wv

Description

Provides a series of tools to compute and plot quantities related to classical and robust wavelet variance for time series and regular lattices. More details can be found, for example, in Serroukh, A., Walden, A.T., & Percival, D.B. (2000) <doi:10.2307/2669537> and Guerrier, S. & Molinari, R. (2016) <arXiv:1607.05858>.

Author(s)

Maintainer: Stéphane Guerrier stef.guerrier@gmail.com

Authors:

• James Balamuta

• Justin Lee

• Roberto Molinari

• Yuming Zhang

• Mucyo Karemera

Other contributors:

• Nathanael Claussen [contributor]

• Haotian Xu [contributor]

• Lionel Voirol [contributor]

See Also

Useful links:

• https://github.com/SMAC-Group/wv

• Report bugs at https://github.com/SMAC-Group/wv/issues

Auto-Covariance and Correlation Functions

Description

The acf function computes the estimated autocovariance or autocorrelation for both univariate and multivariate cases.

Usage

.acf(x, lagmax = 0L, cor = TRUE, demean = TRUE)

Arguments

Auto-Covariance and Correlation Functions

Description

The ACF function computes the estimated autocovariance or autocorrelation for both univariate and multivariate cases.

Usage

ACF(x, lagmax = 0, cor = TRUE, demean = TRUE)

Arguments

Details

lagmax default is 10*log10(N/m) where N is the number of observations and m is the number of series being compared. If lagmax supplied is greater than the number of observations, then one less than the total will be taken.

Value

An array of dimensions N \times S \times S.

Author(s)

Yunxiang Zhang

Examples

# Get Autocorrelation
m = ACF(datasets::AirPassengers)

# Get Autocovariance and do not remove trend from signal
m = ACF(datasets::AirPassengers, cor = FALSE, demean = FALSE)

Compute Theoretical ACF for an ARMA Process

Description

Compute the theoretical autocorrelation function for an ARMA process.

Usage

ARMAacf_cpp(ar,ma,lag_max)

Arguments

Details

This is an implementaiton of the ARMAacf function in R. It is approximately 40x times faster. The benchmark was done on iMac Late 2013 using vecLib as the BLAS.

Value

x A matrix listing values from 1...nx in one column and 1...1, 2...2,....,n...n, in the other

Author(s)

James J Balamuta

Converting an ARMA Process to an Infinite MA Process

Description

Takes an ARMA function and converts it to an infinite MA process.

Usage

ARMAtoMA_cpp(ar, ma, lag_max)

Arguments

Details

This function is a port of the base stats package's ARMAtoMA. There is no significant speed difference between the two.

Value

A column vector containing coefficients

Author(s)

James J Balamuta

Helper Function for ARMA to WV Approximation

Description

Indicates where the minimum ARMAacf value is and returns that as an index.

Usage

acf_sum(ar, ma, last_tau, alpha = 0.99)

Arguments

Value

A vec containing the wavelet variance of the ARMA process.

See Also

arma_to_wv_app

Wavelet variance of IMU Data from an ADIS 16405 sensor

Description

This data set contains wavelet variance of gyroscope and accelerometer data from an ADIS 16405 sensor.

Usage

adis_wv

Format

A list of the following elements:

• "sensor": Name of the sensor.

• "freq": The frequency at which the error signal is measured.

• "n": Sample size of the data.

• "type": The types of sensors considered in the data.

• "axis": The axes of sensors considered in the data.

• "wvar": A list containing the computed wavelet variance based on the data.

Source

The IMU data comes from Department of Geomatics Engineering, University of Calgary.

AR(1) process to WV

Description

This function computes the Haar WV of an AR(1) process

Usage

ar1_to_wv(phi, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv.

Value

A vec containing the wavelet variance of the AR(1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order 1 (AR(1)) process has a Haar Wavelet Variance given by:

\frac{{2{\sigma ^2}\left( {4{\phi ^{\frac{{{\tau _j}}}{2} + 1}} - {\phi ^{{\tau _j} + 1}} - \frac{1}{2}{\phi ^2}{\tau _j} + \frac{{{\tau _j}}}{2} - 3\phi } \right)}}{{{{\left( {1 - \phi } \right)}^2}\left( {1 - {\phi ^2}} \right)\tau _j^2}}

See Also

arma_to_wv, arma11_to_wv

ARMA(1,1) to WV

Description

This function computes the WV (haar) of an Autoregressive Order 1 - Moving Average Order 1 (ARMA(1,1)) process.

Usage

arma11_to_wv(phi, theta, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv

Value

A vec containing the wavelet variance of the ARMA(1,1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order 1 and Moving Average Order 1 (ARMA(1,1)) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = - \frac{{2{\sigma ^2}\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)}}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}

See Also

arma_to_wv

ARMA process to WV

Description

This function computes the Haar Wavelet Variance of an ARMA process

Usage

arma_to_wv(ar, ma, sigma2, tau)

Arguments

Details

The function is a generic implementation that requires a stationary theoretical autocorrelation function (ACF) and the ability to transform an ARMA(p,q) process into an MA(\infty) (e.g. infinite MA process).

Value

A vec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Order p and Moving Average Order q (ARMA(p,q)) process has a Haar Wavelet Variance given by:

\frac{{{\tau _j}\left[ {1 - \rho \left( {\frac{{{\tau _j}}}{2}} \right)} \right] + 2\sum\limits_{i = 1}^{\frac{{{\tau _j}}}{2} - 1} {i\left[ {2\rho \left( {\frac{{{\tau _j}}}{2} - i} \right) - \rho \left( i \right) - \rho \left( {{\tau _j} - i} \right)} \right]} }}{{\tau _j^2}}\sigma _X^2

where \sigma _X^2 is given by the variance of the ARMA process. Furthermore, this assumes that stationarity has been achieved as it directly

See Also

ARMAtoMA_cpp, ARMAacf_cpp, and arma11_to_wv

ARMA process to WV Approximation

Description

This function computes the (haar) WV of an ARMA process

Usage

arma_to_wv_app(ar, ma, sigma2, tau, alpha = 0.9999)

Arguments

Details

This function provides an approximation to the arma_to_wv as computation times were previously a concern. However, this is no longer the case and, thus, this has been left in for the curious soul to discover...

Value

A vec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Order p and Moving Average Order q (ARMA(p,q)) process has a Haar Wavelet Variance given by:

\frac{{{\tau _j}\left[ {1 - \rho \left( {\frac{{{\tau _j}}}{2}} \right)} \right] + 2\sum\limits_{i = 1}^{\frac{{{\tau _j}}}{2} - 1} {i\left[ {2\rho \left( {\frac{{{\tau _j}}}{2} - i} \right) - \rho \left( i \right) - \rho \left( {{\tau _j} - i} \right)} \right]} }}{{\tau _j^2}}\sigma _X^2

where \sigma _X^2 is given by the variance of the ARMA process. Furthermore, this assumes that stationarity has been achieved as it directly

Haar Wavelet Derivation Information

For more information, please see: blog post on SMAC group website.

See Also

ARMAtoMA_cpp, ARMAacf_cpp, acf_sum and arma_to_wv

Calculate Theoretical Allan Variance for Stationary First-Order Autoregressive (AR1) Process

Description

This function allows us to calculate the theoretical allan variance for stationary first-order autoregressive (AR1) process.

Usage

av_ar1(n, phi, sigma2)

Arguments

Value

A double indicating the theoretical allan variance for AR1 process.

Note

This function is based on the calculation of the theoretical allan variance for stationary AR1 process raised in "Allan Variance of Time Series Models for Measurement Data" by Nien Fan Zhang.) This calculation is fundamental and necessary for the study in "A Study of the Allan Variance for Constant-Mean Non-Stationary Processes" by Xu et al. (IEEE Signal Processing Letters, 2017).

Author(s)

Yuming Zhang

Examples

av1 = av_ar1(n = 5, phi = 0.9, sigma2 = 1)
av2 = av_ar1(n = 8, phi = 0.5, sigma2 = 2)

Calculate Theoretical Allan Variance for Stationary White Noise Process

Description

This function allows us to calculate the theoretical allan variance for stationary white noise process.

Usage

av_wn(sigma2, n)

Arguments

Value

A double indicating the theoretical allan variance for the white noise process.

Note

This function is based on the calculation of the theoretical allan variance for stationary white noise process raised in "Allan Variance of Time Series Models for Measurement Data" by Nien Fan Zhang. This calculation is fundamental and necessary for the study in "A Study of the Allan Variance for Constant-Mean Non-Stationary Processes" by Xu et al. (IEEE Signal Processing Letters, 2017).

Author(s)

Yuming Zhang

Examples

av1 = av_wn(sigma2 = 1, n = 5)
av2 = av_wn(sigma2 = 2, n = 8)

Computes the MO/DWT wavelet variance for multiple processes

Description

Calculates the MO/DWT wavelet variance

Usage

batch_modwt_wvar_cpp(
  signal,
  nlevels,
  robust,
  eff,
  alpha,
  ci_type,
  strWavelet,
  decomp
)

Arguments

Details

This function processes the decomposition of multiple signals quickly

Value

A field<mat> with the structure:

• "variance"Wavelet Variance

• "low"Lower CI

• "high"Upper CI

Time Series Convolution Filters

Description

Applies a convolution filter to a univariate time series.

Usage

cfilter(x, filter, sides, circular)

Arguments

Details

This is a port of the cfilter function harnessed by the filter function in stats. It is about 5-7 times faster than R's base function. The benchmark was done on iMac Late 2013 using vecLib as the BLAS.

Value

A column vec that contains the results of the filtering process.

Author(s)

James J Balamuta

Generate eta3 confidence interval

Description

Computes the eta3 CI

Usage

ci_eta3(y, dims, alpha_ov_2)

Arguments

Value

A matrix with the structure:

• Column 1Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Generate eta3 robust confidence interval

Description

Computes the eta3 robust CI

Usage

ci_eta3_robust(wv_robust, wv_ci_class, alpha_ov_2, eff)

Arguments

Details

Within this function we are scaling the classical

Value

A matrix with the structure:

• Column 1Robust Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Generate a Confidence interval for a Univariate Time Series

Description

Computes an estimate of the multiscale variance and a chi-squared confidence interval

Usage

ci_wave_variance(
  signal_modwt_bw,
  wv,
  type = "eta3",
  alpha_ov_2 = 0.025,
  robust = FALSE,
  eff = 0.6
)

Arguments

Details

This function can be expanded to allow for other confidence interval calculations.

Value

A matrix with the structure:

• Column 1Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Comparison Between Multiple Wavelet Variances

Description

Displays plots of multiple wavelet variances of different time series accounting for CI values.

Usage

compare_wvar(
  ...,
  split = FALSE,
  add_legend = TRUE,
  units = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  col_wv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  legend_position = NULL,
  ci_wv = NULL,
  point_cex = NULL,
  point_pch = NULL,
  names = NULL,
  cex_labels = 0.8,
  x_range = NULL,
  y_range = NULL
)

Arguments

Author(s)

Stephane Guerrier and Justin Lee

Examples

set.seed(999)
n = 10^4
Xt = arima.sim(n = n, list(ar = 0.10))
Yt = arima.sim(n = n, list(ar = 0.35))
Zt = arima.sim(n = n, list(ar = 0.70))
Wt = arima.sim(n = n, list(ar = 0.95))

wv_Xt = wvar(Xt)
wv_Yt = wvar(Yt)
wv_Zt = wvar(Zt)
wv_Wt = wvar(Wt)

compare_wvar(wv_Xt, wv_Yt, wv_Zt, wv_Wt)

Combined Plot Comparison Between Multiple Wavelet Variances

Description

This is a helper function for the compare_var() function. This method accepts the same set of arguments as compare_wvar and returns a single plot that compares multiple wavelet variances of different time series accounting for CI values.

Usage

compare_wvar_no_split(graph_details)

Arguments

Author(s)

Stephane Guerrier, Justin Lee, and Nathanael Claussen

Multi-Plot Comparison Between Multiple Wavelet Variances

Description

This is a helper function for the compare_var() function. This method accepts the same set of arguments as compare_wvar and returns a comparision of multiple wavelet variances of different time series accounting for CI values as a set of different plots.

Usage

compare_wvar_split(graph_details)

Arguments

Author(s)

Stephane Guerrier, Justin Lee, and Nathanael Claussen

Create a wvar object

Description

Structures elements into a wvar object

Usage

create_wvar(
  obj,
  decomp,
  filter,
  robust,
  eff,
  alpha,
  scales,
  unit,
  mean_diff,
  N,
  ranged,
  J
)

Arguments

Value

A list with the structure:

• "variance": Wavelet variance

• "ci_low": Lower CI

• "ci_high": Upper CI

• "robust": Robust active

• "eff": Efficiency level for robust calculation

• "alpha": p value used for CI

• "unit": String representation of the unit

• "mean_diff": Empirical mean of the first difference

• "N": Length of the time series

• "ranged": Scaled range of the data, i.e. (max(x) - min(x))/length(x)

• "J": Number of scales

Each Models Process Decomposed to WV

Description

This function computes each process to WV (haar) in a given model.

Usage

decomp_theoretical_wv(theta, desc, objdesc, tau)

Arguments

Value

A mat containing the wavelet variance of each process in the model

Decomposed WV to Single WV

Description

This function computes the combined processes to WV (haar) in a given model.

Usage

decomp_to_theo_wv(decomp)

Arguments

Value

A vec containing the wavelet variance of the process for the overall model

Discrete Fourier Transformation for Autocovariance Function

Description

Calculates the autovariance function (ACF) using Discrete Fourier Transformation.

Usage

dft_acf(x)

Arguments

Details

This implementation is 2x as slow as Rs. Two issues: 1. memory resize and 2. unoptimized fft algorithm in arma. Consider piping back into R and rewrapping the object. (Decrease of about 10 microseconds.)

Value

A vec containing the ACF.

Lagged Differences in Armadillo

Description

Returns the ith difference of a time series of rth lag.

Usage

diff_cpp(x, lag, differences)

Arguments

Value

A vector containing the differenced time series.

Author(s)

James J Balamuta

Drift to WV

Description

This function compute the WV (haar) of a Drift process

Usage

dr_to_wv(omega, tau)

Arguments

Value

A vec containing the wavelet variance of the drift.

Process Haar Wavelet Variance Formula

The Drift (DR) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {{\omega }} \right) = \frac{{\tau _j^2{\omega ^2}}}{{16}}

Discrete Wavelet Transform

Description

Calculation of the coefficients for the discrete wavelet transformation

Usage

dwt(x, nlevels = floor(log2(length(x))), filter = "haar")

Arguments

Details

Performs a level J decomposition of the time series using the pyramid algorithm. The default J is determined by floor\left(log_2 \left(length\left(x\right)\right)\right)

Value

A field<vec> that contains the wavelet coefficients for each decomposition level

Author(s)

James Balamuta, Justin Lee and Stephane Guerrier

Examples

set.seed(999)
x = rnorm(2^8)
ret = dwt(x)

summary(ret)

plot(ret)

Discrete Wavelet Transform

Description

Calculation of the coefficients for the discrete wavelet transformation.

Usage

dwt_cpp(x, filter_name, nlevels)

Arguments

Details

Performs a level J decomposition of the time series using the pyramid algorithm

Value

y A field<vec> that contains the wavelet coefficients for each decomposition level

Author(s)

James Balamuta and Justin Lee

Wavelet variance of IMU Data from IMAR Gyroscopes

Description

This data set contains wavelet variance of IMAR gyroscopes data.

Usage

imar_wv

Format

A list of the following elements:

• "sensor": Name of the sensor.

• "freq": The frequency at which the error signal is measured.

• "n": Sample size of the data.

• "type": The types of sensors considered in the data.

• "axis": The axes of sensors considered in the data.

• "wvar": A list containing the computed wavelet variance based on the data.

Source

The IMU data comes from Geodetic Engineering Laboratory (TOPO) and Swiss Federal Institute of Technology Lausanne (EPFL).

Discrete Intergral: Inverse Difference

Description

Takes the inverse difference (e.g. goes from diff() result back to previous vector)

Usage

intgr_vec(x, xi, lag)

diff_inv_values(x, lag, d, xi)

diff_inv(x, lag, d)

Arguments

Wavelet variance of IMU Data from a KVH1750 IMU sensor

Description

This data set contains wavelet variance of gyroscope and accelerometer data from an KVH1750 sensor.

Usage

kvh1750_wv

Format

A list of the following elements:

• "sensor": Name of the sensor.

• "freq": The frequency at which the error signal is measured.

• "n": Sample size of the data.

• "type": The types of sensors considered in the data.

• "axis": The axes of sensors considered in the data.

• "wvar": A list containing the computed wavelet variance based on the data.

Source

The IMU data comes from Department of Geomatics Engineering, University of Calgary.

Wavelet variance of IMU Data from a LN200 sensor

Description

This data set contains wavelet variance of LN200 gyroscope and accelerometer data.

Usage

ln200_wv

Format

A list of the following elements:

• "sensor": Name of the sensor.

• "freq": The frequency at which the error signal is measured.

• "n": Sample size of the data.

• "type": The types of sensors considered in the data.

• "axis": The axes of sensors considered in the data.

• "wvar": A list containing the computed wavelet variance based on the data.

Source

The IMU data comes from Geodetic Engineering Laboratory (TOPO) and Swiss Federal Institute of Technology Lausanne (EPFL).

Moving Average Order 1 (MA(1)) to WV

Description

This function computes the WV (haar) of a Moving Average order 1 (MA1) process.

Usage

ma1_to_wv(theta, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv.

Value

A vec containing the wavelet variance of the MA(1) process.

Process Haar Wavelet Variance Formula

The Moving Average Order 1 (MA(1)) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{\left( {{{\left( {\theta + 1} \right)}^2}{\tau _j} - 6\theta } \right){\sigma ^2}}}{{\tau _j^2}}

See Also

arma_to_wv, arma11_to_wv

Mean of the First Difference of the Data

Description

The mean of the first difference of the data

Usage

mean_diff(x)

Arguments

Value

A double that contains the mean of the first difference of the data.

Maximum Overlap Discrete Wavelet Transform

Description

Calculates the coefficients for the discrete wavelet transformation

Usage

modwt(x, nlevels = floor(log2(length(x) - 1)), filter = "haar")

Arguments

Details

Performs a level J decomposition of the time series using the pyramid algorithm. The default J is determined by floor\left(log_2 \left(length\left(x\right)\right)\right)

Value

A field<vec> that contains the wavelet coefficients for each decomposition level

Author(s)

James Balamuta, Justin Lee and Stephane Guerrier

Examples

set.seed(999)
x = rnorm(100)
ret = modwt(x)

summary(ret)

plot(ret)

Maximum Overlap Discrete Wavelet Transform

Description

Calculation of the coefficients for the discrete wavelet transformation

Usage

modwt_cpp(x, filter_name, nlevels)

Arguments

Details

Performs a level J decomposition of the time series using the pyramid algorithm. Use this implementation to supply custom parameters instead of modwt(x), which serves as a wrapper function.

Value

y A field<vec> that contains the wavelet coefficients for each decomposition level

Author(s)

James Balamuta and Justin Lee

Computes the (MODWT) wavelet variance

Description

Calculates the (MODWT) wavelet variance

Usage

modwt_wvar_cpp(
  signal,
  nlevels,
  robust,
  eff,
  alpha,
  ci_type,
  strWavelet,
  decomp
)

Arguments

Details

This function powers the wvar object. It is also extendable...

Value

A mat with the structure:

• "variance"Wavelet Variance

• "low"Lower CI

• "high"Upper CI

Wavelet variance of IMU Data from a navchip sensor

Description

This data set contains wavelet variance of gyroscope and accelerometer data from a navchip sensor.

Usage

navchip_wv

Format

A list of the following elements:

• "sensor": Name of the sensor.

• "freq": The frequency at which the error signal is measured.

• "n": Sample size of the data.

• "type": The types of sensors considered in the data.

• "axis": The axes of sensors considered in the data.

• "wvar": A list containing the computed wavelet variance based on the data.

Source

The IMU data of the navchip sensor comes from Geodetic Engineering Laboratory (TOPO) and Swiss Federal Institute of Technology Lausanne (EPFL).

Replicate a Vector of Elements n times

Description

This function takes a vector and replicates all of the data n times

Usage

num_rep(x, n)

Arguments

Value

A vec with repeated elements of the initial supplied vector.

Auto-Covariance and Correlation Functions

Description

The acf function computes the estimated autocovariance or autocorrelation for both univariate and multivariate cases.

Usage

## S3 method for class 'auto_corr'
plot(x, show.ci = TRUE, alpha = 0.05, main = NULL, ...)

Arguments

Value

An array of dimensions N \times S \times S.

Author(s)

Yunxiang Zhang

Examples

# Calculate the Autocorrelation
m = ACF(datasets::AirPassengers)

# Plot with 95% CI
plot(m) 

# Plot with 90% CI
plot(m, ci = 0.90) 

# Plot without 95% CI
plot(m, show.ci = FALSE)

Plot Discrete Wavelet Transform

Description

Plots results of the dwt list in which additional parameters can be specified

Usage

## S3 method for class 'dwt'
plot(x, index = NULL, couleur = NULL, ...)

Arguments

Author(s)

Justin Lee and Stephane Guerrier

Examples

# Simulate a Gaussian white noise
n = 10^3
Xt = rnorm(n)

# dwt
Yt = dwt(Xt)

# Graph examples
plot(Yt)
plot(Yt, index = c(1,4,5,6,8,2))
plot(Yt, index = c(1,4,5,6), couleur = "blue")
plot(Yt, index = c(1,4,5,6), couleur = rep(c("blue","yellow"),2))

Plot Wavelet Variance based on IMU Data

Description

Displays a plot of wavelet variance accounting for CI values and supplied efficiency.

Usage

## S3 method for class 'imu_wvar'
plot(
  x,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  col_wv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  ci_wv = NULL,
  point_cex = NULL,
  point_pch = NULL,
  ...
)

Arguments

Value

Plot of wavelet variance and confidence interval for each scale.

Author(s)

Stephane Guerrier and Yuming Zhang

Examples

data("kvh1750_wv")
plot(kvh1750_wv)

Plot Maximum Overlap Discrete Wavelet Transform

Description

Plots results of the modwt list in which additional parameters can be specified

Usage

## S3 method for class 'modwt'
plot(x, index = NULL, couleur = NULL, ...)

Arguments

Author(s)

Justin Lee and Stephane Guerrier

Examples

# Simulate a Gaussian white noise
n = 10^3
Xt = rnorm(n)

# MODWT
Yt = modwt(Xt)

# Graph examples
plot(Yt)
plot(Yt, index = c(1,4,5,6,8,2))
plot(Yt, index = c(1,4,5,6), couleur = "blue")
plot(Yt, index = c(1,4,5,6), couleur = rep(c("blue","yellow"),2))

Plot Cross Covariance Pair

Description

Plots results of the a wccv_pair list in which additional parameters can be specified

Usage

## S3 method for class 'wccv_pair'
plot(
  x,
  theo.wccv = NULL,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  units = NULL,
  col_wccv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  ...
)

Author(s)

Justin Lee, Haotian Xu, and Stephane Guerrier

Examples

n = 10^5
Xt = cumsum(rnorm(n, 0, 0.01))
Wt = Xt + rnorm(n)
Yt = Xt + rnorm(n)
wcov = wccv_pair(Wt, Yt)
plot(wcov)

Plot Wavelet Variance

Description

Displays a plot of wavelet variance accounting for CI values and supplied efficiency.

Usage

## S3 method for class 'wvar'
plot(
  x,
  units = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  col_wv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  legend_position = NULL,
  ci_wv = NULL,
  point_cex = NULL,
  point_pch = NULL,
  ...
)

Arguments

Value

Plot of wavelet variance and confidence interval for each scale.

Author(s)

Stephane Guerrier, Nathanael Claussen, and Justin Lee

Examples

set.seed(999)
n = 10^4
Xt = rnorm(n)
wv = wvar(Xt)
plot(wv)
plot(wv, main = "Simulated white noise", xlab = "Scales")
plot(wv, units = "sec", legend_position = "topright")
plot(wv, col_wv = "darkred", col_ci = "pink")

Print Discrete Wavelet Transform

Description

Prints the results of the modwt list

Usage

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

Arguments

Value

Prints the dwt decomposition

Author(s)

James Balamuta and Nathanael Claussen

Examples

set.seed(999)
x = rnorm(2^8)
print(dwt(x))

Print Maximum Overlap Discrete Wavelet Transform

Description

Prints the results of the modwt list

Usage

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

Arguments

Value

Prints the modwt decomposition

Author(s)

James Balamuta and Nathanael Claussen

Examples

set.seed(999)
x = rnorm(100)
print(modwt(x))

Print Wavelet Variances

Description

Displays the summary table of wavelet variance.

Usage

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

Arguments

Value

Summary table

Author(s)

James Balamuta

Examples

set.seed(999)
x = rnorm(100)
out = wvar(x)
print( out )

Quantisation Noise (QN) to WV

Description

This function compute the Haar WV of a Quantisation Noise (QN) process

Usage

qn_to_wv(q2, tau)

Arguments

Value

A vec containing the wavelet variance of the QN.

Process Haar Wavelet Variance Formula

The Quantization Noise (QN) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {{Q^2}} \right) = \frac{{6{Q^2}}}{{\tau _j^2}}

Find Quantiles

Description

Attempts to find quantiles

Usage

quantile_cpp(x, probs)

Arguments

Value

A vector containing the quantiles

Author(s)

James J Balamuta

Time Series Recursive Filters

Description

Applies a recursive filter to a univariate time series.

Usage

rfilter(x, filter, init)

Arguments

Details

Note: The length of 'init' must be equal to the length of 'filter'. This is a port of the rfilter function harnessed by the filter function in stats. It is about 6-7 times faster than R's base function. The benchmark was done on iMac Late 2013 using vecLib as the BLAS.

Value

x A column vector with its contents reversed.

Author(s)

James J Balamuta

Comparison between classical and robust Wavelet Variances

Description

Displays a plot of the wavelet variances (classical and robust) for a given time series accounting for CI values.

Usage

robust_eda(
  x,
  eff = 0.6,
  units = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  col_wv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  legend_position = NULL,
  ...
)

Arguments

Value

Plot of wavelet variance and confidence interval for each scale.

Author(s)

Stephane Guerrier, Nathanael Claussen, and Justin Lee

Examples

set.seed(999)
n = 10^4
Xt = rnorm(n)
wv = wvar(Xt)

plot(wv)
plot(wv, main = "Simulated white noise", xlab = "Scales")
plot(wv, units = "sec", legend_position = "topright")
plot(wv, col_wv = "darkred", col_ci = "pink")

Random Walk to WV

Description

This function compute the WV (haar) of a Random Walk process

Usage

rw_to_wv(gamma2, tau)

Arguments

Value

A vec containing the wavelet variance of the random walk.

Process Haar Wavelet Variance Formula

The Random Walk (RW) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {{\gamma ^2}} \right) = \frac{{\left( {\tau _j^2 + 2} \right){\gamma ^2}}}{{12{\tau _j}}}

Calculates Length of Seasonal Padding

Description

Computes the total phi and total theta vector length.

Usage

sarma_calculate_spadding(np, nq, nsp, nsq, ns)

Arguments

Value

A vec with rows:

p

Number of phi parameters

q

Number of theta parameters

See Also

sarma_components

Determine parameter expansion based upon objdesc

Description

Calculates the necessary vec space needed to pad the vectors for seasonal terms.

Usage

sarma_components(objdesc)

Arguments

Value

A vec with the structure:

np

Number of Non-Seasonal AR Terms

nq

Number of Non-Seasonal MA Terms

nsp

Number of Seasonal AR Terms

nsq

Number of Seasonal MA Terms

ns

Number of Seasons (e.g. 12 is year)

p

Total number of phi terms

q

Total number of theta terms

Expand Parameters for an SARMA object

Description

Creates an expanded PHI and THETA vector for use in other objects.

Usage

sarma_expand(params, objdesc)

Arguments

Details

The objdesc is assumed to have the structure of:

• AR(p)

• MA(q)

• SAR(P)

• SMA(Q)

• Seasons

Value

A field<vec> of size two as follows:

• AR values

• THETA values

(Internal) Expand the SARMA Parameters

Description

(Internal) Expand the SARMA Parameters

Usage

sarma_expand_unguided(params, np, nq, nsp, nsq, ns, p, q)

Arguments

Value

A field<vec> that contains the expansion.

Create the ts.model obj.desc given split values

Description

Computes the total phi and total theta vector length.

Usage

sarma_objdesc(ar, ma, sar, sma, s, i, si)

Arguments

Value

A vec with rows:

np

Number of Non-Seasonal AR Terms

nq

Number of Non-Seasonal MA Terms

nsp

Number of Seasonal AR Terms

nsq

Number of Seasonal MA Terms

nsigma

Number of Variances (always 1)

s

Season Value

i

Number of non-seasonal differences

si

Number of Seasonal Differences

Efficient way to merge items together

Description

Efficient way to merge items together

Usage

sarma_params_construct(ar, ma, sar, sma)

Computes the MODWT scales

Description

Calculates the MODWT scales

Usage

scales_cpp(nb_level)

Arguments

Details

Used in wvar object.

Value

A vec that contains 2^1, ... , 2^J

Generate a sequence of values

Description

Creates a vector containing a sequence of values starting at the initial point and going to the terminal point.

Usage

seq_cpp(a, b)

Arguments

Value

A vector containing values moving from a to b. There are no restrictions on A's range.

Author(s)

James J Balamuta

Generate a sequence of values based on supplied number

Description

Creates a vector containing a sequence of values starting at 1 and going to the terminal point.

Usage

seq_len_cpp(n)

Arguments

Value

A vector containing values moving from 1 to n.

Author(s)

James J Balamuta

Haar filter for a spatial case

Description

Haar filter for a spatial case

Usage

sp_hfilter(jscale)

Arguments

Compute the Spatial Wavelet Coefficients

Description

Compute the Spatial Wavelet Coefficients

Usage

sp_modwt_cpp(X, J1, J2)

Arguments

Details

By default this function will return the wavelet coefficient in addition to the wavelet

Value

A list of vectors containing the wavelet coefficients.

Summary Discrete Wavelet Transform

Description

Prints DWT object in a concise format

Usage

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

Arguments

Value

Prints the dwt matrix decomposition

Author(s)

Nathanael Claussen and Justin Lee

Examples

set.seed(999)
x = rnorm(2^8)
summary(dwt(x))

Summary Maximum Overlap Discrete Wavelet Transform

Description

Prints MODWT object in a concise format

Usage

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

Arguments

Value

Prints the modwt matrix decomposition

Author(s)

Nathanael Claussen and Justin Lee

Examples

set.seed(999)
x = rnorm(100)
summary(modwt(x))

Summary of Wavelet Variances

Description

Displays the summary table of wavelet variance accounting for CI values and supplied efficiency.

Usage

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

Arguments

Value

Summary table and other properties of the object.

Author(s)

James Balamuta

Examples

set.seed(999)
x = rnorm(100)
ret = wvar(x)
summary(ret)

Model Process to WV

Description

This function computes the summation of all Processes to WV (haar) in a given model

Usage

theoretical_wv(theta, desc, objdesc, tau)

Arguments

Value

A vec containing the wavelet variance of the model.

Convert Unit of Time Series Data

Description

Manipulate the units of time to different ones

Usage

unitConversion(x, from.unit, to.unit)

Arguments

Details

The supported units are "ns"(nanosecond), "ms"(millisecond), "sec", "min", "hour", "day", "month", and "year". Make sure from.unit and to.unit are not NULL before it is passed to this function.

Value

A list with the following structure:

• "x": Data

• "converted": A boolean indicating whether conversion is made

Examples

x = seq(60, 3600, 60)
unitConversion(x, 'sec', 'min')
y = 1:10
unitConversion(y, 'hour', 'sec')

Generate a Wave Variance for a Univariate Time Series

Description

Computes an estimate of the wave variance

Usage

wave_variance(signal_modwt_bw, robust = FALSE, eff = 0.6)

Arguments

Value

A vec that contains the wave variance.

Cross Covariance of Matrix

Description

Calculates the Cross-covariance between multiple wavelet transfomations (dwt or modwt)

Usage

wccv(x, decomp = "modwt", filter = "haar", nlevels = NULL)

Arguments

Details

If nlevels is not specified, it is set to \left\lfloor {{{\log }_2}\left( {length\left( x \right)} \right)} \right\rfloor

Value

Returns a matrix of lists of all the possible pair cross-covariance, variance of each wavelet cross-covariance and its 95

Author(s)

Justin Lee

Mapping to log10 scale

Description

Map x to the value in log10 scale

Usage

wccv_get_y(x, tick_y_min, tick_y_step)

Arguments

Details

tick_y_min is usually chosen as floor(min(log10(abs(x))))

Value

A field<vec> that contains values in log10 scale.

Author(s)

James Balamuta and Justin Lee

Examples

x = 2^(-1:-9)
y.min = floor(min(log10(abs(x))))
y.step = 2
wccv_get_y(x, y.min, y.step)

Cross Covariance of a TS Pair

Description

Calculates the Cross-covariance between two wavelet transfomations (dwt or modwt)

Usage

wccv_pair(x, y, decomp = "modwt", filter = "haar", nlevels = NULL)

Arguments

Details

If nlevels is not specified, it is set to \left\lfloor {{{\log }_2}\left( {length\left( x \right)} \right)} \right\rfloor

Value

Returns a list of a matrix containing cross-covariance, variance of each wavelet cross-covariance and its 95

Author(s)

Justin Lee

Gaussian White Noise to WV

Description

This function compute the Haar WV of a Gaussian White Noise process

Usage

wn_to_wv(sigma2, tau)

Arguments

Value

A vec containing the wavelet variance of the white noise.

Process Haar Wavelet Variance Formula

The Gaussian White Noise (WN) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {{\sigma ^2}} \right) = \frac{{{\sigma ^2}}}{{\tau _j^2}}

Wavelet Variance

Description

Calculates the (MO)DWT wavelet variance

Usage

wvar(x, ...)

## S3 method for class 'lts'
wvar(
  x,
  decomp = "modwt",
  filter = "haar",
  nlevels = NULL,
  alpha = 0.05,
  robust = FALSE,
  eff = 0.6,
  to.unit = NULL,
  ...
)

## S3 method for class 'gts'
wvar(
  x,
  decomp = "modwt",
  filter = "haar",
  nlevels = NULL,
  alpha = 0.05,
  robust = FALSE,
  eff = 0.6,
  to.unit = NULL,
  ...
)

## S3 method for class 'ts'
wvar(
  x,
  decomp = "modwt",
  filter = "haar",
  nlevels = NULL,
  alpha = 0.05,
  robust = FALSE,
  eff = 0.6,
  to.unit = NULL,
  ...
)

## S3 method for class 'imu'
wvar(
  x,
  decomp = "modwt",
  filter = "haar",
  nlevels = NULL,
  alpha = 0.05,
  robust = FALSE,
  eff = 0.6,
  to.unit = NULL,
  ...
)

## Default S3 method:
wvar(
  x,
  decomp = "modwt",
  filter = "haar",
  nlevels = NULL,
  alpha = 0.05,
  robust = FALSE,
  eff = 0.6,
  freq = 1,
  from.unit = NULL,
  to.unit = NULL,
  ...
)

Arguments

Details

The default value of nlevels will be set to \left\lfloor {{{\log }_2}\left( {length\left( x \right)} \right)} \right\rfloor, unless otherwise specified.

Value

A list with the structure:

• "variance": Wavelet Variance

• "ci_low": Lower CI

• "ci_high": Upper CI

• "robust": Robust active

• "eff": Efficiency level for Robust calculation

• "alpha": p value used for CI

• "unit": String representation of the unit

Author(s)

James Balamuta, Justin Lee and Stephane Guerrier

Examples

set.seed(999)
x = rnorm(100)

# Default
wvar(x)

# Robust
wvar(x, robust = TRUE, eff=0.3)

# Classical
wvar(x, robust = FALSE, eff=0.3)

# 90% Confidence Interval 
wvar(x, alpha = 0.10)

Computes the (MODWT) wavelet variance

Description

Calculates the (MODWT) wavelet variance

Usage

wvar_cpp(signal_modwt_bw, robust, eff, alpha, ci_type)

Arguments

Details

This function does the heavy lifting with the signal_modwt_bw

Value

A mat with the structure:

• "variance"Wavelet Variance

• "low"Lower CI

• "high"Upper CI