Abstract
The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.
A Appendix
In this appendix, we compare the performance of two subroutines shown in Figure 2, for generating sets of N uniformly distributed points on the unit n-sphere centered at the origin. We generate the normal variate vectors using the built-in function randn() in MATLAB R2018a(9.4).
Subroutine 1 creates a vector variable
Table 6 shows the elapsed time to generate sets of
The run-time reported in our tables refer to the time it took for executing our programs on a computer equipped with Intel(R) Core(TM) i7-8565U CPU @ 1.80 GHz 1.99 GHz and 16 GB of RAM.
Subroutines for generating a uniformly distributed set of N points on a unit sphere.
Elapsed time for generating a uniformly distributedset of N points on a unit sphere using subroutines 1 and 2for two different dimensions.
| n | N | Elapsed time(1) | Elapsed time(2) |
| 5 | 250.762282 | 0.090486 | |
| 40 | 2812.187933 | 0.466752 |
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