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The density estimates obtained from using our methods have a clear improvement in maintaining the boundary of the valid region.
The Weighted TV MPLE method does not perform as well for sparse data sets and fails to keep the boundary of the valid region.
Once we have determined a valid region, we wish to align the level curves of the density function with the boundary of the valid region.
In doing so, the boundary of the valid region may attain density values too large in comparison to the rest of the image when the size of the image is very large.
To remedy this, we may take the resulting image from the algorithm and set the boundary of the valid region to zero and rescale the image to have a sum of one.
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However, our methods prohibit the density estimates from spreading through the boundaries of the valid region.
When estimates are on the boundaries of the valid parameter space, the estimates of the Newton-Raphson method will not converge to the maximum likelihood estimates [ 24].
We are concerned with two key problems of adaptive control design: one is to find the feasible conditions and the valid boundary of the time-varying feedback scheme; another one lies in the improved generalization of the homogeneous idea in the global adaptive stabilization of high-order uncertain nonlinear systems.
The centroid of a set of sampling points is defined as the point that minimizes the sum of squared geodesic distances between itself and each point in the set and it is calculated from all the initial records except those that have zero coordinates or coordinates that fall out of the valid boundaries for the coordinate reference system WGS84.
Reflection at the boundary was used to avoid proposals outside of the valid range (e.g. m ≤ 0).
To avoid proposals lying outside of the valid interval (e.g. negative values) we use reflection at the boundary.
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