# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 5 Quantitative and Comparative Assessment
The GMM is a parametric technique that represents the PDF as a combination of weighted Gaussian components. It estimates the parameters using methods such as expectation-maximization, providing an effective approach for clustering and density estimation. While the GMM performs well when data adhere to Gaussian assumptions, it may face challenges with non-Gaussian distributions. In this study, a three-component Gaussian model was employed, a configuration commonly observed in many applications [33; 34; 35]. The QHBS combines histogram-based density estimation with smoothing techniques. It divides the data into segments, or bins, estimates the density within each bin, and then applies smoothing to reduce abrupt changes in the data distribution. For this study, the number of bins was set to 20 to balance detail and smoothness [36]. To evaluate the performance of our proposed approach, we utilized six metrics: the **Wasserstein Distance**, the **Bhattacharyya Distance**, the **Correlation Coefficient**, the **Kullback-Leibler (KL) Divergence**, the **L1 Distance**, and the **L2 Distance**. The **Wasserstein Distance** evaluates the cost of transforming one probability distribution into another by examining variations in their distributions, making it responsive to the distribution's shape and useful for assessing alignment. A score of zero for the Wasserstein Distance signifies similarity between distributions. The **Bhattacharyya Distance** measures the overlap between two distributions, helping us understand how well two methods approximate densities by directly reflecting their probabilistic similarity. A smaller Bhattacharyya Distance suggests a better approximation quality, with zero indicating a perfect overlap. The **Correlation Coefficient** assesses the linear relationship between distributions, providing a clear interpretation of their alignment. A Correlation Coefficient of 1 signifies perfect linear correlation, whereas a value of 0 indicates no linear relationship between them. The **KL Divergence** measures the difference between two distributions, revealing the drawbacks of approximating one using the other. It is useful for identifying subtle variations in the distribution of probability mass. A lower KL Divergence is preferred, as it signifies greater similarity between distributions, with zero indicating identical distributions. The **L1 Distance** (also known as Manhattan or Taxicab Distance) calculates the sum of absolute differences between corresponding elements in two vectors or distributions. It is particularly effective in high-dimensional spaces and provides an intuitive measure of dissimilarity. A smaller L1 Distance indicates greater similarity between distributions, with zero representing identical distributions. The **L2 Distance** (commonly referred to as Euclidean Distance) measures the straight-line distance between two points in a multidimensional space using the Pythagorean theorem. It is widely used for numerical data and provides a geometric interpretation of similarity. Like L1, an L2 Distance of zero indicates identical distributions, while larger values signify greater dissimilarity. These measures collectively provide a comprehensive foundation for assessing the accuracy, reliability, and alignment of density estimation methods across various dimensions and scenarios.