Density plots, particularly those generated using Kernel Density Estimation (KDE), can extend beyond the observed minimum and maximum values of your data because of the fundamental properties of the kernel function used in the estimation process.
Understanding Density Plots and KDE
A density plot visually represents the distribution of a dataset. Instead of using discrete bins like a histogram, it uses a smooth curve to show where data points are concentrated. Kernel Density Estimation (KDE) is a common method for creating these smooth density curves. It works by placing a "kernel" (a small, smooth function, like a bump) over each data point and then summing these kernels to get the overall density estimate.
The Kernel's Role Beyond Data Boundaries
The primary reason the density plot can extend past your data's range is directly related to the kernel function itself. As stated in the reference:
The density can extend over data boundaries because the kernel used is positive over the entire real axis.
Most standard kernel functions (like the Gaussian or Epanechnikov kernels) are defined and positive for all possible real numbers, even values far away from any data point. When you sum up these positive kernel contributions across all data points, the resulting density estimate remains positive over a broader range than the data itself occupies.
Imagine placing a small, positive bump centered at each data point. Even at locations slightly below the minimum data value or slightly above the maximum, these bumps still have a tiny positive height. When you add up these tiny positive heights from all the data points, the total estimated density at those "out-of-range" locations is still greater than zero, causing the curve to extend outwards.
The reference also notes:
If you change the kernel to rectangular or triangular the density estimate will reach zero at some distant points but again it won't respect the data minimum and maximum.
Even with kernels that become zero beyond a certain point (like rectangular or triangular ones), the resulting smooth density function doesn't necessarily drop precisely to zero exactly at the data's minimum or maximum boundary. The smoothing process based on the kernel function inherently creates a distribution that is continuous and can extend beyond the strict data limits.
Practical Implications
- Visual Representation: Density plots provide a smoothed estimate of the underlying probability distribution, which might extend beyond the observed sample due to inherent variability and the smoothing technique.
- Mathematical Property: This extension is a mathematical consequence of using kernels that are positive or non-zero beyond the data range.
- Not an Error: It is generally not an error in the plot but a characteristic of the KDE method.
In essence, the kernel smoothing process extrapolates slightly based on the data points, resulting in a curve that smoothly transitions towards zero outside the observed data range, rather than abruptly stopping.
