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2d plot veusz8/18/2023 ![]() ![]() Its edges are straight lines which connect the outermost points. The convex hull of points in 2D is the smallest convex polygon that encloses all the points. Polygons: convex hulls The convex hull of a set of points. This is the main reason the number of tests for the polygon algorithm is much higher. 2Īnother issue is that it is harder to account for all cases.Įdge cases can be isolated directly in the mathematics for ellipses, but for polygons it requires actually testing each case. This means operations require looping through all the edges and so are of order $O(n)$ or higher, such as finding the area, the perimeter or whether or not a point lies within the polygon.Ĭompare this to ellipses which are defined by a single equation, so all the previous operations can be found directly via formulas in $O(1)$ time. ![]() This method returns an IoU of 0 for any distribution that does not intersect it so we have no information about the third distribution.Īnother difficulty with the polygon method is that polygons are described by $n$ edges. For example, in the following image is distribution 2 or distribution 3 more similar to distribution 1? While further statistical techniques can be applied to mitigate the outlier problem, a fundamental issue is that it does not balance the contributions of means and variance well. It has a time complexity $O(nh)$ for $n$ points and $h$ points on each convex hull.Ĭalculating the intersections once they are found is much faster: $O(h_1 h_2)$.Ī problem with this approach is that outliers can have a dramatic effect: The outlier on the top left greatly reduces the IoU while the outlier on the far right greatly increases it. This gives a value that varies between 0 and 1.įinding the convex hulls is the bottleneck in this algorithm. The widget above allows you to adjust these five variables to create many variants of this distribution.Ī bivariate normal distribution with correlation $\rho$ can be constructed from two normal distributions $Z_1$ and $Z_2$ as follows:
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