Fraction of catchment covered by forest | - | forest | 0.01 | 0.59 | 1 | 0.35 |

Rainfall intensity (6-h duration and 2-year ARI) | mm/h | I_{6,2} | 24.6 | 34.29 | 46.7 | 5.27 |

Mean annual rainfall | mm | rain | 484.39 | 931.64 | 1760.81 | 319.01 |

Mean annual potential evapotranspiration | mm | evap | 925.9 | 1035.47 | 1155.3 | 42.80 |

Equation | Predictor Variables | Regression Coefficient (β) | Standard Error | Standard Error of Estimate (SEE) | R^{2} | D.F |
---|---|---|---|---|---|---|

log Q_{2} | (constant) | −2.42 | 0.52 | 0.22 | 0.69 | 110 |

log (area) | 0.68 | 0.04 | ||||

log (I_{6,2}) | 1.48 | 0.33 | ||||

log (sden) | 0.39 | 0.15 | ||||

log Q_{5} | (constant) | −1.60 | 0.57 | 0.23 | 0.67 | 109 |

log (area) | 0.68 | 0.05 | ||||

log (I_{6,2}) | 1.74 | 0.41 | ||||

log (rain) | −0.29 | 0.19 | ||||

log (sden) | 0.31 | 0.16 | ||||

log Q_{10} | (constant) | −1.25 | 0.62 | 0.25 | 0.63 | 110 |

log (area) | 0.66 | 0.05 | ||||

log (I_{6,2}) | 2.14 | 0.43 | ||||

log (rain) | −0.53 | 0.20 | ||||

log Q_{20} | (constant) | −1.00 | 0.66 | 0.27 | 0.61 | 110 |

log (area) | 0.66 | 0.05 | ||||

log (I_{6,2}) | 2.30 | 0.46 | ||||

log (rain) | −0.66 | 0.21 | ||||

log Q_{50} | (constant) | −0.79 | 0.73 | 0.30 | 0.57 | 110 |

log (area) | 0.66 | 0.06 | ||||

log (I_{6,2}) | 2.45 | 0.51 | ||||

log (rain) | −0.76 | 0.23 | ||||

log Q_{100} | (constant) | −0.70 | 0.78 | 0.32 | 0.53 | 110 |

log (area) | 0.66 | 0.06 | ||||

log (I_{6,2}) | 2.54 | 0.54 | ||||

log (rain) | −0.81 | 0.25 |

Flood Quantile | Predictor Variables | Deviance Explained (%) | Generalized Cross Validation Statistic (GCV) | R^{2} | F Value |
---|---|---|---|---|---|

Q_{2} | area | 73.70 | 501.61 | 0.69 | 30.199 |

I_{6,2} | 5.37 | ||||

evap | 7.59 | ||||

sden | 6.07 | ||||

Q_{5} | area | 71.3 | 3201.90 | 0.66 | 26.69 |

I_{6,2} | 4.898 | ||||

rain | 3.073 | ||||

evap | 6.278 | ||||

sden | 4.492 | ||||

Q_{10} | area | 67.60 | 8437.80 | 0.62 | 23.46 |

I_{6,2} | 4.67 | ||||

rain | 6.91 | ||||

evap | 5.02 | ||||

sden | 3.15 | ||||

Q_{20} | area | 62.20 | 18974.00 | 0.56 | 17.39 |

I_{6,2} | 4.41 | ||||

rain | 8.95 | ||||

evap | 3.99 | ||||

Q_{50} | area | 56.20 | 45823.00 | 0.50 | 9.96 |

I_{6,2} | 8.56 | ||||

rain | 12.12 | ||||

evap | 3.31 | ||||

Q_{100} | area | 48.40 | 82994.00 | 0.44 | 17.32 |

I_{6,2} | 11.53 | ||||

rain | 10.87 | ||||

evap | 2.46 |

Flood Quantile | Combined Group | Group (A1) | Group (A2) | Group (B1) | Group (B2) | |||||
---|---|---|---|---|---|---|---|---|---|---|

log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | |

Q_{2} | 0.69 | 0.69 | 0.74 | 0.83 | 0.69 | 0.75 | 0.78 | 0.90 | 0.65 | 0.712 |

Q_{5} | 0.67 | 0.66 | 0.72 | 0.79 | 0.55 | 0.676 | 0.74 | 0.83 | 0.57 | 0.626 |

Q_{10} | 0.63 | 0.62 | 0.70 | 0.73 | 0.48 | 0.554 | 0.71 | 0.78 | 0.48 | 0.506 |

Q_{20} | 0.61 | 0.56 | 0.68 | 0.67 | 0.43 | 0.506 | 0.69 | 0.71 | 0.42 | 0.456 |

Q_{50} | 0.57 | 0.50 | 0.65 | 0.58 | 0.32 | 0.437 | 0.65 | 0.60 | 0.39 | 0.322 |

Q_{100} | 0.53 | 0.44 | 0.62 | 0.51 | 0.27 | 0.36 | 0.62 | 0.55 | 0.32 | 0.30 |

Overall | 0.62 | 0.58 | 0.69 | 0.69 | 0.46 | 0.55 | 0.70 | 0.73 | 0.47 | 0.49 |

**Table 5.**Median RE values (%) for the GAM and log-log linear model based RFFA techniques for ten cases.

Flood Quantile | Combined Group | Group (A1) | Group (A2) | Group (B1) | Group (B2) | |||||
---|---|---|---|---|---|---|---|---|---|---|

log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | |

Q_{2} | 18.73 | 34.81 | 29.56 | 22.52 | 23.10 | 39.31 | 30.33 | 16.80 | 25.82 | 33.24 |

Q_{5} | 32.88 | 33.88 | 28.60 | 33.10 | 34.69 | 41.46 | 28.20 | 28.92 | 31.97 | 41.11 |

Q_{10} | 19.36 | 33.75 | 27.47 | 31.96 | 40.54 | 40.29 | 27.37 | 34.46 | 33.05 | 38.17 |

Q_{20} | 34.51 | 34.05 | 30.74 | 39.53 | 43.02 | 42.35 | 29.37 | 42.47 | 36.69 | 45.82 |

Q_{50} | 40.41 | 42.67 | 33.25 | 40.12 | 53.10 | 49.59 | 37.42 | 42.08 | 39.29 | 31.38 |

Q_{100} | 40.99 | 49.09 | 37.05 | 53.38 | 59.94 | 49.37 | 37.00 | 45.90 | 42.63 | 39.04 |

Overall | 31.15 | 38.04 | 31.11 | 36.77 | 42.40 | 43.73 | 31.61 | 35.10 | 34.91 | 38.13 |

**Table 6.**Median Q

_{pred}/Q

_{obs}ratio values for the GAM and log-log linear model based RFFA techniques for 10 cases.

Flood Quantile | Combined | Group (A1) | Group (A2) | Group (B1) | Group (B2) | |||||
---|---|---|---|---|---|---|---|---|---|---|

log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | log-log Linear Model | GAM | |

Q_{2} | 1.03 | 1.07 | 1.04 | 1.01 | 1.00 | 1.13 | 1.01 | 1.05 | 1.04 | 1.10 |

Q_{5} | 1.00 | 1.02 | 0.95 | 1.03 | 0.99 | 1.04 | 0.98 | 1.00 | 1.03 | 0.95 |

Q_{10} | 0.97 | 1.04 | 0.94 | 1.06 | 0.98 | 0.83 | 0.96 | 1.02 | 0.92 | 1.04 |

Q_{20} | 1.00 | 1.12 | 0.97 | 1.10 | 1.01 | 0.84 | 1.01 | 1.06 | 0.94 | 0.98 |

Q_{50} | 0.98 | 1.12 | 1.02 | 1.16 | 0.95 | 0.86 | 1.05 | 1.14 | 0.94 | 0.98 |

Q_{100} | 0.94 | 1.12 | 1.02 | 1.12 | 0.95 | 1.14 | 1.09 | 1.13 | 0.90 | 1.01 |

Overall | 0.99 | 1.08 | 0.99 | 1.08 | 0.98 | 0.97 | 1.01 | 1.07 | 0.96 | 1.01 |

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**MDPI and ACS Style**

Noor, F.; Laz, O.U.; Haddad, K.; Alim, M.A.; Rahman, A.
Comparison between Quantile Regression Technique and Generalised Additive Model for Regional Flood Frequency Analysis: A Case Study for Victoria, Australia. *Water* **2022**, *14*, 3627.
https://doi.org/10.3390/w14223627

**AMA Style**

Noor F, Laz OU, Haddad K, Alim MA, Rahman A.
Comparison between Quantile Regression Technique and Generalised Additive Model for Regional Flood Frequency Analysis: A Case Study for Victoria, Australia. *Water*. 2022; 14(22):3627.
https://doi.org/10.3390/w14223627

**Chicago/Turabian Style**

Noor, Farhana, Orpita U. Laz, Khaled Haddad, Mohammad A. Alim, and Ataur Rahman.
2022. "Comparison between Quantile Regression Technique and Generalised Additive Model for Regional Flood Frequency Analysis: A Case Study for Victoria, Australia" *Water* 14, no. 22: 3627.
https://doi.org/10.3390/w14223627