Package 'CVThresh' reference manual

Title:	Level-Dependent Cross-Validation Thresholding
Description:	The level-dependent cross-validation method is implemented for the selection of thresholding value in wavelet shrinkage. This procedure is implemented by coupling a conventional cross validation with an imputation method due to a limitation of data length, a power of 2. It can be easily applied to classical leave-one-out and k-fold cross validation. Since the procedure is computationally fast, a level-dependent cross validation can be performed for wavelet shrinkage of various data such as a data with correlated errors.
Authors:	Donghoh Kim <[email protected]>, Hee-Seok Oh <[email protected]>
Maintainer:	Donghoh Kim <[email protected]>
License:	GPL (>= 2)
Version:	1.1.2
Built:	2024-11-12 02:54:13 UTC
Source:	https://github.com/cran/CVThresh

Imputation by wavelet

Description

This function performs imputation for test dataset of cross-validation given test dataset index and initial values.

Usage

cvimpute.by.wavelet(y, impute.index, impute.tol=0.1^3, 
    impute.maxiter=100, impute.vscale="independent",
    filter.number=10, family="DaubLeAsymm", ll=3)
cvimpute.by.wavelet(y, impute.index, impute.tol=0.1^3, 
    impute.maxiter=100, impute.vscale="independent",
    filter.number=10, family="DaubLeAsymm", ll=3)

Arguments

`y`	observation
`impute.index`	test dataset index for cross-validation
`impute.tol`	tolerance for imputation
`impute.maxiter`	maximum iteration for imputation
`impute.vscale`	specifies whether variance is adjusted level-by-level or not. “level" or “independent"
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`family`	specifies the family of wavelets “DaubExPhase" or “DaubLeAsymm" (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

In wavelet context, test dataset of cross-validation is missing values. Based on h-likelihood concept and penalized least squares, this function performs imputation by wavelet for missing dataset, given the missing dataset. Lee and Nelder (1996, 2001) introduced the hierarchical likelihood as an extended likelihood for general models that include unobserved random variables such as missing. Following Lee and Nelder (1996, 2001), treat the missing values as random parameters and it has been known that a wavelet shrinkage estimator can be formulated by penalized least squares problem (Antoniadis and Fan, 2001). This arguments lead to the iterative algorithm for imputing the missing values based on wavelet shrinkage.

Value

Imputed values according to cross-validation scheme.

References

Antoniadis, A. and Fan, J. (2001) Regularization of wavelet approximations. Journal of the American Statistical Association, 96, 939–962.

Lee, Y. and Nelder, J.A. (1996) Hierarchical generalised linear models (with discussion). Journal of the Royal Statistical Society Ser. B, 58, 619–678.

Lee, Y. and Nelder, J.A. (2001) Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika, 88, 987–1006.

Examples

# 8-fold cross-validation scheme with block size 2
set.seed(1)
cv.index <- cvtype(n=1024, cv.bsize=2, cv.kfold=8, cv.random=TRUE)$cv.index

# Generate 1024 observation from Heavisine function
snr <- 5
testdata <- heav(1024)
x <- testdata$x
y <- testdata$meanf + rnorm(1024, 0, testdata$sdf / snr)

# Impute by wavelet
yimpute <- cvimpute.by.wavelet(y=y, impute.index=cv.index)$yimpute

# Compare imputed values and observations
par(mar=0.1+c(4,4,2,1))
plot(y, yimpute, xlab="Observations", ylab="Imputed Values",
     main="Piecewise Polynomial", cex=0.5);abline(0,1)
# 8-fold cross-validation scheme with block size 2
set.seed(1)
cv.index <- cvtype(n=1024, cv.bsize=2, cv.kfold=8, cv.random=TRUE)$cv.index

# Generate 1024 observation from Heavisine function
snr <- 5
testdata <- heav(1024)
x <- testdata$x
y <- testdata$meanf + rnorm(1024, 0, testdata$sdf / snr)

# Impute by wavelet
yimpute <- cvimpute.by.wavelet(y=y, impute.index=cv.index)$yimpute

# Compare imputed values and observations
par(mar=0.1+c(4,4,2,1))
plot(y, yimpute, xlab="Observations", ylab="Imputed Values",
     main="Piecewise Polynomial", cex=0.5);abline(0,1)

Imputation for two-dimensional data by wavelet

Description

This function performs imputation for two-dimensional test dataset of cross-validation given test dataset index and initial values.

Usage

cvimpute.image.by.wavelet(images, impute.index1, impute.index2, 
   impute.tol=0.1^3, impute.maxiter=100, filter.number=2, ll=3)
cvimpute.image.by.wavelet(images, impute.index1, impute.index2, 
   impute.tol=0.1^3, impute.maxiter=100, filter.number=2, ll=3)

Arguments

`images`	noisy image
`impute.index1`	test dataset row index according to cross-validation scheme
`impute.index2`	test dataset column index according to cross-validation scheme
`impute.tol`	tolerance for imputation
`impute.maxiter`	maximum iteration for imputation
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

Value

Imputed values according to cross-validation scheme.

References

Antoniadis, A. and Fan, J. (2001) Regularization of wavelet approximations. Journal of the American Statistical Association, 96, 939–962.

Lee, Y. and Nelder, J.A. (1996) Hierarchical generalised linear models (with discussion). Journal of the Royal Statistical Society Ser. B, 58, 619–678.

Lee, Y. and Nelder, J.A. (2001) Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika, 88, 987–1006.

Level-Dependent Cross-Validation Approach for Wavelet Thresholding

Description

This package carries out level-dependent cross-validation method for the selection of thresholding value in wavelet shrinkage. This procedure is implemented by coupling a conventional cross validation with an imputation method due to a limitation of data length, a power of 2. It can be easily applied to classical leave-one-out and k-fold cross validation. Since the procedure is computationally fast, a level-dependent cross validation can be performed for wavelet shrinkage of various data such as a data with correlated errors.

Generating test dataset index for cross-validation

Description

This function generates test dataset index for cross-validation.

Usage

cvtype(n, cv.bsize=1, cv.kfold, cv.random=TRUE)
cvtype(n, cv.bsize=1, cv.kfold, cv.random=TRUE)

Arguments

`n`	the number of observation
`cv.bsize`	block size of cross-validation
`cv.kfold`	the number of fold of cross-validation
`cv.random`	whether or not random cross-validation scheme should be used. Set cv.random=TRUE for random cross-validation scheme

Details

This function provides index of test dataset according to various cross-validation scheme. One may construct K test datasets in a way that each testset consists of blocks of b consecutive data. Set cv.bsize = b for this. To select each fold at random, set cv.random = TRUE.

Value

matrix of which row is test dataset index for cross-validation.

Examples

# Traditional 4-fold cross-validation for 100 observations
cvtype(n=100, cv.bsize=1, cv.kfold=4, cv.random=FALSE)
# Random 4-fold cross-validation with block size 2 for 100 observations
cvtype(n=100, cv.bsize=2, cv.kfold=4, cv.random=TRUE)
# Traditional 4-fold cross-validation for 100 observations
cvtype(n=100, cv.bsize=1, cv.kfold=4, cv.random=FALSE)
# Random 4-fold cross-validation with block size 2 for 100 observations
cvtype(n=100, cv.bsize=2, cv.kfold=4, cv.random=TRUE)

Generating test dataset index of two-dimensional data for cross-validation

Description

This function generates test dataset index of two-dimensional data for cross-validation

Usage

cvtype.image(n, cv.bsize=c(1,1), cv.kfold)
cvtype.image(n, cv.bsize=c(1,1), cv.kfold)

Arguments

`n`	the size of image
`cv.bsize`	two-dimensional block size of cross-validation
`cv.kfold`	the number of fold of cross-validation

Details

This function provides indexes of two-dimensional cross-validation scheme. Only random cross-validation scheme is provided.

Value

Two matrix representing test dataset index of each dimension for cross-validation.

`cv.index1`	each row is test dataset index of one dimension
`cv.index2`	each row is test dataset index of the other dimension

Examples

# Two-dimensional 4-fold cross-validation with block size 2*2
out <- cvtype.image(n=c(256,256), cv.bsize=c(2,2), cv.kfold=4)
# Two-dimensional 4-fold cross-validation with block size 2*2
out <- cvtype.image(n=c(256,256), cv.bsize=c(2,2), cv.kfold=4)

Wavelet reconstruction by level-dependent Cross-Validation

Description

This function reconstructs the noise data by level-dependent cross-validation wavelet shrinkage.

Usage

cvwavelet(y=y, ywd=ywd, cv.optlevel, cv.bsize=1, cv.kfold, 
    cv.random=TRUE, cv.tol=0.1^3, cv.maxiter=100,
    impute.vscale="independent", impute.tol=0.1^3, impute.maxiter=100,
    filter.number=10, family="DaubLeAsymm", thresh.type ="soft", ll=3)
cvwavelet(y=y, ywd=ywd, cv.optlevel, cv.bsize=1, cv.kfold, 
    cv.random=TRUE, cv.tol=0.1^3, cv.maxiter=100,
    impute.vscale="independent", impute.tol=0.1^3, impute.maxiter=100,
    filter.number=10, family="DaubLeAsymm", thresh.type ="soft", ll=3)

Arguments

`y`	observation
`ywd`	DWT object
`cv.optlevel`	thresholding levels
`cv.bsize`	block size of cross-validation
`cv.kfold`	the number of fold of cross-validation
`cv.random`	whether or not random cross-validation scheme should be used. Set cv.random=TRUE for random cross-validation scheme
`cv.tol`	tolerance for cross-validation
`cv.maxiter`	maximum iteration for cross-validation
`impute.vscale`	specifies whether variance is adjusted level-by-level or not. “level" or “independent"
`impute.tol`	tolerance for imputation
`impute.maxiter`	maximum iteration for imputation
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`family`	specifies the family of wavelets “DaubExPhase" or “DaubLeAsymm" (argument of WaveThresh)
`thresh.type`	specifies the type of thresholding “hard" or “soft" (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

This function performs level-dependent cross-validation wavelet shrinkage.

Value

`y`	observations
`yimpute`	imputed values by provided cross-validation scheme
`yc`	reconstruction by level-dependent cross-validation wavelet shrinkage
`cvthresh`	threshold values by level-dependent cross-validation

Examples

data(ipd)
y <- as.numeric(ipd); n <- length(y); nlevel <- log2(n)
ywd <- wd(y)
#out <- cvwavelet(y=y, ywd=ywd, cv.optlevel=c(3:(nlevel-1)), 
#                     cv.bsize=2, cv.kfold=4)

#ts.plot(ts(out$yc, start=1229.98, deltat=0.02, frequency=50),
#   main="Level-dependent Cross Validation", xlab = "Seconds", ylab="")
data(ipd)
y <- as.numeric(ipd); n <- length(y); nlevel <- log2(n)
ywd <- wd(y)
#out <- cvwavelet(y=y, ywd=ywd, cv.optlevel=c(3:(nlevel-1)), 
#                     cv.bsize=2, cv.kfold=4)

#ts.plot(ts(out$yc, start=1229.98, deltat=0.02, frequency=50),
#   main="Level-dependent Cross Validation", xlab = "Seconds", ylab="")

Cross-Validation Wavelet Shrinkage after imputation

Description

This function performs level-dependent cross-validation wavelet shrinkage given the cross-validation scheme and imputation values.

Usage

cvwavelet.after.impute(y, ywd, yimpute,
    cv.index, cv.optlevel, cv.tol=0.1^3, cv.maxiter=100,
    filter.number=10, family="DaubLeAsymm", thresh.type="soft", ll=3)
cvwavelet.after.impute(y, ywd, yimpute,
    cv.index, cv.optlevel, cv.tol=0.1^3, cv.maxiter=100,
    filter.number=10, family="DaubLeAsymm", thresh.type="soft", ll=3)

Arguments

`y`	observation
`ywd`	DWT object
`yimpute`	imputed values according to cross-validation scheme
`cv.index`	test dataset index according to cross-validation scheme
`cv.optlevel`	thresholding levels
`cv.tol`	tolerance for cross-validation
`cv.maxiter`	maximum iteration for cross-validation
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`family`	specifies the family of wavelets “DaubExPhase" or “DaubLeAsymm" (argument of WaveThresh)
`thresh.type`	specifies the type of thresholding “hard" or “soft" (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

Calculating the threshold values and reconstructing noisy data $y$ , given the index of each testdata, imputed values according to cross-validation scheme and discrete wavelet transform of $y$ .

Value

Reconstruction and thresholding values by level-dependent cross-validation

`yc`	reconstruction
`cvthresh`	thresholding values by level-dependent cross-validation

Examples

data(ipd)
y <- as.numeric(ipd); n <- length(y); nlevel <- log2(n)

set.seed(1)
cv.index <- cvtype(n=n, cv.bsize=2, cv.kfold=4, cv.random=TRUE)$cv.index
yimpute <- cvimpute.by.wavelet(y=y, impute.index=cv.index)$yimpute

ywd <- wd(y)

#out <- cvwavelet.after.impute(y=y, ywd=ywd, yimpute=yimpute,
#cv.index=cv.index, cv.optlevel=c(3:(nlevel-1)))

#ts.plot(ts(out$yc, start=1229.98, deltat=0.02, frequency=50),
#   main="Level-dependent Cross Validation", xlab = "Seconds", ylab="")

##### Specifying thresholding structure
# cv.optlevel <- c(3) # Threshold (level 3 to finest level) at the same time.
# cv.optlevel <- c(3, 5) # Threshold two groups of resolution levels,
                         # (level 3, 4) and  (level 5 to finest level).
# cv.optlevel <- c(3,4,5,6,7,8) # Threshold each resolution level 3 to 8.
data(ipd)
y <- as.numeric(ipd); n <- length(y); nlevel <- log2(n)

set.seed(1)
cv.index <- cvtype(n=n, cv.bsize=2, cv.kfold=4, cv.random=TRUE)$cv.index
yimpute <- cvimpute.by.wavelet(y=y, impute.index=cv.index)$yimpute

ywd <- wd(y)

#out <- cvwavelet.after.impute(y=y, ywd=ywd, yimpute=yimpute,
#cv.index=cv.index, cv.optlevel=c(3:(nlevel-1)))

#ts.plot(ts(out$yc, start=1229.98, deltat=0.02, frequency=50),
#   main="Level-dependent Cross Validation", xlab = "Seconds", ylab="")

##### Specifying thresholding structure
# cv.optlevel <- c(3) # Threshold (level 3 to finest level) at the same time.
# cv.optlevel <- c(3, 5) # Threshold two groups of resolution levels,
                         # (level 3, 4) and  (level 5 to finest level).
# cv.optlevel <- c(3,4,5,6,7,8) # Threshold each resolution level 3 to 8.

Wavelet reconstruction of image by level-dependent Cross-Validation

Description

This function reconstructs image by level-dependent cross-validation wavelet shrinkage.

Usage

cvwavelet.image(images, imagewd,
    cv.optlevel, cv.bsize=c(1,1), cv.kfold, cv.tol=0.1^3, cv.maxiter=100,
    impute.tol=0.1^3, impute.maxiter=100, filter.number=2, ll=3)
cvwavelet.image(images, imagewd,
    cv.optlevel, cv.bsize=c(1,1), cv.kfold, cv.tol=0.1^3, cv.maxiter=100,
    impute.tol=0.1^3, impute.maxiter=100, filter.number=2, ll=3)

Arguments

`images`	noisy image
`imagewd`	two-dimensional wavelet transform
`cv.optlevel`	thresholding level
`cv.bsize`	block size of cross-validation
`cv.kfold`	the number of fold of cross-validation
`cv.tol`	tolerance for cross-validation
`cv.maxiter`	maximum iteration for cross-validation
`impute.tol`	tolerance for imputation
`impute.maxiter`	maximum iteration for imputation
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

This function performs level-dependent cross-validation wavelet shrinkage for two-dimensional data.

Value

`imagecv`	reconstruction of image by level-dependent cross-validation wavelet shrinkage
`cvthresh`	threshold values by level-dependent cross-validation

Examples

 
# Generate Noisy Lennon Image
data(lennon)
sdimage <- sd(as.numeric(lennon))
nlennon <- ncol(lennon); nlevel <- log2(ncol(lennon))
optlevel <- c(3:(nlevel-1))
set.seed(55)
lennonnoise <- lennon+matrix(rnorm(nlennon^2, 0, sdimage), nlennon, nlennon)

# Level-dependent Cross-validation Thresholding
lennonwd <- imwd(lennonnoise)
#lennoncv <- cvwavelet.image(images=lennonnoise, imagewd=lennonwd,
#      cv.optlevel=optlevel, cv.bsize=c(1,1), cv.kfold=10)$imagecv
#image(lennoncv, axes=FALSE, col=gray(100:0/100), 
#   main="Level-dependent CV")
# Generate Noisy Lennon Image
data(lennon)
sdimage <- sd(as.numeric(lennon))
nlennon <- ncol(lennon); nlevel <- log2(ncol(lennon))
optlevel <- c(3:(nlevel-1))
set.seed(55)
lennonnoise <- lennon+matrix(rnorm(nlennon^2, 0, sdimage), nlennon, nlennon)

# Level-dependent Cross-validation Thresholding
lennonwd <- imwd(lennonnoise)
#lennoncv <- cvwavelet.image(images=lennonnoise, imagewd=lennonwd,
#      cv.optlevel=optlevel, cv.bsize=c(1,1), cv.kfold=10)$imagecv
#image(lennoncv, axes=FALSE, col=gray(100:0/100), 
#   main="Level-dependent CV")

Cross-Validation Wavelet Shrinkage for two-dimensional data after imputation

Description

This function performs level-dependent cross-validation wavelet shrinkage for two-dimensional data given the cross-validation scheme and imputation values.

Usage

cvwavelet.image.after.impute(images, imagewd, imageimpute,
   cv.index1=cv.index1, cv.index2=cv.index2,
   cv.optlevel=cv.optlevel, cv.tol=cv.tol, cv.maxiter=cv.maxiter,
   filter.number=2, ll=3)
cvwavelet.image.after.impute(images, imagewd, imageimpute,
   cv.index1=cv.index1, cv.index2=cv.index2,
   cv.optlevel=cv.optlevel, cv.tol=cv.tol, cv.maxiter=cv.maxiter,
   filter.number=2, ll=3)

Arguments

`images`	noisy image
`imagewd`	two-dimensional wavelet transform
`imageimpute`	two-dimensional imputed values according to cross-validation scheme
`cv.index1`	test dataset row index according to cross-validation scheme
`cv.index2`	test dataset column index according to cross-validation scheme
`cv.optlevel`	thresholding levels
`cv.tol`	tolerance for cross-validation
`cv.maxiter`	maximum iteration for cross-validation
`filter.number`	specifies the smoothness of wavelet in the decomposition (argument of WaveThresh)
`ll`	specifies the lowest level to be thresholded

Details

Calculating thresholding values and reconstructing noisy image given cross-validation scheme and imputation.

Value

Reconstruction of images and thresholding values by level-dependent cross-validation

`imagecv`	reconstruction of images
`cvthresh`	thresholding values by level-dependent cross-validation

Doppler function

Description

This function generates Doppler function values for $n$ equally spaced points in $[0,1]$ .

Usage

dopp(norx=1024)
dopp(norx=1024)

Arguments

norx

the number of data or x values in [0, 1]

Details

Doppler function is introduced by Donoho and Johnstone (1994) and is useful test function evaluating a wavelet shrinkage method.

Value

Doppler function values $f(\frac{i}{n}), i=1,\ldots,n$ and its variability $||f|| = \frac{\sum_{i=1}^n (f_i - \bar f)^2}{n-1}$ where $\bar f = \frac{\sum_{i=1}^n f_i}{n}$ .

References

Donoho, D.L. and Johnstone, I.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425–455.

Examples

testdopp <- dopp(1024)
plot(testdopp$x, testdopp$meanf, xlab="", ylab="", 
     main="Plot of Doppler function", type="l")
testdopp <- dopp(1024)
plot(testdopp$x, testdopp$meanf, xlab="", ylab="", 
     main="Plot of Doppler function", type="l")

fg1 function

Description

This function generates fg1 function values for $n$ equally spaced points in $[0,1]$ .

Usage

fg1(norx=1024)
fg1(norx=1024)

Arguments

norx

the number of data or x values in [0, 1]

Details

A smooth function fg1 is introduced by Fan and Gijbels (1995) and is useful test function evaluating a wavelet shrinkage method.

Value

fg1 function values $f(\frac{i}{n}), i=1,\ldots,n$ and its variability $||f|| = \frac{\sum_{i=1}^n (f_i - \bar f)^2}{n-1}$ where $\bar f = \frac{\sum_{i=1}^n f_i}{n}$ .

References

Fan, J. and Gijbels, I. (1995) Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. Journal of the Royal Statistical Society Ser. B 57, 371–394.

Examples

testfg1 <- fg1(1024)
plot(testfg1$x, testfg1$meanf, xlab="", ylab="", 
     main="Plot of fg1 function", type="l")
testfg1 <- fg1(1024)
plot(testfg1$x, testfg1$meanf, xlab="", ylab="", 
     main="Plot of fg1 function", type="l")

Heavisine function

Description

This function generates Heavisine function values for $n$ equally spaced points in $[0,1]$ .

Usage

heav(norx=1024)
heav(norx=1024)

Arguments

norx

the number of data or x values in [0, 1]

Details

Heavisine function is introduced by Donoho and Johnstone (1994) and is useful test function evaluating a wavelet shrinkage method.

Value

Heavisine function values $f(\frac{i}{n}), i=1,\ldots,n$ and its variability $||f|| = \frac{\sum_{i=1}^n (f_i - \bar f)^2}{n-1}$ where $\bar f = \frac{\sum_{i=1}^n f_i}{n}$ .

References

Donoho, D.L. and Johnstone, I.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425–455.

Examples

testheav <- heav(1024)
plot(testheav$x, testheav$meanf, xlab="", ylab="", 
     main="Plot of Heavisine function", type="l")
testheav <- heav(1024)
plot(testheav$x, testheav$meanf, xlab="", ylab="", 
     main="Plot of Heavisine function", type="l")

Inductance plethysmography data

Description

4,096 observations of inductance plethysmography data regularly sampled at 50Hz starting at 1229.98 seconds.

Usage

data(ipd)
data(ipd)

Format

time series.

Source

This data set contains 4,096 observations of inductance plethysmography data regularly sampled at 50Hz starting at 1229.98 seconds. The data were collected in an investigation of the recovery of patients after general anesthesia.

The data set was used in Nason (1996) to illustrate cross-validation method for threshold selection. See the reference; Nason, G.P. (1996) Wavelet shrinkage by cross-validation. Journal of the Royal Statistical Society Ser. B 58, 463–479.

Piecewise polynomial function

Description

This function generates Piecewise polynomial function values for $n$ equally spaced points in $[0,1]$ .

Usage

ppoly(norx=1024)
ppoly(norx=1024)

Arguments

norx

the number of data or x values in [0, 1]

Details

Piecewise polynomial function with the discontinuity is introduced by Nason and Silverman (1994) and is useful test function evaluating a wavelet shrinkage method.

Value

Piecewise polynomial function values $f(\frac{i}{n}), i=1,\ldots,n$ and its variability $||f|| = \frac{\sum_{i=1}^n (f_i - \bar f)^2}{n-1}$ where $\bar f = \frac{\sum_{i=1}^n f_i}{n}$ .

References

Nason, G.P. and Silverman, B.W. (1994) The discrete wavelet transform in S. Journal of Computational and Graphical Statistics, 3, 163–191.

Examples

testpoly <- ppoly(1024)
plot(testpoly$x, testpoly$meanf, xlab="", ylab="", 
     main="Plot of Piecewise polynomial function", type="l")
testpoly <- ppoly(1024)
plot(testpoly$x, testpoly$meanf, xlab="", ylab="", 
     main="Plot of Piecewise polynomial function", type="l")

Package 'CVThresh'

Help Index

Imputation by wavelet

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Imputation for two-dimensional data by wavelet

Description

Usage

Arguments

Details

Value

References

See Also

Level-Dependent Cross-Validation Approach for Wavelet Thresholding

Description

Generating test dataset index for cross-validation

Description

Usage

Arguments

Details

Value

See Also

Examples

Generating test dataset index of two-dimensional data for cross-validation

Description

Usage

Arguments

Details

Value

See Also

Examples

Wavelet reconstruction by level-dependent Cross-Validation

Description

Usage

Arguments

Details

Value

See Also

Examples

Cross-Validation Wavelet Shrinkage after imputation

Description

Usage

Arguments

Details

Value

See Also

Examples

Wavelet reconstruction of image by level-dependent Cross-Validation

Description

Usage

Arguments

Details

Value

See Also

Examples

Cross-Validation Wavelet Shrinkage for two-dimensional data after imputation

Description

Usage

Arguments

Details

Value

See Also

Doppler function

Description

Usage

Arguments

Details

Value

References

See Also

Examples

fg1 function

Description

Usage