| Title: | Imputation Methods for Multivariate Locally Stationary Time Series |
|---|---|
| Description: | Implementation of imputation techniques based on locally stationary wavelet time series forecasting methods from Wilson, R. E. et al. (2021) <doi:10.1007/s11222-021-09998-2>. |
| Authors: | Rebecca Wilson [aut], Matt Nunes [aut, cre], Idris Eckley [ctb, ths], Tim Park [ctb] |
| Maintainer: | Matt Nunes <[email protected]> |
| License: | GPL-2 |
| Version: | 0.1.1 |
| Built: | 2024-12-08 06:49:48 UTC |
| Source: | CRAN |
Implementation of imputation techniques based on locally stationary wavelet time series forecasting methods from Wilson, R. E. et al. (2021) <doi:10.1007/s11222-021-09998-2>.
The DESCRIPTION file:
| Package: | mvLSWimpute |
| Type: | Package |
| Title: | Imputation Methods for Multivariate Locally Stationary Time Series |
| Version: | 0.1.1 |
| Date: | 2022-08-15 |
| Author: | Rebecca Wilson [aut], Matt Nunes [aut, cre], Idris Eckley [ctb, ths], Tim Park [ctb] |
| Authors@R: | c(person("Rebecca", "Wilson", role = "aut"), person("Matt", "Nunes", role=c("aut","cre"), email="[email protected]"), person("Idris", "Eckley", role=c("ctb","ths")), person("Tim","Park", role="ctb")) |
| Maintainer: | Matt Nunes <[email protected]> |
| Description: | Implementation of imputation techniques based on locally stationary wavelet time series forecasting methods from Wilson, R. E. et al. (2021) <doi:10.1007/s11222-021-09998-2>. |
| License: | GPL-2 |
| Depends: | wavethresh, mvLSW |
| Imports: | binhf, xts, zoo, imputeTS, utils |
| NeedsCompilation: | no |
| Packaged: | 2022-08-15 14:32:45 UTC; matt |
| Repository: | CRAN |
| Date/Publication: | 2022-08-16 10:20:02 UTC |
| Config/pak/sysreqs: | libicu-dev libjpeg-dev libpng-dev libxml2-dev libssl-dev |
Index of help topics:
correct_per Function to smooth the raw wavelet periodogram
form_lacv_forward Function to form the local autocovariance array
for the forecasting / backcasting step.
haarWT Function to apply the (univariate) Haar wavelet
transform
mvLSWimpute-package Imputation Methods for Multivariate Locally
Stationary Time Series
mv_impute Function to apply the mvLSWimpute method and
impute missing values in a multivariate locally
stationary time series
pdef Function to regularise the LWS matrix.
pred_eq_forward Function to form the prediction equations for
the forecasting / backcasting step.
smooth_per Function to smooth the raw wavelet periodogram
using the default 'mvLSW' routine.
spec_estimation Function to estimate the Local Wavelet Spectral
matrix for a multivariate locally stationary
time series containing missing values
The main routine of the package is mv_impute which performs forward or forward and backward imputation of locally stationary multivariate time series, using one-step ahead forecasting (and backcasting).
Rebecca Wilson [aut], Matt Nunes [aut, cre], Idris Eckley [ctb, ths], Tim Park [ctb]
Maintainer: Matt Nunes <[email protected]>
Wilson, R. E., Eckley, I. A., Nunes, M. A. and Park, T. (2021) A wavelet-based approach for imputation in nonstationary multivariate time series. _Statistics and Computing_ *31* Article 18, doi:10.1007/s11222-021-09998-2.
This function corrects the raw wavelet periodogram, similar to the mvEWS function in the mvLSW package, except acting on the raw periodogram directly. See
mvEWS for more details. Note: this function is not really intended to be used separately, but internally within the spec_estimation function.
correct_per(RawPer)correct_per(RawPer)
RawPer |
Raw wavelet periodogram that is to be corrected, can be either a 4D array or a |
The raw wavelet periodogram as an estimator for the local wavelet spectrum (LWS) is biased, and thus needs to be corrected. This is done using a correction (debiasing) matrix, formed from the inner product of autocorrelation wavelets, see Park et al. (2014), Taylor et al. (2019) for more details. This function performs this bias-correction.
Returns a mvLSW object containing the smoothed EWS of a multivariate locally stationary time series.
Rebecca Wilson
Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package.
_Journal of Statistical Software_ *90*(11) pp. 1-16, doi:10.18637/jss.v090.i11.
Park, T., Eckley, I. and Ombao, H.C. (2014) Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. _IEEE Transactions on Signal Processing_ *62*(20) pp. 5240-5250.
## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram, reshaping array as necessary tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now correct correctedper = correct_per(RawPer)## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram, reshaping array as necessary tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now correct correctedper = correct_per(RawPer)
This function generates the local autocovariance (LACV) array that is used in the forecasting / backcasting step to form the prediction equations.
form_lacv_forward(spectrum, index, p.len = 2) form_lacv_backward(spectrum, index, p.len = 2)form_lacv_forward(spectrum, index, p.len = 2) form_lacv_backward(spectrum, index, p.len = 2)
spectrum |
Local wavelet spectral matrix for which we wish to form the local autocovariance array. |
index |
Time index of the missing data which we wish to impute. |
p.len |
Number of terms to include in the clipped predictor when forecasting / backcasting. |
In order to form the one-step ahead predictor for use in the imputation algorithm of Wilson et al. (2021), one needs the local autocovariance (LACV). This is computed using the relationship between the LACV and the local wavelet spectrum (LWS). See equations (4) and (5) in Wilson et al. (2021) for more details.
Returns the local autocovariance array that can be used as an input to the pred_eq_forward or pred_eq_backward function.
Rebecca Wilson
Wilson, R. E., Eckley, I. A., Nunes, M. A. and Park, T. (2021) A wavelet-based approach for imputation in nonstationary multivariate time series.
_Statistics and Computing_ *31* Article 18, doi:10.1007/s11222-021-09998-2.
Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package.
_Journal of Statistical Software_ *90*(11) pp. 1-16, doi:10.18637/jss.v090.i11.
Park, T., Eckley, I. and Ombao, H.C. (2014) Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. _IEEE Transactions on Signal Processing_ *62*(20) pp. 5240-5250.
pred_eq_forward, pred_eq_backward
## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X) out <- form_lacv_forward(spec$spectrum, missing.index[1], p.len=2)## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X) out <- form_lacv_forward(spec$spectrum, missing.index[1], p.len=2)
This function applies the (univariate) Haar wavelet transform. For a time series containing missing values, the wavelet coefficients are generating and any NAs remain intact.
haarWT(data)haarWT(data)
data |
Input univariate time series. |
Returns a list containing the following elements:
C |
Matrix containing the smooth coefficients for the transform. |
D |
Matrix containing the detail coefficients for the transform. |
set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # compute the haar wavelet coefficients of the first time series component: Xwt1 = haarWT(X[, 1])set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # compute the haar wavelet coefficients of the first time series component: Xwt1 = haarWT(X[, 1])
This function applies the mvLSWimpute method to impute missing values in a multivariate locally stationary time series. The imputation can be based on forecasts only or use information from both a forecasting and backcasting step.
mv_impute(data, p = 2, type = "forward", index = NULL)mv_impute(data, p = 2, type = "forward", index = NULL)
data |
Input multivariate time series, matrix of dimension TxP where P is the number of channels and T is the length of the series. |
p |
The number of terms to include in the clipped predictor when carrying out one step ahead forecasting/backcasting. |
type |
The type of imputation to carry out, either |
index |
The set of time indices containing missing values, this is |
Returns a list containing the following elements:
ImputedData |
Matrix containing the imputed time series. |
missing.index |
Vector containing the set of time indices that have missing values. |
As with other time series imputation methods, mv_impute requires some data values at the start of the series. In this case, we need 5 time points.
Rebecca Wilson
Wilson, R. E., Eckley, I. A., Nunes, M. A. and Park, T. (2021) A wavelet-based approach for imputation in nonstationary multivariate time series. _Statistics and Computing_ *31* Article 18, doi:10.1007/s11222-021-09998-2.
set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some fake missing data, taking care not to have missingness hear the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <- NA newdata = mv_impute(X)set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some fake missing data, taking care not to have missingness hear the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <- NA newdata = mv_impute(X)
This function regularises each EWS matrix to ensure that they are strictly positive definite, similar to the mvEWS function in the mvLSW package, except acting on a (bias-corrected) periodogram directly. See
mvEWS for more details. Note: this function is not really intended to be used separately, but internally within the spec_estimation function.
pdef(spec, W = 1e-10)pdef(spec, W = 1e-10)
spec |
EWS matrix that is to be regularised, can be either a 4D array or a |
W |
Tolerance in applying matrix regularisation to ensure each EWS matrix to be strictly positive definite. This is |
Returns a mvLSW object containing the regularised EWS of a multivariate locally stationary time series.
Rebecca Wilson
Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package.
_Journal of Statistical Software_ *90*(11) pp. 1-16, doi:10.18637/jss.v090.i11.
Park, T., Eckley, I. and Ombao, H.C. (2014) Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. _IEEE Transactions on Signal Processing_ *62*(20) pp. 5240-5250.
set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now correct correctedper = correct_per(RawPer) # now regularize newper = pdef(correctedper)set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now correct correctedper = correct_per(RawPer) # now regularize newper = pdef(correctedper)
This function generates the prediction equations (B matrix and RHS matrix) for one step ahead prediction.
pred_eq_forward(lacv.array, p = 2, index) pred_eq_forward(lacv.array, p = 2, index)pred_eq_forward(lacv.array, p = 2, index) pred_eq_forward(lacv.array, p = 2, index)
lacv.array |
The local autocovariance array from which we want to form the prediction equations, can be obtained as the output of the |
p |
Number of terms to include in the clipped predictor when forecasting / backcasting. |
index |
Time index of the missing data which we wish to impute. |
The one-step ahead predictor is formed as a linear combination of the time series. The coefficients involved in optimal predictor (in the sense of minimising the mean square prediction error) are obtained by solving a matrix equation formed using parts of the (estimated) local autocovariance array. This function forms the matrices involved in the equation used to find the optimal linear predictor. See equation (6) in Wilson et al. (2021) or Section 3.3 in Fryzlewicz et al. (2003) for more details.
Returns a list containing the following elements:
B |
The left-hand side of the matrix equation to compute the optimal one-step ahead predictor, which is essentially used to approximate the MSPE for a particular set of coefficients used in a predictor. |
RHS |
The right hand side of the matrix equation used to compute the optimal one-step ahead predictor. |
Rebecca Wilson
Wilson, R. E., Eckley, I. A., Nunes, M. A. and Park, T. (2021) A wavelet-based approach for imputation in nonstationary multivariate time series.
_Statistics and Computing_ *31* Article 18, doi:10.1007/s11222-021-09998-2.
Fryzlewicz, P. van Bellegem, S. and von Sachs, R. (2003) Forecasting non-stationary time series by wavelet process modelling. _Annals of the Institute of Statistical Mathematics_ *55* (4), pp. 737-764.
form_lacv_forward, pred_eq_backward
## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X) # obtain the LACV lacvfor <- form_lacv_forward(spec$spectrum, missing.index[1], p.len = 2) # form matrix equation terms mspeterms = pred_eq_forward(lacvfor, p = 2, missing.index[1]) # compute the optimal coefficients in the linear predictor: predcoeffs = solve(mspeterms$B, mspeterms$RHS)## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X) # obtain the LACV lacvfor <- form_lacv_forward(spec$spectrum, missing.index[1], p.len = 2) # form matrix equation terms mspeterms = pred_eq_forward(lacvfor, p = 2, missing.index[1]) # compute the optimal coefficients in the linear predictor: predcoeffs = solve(mspeterms$B, mspeterms$RHS)
mvLSW routine.This function smooths the raw wavelet periodogram, similar to the mvEWS function in the mvLSW package, except acting on the raw periodogram directly. See
mvEWS for more details. Note: this function is not really intended to be used separately, but internally within the spec_estimation function.
smooth_per(RawPer, type = "all", kernel.name="daniell", optimize = FALSE, kernel.param = NULL, smooth.Jset = NULL)smooth_per(RawPer, type = "all", kernel.name="daniell", optimize = FALSE, kernel.param = NULL, smooth.Jset = NULL)
RawPer |
Raw wavelet periodogram that is to be smoothed, can be either a 4D array or a |
type |
Determines the type of smoothing to be performed, if |
kernel.name |
Name of the smoothing kernel to be applied. |
optimize |
Should the smoothing be optimized. If |
kernel.param |
Value of the smoothing kernel parameter to be applied. |
smooth.Jset |
Vector indicating which levels should be used in the calculation of the optimal kernel parameter. By default all levels are used. |
Returns a mvLSW object containing the smoothed EWS of a multivariate locally stationary time series.
Rebecca Wilson
Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package.
_Journal of Statistical Software_ *90*(11) pp. 1-16, doi:10.18637/jss.v090.i11.
Park, T., Eckley, I. and Ombao, H.C. (2014) Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. _IEEE Transactions on Signal Processing_ *62*(20) pp. 5240-5250.
## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) #sqrv <- function(d) return( d %*% t(d) ) #RawPer = array(apply(D, c(2, 3), sqrv), dim = c(2, 2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now smooth smoothper = smooth_per(RawPer)## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # form periodogram tmp = apply(X, 2, function(x){haarWT(x)$D}) D = array(t(tmp), dim = c(2, 2^8, 8)) #sqrv <- function(d) return( d %*% t(d) ) #RawPer = array(apply(D, c(2, 3), sqrv), dim = c(2, 2, 2^8, 8)) RawPer = array(apply(D, c(2, 3), tcrossprod), dim = c(2, 2, 2^8, 8)) RawPer = aperm(RawPer, c(1, 2, 4, 3)) # now smooth smoothper = smooth_per(RawPer)
This function estimates the LWS matrix for a multivariate locally stationary time series containing missing values. If the input time series does not contain missing values then spectral estimation is carried out using routines from the mvLSW package.
spec_estimation(data, interp = "linear")spec_estimation(data, interp = "linear")
data |
Input multivariate time series, matrix of dimension TxP where P is the number of channels and T is the length of the series. |
interp |
Method of interpolation of NAs in spectrum. Can be |
Returns a mvLSW object containing the estimated LWS matrix.
For some series with a lot of missing values, the linear interpolation will result in zero periodogram values (due to the form of the Haar filters). This may not be desirable, so a higher order (spline) interpolation function may be better.
correct_per, smooth_per, mvEWS, na_interpolation
## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X)## Sample bivariate locally stationary time series set.seed(1) X <- matrix(rnorm(2 * 2^8), ncol = 2) X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1]) X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1] X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4 X <- as.ts(X) # create some missing values, taking care to provide some data at the start of the series missing.index = sort(sample(10:2^8, 30)) X[missing.index, ] <-NA # estimate the spectrum spec = spec_estimation(X)