Distance for circular histograms

[mnb96]

Hello,
I know that several definitions of distances between pdf's were definied, for example: Bhattacharyya distance.
I would like to know if there exist some definitions of distance between "circular" histograms.

A practical example would emerge when considering histograms deriving from sets of samples of wind-direction angle of the kind $$\Theta_n = \{ \theta(1),\ldots, \theta(n) \}$$.

A "good" definition should properly take into account the periodicity, hence to classify the following two histograms as very close to each other:

$$H_{1}(\theta)=\begin{cases} 1 & \theta=0^{\circ}\\ 0 & otherwise\end{cases}$$ $$H_{2}(\theta)=\begin{cases} 1 & \theta=359^{\circ}\\ 0 & otherwise\end{cases}$$

 

[Valkarie]

Yes, there are several definitions of distance between circular histograms. One such definition is the Jensen-Shannon divergence, which is a symmetric and bounded measure of the similarity between two probability distributions. Other options include Kullback-Leibler divergence, Bhattacharyya distance, and Hellinger distance. All of these measures can be applied to circular histograms.