An Introduction to CNVreg
Package to Perform Copy Number Variant Association Analysis with
Penalied Regression
# load the package
library("CNVreg")Introduction
The CNVreg package provides functions to perform copy number variants (CNV) association analysis with penalized regression model.
This package converts CNVs over a genomic region as a piecewise constant curve to capture the dosage and length of CNVs. The association analysis is then evaluated by regressing outcome traits on all CNV fragments in the region while adjusting for covariates. The corresponding CNV effects are obtained at each genome position. The penalized regression model with Lasso and weighted fusion penalties would perform variable selection and encourage adjacent CNVs to share similar effect size.
This package has 3 main functions:
prep(): Data preprocessing and format conversion – required to prepare data for analysis.cvfit_WTSMTH(): Model fitting and effect estimate with cross-validation(CV). The CV procedure is to tune an optimal model by selecting the best pair of candidate tuning parameters.fit_WTSMTH(): Model fitting and effect estimate with a given pair of tuning parameters.
All functions use an example data included in the CNVreg
package.
Data
The CNVCOVY dataset included in this package contains a
small sample of data for demonstration purposes. It has 4 separate data
files: copy number variants data in CNV, covariate data in
Cov, and outcome traits Y_QT (quantitative)
and Y_BT(Binary).
CNV: A data frame describing CNV data in PLINK format with 5 variablesID,CHR,BP1,BP2, andTYPE.Cov: A data frame with 3 variables:ID,Sex, andAge.Y_QTandY_BT: each is a data frame for outcome traits.Y_QTcontains a quantitative trait.Y_BTcontains a binary trait. Both have 2 variables:IDandY.
Here is how you can load and view the summary of the datasets:
# load the example dataset
data("CNVCOVY", package = "CNVreg")Click to see summary information of each object loaded
Toy Datasets
CNV
# view the dataset
summary(CNV)
#> ID CHR BP1 BP2
#> Length:2680 Min. :1 Min. :118956400 Min. :118956600
#> Class :character 1st Qu.:1 1st Qu.:175325200 1st Qu.:175325800
#> Mode :character Median :1 Median :203709300 Median :203709400
#> Mean :1 Mean :191999781 Mean :192000245
#> 3rd Qu.:1 3rd Qu.:229563000 3rd Qu.:229564400
#> Max. :1 Max. :238592100 Max. :238593100
#> TYPE
#> Min. :0.0000
#> 1st Qu.:1.0000
#> Median :1.0000
#> Mean :0.9828
#> 3rd Qu.:1.0000
#> Max. :3.0000COV
# view the dataset
summary(Cov)
#> ID Sex Age
#> Length:900 Min. :0.0000 Min. :50.00
#> Class :character 1st Qu.:0.0000 1st Qu.:59.00
#> Mode :character Median :1.0000 Median :70.00
#> Mean :0.5011 Mean :69.82
#> 3rd Qu.:1.0000 3rd Qu.:80.00
#> Max. :1.0000 Max. :89.00Y_QT
# view the dataset
summary(Y_QT)
#> ID Y
#> Length:900 Min. :-4.8950
#> Class :character 1st Qu.:-2.3411
#> Mode :character Median :-1.3730
#> Mean : 0.4128
#> 3rd Qu.: 3.5676
#> Max. :16.7008Y_BT
# view the dataset
summary(Y_BT)
#> ID Y
#> Length:900 Min. :0.0000
#> Class :character 1st Qu.:0.0000
#> Mode :character Median :0.0000
#> Mean :0.4056
#> 3rd Qu.:1.0000
#> Max. :1.0000Briefly, the dataset has the CNV (2680 records), covariates
(Sex and Age), and outcome traits for 900
individuals.
Data preprocessing
The prep() function converts an individual’s CNV events
within a genomic region to fragments, and filters out rare events. It
analyzes the adjacency relationship between CNV fragments and prepares
different weight options for the penalized regression analysis.
Input of function prep()
The function prep() has 4 inputs.
CNV: takes a data frame describing CNV in PLINK format with 5 variables:ID,CHR,BP1,BP2, andTYPE.Y: takes a data frame describing outcomes with 2 variables:IDandY.Z: takes a data frame describing covariates, ifZis provided, one variable must beID, other variables can be any covariates of interest.rare.out: takes a number in [0, 0.5). A default value is 0.05, which excludes CNVs with frequency < \(5\%\)
Output of function prep()
The output of the prep() function has a specially
designed “WTsmth.data” format for easy application in the next step for
CNV association analysis. It has 6 components.
design: a matrix of the CNV fragments in n by p dimensions, where n is the number of samples and p is the total number of CNV fragments. Rownames are sample ID, and the order of rownames is the same as the rownames of the outcome file (frag_data_QT$Y).Z: a matrix of covariates with sample ID as rownames. The rownames are in the same order as in the outcome file (frag_data_QT$Y).Y: a matrix of 1 column with sample ID as rownames. The rownames are in the same order as in the CNV design matrixfrag_data_QT$designand covariatesfrag_data_QT$Z.weight.structure: a matrix that describes the adjacency structure of CNV fragments. The matrix is sparse and most values are zero, while non-zero values represent two adjacent CNV fragments that are overlapped by at least one CNV event in the population.weight.options: we provide 6 different options of weights that encourage differential information sharing based on the relationship between adjacent CNV fragments. Equal weighteql, Cosine-similarity based weightwcs, Inverse frequency weightwif, and combining these 3 weights with the frequency (k) of any CNV events within each CNV-active region (“keql”, “kwcs”, and “kwif”). Refer to the user manual for more information.CNVR.infosummarizes the positions of all CNV fragments and their adjacency information. Each row represents a CNV fragment and the fragment names match the column names infrag_data_QT$design.
Application of prep() function
Here is how you can use the prep() function to
preprocess CNV, Cov and an outcome trait. The
outcome trait can be a continuous trait Y_QT or a binary
trait Y_BT. The command syntax of the prep()
function is the same for continuous and binary outcomes. We showcase two
application examples, which will be used in the next step to illustrate
CNV association analysis with a continuous outcome and a binary
outcome.
Application 1: Process CNV, Cov, and a
continuous trait Y_QT
# data preprocessing for a quantitative(continuous) outcome Y_QT
frag_data_QT <- prep(CNV = CNV, Y = Y_QT, Z = Cov, rare.out = 0.05)
Click to see the summary details of the output frag_data_QT
# Format of `prep()` funtion output
str(frag_data_QT)
#> List of 6
#> $ design :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
#> .. ..@ i : int [1:5552] 9 13 15 21 24 38 55 57 62 66 ...
#> .. ..@ p : int [1:20] 0 177 320 465 608 752 1189 1914 2517 3081 ...
#> .. ..@ Dim : int [1:2] 900 19
#> .. ..@ Dimnames:List of 2
#> .. .. ..$ : chr [1:900] "U1" "U10" "U100" "U101" ...
#> .. .. ..$ : chr [1:19] "del1" "del3" "del4" "del5" ...
#> .. ..@ x : num [1:5552] 200 200 200 200 200 200 200 200 200 200 ...
#> .. ..@ factors : list()
#> $ Z :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
#> .. ..@ i : int [1:1351] 3 4 5 6 7 9 10 12 14 15 ...
#> .. ..@ p : int [1:3] 0 451 1351
#> .. ..@ Dim : int [1:2] 900 2
#> .. ..@ Dimnames:List of 2
#> .. .. ..$ : chr [1:900] "U1" "U10" "U100" "U101" ...
#> .. .. ..$ : chr [1:2] "Sex" "Age"
#> .. ..@ x : num [1:1351] 1 1 1 1 1 1 1 1 1 1 ...
#> .. ..@ factors : list()
#> $ Y : Named num [1:900] -1.816 -0.321 -2.624 -0.688 3.288 ...
#> ..- attr(*, "names")= chr [1:900] "U1" "U10" "U100" "U101" ...
#> $ weight.structure:Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
#> .. ..@ i : int [1:22] 1 1 2 2 3 3 4 4 5 5 ...
#> .. ..@ p : int [1:20] 0 0 1 3 5 6 7 9 11 12 ...
#> .. ..@ Dim : int [1:2] 15 19
#> .. ..@ Dimnames:List of 2
#> .. .. ..$ : NULL
#> .. .. ..$ : NULL
#> .. ..@ x : num [1:22] -1 1 -1 1 -1 1 -1 1 -1 1 ...
#> .. ..@ factors : list()
#> $ weight.options : num [1:6, 1:15] 0 0 0 0 0 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:6] "eql" "keql" "wcs" "kwcs" ...
#> .. ..$ : NULL
#> $ CNVR.info :'data.frame': 19 obs. of 7 variables:
#> ..$ grid.id : int [1:19] 1 3 4 5 6 8 9 10 11 13 ...
#> ..$ CNV.id : int [1:19] 1 2 2 2 2 3 3 3 3 4 ...
#> ..$ freq : int [1:19] 177 143 145 143 144 437 725 603 564 640 ...
#> ..$ CHR : int [1:19] 1 1 1 1 1 1 1 1 1 1 ...
#> ..$ CNV.start: num [1:19] 1.19e+08 1.21e+08 1.21e+08 1.21e+08 1.21e+08 ...
#> ..$ CNV.end : num [1:19] 1.19e+08 1.21e+08 1.21e+08 1.21e+08 1.21e+08 ...
#> ..$ deldup : chr [1:19] "del" "del" "del" "del" ...
#> - attr(*, "class")= chr "WTsmth.data"Application 2: Process CNV, Cov, and a
binary trait Y_BT
# data preprocessing with a binary trait
frag_data_BT <- prep(CNV = CNV, Y = Y_BT, Z = Cov, rare.out = 0.05)There is an alternative way to prepare data when performing CNV
association analysis with the same set of CNVs for multiple outcome
traits. Since we have the same CNV data, Cov
data, and a different outcome trait Y_BT, we can manually
format Y_BT to match the format in
frag_data_QT$Y.
Click to see alternative usage.
## copy frag_data_QT
frag_data_BT <- frag_data_QT
### replace Y with Y_BT in the correct format: ordered named vector
### order the sample in Y_BT as in frag_data_QT$Y
rownames(Y_BT) <- Y_BT$ID
frag_data_BT$Y <- Y_BT[names(frag_data_QT$Y), "Y"] |> drop()
names(frag_data_QT$Y) <- rownames(frag_data_QT$Y) Directly replace frag_data_QT$Z is also possible, keep
in mind to use the correct variable names and sample ID order.
Analysis with cross-validation (CV)
The cvfit_WTSMTH() function analyzes the association
between a continuous/binary trait value and CNV while
adjusting for the covariates Cov.
CNV association analysis for a continuous outcome
We already have frag_data_QT prepared in the
prep() step. We can fit a model to perform CNV association
analysis for a continuous outcome using CV to fine-tune tuning
parameters and fit an optimal model with the selected parameters.
QT_TUNE <- withr::with_seed(
12345,
cvfit_WTSMTH(data = frag_data_QT,
lambda1 = seq(-8, -3, 1),
lambda2 = seq(12, 25, 2),
weight = "eql",
family = "gaussian",
cv.control = list(n.fold = 5L,
n.core = 1L,
stratified = FALSE),
verbose = FALSE))
#> Warning: executing %dopar% sequentially: no parallel backend registeredInput of function cvfit_WTSMTH()
data: Thecvfit_WTSMTH()function takes adatainWTsmth.dataformat, which is the output ofprep()function, as one of the major inputs, for example,frag_data_QTprepared for the continuous trait andfrag_data_BTprepared for the binary trait.lambda1andlambda2: take the candidate tuning parameters that control variable selection (lambda1) and effect smoothness (lambda2). Provided values will be transformed to 2^(lambda1) and 2^(lambda2). We provide default values for both vectors. The user can customize the range and step_size of the candidate tuning parameters. In most cases, the user will need to run the function more than one time to adjust the range and step_size of tuning parameters to locate to a reasonable range according to the previous round of model fitting.weight: it has six different options as described earlier. Since we only have a small dataset, varying theweightoptions will not have much influence on the model fitting results. In real CNV data with different similarity patterns and CNV frequencies, varying theweightoption are expected to have different effects.family: has two options:gaussianfor a continuous outcome, andbinomialfor a binary outcome.cv.control: This function also supports parallel computing and change of n-folds in CV by adjusting thecv.controllist.n.foldcontrols the number of folds used in CV.n.corecontrols the the number of cores used in parallel computing.stratifiedonly has control for a binary outcome. We will skip it here and describe it in the binary section.
verbose: If chooseverbose= TRUE, it will print a message about where the program is currently working on.
Output of function cvfit_WTSMTH()
The output of the cvfit_WTSMTH() function is a list
object containing 3 elements: Loss,
lambda.selected, and coef.
LossThe
Losskeeps track of the average validation loss in CV for each pair of candidate tuning parameters \(\lambda_{1}\) and \(\lambda_{2}\). In the following table, the minimum loss is highlighted and the corresponding \(\lambda_{1}\) and \(\lambda_{2}\) values are selected to fit a final model.In this simulated data, the variation of loss for different \(\lambda_{2}\) with the same \(\lambda_{1}\) is not very large. One reason is that \(\lambda_{2}\) controls the effect smoothness between adjacent CNVs, and the simulation data only has a small number of CNVs in adjacent that share effects to other CNVs. The effect of changing \(\lambda_{2}\) seems not prominence in this case. When we have more CNVs in adjacent and share effects, it should have larger variance across \(\lambda_{2}\).
| Lambda2 | -8 | -7 | -6 | -5 | -4 | -3 |
|---|---|---|---|---|---|---|
| 12 | 1.004534 | 1.003579 | 0.998444 | 0.993686 | 0.997349 | 1.02374 |
| 14 | 1.008237 | 1.007236 | 1.002091 | 0.997715 | 1.002085 | 1.02904 |
| 16 | 1.005668 | 1.004547 | 0.999433 | 0.995208 | 0.999796 | 1.02829 |
| 18 | 1.001295 | 1.000519 | 0.995631 | 0.991565 | 0.996885 | 1.02515 |
| 20 | 0.999942 | 0.998498 | 0.993783 | 0.990641 | 0.996864 | 1.02829 |
| 22 | 0.998218 | 0.996816 | 0.992557 | 0.991599 | 1.000824 | 1.04408 |
| 24 | 0.998674 | 0.995685 | 0.992663 | 0.995447 | 1.015663 | 1.10326 |
selected.lambda
The selected.lambda is the optimal tuning parameters
from the candidate lists that has the lowest loss, which can be
confirmed with the Loss table.
# selected optimal tuning parameters with minimum loss
QT_TUNE$selected.lambda
#> [1] -5 20coef
The coef shows the estimated beta coefficients at the
selected tuning parameters. It has (intercept), CNV
fragments (with detailed positions/type information), and covariate
effects. In this small example, we can print all coefficient estimate,
but you can modify the code to show only non-zero ones.
Here lists the coefficients for (Intercept) and
covariates. The characteristics for CNV (CHR, CNV.start, CNV.end, and
deldup) are left as NAs intentionally in the original
output. Here we only show the effect estimate.
##coefficients of intercept and covariates
QT_TUNE$coef[c(1, 21:22), c("Vnames", "coef") ]
#> Vnames coef
#> 1 (Intercept) -1.99855
#> 21 Sex 0.00000
#> 22 Age 0.00000Here lists the coefficients for CNVs and the corresponding plots.
We highlight the regions with adjacent CNVs. From the coefficient estimates, the model selects several non-zero CNVs (data points). Among the data points, the red ones have stronger effect than the black dots. The black dots are likely noise.
We also zoom in on two highlighted regions with strong signals and adjacent CNVs to show the effect smoothness within the regions.
The results illustrate the variable selection and effect smoothness of the penalized regression method for CNV association analysis.
Click to see CNV coefficient estimates.
# estimated coefficents for CNV
QT_TUNE$coef[2:20, ]
#> Vnames CHR CNV.start CNV.end deldup coef
#> 2 del1 1 118956400 118956600 del 0.0000000000
#> 3 del3 1 121299300 121299500 del 0.0048659084
#> 4 del4 1 121299500 121299700 del 0.0046967844
#> 5 del5 1 121299700 121299800 del 0.0046911466
#> 6 del6 1 121299800 121300400 del 0.0051456642
#> 7 del8 1 175325200 175325400 del -0.0001740566
#> 8 del9 1 175325400 175325500 del 0.0000000000
#> 9 del10 1 175325500 175325600 del 0.0000000000
#> 10 del11 1 175325600 175325800 del 0.0000000000
#> 11 del13 1 203709300 203709400 del -0.0002731880
#> 12 del17 1 229563000 229563200 del 0.0042092931
#> 13 del18 1 229563200 229563500 del 0.0046547584
#> 14 del19 1 229563500 229563900 del 0.0048148656
#> 15 del20 1 229563900 229564400 del 0.0057050100
#> 16 del23 1 235735000 235735100 del 0.0000000000
#> 17 del25 1 238591800 238592100 del 0.0000000000
#> 18 del26 1 238592100 238592900 del 0.0000000000
#> 19 del27 1 238592900 238593100 del 0.0000000000
#> 20 dup15 1 212455200 212455300 dup 0.0000000000
# non-zero coefficients
# QT_TUNE$coef[which(abs(QT_TUNE$coef$coef)>0), ] 
CNV association analysis for a binary outcome
BT_TUNE <- withr::with_seed(
12345,
cvfit_WTSMTH(frag_data_BT,
lambda1 = seq(-5.25, -4.75, 0.25),
lambda2 = seq(2, 8, 2),
weight = "eql",
family = "binomial",
cv.control = list(n.fold = 5L,
n.core = 1L,
stratified = FALSE),
iter.control = list(max.iter = 8L,
tol.beta = 10^(-3),
tol.loss = 10^(-6)),
verbose = FALSE))
Input of function cvfit_WTSMTH()
The CNV association analysis for a bianry outcome has similar inputs as for a continuous outcome. The differences are:
The output
frag_data_BTfrom theprep()step has a binary traitY_BTready for CNV association analysis with a binary outcome.Choose
family= “binomial” for a binary trait.There are a few more options specifically designed for the binary trait.
stratifiedwithin thecv.controllist: If one category of the binary outcome is considered “rare”,stratified= TRUE is recommended to make sure the data splits are having the same proportion of cases and controls in each fold.iter.control: For a binary outcome, we can also adjust theiter.controllist with desired threshold that is deemed converged for coefficient estimate of a binary outcome. Refer to the user manual for more details.
Output of function cvfit_WTSMTH()
The output of the cvfit_WTSMTH() function has the same
list object containing 3 elements: Loss,
lambda.selected, and coef.
Loss
TheLoss` keeps track of the average validation loss in
CV for each pair of candidate tuning parameters \(\lambda_{1}\) and \(\lambda_{2}\). In the following table, the
minimum loss is highlighted and the corresponding \(\lambda_{1}\) and \(\lambda_{2}\) values are selected to fit a
final model.
Since the regression process for a binary trait takes longer time to converge, here we only use a short list of candidate tuning parameters for illustration purpose.
| Lambda2 | -5.25 | -5 | -4.75 |
|---|---|---|---|
| 2 | 0.281147 | 0.280297 | 0.279914 |
| 4 | 0.279573 | 0.278999 | 0.278696 |
| 6 | 0.278115 | 0.277811 | 0.277819 |
| 8 | 0.278173 | 0.278394 | 0.279038 |
selected.lambda
The selected.lambda are the optimal tuning parameters
from the candidate lists that has the lowest loss, which can be
confirmed with the Loss table.
# selected optimal tuning parameters with minimum loss
BT_TUNE$selected.lambda
#> [1] -5 6coef
The estimated beta coefficients coef at the selected
tuning parameters. It has (intercept), CNV fragments (with
detailed positions/type information), and covariate effects. In this
small data example, we can print all coefficient estimate, but you can
modify the code to show non-zero ones or the first few ones.
Here lists the coefficients for (Intercept) and
covariates.
BT_TUNE$coef[c(1, 21:22), c("Vnames", "coef") ]
#> Vnames coef
#> 1 (Intercept) -1.953386
#> 21 Sex 0.000000
#> 22 Age 0.000000Here lists the coefficients for CNVs and the corresponding plots.
Click to see CNV coefficient estimates.
BT_TUNE$coef[2:20, ]
#> Vnames CHR CNV.start CNV.end deldup coef
#> 2 del1 1 118956400 118956600 del 0.000000e+00
#> 3 del3 1 121299300 121299500 del 2.879470e-03
#> 4 del4 1 121299500 121299700 del 3.530621e-03
#> 5 del5 1 121299700 121299800 del 3.023385e-03
#> 6 del6 1 121299800 121300400 del 4.925100e-03
#> 7 del8 1 175325200 175325400 del 0.000000e+00
#> 8 del9 1 175325400 175325500 del 0.000000e+00
#> 9 del10 1 175325500 175325600 del 0.000000e+00
#> 10 del11 1 175325600 175325800 del 4.010457e-05
#> 11 del13 1 203709300 203709400 del 0.000000e+00
#> 12 del17 1 229563000 229563200 del 1.650097e-03
#> 13 del18 1 229563200 229563500 del 2.782514e-03
#> 14 del19 1 229563500 229563900 del 4.412988e-03
#> 15 del20 1 229563900 229564400 del 5.905259e-03
#> 16 del23 1 235735000 235735100 del 0.000000e+00
#> 17 del25 1 238591800 238592100 del 0.000000e+00
#> 18 del26 1 238592100 238592900 del 0.000000e+00
#> 19 del27 1 238592900 238593100 del 0.000000e+00
#> 20 dup15 1 212455200 212455300 dup 0.000000e+00
Analysis with fixed tuning parameters
The user can use the function fit_WTSMTH() to reproduce
the regression result with the selected pair of parameters.
The fit_WTSMTH() function and the
cvfit_WTSMTH() function uses the same analytical methods to
perform CNV association analysis with penalized regression. Unlike the
cvfit_WTSMTH() function that will fine-tune the parameters
and select the optimal combination of \(\lambda_{1}\) and \(\lambda_{2}\) from a series of candidates,
the fit_WTSMTH() function takes a user-specified value for
\(\lambda_{1}\) and \(\lambda_{2}\) and estimate the coefficients
for the given pair of parameters. Although, it is much faster to perform
fit_WTSMTH(), we do recommend the user to stick with the
parameter tuning procedure with cvfit_WTSMTH() and find the
best parameters and the corresponding model.
If we already have a selected the best pair of parameters \(\lambda_{1}\) and \(\lambda_{2}\) from the CV procedure, we can refit the regression model, and the coefficient estimate is the same as in the fine-tuned model.
# we know the optimal tuning parameters and directly apply it to reproduce the analysis for a continuous outcome.
QT_fit <- fit_WTSMTH(frag_data_QT,
lambda1 = -5,
lambda2 = 20,
weight="eql",
family="gaussian")Input of function fit_WTSMTH()
The input variables of function fit_WTSMTH() is similar
to that of cvfit_WTSMTH().
data: it also takesdatainWTsmth.dataformat as prepared in theprep()step.lambda1andlambda2: each takes one numeric value. Provided values will be transformed to 2^(lambda1) and 2^(lambda2).weight: it takes one of the options from {eql,keql,wcs,kwcs,wif,kwif} as mentioned earlier.family: it takesgaussianfor a continuous outcome andbinomialfor a binary outcome.iter.control: for a binary outcome,iter.controlcontrols iterative coefficient estimate procedure as described earlier.
Output of fit_WTSMTH()
Refit the model with the selected pair of tuning parameters for a
continuous outcome Y_QT. The coefficient estimate from
fit_WTSMTH() is the same as the coefficient estimate of
cvfit_WTSMTH().
Here lists the coefficients for (Intercept) and
covariates.
QT_fit[c(1, 21:22), c("Vnames", "coef") ]
#> Vnames coef
#> 1 (Intercept) -1.99855
#> 21 Sex 0.00000
#> 22 Age 0.00000Here lists the coefficients for CNVs and the corresponding plots.
Click to see CNV coefficient estimates.
QT_fit[2:20, ]
#> Vnames CHR CNV.start CNV.end deldup coef
#> 2 del1 1 118956400 118956600 del 0.0000000000
#> 3 del3 1 121299300 121299500 del 0.0048659084
#> 4 del4 1 121299500 121299700 del 0.0046967844
#> 5 del5 1 121299700 121299800 del 0.0046911466
#> 6 del6 1 121299800 121300400 del 0.0051456642
#> 7 del8 1 175325200 175325400 del -0.0001740566
#> 8 del9 1 175325400 175325500 del 0.0000000000
#> 9 del10 1 175325500 175325600 del 0.0000000000
#> 10 del11 1 175325600 175325800 del 0.0000000000
#> 11 del13 1 203709300 203709400 del -0.0002731880
#> 12 del17 1 229563000 229563200 del 0.0042092931
#> 13 del18 1 229563200 229563500 del 0.0046547584
#> 14 del19 1 229563500 229563900 del 0.0048148656
#> 15 del20 1 229563900 229564400 del 0.0057050100
#> 16 del23 1 235735000 235735100 del 0.0000000000
#> 17 del25 1 238591800 238592100 del 0.0000000000
#> 18 del26 1 238592100 238592900 del 0.0000000000
#> 19 del27 1 238592900 238593100 del 0.0000000000
#> 20 dup15 1 212455200 212455300 dup 0.0000000000
However, if we choose a random pair of tuning parameters,
the function will not have optimal variable selection and effect
smoothness performance.
In summary, the cvfit_WTSMTH() function is recommended
for model fitting with CV. fit_WTSMTH() can be used for
testing purpose or to reproduce the result.