Description of STKDE+

Spatial-Temporal Kernel Density Estimation PLUS (STKDE+) stems from the KDE+ method (Bíl et al., 2013) determining locations (parts of lines) with a statistically significant concentration of points (e. g. collisions, traffic crashes). These locations, which we refer to as hotspots, are significantly different from a pattern arising from random uniform distribution. The presence of hotspots therefore shows the least likely arrangement of points along the line section. The STKDE+ method extends the KDE+ framework by including the temporal dimension through a moving window approach. As a result, the hotspots are identified not only in space but also in time. It allows a user to observe the temporal evolution of hotspots.

Two feature classes are expected as inputs:
(1) Lines representing sections of the network. They should be divided by intersections.
(2) Points along lines containing a Date field specifying the date of occurrence of the event.

The outputs are:
(1) spatiotemporal (STKDE+) graphs of points and resulting KDE+ hotspots in space (horizontal axis) and time (vertical axis)
(2) a table with the positions of located points along lines
(3) the resulting feature class containing the KDE+ hotspots.

What is depicted in the STKDE+ graphs?
The black dots indicate input points (collisions, traffic crashes) in space (horizontal axis) and time (vertical axis) on a single line section. The gray rectangles represent the KDE+ hotspots while the red rectangles highlight the most important hotspots which are less likely, in comparison with the hotspots depicted in gray, false alarms. In the temporal dimension, the hotspots remain at the same location as long as the input data (i. e. the points within the sliding window) remains unchanged.

How can I use the STKDE+ graphs?
The STKDE+ graphs help with investigation of the spatiotemporal pattern of points and hotspots identified by means of the KDE+ method. Three elementary forms of hotspots, in terms of their temporal behavior, reflect the following safety conditions along roads:
- hotspot disappearance (e.g., a safety measure was successfully applied at the place of the hotspot)
- hotspot emergence (safety conditions deteriorated in comparison with the previous period)
- hotspot stability (a dangerous place with no or unsuccessfully applied safety measures).

The feature class containing all the hotspots is one of the additional outputs of the STKDE+ analysis for subsequent work in GIS.

The theoretical background of the STKDE+ method can be found in:
Bíl, M., Andrášik, R., Sedoník, J., 2019. A detailed spatiotemporal analysis of traffic crash hotspots. Applied Geography 107, 82–90.

License agreement
STKDE+ software is licensed:
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Creative Commons License
The STKDE+ software by CDV - Transport Research Centre is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. Based on a work at

If you make use of this software in your academic work, please cite:
STKDE+ toolbox for ArcGIS can be downloaded here.