# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 2 A Stochastic Risk Analysis for Extreme Wind Speed Data under the IRD Model  
For the purpose of developing a stochastic risk analysis for extreme wind speed data, a stochastic approach established in some literature on extremal processes for safety applications [25] is adopted, also referring to specific application of such extremal processes to wind data [28; 29; 30]. Let a threshold value \(z\) be assigned to the EWS values for safety reasons. If \(z\) is large enough, the occurrences of wind speed values which exceed such thresholds (or \"exceedances\" in the following), are rare events which, under general hypotheses, can be described by the well-known Poisson process, i.e., by the probability law given, as well known, by the following [25; 31]:  
\[p(k,t)\equiv P[\text{N}(t)=k]=e^{-\phi t}\frac{\left(\phi t\right)^{k}}{k!},\ k=0,1,\ldots \tag{5}\]  
where  
* N(\(t\)) is a process which counts the number of EWS values of exceeding \(z\) in the time interval (0,\(t\));
* \(p(k,t)\) is the probability that the number of exceedances in (0,\(t\)), N(\(t\)), and attains the nonnegative integer value \(k\);
* \(\phi>0\) is the mean frequency of the process N(\(t\)), i.e., the expected number of exceedances, over the threshold value \(z\) for a unit time or \(\phi=E[\text{N}(t)]/t\). Indeed, mean (or expected) value and variance of N(\(t\)) are expressed, as well known [25; 31], by the following:  
\[E[\text{N}(t)]=Var[\text{N}(t)]=\phi t \tag{6}\]  
Moreover, a stronger result holds, almost surely (\(a.s.\)) in terms of the so called \"strong law\" of the renewal processes [25; 31]:  
\[\phi=\lim_{t\to 0}\frac{E[\text{N}(t)]}{t}\ \ (a.s.) \tag{7}\]  
The following developments, leading to the analytical expression of the RI, are discussed with more details in Appendix A. Let \(u(t,z)\) denote the probability that \(z\) is never exceeded over (0,\(t\)), i.e., a \"safety\" index. Let the random variable \"EWS amplitude occurring at time \(T_{k}\) of the \(k\)th exceedance\" (\(k=1,2,\ldots\) ) be denoted as \(Z_{k}\). If the RV \(Z_{k}\) is (as typically assumed in the literature [18]) assumed to be statistically independent and identically distributed with the common, time-independent CDF,  
\[F(x)=P(Z_{k}\leq x),\ \forall\ k=1,2,\ldots,n \tag{8}\]  
then, the following compact expression can be obtained for the above function \(u(t,z)\), as shown in Appendix A:  
\[u(t,z)=e^{-\varphi t[1-F(z)]} \tag{9}\]  
Therefore, with the RI defined--as above reported--as the probability of the event that the EWS amplitude \(z\) is exceeded at least once, since such event is complementary of the event \"\(z\) is never exceeded over (0,\(t\))\", the above defined RI is expressed in terms of the function \(u(t,z)\) in (9) by the following:  
\[R(t,z)=1-u(t,z)=1-e^{-\varphi t[1-F(z)]} \tag{10}\]  
By its very definition and clearly shown in its expression, such RI is intrinsically dynamic, being dependent on the chosen value of time horizon. If the above limit value \(z\) is chosen as the maximum value of the EWS that the structure can resist, then the RI is an efficient measure of the failure probability the structure. As discussed in [30], this kind of measure appears to be very significant for predicting the risk values associated with gust occurrence and their effects on real wind power systems. In this framework, this paper discusses the problem of the statistical estimation of wind probabilistic distribution with the primary purpose of an efficient assessment of wind tower reliability and safety and of wind power generation statistics. This is performed by assuming an IRD model as the reference model of the CDF \(F(z)\) in (10) and then by performing an estimation of the RI based upon the Poisson exceedances process and the above IRD model. Let us first briefly illustrate the use of the IRD model for the EWS statistics and then the expression of the RI in (10) in terms of such IRD model.