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## Useful Characteristics of Microsphere Projection |

3. MS Projection is guaranteed to preserve monotonic and
strict
monotonic behavior over any set or subset of sample
points. For example, if the set or subset of sample points is increasing
or strictly increasing over a range, then the interpolation is
guaranteed to be increasing or strictly
increasing over the same range.4. MS Projection demonstrates no oscillatory behavior between
sample points, unlike functional approximations which
are designed to
preserve high differentiability.5. MS Projection provides a stable extrapolation ability.
Functional approximations tend to produce extremely volatile extrapolation
results beyond the range of the data points. This
can cause serious issues in higher dimensions where the differentiation
between interpolation and extrapolation within the volume is difficult
to determine. Because MS Projection provides
a stable extrapolation, it has considerable benefits over functional
approximations when visualizing higher dimensional data. |

## Not-So Useful Characteristics of Microsphere Projection |

^{1}, but furhter research must be done.2. Depending on the nature of the problem, the fact that
MS Projection exhibits the Maximum
Principle can be an issue. That the interpolation
method is unable to interpolate a value beyond the minimum and maximum sampled values can cause problems depending on the context.3. Depending on the size of the data set
and other considerations, MS Projection can require more computation
time than some of the
other interpolation algorithms.
Though the overall runtime is O(P*N), this set of calculations must
be run every time a point is to be interpolated. Radial
Basis Function (RBF) interpolations such as Thin-Plate
Spline, Multiquadric, and
Volume Spline are all O(N^{2}) (using Gaussian elimination)
for the first interpolation and O(N) (with a very small overhead)
for subsequent
interpolations.4. As interpolation location approaches a sample point, the
first derivative in
all dimensions approaches 0 when p>1. In most contexts
this is undesirable behavior; however it is necessary if we wish
to preserve
the Maximum
Principle in conjunction with differentiability.Definitions:p is the inverse distance propagation constant (p=1 is linear)N is the number of samples of "real" dataP is a precision constant representing the number of faces of the microsphere |