Numerical optimization for finding appropriate smoothing levels.

Numerical optimization of an objective function \(G\) is carried out to find appropriate signal-dependent smoothing levels (\(\lambda\)'s). This is easier than visual inspection via the signal-dependent tapering function in TaperingPlot.

MinLambda(Xmu, mm, nn, nGrid, nLambda = 2, lambda, sphere = FALSE)

Arguments

Xmu	Posterior mean of the input object as a vector.
mm	Number of rows of the original input object.
nn	Number of columns of the original input object.
nGrid	Size of grid where objective function is evaluated (nGrid-by-nGrid). This argument is ignorded if a sequence lambda is specified.
nLambda	Number of lambdas to minimize over. Possible arguments: 2 (default) or 3.
lambda	\(\lambda\)-sequence which is used for optimization. If nothing is provided, lambda <- 10^seq(-3, 10, len = nGrid) is used for data on a grid and lambda <- 10^seq(-6, 1, len = nGrid) is used for spherical data.
sphere	TRUE or FALSE: Is the input object defined on a sphere?

Value

A list with 3 objects:

G Value of objective function \(G\).

lambda Evaluated smoothing parameters \(\lambda\).

minind Index of minimal \(\lambda\)'s. lambda[minind] gives the minimal values.

Details

As signal-dependent tapering functions are quiet irregular, it is hard to find appropriate smoothing values only by visual inspection of the tapering function plot. A more formal approach is the numerical optimization of an objective function.

Optimization can be carried out with 2 or 3 smoothing parameters. As the smoothing parameters 0 and \(\infty\) are always added, this results in a mrbsizeR analysis with 4 or 5 smoothing parameters.

Sometimes, not all features of the input object can be extracted using the smoothing levels proposed by MinLambda. It might then be necessary to include additional smoothing levels.

plot.minLambda creates a plot of the objective function \(G\) on a grid. The minimum is indicated with a white point. The minimum values of the \(\lambda\)'s can be extracted from the output of MinLambda, see examples.

Examples

# Artificial sample data
set.seed(987)
sampleData <- matrix(stats::rnorm(100), nrow = 10)
sampleData[4:6, 6:8] <- sampleData[4:6, 6:8] + 5

# Minimization of two lambdas on a 20-by-20-grid
minlamOut <- MinLambda(Xmu = c(sampleData), mm = 10, nn = 10,
                       nGrid = 20, nLambda = 2)

# Minimal lambda values
minlamOut$lambda[minlamOut$minind]
#> [1]   0.1128838 297.6351442