# Kernel Density Derivative Estimation of Euler Solutions
## 2 Materials and Methods
### Multivariate KDE of the Euler Solution Datasets  
Multivariate KDE is considered a nonparametric approach for calculating probability density distributions of data instances. Nonparametric approaches have the advantages of asymptotic unbiasedness, square consistency, and uniform convergence. Multivariate KDE is a trendy tool for analyzing low-dimensional data instances [75; 76]. Letting \(\mathbf{X}=(\mathbf{X}_{1},\ \ldots\,\mathbf{X}_{i},\ \ldots\,\mathbf{X}_{n})\) be a \(d\)-dimensional random sample drawn from a density function \(F\), the general form of a multivariate kernel density estimator at any given estimation point \(\mathbf{x}=(x_{1},\cdots,\mathbf{x}_{d})\) can be defined as follows [77]:  
\[\overline{F}(\mathbf{x};\mathbf{H})=\sum_{i=1}^{n}K_{\mathbf{H}}(\mathbf{x}-\mathbf{X}_{i}) \tag{5}\]  
where \(\mathbf{H}\) is a symmetric positive definite \(d\times d\) diagonal bandwidth matrix; the multivariate kernel function \(K_{\mathbf{H}}(\mathbf{x})=n^{-1}|\mathbf{H}|^{-1/2}K\big{(}\mathbf{H}^{-1/2}\mathbf{x}\big{)}\) is the inner product of the scaled kernel, which is a rescaling of the unscaled kernel \(K\) by the bandwidth matrix \(\mathbf{H}\)[62]; \(|\mathbf{H}|\) is the matrix determinant of \(\mathbf{H}\) and \(\mathbf{H}^{-1/2}\) is the inverse square root of its matrix. \(\mathbf{X}_{i}=(X_{1i},\cdots,\mathbf{X}_{id})\) is the various combinations of the elements of Euler solutions, such as \(x_{o}\), \((x_{o},z_{o})\), \((x_{o},z_{o},N)\), and \((x_{o},y_{o},z_{o},N)\). Hence, in this paper, \(1\leq d\leq 4\).  
Traditional discrimination techniques cannot evaluate the overall quality of the solutions or determine how dense/compact the clusters are [65]. There is no difference between sparse and dense distributions of Euler solutions in a scatter plot drawn by traditional discrimination techniques. Crucial information about the potential density structure fails to be recovered using density estimation results, such as saddles, peaks, and valleys. In contrast, the density derivative estimation effectively overcomes this problem. Therefore, taking the partial derivative of Equation (5) concerning \(\overline{F}\), the yielded kernel density derivative estimator is:  
\[\
abla\overline{F}(\mathbf{x};\mathbf{H})=\sum_{i=1}^{n}\
abla K_{\mathbf{H}}(\mathbf{x}-\bm {X}_{i}) \tag{6}\]  
where the Nabla Operator \(\
abla\) is used to find higher dimensional derivatives [76] and \(\
abla K_{\mathbf{H}}(\mathbf{x})=n^{-1}|\mathbf{H}|^{-1/2}\
abla K\big{(}\mathbf{H}^{-1/2} \mathbf{x}\big{)}\)[62]. Because \(K\) is not very susceptible to the shape of the estimator, different kernel functions can be applied to yield good results [78]. The most widely used kernel is the zero-mean unit variance Gaussian [79]. Therefore, a Gaussian kernel \(K(\mathbf{x})=(2\pi)^{-1/2}e^{-x^{2}/2}\) is used to estimate a density function in this paper.  
For isolating complex geological sources, we are required to evaluate \(\
abla\overline{F}(\mathbf{x};\mathbf{H})\) on a grid \(\mathbf{\chi}\) of estimation points, which are expressed as \(\mathbf{\chi}_{1,\cdots,k_{k},\cdots,I_{d}}\), for dimension indices \(I_{k}=1,\cdots,M_{k},1\leq k\leq d\) and \(\mathbf{M}=\prod_{k=1}^{d}M_{k}\). Then supposing \(\mathbf{\chi}\) along the \(k\)th direction to be an equally spaced grid and covering precisely the input samples, we define the lower and upper bounds of the grid using the marginal sample minima \(\{\mathbf{a}_{k}\}\) and maxima \(\{\mathbf{b}_{k}\}\), respectively. The \(k\)th bandwidth is denoted by  
\[\mathbf{H}_{k}=(\mathbf{b}_{k}-\mathbf{a}_{k})/(M_{k}-1) \tag{7}\]