Wavelet Methods for Analysing Locally Stationary Time Series

Description

Provides wavelet-based methods for trend, spectrum and autocovariance estimation of locally stationary time series. See TLSW for the main estimation function.

Details

Author(s)

Euan T. McGonigle <e.t.mcgonigle@soton.ac.uk>, Rebecca Killick <r.killick@lancs.ac.uk>, and Matthew Nunes <m.a.nunes@bath.ac.uk>

Maintainer: Euan T. McGonigle <e.t.mcgonigle@soton.ac.uk>

References

Spectral estimation with differencing/nonlinear trend estimator: McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

Spectral estimation in presence of trend/linear trend estimator: McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

LSW processes without trend: Nason, G. P., von Sachs, R., and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(2), 271–292.

lacf estimation without trend: Nason, G. P. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(5), 879–904.

See Also

TLSW,TLSWsim,plot.TLSW

Examples

# simulates an example time series and estimates its trend and evolutionary wavelet spectrum

spec <- matrix(0, nrow = 9, ncol = 512)
spec[1,] <- 1 + sin(seq(from = 0, to = 2 * pi, length = 512))^2

trend <- seq(from = 0, to = 5, length = 512)

set.seed(1)

x <- TLSWsim(trend = trend, spec = spec)

x.TLSW <- TLSW(x)

summary(x.TLSW)

plot(x.TLSW)

Lagged Autocorrelation Wavelet Inner Product Calculation

Description

Internal function for computing the matrix of lagged autocorrelation wavelet inner products. This is not intended for general use by regular users of the package.

Usage

Atau.mat.calc(J, filter.number = 1, family = "DaubExPhase", lag = 1)

Arguments

Details

Computes the lagged inner product matrix of the discrete non-decimated autocorrelation wavelets. This matrix is used in the calculation to correct the wavelet periodogram of the differenced time series. With lag = \tau, the matrix returned is the matrix A^\tau in McGonigle et al. (2022).

Value

A J-dimensional square matrix giving the lagged inner product autocorrelation wavelet matrix.

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

Nason, G. P., von Sachs, R., and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(2), 271–292.

See Also

TLSW

Cross Autocorrelation Wavelet Inner Product Matrix Calculation

Description

Internal function to compute the cross autocorrelation matrix of inner products. This is not intended for general use by regular users of the package.

Usage

Cmat.calc(
  J,
  gen.filter.number = 1,
  an.filter.number = 1,
  gen.family = "DaubExPhase",
  an.family = "DaubExPhase"
)

Arguments

Details

Computes the cross inner product matrix of the discrete non-decimated autocorrelation wavelets. This matrix is used to correct the wavelet periodogram analysed using a different wavelet to the wavelet that is assumed to generate the time series. The matrix returned is the one denoted C^{(0,1)} in McGonigle et al. (2022).

Value

A J-dimensional square matrix giving the cross inner product autocorrelation wavelet matrix.

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

See Also

TLSW

Estimate Trend and Spectrum of Trend Locally Stationary Wavelet Process

Description

Using wavelet-based methods, this function estimates the trend and evolutionary wavelet spectrum (EWS) of a nonstationary time series.

Two methods are implemented (see references), the direct estimator (T.est.type="linear" and S.do.diff=FALSE), and the difference estimator (T.est.type="nonlinear") and S.do.diff=TRUE) The defaults give the direct estimator.

All the defaults are set carefully. Key times to change defaults are

• if the data contains "cusps", then the difference estimator is preferred.

• to assess stability of the estimate to the wavelet, change the wavelet number T.filter.number and S.filter.number and/or the wavelet type T.family and S.family, see details.

The arguments affecting trend are preceded by T. and those affecting spectral estimation are preceded by S..

Usage

TLSW(
  x,
  do.trend.est = TRUE,
  do.spec.est = TRUE,
  T.est.type = "linear",
  T.filter.number = 4,
  T.family = "DaubExPhase",
  T.transform = "nondec",
  T.boundary.handle = TRUE,
  T.max.scale = floor(log2(length(x)) * 0.7),
  T.thresh.type = "hard",
  T.thresh.normal = TRUE,
  T.CI = FALSE,
  T.sig.lvl = 0.05,
  T.reps = 200,
  T.CI.type = "normal",
  T.lacf.max.lag = floor(10 * (log10(length(x)))),
  S.filter.number = 4,
  S.family = "DaubExPhase",
  S.smooth = TRUE,
  S.smooth.type = "mean",
  S.binwidth = floor(6 * sqrt(length(x))),
  S.max.scale = floor(log2(length(x)) * 0.7),
  S.boundary.handle = TRUE,
  S.inv.mat = NULL,
  S.do.diff = FALSE,
  S.lag = 1,
  S.diff.number = 1,
  gen.filter.number = S.filter.number,
  gen.family = S.family
)

Arguments

Details

The fitted trend LSW process X_{t,n} , t = 0, \ldots , n-1, and n = 2^J is a doubly-indexed stochastic process with the following representation in the mean square sense:

X_{t} = T_t + \varepsilon_t = T_t + \sum_{j = 1}^{\infty} \sum_{k \in \mathbb{Z}} w_{j,k;n} \psi_{j,k} (t) \xi_{j,k} ,

where \{\xi_{j,k} \} is a random, uncorrelated, zero-mean orthonormal increment sequence, \{w_{j,k;n} \} is a set of amplitudes, and \{\psi_{j, k} \}_{j,k} is a set of discrete non-decimated wavelets. The trend component T_t := T(t/n) is assumed to be a general smooth (Holder) continuous function. See the referenced papers for full details of the model. The key considerations for users are:

• The model assumes smooth trend and spectral components. The larger the T.filter.number the smoother the assumption on the underlying trend and similarly for S.filter.number and the spectral estimate.

• The choice of wavelet (smoothness assumption) does affect the estimation so one should check the robustness of their conclusions to the choice of wavelet (T.filter.number and S.filter.number). This is akin to selecting the kernel in nonparametric modelling.

• The underlying methods are designed for signals of length n=2^J and so modifications are made to signals which are not of this form. A natural approach is to extend the data (at both ends) and the default approach does this by reflection with a trend correction to avoid discontinuities.

Value

An object of class "TLSW", a list that contains the following components:

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022a). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

McGonigle, E. T., Killick, R., and Nunes, M. (2022b). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

See Also

plot.TLSW, summary.TLSW, print.TLSW, wd, ewspec3

Examples

# simulates an example time series and estimates its trend and evolutionary wavelet spectrum

spec <- matrix(0, nrow = 10, ncol = 2^10)

spec[1, ] <- seq(from = 1, to = 10, length = 1024)

trend <- sin(pi * (seq(from = 0, to = 4, length = 1024)))

set.seed(1)

x <- TLSWsim(trend = trend, spec = spec)

plot.ts(x)

x.TLSW <- TLSW(x)

summary(x.TLSW)

plot(x.TLSW) # by default plots both the trend and spectrum estimates

Compute Localised Autocovariance Estimate of a TLSW Object

Description

Computes the local autocovariance and autocorrelation estimates, given an input of an object of class TLSW containing the estimated spectrum. Provides the same functionality as the function lacf from the locits package, but user provides an object of class TLSW as the main argument.

Usage

TLSWlacf(x.TLSW, lag.max = NULL)

Arguments

Value

An object of class lacf which contains the following components:

• lacf: a matrix containing the estimate of the local autocovariance. Columns represent lags (beginning at lag 0), and rows represent time points.

• lacr: a matrix containing the estimate of the local autocorrelation. Columns represent lags (beginning at lag 0), and rows represent time points.

• name: the name of the time series (if applicable).

• date: the date the function was executed.

• SmoothWP: The smoothed, un-corrected raw wavelet periodogram of the input data.

• S: the spectral estimate used to compute the local autocovariance.

• J: the number of total wavelet scales.

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

Nason, G. P. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(5), 879–904.

Nason, G. P. (2016). locits: Tests of stationarity and localized autocovariance. R package version 1.7.3.

See Also

lacf

Examples

## ---- computes estimate of local autocovariance function


spec <- matrix(0, nrow = 9, ncol = 512)
spec[2, ] <- 1 + sin(seq(from = 0, to = 2 * pi, length = 512))^2

trend <- seq(from = 0, to = 10, length = 512)

set.seed(123)

x <- TLSWsim(trend = trend, spec = spec)

## ---- first estimate the spectrum:

x.TLSW <- TLSW(x)

#---- estimate the lacf:

lacf.est <- TLSWlacf(x.TLSW)

#---- plot the variance (lag 0 lacf) over time:

plot.ts(lacf.est$lacf[, 1], ylab = "Variance")

Simulate Trend Locally Stationary Wavelet Process

Description

Simulates a trend locally stationary wavelet process with a given trend function and spectrum. Extends the LSWsim function from the wavethresh package.

Usage

TLSWsim(
  trend,
  spec,
  filter.number = 4,
  family = "DaubExPhase",
  innov.func,
  ...
)

Arguments

Value

A n-length vector containing a TLSW process simulated from the trend and spectral description given by the trend and spec arguments.

See Also

LSWsim

Examples

#---- simulate with numeric trend, and spec a wd object as in wavethresh-----

spec <- wavethresh::cns(1024)

spec <- wavethresh::putD(spec, level = 8, seq(from = 2, to = 8, length = 1024))

trend <- sin(pi * (seq(from = 0, to = 4, length = 1024)))

x <- TLSWsim(trend = trend, spec = spec)

plot.ts(x)

#---- simulate with numeric trend, and spec a matrix, with non-dyadic n-----

spec <- matrix(0, nrow = 9, ncol = 1000)

spec[1, ] <- seq(from = 1, to = 10, length = 1000)

trend <- sin(pi * (seq(from = 0, to = 4, length = 1000)))

x <- TLSWsim(trend = trend, spec = spec)

plot.ts(x)

#---- simulate with functional trend, and spec a list of functions-----

spec <- vector(mode = "list", length = 10)

spec[[1]] <- function(u) {
  1 + 9 * u
}

trend <- function(u) {
  sin(pi * u)
}

x <- TLSWsim(trend = trend, spec = spec)

plot.ts(x)

Bioluminescence of C. Elegens

Description

This dataset gives the time series of bioluminescence of an experiment monitoring C. Elegens as they feed and forage. The observations are taken 6-minutes apart with no missing data.

Usage

celegensbio

Format

A vector of length 623.

Source

Experiment from Alexandre Benedetto's research group at Lancaster University.

Estimation of Evolutionary Wavelet Spectrum of Non-Zero Mean Time Series via Differencing

Description

Internal function to estimate the evolutionary wavelet spectrum (EWS) of a time series that may include a trend component. The estimate is computed by taking the non-decimated wavelet transform of the first differenced time series data, squaring it; smoothing using a running mean and then correcting for bias using the appropriate correction matrix. Inherits the smoothing functionality from the ewspec3 function in the R package locits. This function is not intended for general use by regular users of the package.

Usage

ewspec.diff(
  x,
  lag = 1,
  filter.number = 4,
  family = "DaubExPhase",
  binwidth = floor(2 * sqrt(length(x))),
  diff.number = 1,
  max.scale = floor(log2(length(x)) * 0.7),
  S.smooth = TRUE,
  smooth.type = "epan",
  boundary.handle = FALSE,
  AutoReflect = FALSE,
  supply.inv.mat = FALSE,
  inv.mat = NULL
)

Arguments

Details

Computes an estimate of the evolutionary wavelet spectrum of a time series that displays nonstationary mean and autocovariance. The estimation procedure is as follows:

1. The time series is differenced to remove the trend.

2. The squared modulus of the non-decimated wavelet transform is computed, known as the raw wavelet periodogram. This is returned by the function.

3. The raw wavelet periodogram is smoothed using a running mean smoother.

4. The smoothed periodogram is bias corrected using the inverse of the bias matrix.

The final estimate, stored in the S component, can be plotted using the plot function, please see the example below.

Value

A list object, containing the following fields:

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448. wavethresh::plot.wd(spec.est$S)

See Also

TLSW, ewspec3

Estimation of Evolutionary Wavelet Spectrum for Non-Zero Mean Time Series

Description

Internal function to compute the evolutionary wavelet spectrum (EWS) estimate from a time series that may include a trend component. The estimate is computed by taking the non-decimated wavelet transform of the time series data, squaring it, smoothing using a running mean, and then correction for bias using the appropriate correction matrix. This function is not intended for general use by regular users of the package.

Usage

ewspec.trend(
  x,
  an.filter.number = 4,
  an.family = "DaubExPhase",
  gen.filter.number = an.filter.number,
  gen.family = an.family,
  binwidth = floor(2 * sqrt(length(x))),
  max.scale = floor(log2(length(x)) * 0.7),
  S.smooth = TRUE,
  smooth.type = "epan",
  AutoReflect = TRUE,
  supply.inv.mat = FALSE,
  inv.mat = NULL,
  boundary.handle = TRUE
)

Arguments

Details

Estimates the evolutionary wavelet spectrum of a time series that displays a smooth mean and nonstationary autocovariance. The estimation procedure is as follows:

1. The squared modulus of the non-decimated wavelet transform is computed, known as the raw wavelet periodogram. This is returned by the function.

2. The raw wavelet periodogram is smoothed using a running mean smoother.

3. The smoothed periodogram is bias corrected using the inverse of the bias matrix. The correction is applied across the finest max.scale scales. If the analysing wavelet and generating wavelet are different, this is given by the inverse of the C matrix defined in McGonigle et al. (2022). If they are the same, this is the inverse of the A matrix, defined in Nason et al. (2000). If you are unsure on the filter and wavelet choices, it is recommended to use the same wavelet for generating and analysing purposes.

The final estimate, stored in the S component, can be plotted using the plot function, please see the example below.

Value

A list object, containing the following fields:

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

Nason, G. P., von Sachs, R., and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(2), 271–292.

See Also

TLSW, wd

Calculate Boundary Extended Time Series

Description

Internal function to calculate the boundary extended time series, to be used within the TLSW function. Not recommended for general usage.

Usage

get.boundary.timeseries(x, type = "TLSW")

Arguments

Value

A vector of 4 times the length of the input vector.

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022a). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

McGonigle, E. T., Killick, R., and Nunes, M. (2022b). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

See Also

TLSW

Plot Trend and/or Spectrum Information in a TLSW Object

Description

Plots information contained within a TLSW object. Depending on the plot.type option this will produce a plot of the data with trend estimate overlayed, a plot of the spectral estimate, or both (default). If the TLSW object does not contain trend or spectral estimates and these are requested a warning will be given.

Usage

## S3 method for class 'TLSW'
plot(
  x,
  plot.type = c("trend", "spec"),
  trend.plot.args,
  spec.plot.args,
  plot.CI = TRUE,
  ...
)

Arguments

Details

A TLSW object can be plotted using the standard plot function in R to display the estimated trend function and wavelet spectrum. The estimated trend is visualised using plot.default. Visualisation of the estimated spectrum is based on plot.wd, for which credit belongs to Guy Nason. Graphical parameters for customising the display of the trend or spectrum plots should be given to the trend.plot.args and spec.plot.args arguments respectively. For graphical parameters for the trend plot:

• Parameters related to the overall plot should be provided as they usually would be when using the plot function, in the trend.plot.args list object. For example, to change the title of the plot to "Plot", use main = "Plot".

• Parameters affecting the display of the estimated trend line should begin with the prefix "T.". For example, to set the colour of the trend line to blue, use T.col = "blue".

• Parameters affecting the display of the confidence interval lines should begin with the prefix "CI.". For example, to set the line width of the confidence interval to 2, use CI.lwd = 2.

• Parameters affecting the display of the polygon drawn by the confidence interval should begin with the prefix "poly.". For example, to set the colour of the confidence interval region to green, use poly.col = "green".

Value

No return value, called for side effects

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

See Also

TLSW, summary.TLSW, print.TLSW, plot.wd

Examples

# Simulates an example time series and estimates its trend and evolutionary wavelet spectrum.
# Then plots both estimates.

spec <- matrix(0, nrow = 9, ncol = 512)

spec[1, ] <- 4 + 4 * sin(seq(from = 0, to = 2 * pi, length = 512))^2

trend <- seq(from = 0, to = 10, length = 512) + 2 * sin(seq(from = 0, to = 2 * pi, length = 512))

set.seed(1)

x <- TLSWsim(trend = trend, spec = spec)

x.TLSW <- TLSW(x)

plot(x.TLSW, trend.plot.args = list(
  ylab = "Simulated Data", T.col = 4,
  T.lwd = 2, T.lty = 2
))

Print an Object of Class TLSW

Description

Prints a TLSW object, alongside summary information. The first part prints details of the class, specifically the names of elements within. Then prints out the summary, which gives information about a TLSW object. If spectral estimation was performed, then the type of smoothing and binwidth is printed, along with the differencing performed if it is used, the maximum wavelet scale analysed, and whether or not boundary handling was used. If trend estimation is performed, then the type of wavelet thresholding and transform used is printed, as well as the maximum wavelet scale used, whether or not boundary handling was used, and the significance of the confidence interval if it was calculated.

Usage

## S3 method for class 'TLSW'
print(x, ...)

Arguments

Value

No return value, called for side effects

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

See Also

TLSW, summary.TLSW

Examples

# simulates an example time series and estimates its trend and evolutionary wavelet spectrum

spec <- wavethresh::cns(512)
spec <- wavethresh::putD(spec, level = 8, 1 + sin(seq(from = 0, to = 2 * pi, length = 512))^2)

trend <- seq(from = 0, to = 5, length = 512)

set.seed(1)

x <- TLSWsim(trend = trend, spec = spec)

x.TLSW <- TLSW(x)

print(x.TLSW)

Summary of Output Provided by the TLSW Function

Description

Summary method for objects of class TLSW.

Usage

## S3 method for class 'TLSW'
summary(object, ...)

Arguments

Details

Prints out information about a TLSW object. If spectral estimation was performed, then the type of smoothing and binwidth is printed, along with the differencing performed if it is used, the maximum wavelet scale analysed, and whether or not boundary handling was used. If trend estimation is performed, then the type of wavelet thresholding and transform used is printed, as well as the maximum wavelet scale used, whether or not boundary handling was used, and the significance of the confidence interval if it was calculated.

Value

No return value, called for side effects

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

See Also

TLSW, print.TLSW

Examples

# simulates an example time series and estimates its trend and evolutionary wavelet spectrum

spec <- matrix(0, nrow = 10, ncol = 2^10)

spec[1, ] <- seq(from = 1, to = 10, length = 1024)

trend <- sin(pi * (seq(from = 0, to = 4, length = 1024)))

set.seed(1)

x <- TLSWsim(trend = trend, spec = spec)

x.TLSW <- TLSW(x)

summary(x.TLSW)

Wavelet Thresholding Trend Estimation of Time Series

Description

Internal function to compute the wavelet thresholding trend estimate for a time series that may be second-order nonstationary. The function calculates the wavelet transform of the time series, thresholds the coefficients based on an estimate of their variance, and inverts to give the trend estimate. This function is not intended for general use by regular users of the package.

Usage

wav.diff.trend.est(
  x,
  spec.est,
  filter.number = 4,
  family = "DaubExPhase",
  thresh.type = "hard",
  normal = TRUE,
  transform.type = "nondec",
  max.scale = floor(0.7 * log2(length(x))),
  boundary.handle = FALSE,
  T.CI = FALSE,
  reps = 199,
  sig.lvl = 0.05,
  confint.type = "normal",
  ...
)

Arguments

Details

Estimates the trend function of a locally stationary time series, by incorporating the evolutionary wavelet spectrum estimate in a wavelet thresholding procedure. To use this function, first compute the spectral estimate of the time series, using the function ewspec.diff.

The function works as follows:

1. The wavelet transform of the time series is calculated.

2. The wavelet coefficients at scale j and location k are individually thresholded using the universal threshold \hat{\sigma}_{j,k}\sqrt{2 \log n}, where \hat{\sigma}_{j,k}^2 is an estimate of their variance. The variance estimate is calculated using the spectral estimate, supplied by the user in the spec argument.

3. The inverse wavelet transform is applied to obtain the final estimate.

Value

A list object containing the following fields:

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Modelling time-varying first and second-order structure of time series via wavelets and differencing. Electronic Journal of Statistics, 6(2), 4398-4448.

See Also

TLSW

Linear Wavelet Thresholding Trend Estimation of Time Series

Description

Internal function to compute the linear wavelet thresholding trend estimate for a time series that may be second-order nonstationary. The function calculates the wavelet transform of the time series, sets to zero the non-boundary coefficients, then inverts the transform to obtain the estimate. This function is not intended for general use by regular users of the package.

Usage

wav.trend.est(
  x,
  filter.number = 4,
  family = "DaubLeAsymm",
  max.scale = floor(log2(length(x)) * 0.7),
  transform.type = "nondec",
  boundary.handle = FALSE,
  T.CI = FALSE,
  sig.lvl = 0.05,
  lag.max = floor(10 * (log10(length(x)))),
  confint.type = "normal",
  reps = 199,
  spec.est = NULL,
  ...
)

Arguments

Value

A list object containing the following fields:

References

McGonigle, E. T., Killick, R., and Nunes, M. (2022). Trend locally stationary wavelet processes. Journal of Time Series Analysis, 43(6), 895-917.

See Also

TLSW

Z-Axis Acceleration for Human Activity Monitoring

Description

This dataset is a section of data from Experiment 3, User 2, based on accelerometer readings from a smartphone (Reyes-Ortiz, Oneto, Sama, Parra, and Anguita (2016)), obtained from the UCI data repository (Kelly, Longjohn, and Nottingham (2024)). The data gives the time series of the acceleration along the Z-axis of an experiment participant as they perform the activities of walking up and downstairs several times.

Usage

  z.acc

Format

A vector of length 6000.

Source

Kelly M., Longjohn R., and Nottingham, K. (2024). The UCI Machine Learning Repository. doi:10.24432/C54G7M.

References

Reyes-Ortiz, J. L., Oneto, L., Sama, A., Parra, X., and Anguita, D. (2016). Transition-Aware Human Activity Recognition Using Smartphones. Neurocomputing, 171, 754–767.

See Also

z.labels

Activity Labels for Human Activity Monitoring

Description

This dataset gives the labelled activities recorded during the time period of observations given in the data object z.acc.

Usage

  z.labels

Format

A data frame with 6 rows and 3 variables:

activity

The activity recorded, either "downstairs" or "upstairs", corresponding to walking downstairs and upstairs respectively.

start

the starting time of the activity.

end

the ending time of the activity.

Source

Kelly M., Longjohn R., and Nottingham, K. (2024). The UCI Machine Learning Repository. doi:10.24432/C54G7M.

References

Reyes-Ortiz, J. L., Oneto, L., Sama, A., Parra, X., and Anguita, D. (2016). Transition-Aware Human Activity Recognition Using Smartphones. Neurocomputing, 171, 754–767.

See Also

z.acc