We present a comprehensive analysis of the physical parameters and relationships of umbral dots (UDs), which assists in our understanding of the physical properties of the Sun. This study is based on a detailed analysis of UDs detected in 12 umbras belonging to 10 different sunspots using high-resolution data recorded by the Goode Solar Telescope at the Big Bear Solar Observatory. We obtained the physical parameters (brightness, diameter, eccentricity, lifetime, dynamic velocity) of each UD and calculated correlation coefficients using linear and nonlinear approaches to reveal the relationships between these parameters. We found that: i) the diameter of UDs are varying between 92.2 km to 246.5 km, the eccentricity varies between 0.02 to 0.65, the lifetime of UDs vary from 0.75 to 120.00 min and the dynamic velocities are varying from 0.01 km/s to 3.80 km/s. ii) The intensity-diameter and diameter-eccentricity show the highest degree of correlation, while the lowest linear correlation was obtained for the diameter-lifetime and the lowest nonlinear correlation was obtained for the eccentricity-lifetime relationships. iii) In general the nonlinear correlation coefficients are higher than the linear correlation without any exception. iv) The linear and nonlinear correlation coefficients are very close to each other in the case of diameter-eccentricity relation. v) While the average diameter, intensity, A. Kilcik