What is the principle of kernel density estimation for line data?

[ChouJay]

ArcMap provides us a Kernel Density Estimation tool for both point data and line data. The principle of kernel density estimation for point features is easy to understand, it is the summation of densities of all points within a bandwidth around the focus point (Silverman (p.76)):

But, how to understand the principle of Kernel Density Estimation for line data? The official website of ArcMap says that "the use of kernel function for lines is adapted from kernel function for point densities" (How Kernel Density works—Help | ArcGIS Desktop ), but there is no information about how this adaption is conducted. Could somebody help me understand the principle of kernel density estimation for line data? Thank you!

[DanPatterson_Retired]

Kernel density estimation - Wikipedia 

And see my links here to Bill Huber's discussions on this topic

https://community.esri.com/thread/175729 

[ChouJay]

Dear Dan Patterson, 

Thank you very much for your reply. Still, the information you provided is about kernel density for points, do you have any idea of the principle of kernel density estimation for line data? Thank you very much!

[DanPatterson_Retired]

Differences between point, line, and kernel density—Help | ArcGIS Desktop 

and

How Line Density works—Help | ArcGIS Desktop 

which describes how density is calculated for lines in more detail

[ChouJay]

There is a big difference between line density and kernel density estimation for lines. Line density in ArcGIS is easy to understand, but there is no information about kernel density estimation for line features. Do you know how kernel density estimations for lines (NOT line density) are calculated in ArcGIS? Thank you very much!

[curtvprice]

The help page you screen-shot for us seems to describe it pretty well. It looks like they do the f-hat calculation for every point along the line. You could verify this by converting your lines to points (probably densifying first) and trying to reproduce the line result.

How Kernel Density works—Help | ArcGIS for Desktop (section: Kernel Density for line features)

If you need more, I recommend calling Esri support and logging an incident.

Another approach I have used (though this doesn't guarantee an instant response, but has the [huge] benefit of helping the entire user community) is to provide feedback on the Esri help page for the tool (there is a link at the bottom of the doc page) and asking for more detail in the help. These notes go directly to the help page author! This help page has not been modified for years so it is reasonable to ask for an update.

[ShabbirRaza]

Can you explain how its generates the surface?