## 3 Extreme Value Analysis Considering Seasonality
### Simplified Approach for Return Period Correlated Values
Figure 3: Regression and residual analysis by using (**a**) nonlinear (Equation (9)) and (**b**) linear (Equation (7)) models for October. Residuals using (**c**) Equation (9) and (**d**) Equation (7). As mentioned before, the regression is first performed by using only the historical observed data, shown by circles in Figure 3a,b. It can be observed that the regression line (solid line) in both cases (i.e., Equations (9) and (7)) predicts increasing \(V_{w}\) for increasing _Hs_. This is consistent with what is reported in the literature. The 95% confidence intervals, which are measures of uncertainty for mean and future values of \(V_{w}\)[33; 34], are also shown in Figure 3a,b (inner and outer dashed lines for mean and future values, respectively). Analysis of residuals versus predicted values indicates that the nonlinear model leads to a better scattering of the data along the horizontal axis (i.e., the model is uncorrelated) and better symmetry in the vertical axis (i.e., the zero mean assumption), indicating that the nonlinear model is better for the values observed in October. Both models roughly exhibit a normal distribution behavior in the residuals, as can be observed in Figure 4a for the model in logarithmic space and 4b for the linear model, where the normal probability papers are plotted using the cumulative probability given by \(i/(n+1)\), where \(n\) is the number of samples (i.e., the number of residuals) and \(i\) corresponds to the number of a datapoint (a residual) in ascending order. Figure 5 is analogous to Figure 3, except that February instead of October is considered (i.e., around cold front season). Opposite to what occurs in the regression analysis for October, the analysis for February is better represented using the linear model (Figure 5b), rather than by using the linearized model in the natural logarithmic space (Figure 5a); this can be inspected in the residuals and normal probability papers in Figure 5c,d and Figure 4c,d, for Equations (9) and (7), respectively. It is found in this study that correlation of significant weight height and wind velocity can be better represented (in terms of classic linear regression) by linear (Equation (7)) or power (Equation (9)) expressions, depending on the month of the year (e.g., hurricane season, cold front season or others). This is not clearly stated in other studies. Nevertheless, it can be observed that both the linear or the nonlinear regression analysis lead to a fairly normal behavior of the residuals, and to a relatively uniform distribution of the variance around the mean, which could be associated with a homoscedastic behavior [33], although this varies depending on the month and type of regression considered (i.e., on using Equation (7) or Equation (9)). The error variance, \(\sigma_{\text{e}}{}^{2}\) is an indication of the uncertainty in the regression, and its squared root is known as the root mean squared error (RMSE). Overall, the residual analyses for all months indicate that the usual assumptions in the regression approach are met [34], and that Equation (7) and Equation (9) can be suitable  
Figure 4: Normal probability papers for residuals. October: (**a**) nonlinear model and (**b**) linear model. alternatives. Although Equation (8) is frequently encountered in codified design and the literature [26] as indicated above, functional forms like those of Equation (7) are also found, as in the standard of the North Atlantic Treaty Organization, Standardized Wave and Wind Environments and Shipboard Reporting of Sea Conditions (NATO-STANAG 4194, see [26]), which can be expressed as:  
\[V_{w}=b+\alpha Hs \tag{10}\]  
where the nomenclature used in the present study applies (for the NATO-STANAG 4194, \(b=0\) and \(\alpha=3.2\)). Equations with the functional form of Equations (8) and (10) are used for open sea and coastal waters; those employed for coastal waters are also linked to the fetch as an important variable [26]. In the present study, the fetch is not considered since the buoy used is located in open sea.