# MinLambda: Numerical optimization for finding appropriate smoothing... In mrbsizeR: Scale Space Multiresolution Analysis of Random Signals

## Description

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

## Usage

 `1` ```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` λ-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?

## 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 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 λ's can be extracted from the output of `MinLambda`, see examples.

## Value

A list with 3 objects:

`G` Value of objective function G.

`lambda` Evaluated smoothing parameters λ.

`minind` Index of minimal λ's. `lambda`[`minind`] gives the minimal values.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```# 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] ```

mrbsizeR documentation built on April 1, 2020, 5:08 p.m.