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>.

Details

The DESCRIPTION file:

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).

Author(s)

Rebecca Wilson [aut], Matt Nunes [aut, cre], Idris Eckley [ctb, ths], Tim Park [ctb]

Maintainer: Matt Nunes <nunesrpackages@gmail.com>

References

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.

Function to smooth the raw wavelet periodogram

Description

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.

Usage

correct_per(RawPer)

Arguments

Details

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.

Value

Returns a mvLSW object containing the smoothed EWS of a multivariate locally stationary time series.

Author(s)

Rebecca Wilson

References

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.

See Also

mvEWS, spec_estimation

Examples

## 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)

Function to form the local autocovariance array for the forecasting / backcasting step.

Description

This function generates the local autocovariance (LACV) array that is used in the forecasting / backcasting step to form the prediction equations.

Usage

form_lacv_forward(spectrum, index, p.len = 2)
form_lacv_backward(spectrum, index, p.len = 2)

Arguments

Details

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.

Value

Returns the local autocovariance array that can be used as an input to the pred_eq_forward or pred_eq_backward function.

Author(s)

Rebecca Wilson

References

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.

See Also

pred_eq_forward, pred_eq_backward

Examples

## 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)

Function to apply the (univariate) Haar wavelet transform

Description

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.

Usage

haarWT(data)

Arguments

Value

Returns a list containing the following elements:

Examples

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])

Function to apply the mvLSWimpute method and impute missing values in a multivariate locally stationary time series

Description

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.

Usage

mv_impute(data, p = 2, type = "forward", index = NULL)

Arguments

Value

Returns a list containing the following elements:

Note

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.

Author(s)

Rebecca Wilson

References

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.

Examples

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)

Function to regularise the LWS matrix.

Description

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.

Usage

pdef(spec, W = 1e-10)

Arguments

Value

Returns a mvLSW object containing the regularised EWS of a multivariate locally stationary time series.

Author(s)

Rebecca Wilson

References

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.

See Also

mvEWS, spec_estimation

Examples

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)

Function to form the prediction equations for the forecasting / backcasting step.

Description

This function generates the prediction equations (B matrix and RHS matrix) for one step ahead prediction.

Usage

pred_eq_forward(lacv.array, p = 2, index)
pred_eq_forward(lacv.array, p = 2, index)

Arguments

Details

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.

Value

Returns a list containing the following elements:

Author(s)

Rebecca Wilson

References

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.

See Also

form_lacv_forward, pred_eq_backward

Examples

## 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)

Function to smooth the raw wavelet periodogram using the default mvLSW routine.

Description

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.

Usage

smooth_per(RawPer, type = "all", kernel.name="daniell", optimize = FALSE, kernel.param =
                 NULL, smooth.Jset = NULL)

Arguments

Value

Returns a mvLSW object containing the smoothed EWS of a multivariate locally stationary time series.

Author(s)

Rebecca Wilson

References

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.

See Also

mvEWS, spec_estimation

Examples

## 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)

Function to estimate the Local Wavelet Spectral matrix for a multivariate locally stationary time series containing missing values

Description

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.

Usage

spec_estimation(data, interp = "linear")

Arguments

Value

Returns a mvLSW object containing the estimated LWS matrix.

Note

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.

See Also

correct_per, smooth_per, mvEWS, na_interpolation

Examples

## 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)