## 2 Methodology
### An Overview of Kernel Density Estimation (KDE)  
The KDE method is an efficient way to address non-parametric data for decision making and to understand data distribution [18]. The non-parametric estimation technique is a fully flexible estimation technique where the true nature of the distribution can be ascertained, and is therefore a flexible estimator. Since the PDF is derived from the data sample, the structure of the data is not predetermined. The details concerning the data structure can easily be disclosed since the non-parametric technique does not presume a distribution pattern. This is also true for each data point, represented by a smooth curve referred to as the kernel function. The kernel function \(f(v)\) of a variable \(v\) is given as follows:  
\[f(v)=\frac{1}{m}{\sum_{m=1}^{m}}K\bigg{(}\frac{v-X_{n}}{h}\bigg{)} \tag{2}\]  
where \(m\) is the sample size, \(h\) is the width commonly called the bandwidth, and \(X_{n}\) is the \(n\)th observation of the data. When all of these curves are integrated with the densities of all of the data points, then it is referred to as the kernel density estimate. The seasonal median rain height for all of the stations was modeled using the four kernels identified below. The bandwidth is optimized by giving a tradeoff between the fluctuation in the data and the smoothness to capture the best window width. This is provided by various types of kernel functions, as provided in Equations (4), (6), (8) and (10). A low value of optimized bandwidth indicates better density estimation. 1. Gaussian Kernel  
The Gaussian kernel gives specific dues which offer very smooth density estimate support for a given set of data. They are most useful if the applied data are normally distributed. The Gaussian non-parametric kernel density estimate is given as follows [29]:  
\[f(v)=\frac{1}{\sqrt{2\pi}}\text{exp}\bigg{(}-\frac{v^{2}}{2}\bigg{)} \tag{3}\]  
The optimal bandwidth \(h\) is expressed as follows:  
\[h=\frac{1.06\times\sigma}{\sqrt[5]{m}} \tag{4}\]  
where \(\sigma\) is the standard deviation of the data. The Gaussian kernel is mostly used for several applications due to its good smoothing capability, though it can over-smooth sharp edges and peaks due to its infinite support. 2. Epanechnikov Kernel  
This kernel was developed by [PERSON] [30] with good performance in terms of bias-variance tradeoff due to its fixed bandwidth and minimization of the mean integrated square error (MISE). The kernel function is given by the Epanechnikov equation as follows [30]:  
\[f(v)=\frac{3}{4}(1-v^{2}),\hskip 28.452756 ptfor\ |v|\leq 1 \tag{5}\]  
The optimal bandwidth is expressed as follows:  
\[h=\frac{2.34\times\sigma}{\sqrt[5]{m}} \tag{6}\]  
3. Triangular Kernel  
This is a finite kernel that has a simple structure, and it is used in linear approximation, which works between the extremes of continuity and local flexibility. The triangular kernel density function \(f(v)\) and \(h\), which gives the optimal bandwidth, are presented in Equations (7) and (8) [31]. \[f(v)=1-|v|,\quad\quad for\ |v|\leq 1 \tag{7}\]  
\[h=\frac{1.06\times\sigma}{\sqrt[3]{m}} \tag{8}\]  
It is easier to apply and faster than the Gaussian kernel but not as continuous as Gaussian, and it has biases near boundary values. 4. Rectangular Kernel  
This is the fastest and simplest kernel density estimation method commonly used for processing large datasets. Although it is has poor smoothing abilities and it is less accurate due to its simple structure, it is preferred for real-time fast data processing. The rectangular density function and the window width are presented in Equations (9) and (10), respectively [31]. \[f(v)=\frac{1}{2},\quad\quad\quad for\ |v|\leq 1 \tag{9}\]  
\[h=\frac{2.34\times\sigma}{\sqrt[3]{m}} \tag{10}\]  
The kernel functions above were applied to the rain height dataset to construct the estimated PDF.