# Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis +
## 4 Density Minimum Volume Set Estimation for the Analysis of Ground Impact Footprints
### Statistical Estimation of a Density Minimum Volume Set with MIS  
In this article, we want to estimate the \(t\)-level set \(\mathbf{L}(t)\) of density \(g\) for a given probability \(\alpha\). The estimation principle is based on the following steps:  
1. Propose an estimate \(\hat{g}\) of \(g\) from a given set of samples \((Z_{1}=\mathcal{M}(U_{1}),...,Z_{N}=\mathcal{M}(U_{N})\).
2. Estimate the threshold \(\hat{t}=\epsilon^{-1}(\alpha)\) with a simple binary search and determine the level set \[\mathbf{L}(\hat{t})=\left\{z\in\mathbb{R}^{2}:\hat{g}(z)\geq\hat{t}\right\}\] (10)  
This estimator \(\mathbf{L}(\hat{t})\) is a plug-in estimator of a minimum volume set [16]. To apply this 2-step procedure, it is necessary first to estimate the unknown density \(g\). This can be done with classical Monte Carlo from samples \(Z_{i}\), \(i=1,\ldots,N\), distributed with the unknown density \(g\), but also with importance sampling. The principle of importance sampling is to modify the sampling distribution of \(Z_{i}\), in order to improve the accuracy of the estimation of \(g\) on some part of its support. A comparison between Monte Carlo and classical importance sampling estimates of \(g\) is indeed performed in [17]. Depending on the value of \(\alpha\), a trade-off should be made. For this purpose, we consider in this article multiple importance sampling [18] that behaves well in the heart of the distribution \(g\), because Monte Carlo and importance sampling samples can be taken into account in the estimation of the density \(g\). Moreover, the estimation of \(\hat{t}\) with binary search is often intractable and cannot be applied in practice, since it requires the estimation of integrals over a multidimensional domain. To avoid this difficulty, one also considers another plug-in estimator of \(t\) described in [19], based on density quantile.  
To estimate the density level set \(\mathbf{L}(\hat{t})\) with multiple importance sampling, the following computational steps are considered in this article:  
1. Generate a set of \(N\) independent and identically (iid) distributed samples \((U_{11},...,U_{1N})\) of density \(f\), and apply the function \(\mathcal{M}\) on these samples to determine a set of samples \((Z_{11}=\mathcal{M}(U_{11}),...,Z_{1N}=\mathcal{M}(U_{1N}))\).
2. Estimate the output density \(\hat{g}^{1}\) from the samples \((Z_{11},...,Z_{1N})\) with multivariate kernel density estimate [20].
3. Estimate the density \(h\) of the samples \(\{U_{1i}|\hat{g}^{1}(Z_{1i})<\gamma\}\) for \(i=1,\ldots,N\) where \(\gamma\) is set by the user.
4. Generate a set of \(N\) iid samples \((U_{21},...,U_{2N})\) from density \(h\), and applies the function \(\mathcal{M}\) on these samples to determine a set of samples \((Z_{21},...,Z_{2N})\)[20].
5. Estimate the density \(\hat{g}^{MIS}\) from the samples \((Z_{11},...,Z_{1N})\) and \((Z_{21},...,Z_{2N})\) with weighted multivariate kernel density estimate. The weight of each \(Z_{ij}\) is \(\frac{f(U_{ij})}{\frac{1}{2}(f(U_{ij}+h(U_{ij}))})\).  
6. Estimate the threshold \(\hat{t}^{MIS}\) as the \((1-\alpha)\)-quantile of the weighted samples \((Z_{11},...,Z_{1N},Z_{21},...,Z_{2N})\).
7. The level set with MIS is then estimated with \[\hat{\mathbf{L}}^{MIS}(t)=\left\{\hat{x}\in\mathbb{R}^{2}:\hat{g}^{MIS}(z)\geq \hat{t}^{MIS}\right\}\] (11)  
The choice of \(\gamma\) can be made with a quantile of the samples \(\hat{g}^{1}(Z_{1i}),...,\hat{g}^{1}(Z_{1N})\). In this article, \(\gamma\) is quantile of level 0.1 as, from our experience, it corresponds to a good trade-off between Monte Carlo samples and extreme samples.