Distances{
Euclidean distances between points are calculated using the function dist. This computation requires the coordinates to be projected. The user is responsible for making sure that this requirement is attained.
}
Mean (squared) shortest distance{
The function objMSSD is used in the optimization of spatial points for sampling. In a previous implementation, objMSSD would calculate the distance matrix at each iteration of the optimization algorithm. This is computationally expensive. Thus, we decided to separate the calculation of the distance matrix, which is done using the function distMat. The user has to square the distance values to get the mean squared shortest distance -- this is done setting exponent = 2 in distMat.Once the distance matrix has been calculated, the algorithm has only to identify the subset of points in the prediction grid. Both distToNearestPoint and nearestPoint perform this operation. The calculation of the mean (squared) shortest distance does not require to know which is the nearest point -- we are only interested in knowing the distance to the nearest point. As such, distToNearestPoint is called internally by objMSSD.
distToNearestPoint and nearestPoint were constructed separately because they are useful for other operations. For instance, distToNearestPoint can be used to build a map of distances to the nearest point, while nearestPoint can be used to define geographic strata.
}
Utopia and nadir points{
Knowledge of the utopia and nadir points can help in the construction of multi-objective optimization problems.
objMSSD is a bi-dimensional criterion because it explicitly takes into account both y and x coordinates. It aims at the spread of points in the geographic space. This is completely different from objPairs and objPoints which are uni-dimensional objective functions. They aim at the spread on points in the variogram space. It is more difficult to calculate the utopia and nadir points of a bi-dimensional criterion.
The utopia ($f^{\circ}_{i}$) point of objMSSD is only known to be larger than zero. The nadir ($f^{max}_{i}$) point is obtained when all points are clustered in one of the corners of the spatial domain. This cannot be calculated and has to be simulated.
One strategy is to first optimize the set of points using objMSSD and then create geographic strata. For the multi-objective optimization one would then have to define an unidimensional criterion aiming at matching the optimal solution obtained by minimizing objMSSD.
One such unidimensional criterion would be the difference between the expected distribution and the observed distribution of points per geographic strata. This criterion would aim at having at least one point per geographic strata. This is similar to what is done when using objPairs or objPoints -- there we use lag distance classes.
A second uni-dimensional criterion would be the difference between the expected MSSD and the observed MSSD. This criterion would aim at having the points coinciding with the optimal solution obtained by minimizing objMSSD. In both cases the utopia point would be exactly zero ($f^{\circ}_{i} = 0$). The nadir point could be easily calculated for the first, but not for the second.
}