Compute Kullback-Leibler divergence.

1 2 | ```
KL.divergence(X, Y, k = 10, algorithm=c("kd_tree", "cover_tree", "brute"))
KLx.divergence(X, Y, k = 10, algorithm="kd_tree")
``` |

`X` |
An input data matrix. |

`Y` |
An input data matrix. |

`k` |
The maximum number of nearest neighbors to search. The default value is set to 10. |

`algorithm` |
nearest neighbor search algorithm. |

If `p(x)`

and `q(x)`

are two continuous probability density functions,
then the Kullback-Leibler divergence of `q`

from `p`

is defined as
*E_p[log p(x)/q(x)]*.

`KL.*`

versions return divergences from `C`

code to `R`

but `KLx.*`

do not.

Return the Kullback-Leibler divergence from `X`

to `Y`

.

Shengqiao Li. To report any bugs or suggestions please email: shli@stat.wvu.edu.

S. Boltz, E. Debreuve and M. Barlaud (2007).
“kNN-based high-dimensional Kullback-Leibler distance for tracking”.
*Image Analysis for Multimedia Interactive Services, 2007. WIAMIS '07. Eighth International Workshop on*.

S. Boltz, E. Debreuve and M. Barlaud (2009).
“High-dimensional statistical measure for region-of-interest tracking”.
*Trans. Img. Proc.*, **18**:6, 1266–1283.

1 2 3 4 5 6 | ```
set.seed(1000)
X<- rexp(10000, rate=0.2)
Y<- rexp(10000, rate=0.4)
KL.divergence(X, Y, k=5)
#theoretical divergence = log(0.2/0.4)+(0.4-0.2)-1 = 1-log(2) = 0.307
``` |

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