Stability of hotspots

The Stability of Hotspots (SoH) is a metric to determine the stability between two different hot spot analyses.

In its downward property (from parent to child, injective) it is defined as: $$ SoH^\downarrow = \frac{ParentsWithChildNodes}{Parents} = \frac{|Parents \cap Children|}{|Parents|} $$

And for its upward property (from child to parent, surjective): $$ SoH^\uparrow = \frac{ChildrenWithParent}{Children} = 1 - \frac{|Children - Parents|}{|Children|} $$

where:

• $ParentsWithChildNodes$ is the number of parents that have at least one child, $Parents$ is the total number of parents,
• $ChildrenWithParent$ is the number of children and $Children$ as the total number of children.

The SoH is defined for a range between 0 and 1, where 1 represents a perfectly stable transformation while 0 would be a transformation with no stability at all.

If $|Children|=0$ or $|Parents|=0$ SoH is 0.

Related demos:

• stability of hotspots
• Urban Heat Islands