Computation of simultaneous credible intervals.

Description

Simultaneous credible intervals for all differences of smooths at neighboring scales z_{i} are computed.

Usage

CImap(smoothVec, mm, nn, prob = 0.95)

Arguments

Details

CImap is an internal function of mrbsizeRgrid and is usually not used independently. The output can be analyzed with the plotting function plot.CImapGrid.

Value

An array with simultaneous credible intervals VmapCI and the dimensions of the original input object, mm and nn.

Examples

# Artificial sample data: 10 observations (5-by-2 object), 10 samples
set.seed(987)
sampleData  <- matrix(stats::rnorm(100), nrow = 10)
sampleData [4:6, ] <- sampleData [4:6, ] + 5

# Calculation of the simultaneous credible intervals
CImap(smoothVec = sampleData , mm = 5, nn = 2, prob = 0.95)

Computation of pointwise and highest pointwise probabilities.

Description

Pointwise (PW) probabilities and highest pointwise (HPW) probabilities of all differences of smooths at neighboring scales are computed.

Usage

HPWmap(smoothVec, mm, nn, prob = 0.95)

Arguments

Details

HPWmap is an internal function of mrbsizeRgrid and is usually not used independently. The output can be analyzed with the plotting function plot.HPWmapGrid.

Value

List with two arrays:

• pw: Pointwise probabilities (VmapPW) including the dimensions of the original input image, mm and nn.

• hpw: Highest pointwise probabilities (VmapHPW) including the dimensions of the original input image, mm and nn.

Examples

# Artificial sample data: 10 observations (5-by-2 object), 10 samples
set.seed(987)
sampleData <- matrix(stats::rnorm(100), nrow = 10)
sampleData[4:6, ] <- sampleData[4:6, ] + 5

# Calculation of the simultaneous credible intervals
HPWmap(smoothVec = sampleData, mm = 5, nn = 2, prob = 0.95)

Numerical optimization for finding appropriate smoothing levels.

Description

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.

Usage

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

Arguments

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.

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.

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]

Plot of tapering functions.

Description

Tapering functions corresponding to the smoothing levels in lambdaSmoother are drawn. This plot helps to assess if the chosen smoothing levels are appropriate.

Usage

TaperingPlot(lambdaSmoother, mm, nn, Xmu, returnseq = FALSE, ...)

Arguments

Details

The tapering functions of the smoothing levels chosen should be generally approximately disjoint. This will produce features which are somewhat orthogonal. With orthogonal features, it is likely that each difference of smooths corresponds to a different pattern in the input image.

Sometimes, not all patterns of the input image can be extracted using smoothing levels whose tapering functions are disjoint. It might then be necessary to include additional smoothing levels, and the disjointedness might not be satisfied anymore. The selection of appropriate smoothing levels with this method therefore requires some user interaction. Still, choosing disjoint tapering functions for finding appropriate smoothing levels is a good starting point.

Better results could be obtained if the structure of the posterior mean of the input is also taken into account. If the posterior mean is available, it can be added with the argument Xmu and moving averages of the absolute values of the signal-dependent tapering functions are drawn. MinLambda offers a more formal approach of optimizing the disjointedness of the tapering functions and can help finding appropriate smoothing levels when the input signal is taken into account.

Value

Plots of the tapering functions for all differences of smooths at neighboring scales are created.

Examples

# Signal-independent tapering function plot for a 30-by-10 object with 
# the smoothing parameter sequence [0, 1, 10, 1000, inf]: 

TaperingPlot(lambdaSmoother = c(1, 10, 1000), mm = 30, nn = 10)


# Signal-dependent tapering function plot for a 30-by-10 object with 
# the smoothing parameter sequence [0, 1, 10, 1000, inf]: 

set.seed(987)
xmuExample <- c(stats::rnorm(300))
TaperingPlot(lambdaSmoother = c(1, 10, 1000), mm = 30, nn = 10, 
             Xmu = xmuExample)

Create a n-by-n discrete cosine transform matrix.

Description

The discrete cosine transform (DCT) matrix for a given dimension n is calculated.

Usage

dctMatrix(n)

Arguments

Details

The function can be used for 1D- or 2D-DCT transforms of data.

• 1D: Let Q be a m-by-n matrix with some data. D is a m-by-m DCT matrix created by dctMatrix(m). Then D %*% Q returns the discrete cosine transform of the columns of Q. t(D) %*% Q returns the inverse DCT of the columns of Q. As D is orthogonal, solve(D) = t(D).

• 2D: Let Q be a m-by-n matrix with some data. D_m is a m-by-m DCT matrix created by dctMatrix(m), D_n a n-by-n DCT matrix created by dctMatrix(n). D_m %*% Q %*% t(D_n) computes the 2D-DCT of Q. The inverse 2D-DCT of Q can be computed via
  t(D_mm) %*% DCT_Q %*% D_n. D_m transforms along columns, D_n along rows. Since D is orthogonal, solve(D) = t(D).

It can be faster to use dctMatrix than using a direct transformation, especially when calculating several DCT's.

Value

The n-by-n DCT matrix.

Examples

D <- dctMatrix(5)

Create a n-by-n discrete Fourier transform matrix.

Description

The discrete Fourier transform (DFT) matrix for a given dimension n is calculated.

Usage

dftMatrix(n)

Arguments

Details

The DFT matrix can be used for computing the discrete Fourier transform of a matrix or vector. dftMatrix(n) %*% testMatrix is the same as apply(testMatrix, MARGIN = 2, FUN = fft).

Value

The n-by-n DFT matrix.

Examples

set.seed(987)
testMatrix <- matrix(sample(1:10, size = 25, replace = TRUE), nrow = 5)
D <- dftMatrix(5)

# Discrete Fourier transform with matrix multiplication:
D %*% testMatrix 

# Discrete Fourier transform with function fft: 
apply(testMatrix, MARGIN = 2, FUN = fft)

Generate eigenvalues of discrete Laplace matrix.

Description

The eigenvalues of a discrete Laplace matrix with dimension (mm, nn) are calculated.

Usage

eigenLaplace(mm, nn)

Arguments

Value

A row vector containing the eigenvalues of the discrete laplace matrix.

Examples

eigval <- eigenLaplace(5, 5)

Generate eigenvalues of precision matrix Q on the surface of a sphere.

Description

The eigenvalues of the precision matrix Q with dimension (mm, nn) and polar angle limits phimin, phimax are calculated.

Usage

eigenQsphere(phimin, phimax, mm, nn)

Arguments

Details

The corresponding function for data on a grid is eigenLaplace.

Value

A list containing 2 elements:

• eigval Row vector containing the eigenvalues of Q.

• eigvec Matrix containing the eigenvectors of Q as columns.

Examples

eig_out <- eigenQsphere(phimin = 180/10, phimax = 180 - 180/10, mm = 10, nn = 20)

Swap the quadrants or halves of a 2d matrix.

Description

fftshift is an R equivalent to the Matlab function fftshift applied on matrices. For more information about fftshift see the Matlab documentation.

Usage

fftshift(inputMatrix, dimension = -1)

Arguments

Details

It is possible to swap the halves or the quadrants of the input matrix. Halves can be swapped along the rows (dimension = 1) or along the columns (dimension = 2). When swapping the quadrants, fftshift swaps the first quadrant with the third and the second quadrant with the fourth (dimension = -1).

Value

Swapped matrix.

Examples

set.seed(987) 
sampleMat <- matrix(sample(1:10, size = 25, replace = TRUE), nrow = 5)

# Swap halves along the rows:
fftshift(sampleMat, dimension = 1)

# Swap halves along the columns:
fftshift(sampleMat, dimension = 2)

# Swap first quadrant with third and second quadrant with fourth:
fftshift(sampleMat, dimension = -1)

Inverse FFT shift of a 2d matrix.

Description

ifftshift is an R equivalent to the Matlab function ifftshift applied on matrices. For more information about ifftshift see the Matlab documentation.

Usage

ifftshift(inputMatrix, dimension = -1)

Arguments

Details

ifftshift is the inverse function to fftshift. For more information see the details of fftshift

Value

Swapped matrix.

Examples

set.seed(987)
sampleMat <- matrix(sample(1:10, size = 25, replace = TRUE), nrow = 5)

# Swap halves along the rows:
ifftshift(sampleMat, dimension = 1)

# Swap halves along the columns:
ifftshift(sampleMat, dimension = 2)

# Swap first quadrant with third and second quadrant with fourth:
ifftshift(sampleMat, dimension = -1)

mrbsizeR: Scale space multiresolution analysis in R.

Description

mrbsizeR contains a method for the scale space multiresolution analysis of spatial fields and images to capture scale-dependent features. The name is an abbreviation for MultiResolution Bayesian SIgnificant ZEro crossings of derivatives in R and the method combines the concept of statistical scale space analysis with a Bayesian SiZer method.

Details

The mrbsizeR analysis can be applied to data on a regular grid and to spherical data. For data on a grid, the scale space multiresolution analysis has three steps:

1. Bayesian signal reconstruction.

2. Using differences of smooths, scale-dependent features of the reconstructed signal are found.

3. Posterior credibility analysis of the differences of smooths created.

In a first step, Bayesian signal reconstruction is used to extract an underlying signal from a potentially noisy observation. Samples of the resulting posterior can be generated and used for the analysis. For finding features on different scales, differences of smooths at neighboring scales are used. This is an important distinction to other scale space methods (which usually use a wide range of smoothing levels without taking differences) and tries to separate the features into distinct scale categories more aggressively. After a successful extraction of the scale-different features, posterior credibility analysis is necessary to assess whether the features found are “really there” or if they are artifacts of random sampling.

For spherical data, no Bayesian signal reconstruction is implemented in mrbsizer. Data samples therefore need to be available beforehand. The analysis procedure can therefore be summarized in two steps:

1. Using differences of smooths, scale-dependent features of the reconstructed signal are found.

2. Posterior credibility analysis of the differences of smooths created.

This method has first been proposed by Holmstrom, Pasanen, Furrer, Sain (2011), see also
http://cc.oulu.fi/~lpasanen/MRBSiZer/.

Major Functions

• TaperingPlot Graphical estimation of useful smoothing levels. Can be used signal-independent and signal-dependent.

• MinLambda Numerical estimation of useful smoothing levels. Takes the underlying signal into account. plot.minLambda can be used for plotting the result.

• rmvtDCT Creates samples on a regular grid from a multivariate t_{\nu}-distribution using a discrete cosine transform (DCT).

• mrbsizeRgrid Interface of the mrbsizeR method for data on a regular grid. Differences of smooths at neighboring scales are created and posterior credibility analysis is conducted. The results can be visualized using plot.smMeanGrid,plot.HPWmapGrid and plot.CImapGrid.

• mrbsizeRsphere Interface of the mrbsizeR method for data on a sphere. Differences of smooths at neighboring scales are created and posterior credibility analysis is conducted. The results can be visualized using plot.smMeanSphere,plot.HPWmapSphere and
  plot.CImapSphere. For data on a sphere, no Bayesian signal reconstruction is implemented. Samples have to be provided instead.

Getting Started

The vignette for this package offers an extensive overview of the functionality and the usage of mrbsizeR.

References

• Holmstrom, L. and Pasanen, L. (2011). MRBSiZer. http://cc.oulu.fi/~lpasanen/MRBSiZer/. Accessed: 2017-03-04.

• Holmstrom, L., Pasanen, L., Furrer, R., and Sain, S. R. (2011). Scale space multiresolution analysis of random signals. Computational Statistics and Data Analysis, 55, 2840-2855. <DOI:10.1016/j.csda.2011.04.011>.

• Holmstrom, L. and Pasanen, L. (2016). Statistical scale space methods. International Statistical Review. <DOI:10.1111/insr.12155>.

DISCLAIMER: The author can not guarantee the correctness of any function or program in this package.

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

# Generate samples from multivariate t-distribution
tSamp <- rmvtDCT(object = sampleData, lambda = 0.2, sigma = 6, nu0 = 15,
                  ns = 1000)  
 
# mrbsizeRgrid analysis
mrbOut <- mrbsizeRgrid(posteriorFile = tSamp$sample, mm = 10, nn = 10, 
                       lambdaSmoother = c(1, 1000), prob = 0.95)

# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, turn_out = TRUE)                    

# Credibility analysis using simultaneous credible intervals
plot(x = mrbOut$ciout, turn_out = TRUE) 

Multiresolution analysis of random signals.

Description

mrbsizeRgrid is the interface of the scale space multiresolution method for data on a regular grid. Here, the differences of smooths as well as the posterior credibility analysis are computed. The output can be analyzed with the plotting functions plot.smMeanGrid, plot.CImapGrid and plot.HPWmapGrid.

Usage

mrbsizeRgrid(
  posteriorFile,
  mm,
  nn,
  lambdaSmoother,
  prob = 0.95,
  smoothOut = FALSE
)

Arguments

Details

mrbsizeRgrid conducts two steps of the scale space multiresolution analysis:

1. Extraction of scale-dependent features from the reconstructed signal. This is done by smoothing at different smoothing levels and taking the difference of smooths at neighboring scales.

2. Posterior credibility analysis of the differences of smooths created. Three different methods are applied: Pointwise probabilities (see HPWmap), highest pointwise probabilities (see HPWmap) and simultaneous credible intervals (see CImap).

The signal can be reconstructed using the build-in multivariate t-distribution sampling rmvtDCT. It is also possible to provide samples generated with other methods, see the parameter posteriorFile and the examples.

For further information and examples, see the vignette.

Value

A list containing the following sublists:

smMean Posterior mean of all differences of smooths created.

hpout Pointwise (PW) and highest pointwise (HPW) probabilities of all differences of smooths created.

ciout Simultaneous credible intervals (CI) of all differences of smooths created.

smoothSamples Samples of differences of smooths at neighboring scales, as column vectors.

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

# Generate samples from multivariate t-distribution
tSamp <- rmvtDCT(object = sampleData, lambda = 0.2, sigma = 6, nu0 = 15,
                  ns = 1000)  
 
# mrbsizeRgrid analysis
mrbOut <- mrbsizeRgrid(posteriorFile = tSamp$sample, mm = 10, nn = 10, 
                       lambdaSmoother = c(1, 1000), prob = 0.95)

Multiresolution analysis of random signals for spherical data.

Description

mrbsizeRSphere is the interface of the scale space multiresolution method for spherical data. Here, the differences of smooths as well as the posterior credibility analysis are computed. The output can be analyzed with the plotting functions plot.smMeanSphere, plot.CImapSphere and plot.HPWmapSphere.

Usage

mrbsizeRsphere(
  posteriorFile,
  mm,
  nn,
  lambdaSmoother,
  prob = 0.95,
  smoothOut = FALSE
)

Arguments

Details

In contrast to mrbsizeRgrid, mrbsizeRsphere does not conduct Bayesian signal reconstruction via sampling from a posterior distribution. Samples of the posterior distribution have to be provided instead.

For further information and examples, see mrbsizeRgrid and the vignette.

Value

A list containing the following sublists:

smMean Posterior mean of all differences of smooths created.

hpout Pointwise (PW) and highest pointwise (HPW) probabilities of all differences of smooths created.

ciout Simultaneous credible intervals (CI) of all differences of smooths created.

smoothSamples Samples of differences of smooths at neighboring scales, as column vectors.

Examples

# Artificial spherical sample data
set.seed(987)
sampleData <- matrix(stats::rnorm(2000), nrow = 200)
sampleData[50:65, ] <- sampleData[50:65, ] + 5

# mrbsizeRsphere analysis
mrbOut <- mrbsizeRsphere(posteriorFile = sampleData, mm = 10, nn = 20,  
                          lambdaSmoother = c(1, 1000), prob = 0.95)

Plot of simultaneous credible intervals.

Description

Maps with simultaneous credible intervals for all differences of smooths at neighboring scales z_{i} are plotted.

Usage

## S3 method for class 'CImapGrid'
plot(
  x,
  color = c("firebrick1", "gainsboro", "dodgerblue3"),
  turnOut = TRUE,
  title,
  aspRatio = 1,
  ...
)

Arguments

Details

The default colors of the maps have the following meaning:

• Blue: Credibly positive pixels.

• Red: Credibly negative pixels.

• Grey: Pixels that are not credibly different from zero.

x corresponds to the ciout-part of the output of mrbsizeRgrid.

Value

Plots of simultaneous credible intervals for all differences of smooths are created.

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

# Generate samples from multivariate t-distribution
tSamp <- rmvtDCT(object = sampleData, lambda = 0.2, sigma = 6, nu0 = 15,
                   ns = 1000)  
 
# mrbsizeRgrid analysis
mrbOut <- mrbsizeRgrid(posteriorFile = tSamp$sample, mm = 10, nn = 10, 
                       lambdaSmoother = c(1, 1000), prob = 0.95)

# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, turnOut = TRUE)                    

# Credibility analysis using simultaneous credible intervals
plot(x = mrbOut$ciout, turnOut = TRUE)

Plotting of simultaneous credible intervals on a sphere.

Description

Maps with simultaneous credible intervals for all differences of smooths at neighboring scales z_{i} are plotted. Continental lines are added.

Usage

## S3 method for class 'CImapSphere'
plot(
  x,
  lon,
  lat,
  color = c("firebrick1", "gainsboro", "dodgerblue3"),
  turnOut = FALSE,
  title,
  ...
)

Arguments

Details

The default colors of the maps have the following meaning:

• Blue: Credibly positive pixels.

• Red: Credibly negative pixels.

• Grey: Pixels that are not credibly different from zero.

x corresponds to the ciout-part of the output of mrbsizeRsphere.

Value

Plots of simultaneous credible intervals for all differences of smooths are created.

Examples

# Artificial spherical sample data
set.seed(987)
sampleData <- matrix(stats::rnorm(2000), nrow = 200)
sampleData[50:65, ] <- sampleData[50:65, ] + 5
lon <- seq(-180, 180, length.out = 20)
lat <- seq(-90, 90, length.out = 10)

# mrbsizeRsphere analysis
mrbOut <- mrbsizeRsphere(posteriorFile = sampleData, mm = 20, nn = 10, 
                         lambdaSmoother = c(0.1, 1), prob = 0.95)
                           
# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, lon = lon, lat = lat,
     color = fields::tim.colors()) 

# Credibility analysis using simultaneous credible intervals
plot(x = mrbOut$ciout, lon = lon, lat = lat)

Plotting of pointwise and highest pointwise probabilities.

Description

Maps with pointwise (PW) probabilities and/or highest pointwise (HPW) probabilities of all differences of smooths at neighboring scales are plotted.

Usage

## S3 method for class 'HPWmapGrid'
plot(
  x,
  plotWhich = "Both",
  color = c("firebrick1", "gainsboro", "dodgerblue3"),
  turnOut = TRUE,
  title,
  aspRatio = 1,
  ...
)

Arguments

Details

The default colors of the maps have the following meaning:

• Blue: Credibly positive pixels.

• Red: Credibly negative pixels.

• Grey: Pixels that are not credibly different from zero.

x corresponds to the hpout-part of the output of mrbsizeRgrid.

Value

Plots of pointwise and/or highest pointwise probabilities for all differences of smooths are created.

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

# Generate samples from multivariate t-distribution
tSamp <- rmvtDCT(object = sampleData, lambda = 0.2, sigma = 6, nu0 = 15,
                   ns = 1000)

# mrbsizeRgrid analysis
mrbOut <- mrbsizeRgrid(posteriorFile = tSamp$sample, mm = 10, nn = 10,
                       lambdaSmoother = c(1, 1000), prob = 0.95)

# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, turnOut = TRUE)

# Credibility analysis using pointwise (PW) maps
plot(x = mrbOut$hpout, plotWhich = "PW", turnOut = TRUE)

# Credibility analysis using highest pointwise probability (HPW) maps
plot(x = mrbOut$hpout, plotWhich = "HPW", turnOut = TRUE)

Plotting of pointwise and highest pointwise probabilities on a sphere.

Description

Maps with pointwise (PW) probabilities and/or highest pointwise (HPW) probabilities of all differences of smooths at neighboring scales are plotted. Continental lines are added.

Usage

## S3 method for class 'HPWmapSphere'
plot(
  x,
  lon,
  lat,
  plotWhich = "Both",
  color = c("firebrick1", "gainsboro", "dodgerblue3"),
  turnOut = FALSE,
  title,
  ...
)

Arguments

Details

The default colors of the maps have the following meaning:

• Blue: Credibly positive pixels.

• Red: Credibly negative pixels.

• Grey: Pixels that are not credibly different from zero.

x corresponds to the hpout-part of the output of mrbsizeRsphere.

Value

Plots of pointwise and/or highest pointwise probabilities for all differences of smooths are created.

Examples

# Artificial spherical sample data
set.seed(987)
sampleData <- matrix(stats::rnorm(2000), nrow = 200)
sampleData[50:65, ] <- sampleData[50:65, ] + 5
lon <- seq(-180, 180, length.out = 20)
lat <- seq(-90, 90, length.out = 10)

# mrbsizeRsphere analysis
mrbOut <- mrbsizeRsphere(posteriorFile = sampleData, mm = 20, nn = 10, 
                         lambdaSmoother = c(0.1, 1), prob = 0.95)
                           
# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, lon = lon, lat = lat,
     color = fields::tim.colors()) 

# Credibility analysis using pointwise (PW) maps
plot(x = mrbOut$hpout, lon = lon, lat = lat, plotWhich = "PW")

# Credibility analysis using highest pointwise probability (HPW) maps
plot(x = mrbOut$hpout, lon = lon, lat = lat, plotWhich = "HPW")

Plot of objective function for finding appropriate smoothing parameters.

Description

The objective function G is plotted on a grid. The minimum is indicated with a white point.

Usage

## S3 method for class 'minLambda'
plot(x, ...)

Arguments

Details

When minimizing over 2 \lambda's, one plot is generated: (\lambda_2 vs \lambda_3). With 3 \lambda's, 3 plots are generated: \lambda_2 vs. \lambda_3, \lambda_2 vs. \lambda_4 and \lambda_3 vs. \lambda_4.

Value

Plot of G on a grid.

Examples

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 = 3)

# Plot of the objective function
plot(x = minLamOut)

Plotting of scale-dependent features.

Description

Scale-dependent features are plotted using differences of smooths at neighboring scales. The features are summarized by their posterior mean.

Usage

## S3 method for class 'smMeanGrid'
plot(
  x,
  color.pallet = fields::tim.colors(),
  turnOut = TRUE,
  title,
  aspRatio = 1,
  ...
)

Arguments

Details

x corresponds to the smmean-part of the output of mrbsizeRgrid.

Value

Plots of the differences of smooths are created.

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

# Generate samples from multivariate t-distribution
tSamp <- rmvtDCT(object = sampleData, lambda = 0.2, sigma = 6, nu0 = 15,
                   ns = 1000)  
 
# mrbsizeRgrid analysis
mrbOut <- mrbsizeRgrid(posteriorFile = tSamp$sample, mm = 10, nn = 10, 
                       lambdaSmoother = c(1, 1000), prob = 0.95)

# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, turnOut = TRUE) 

Plotting of scale-dependent features on a sphere.

Description

Scale-dependent features are plotted using differences of smooths at neighboring scales. The features are summarized by their posterior mean. Continental lines are added to the plots.

Usage

## S3 method for class 'smMeanSphere'
plot(
  x,
  lon,
  lat,
  color.pallet = fields::tim.colors(),
  turnOut = TRUE,
  title,
  ...
)

Arguments

Details

x corresponds to the smmean-part of the output of mrbsizeRsphere.

Value

Plots of the differences of smooths are created.

Examples

# Artificial spherical sample data
set.seed(987)
sampleData <- matrix(stats::rnorm(2000), nrow = 200)
sampleData[50:65, ] <- sampleData[50:65, ] + 5
lon <- seq(-180, 180, length.out = 20)
lat <- seq(-90, 90, length.out = 10)

# mrbsizeRsphere analysis
mrbOut <- mrbsizeRsphere(posteriorFile = sampleData, mm = 20, nn = 10, 
                         lambdaSmoother = c(0.1, 1), prob = 0.95)
                           
# Posterior mean of the differences of smooths
plot(x = mrbOut$smMean, lon = lon, lat = lat,
     color = fields::tim.colors(), turnOut = FALSE) 

Sampling from marginal posterior multivariate t-distribution.

Description

Samples from a marginal posterior multivariate t-distribution with normal-inverse-chi-squared-prior are generated.

Usage

rmvtDCT(object, lambda, sigma, nu0, ns)

Arguments

Details

An eigenvalue decomposition is used for sampling. To speed up computations, a 2D discrete cosine transform (DCT) has been implemented, see dctMatrix. The output is a list containing

1. Samples of the marginal posterior of the input as column vectors.

2. The mean of the marginal posterior of the input as a vector.

Value

A list containing the following elements:

sample Samples of the marginal posterior of the input.

mu Mean of the marginal posterior of the input.

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

# Sampling from a multivariate t-distribution
t_dist_samp <- rmvtDCT(object = sampleData, lambda = 1, sigma = 10,
                       nu0 = 50, ns = 1000)

Generate a tridiagonal matrix.

Description

Generate a tridiagonal matrix with upperDiag as superdiagonal, lowerDiag as subdiagonal and mainDiag as diagonal.

Usage

tridiag(mainDiag, upperDiag, lowerDiag)

Arguments

Value

Tridiagonal matrix.

Examples

set.seed(987)
mainDiag <- sample(100:110, size = 6, replace = TRUE)
upperDiag <- sample(10:20, size = 5, replace = TRUE)
lowerDiag <- sample(1:10, size = 5, replace = TRUE)

tridiag(mainDiag, upperDiag, lowerDiag)  
 

Turn matrix 90 degrees counter-clockwise.

Description

Help function to turn matrix 90 degrees counter-clockwise. turnmat is used as an internal function in some of the plotting functions. Depending on how the data is stored, it can be necessary to turn the matrices 90 degrees counter-clockwise for being able to plot them correctly.

Usage

turnmat(x)

Arguments

Value

Matrix x, turned by 90 degrees counter-clockwise.

Examples

set.seed(987)
sampleMat <- matrix(stats::rnorm(100), nrow = 10)
sampleMatTurn <- turnmat(x = sampleMat)