Package {gmwmx2}

Title:	Estimate Functional and Stochastic Parameters of Linear Models with Correlated Residuals and Missing Data
Version:	0.0.5
Description:	Implements the Generalized Method of Wavelet Moments with Exogenous Inputs estimator (GMWMX) presented in Voirol, L., Xu, H., Zhang, Y., Insolia, L., Molinari, R. and Guerrier, S. (2024) <doi:10.48550/arXiv.2409.05160>. The GMWMX estimator allows to estimate functional and stochastic parameters of linear models with correlated residuals in presence of missing data. The 'gmwmx2' package provides functions to load and plot Global Navigation Satellite System (GNSS) data from the Nevada Geodetic Laboratory and functions to estimate linear model model with correlated residuals in presence of missing data.
License:	AGPL-3
Encoding:	UTF-8
RoxygenNote:	7.3.3
Language:	en-US
Depends:	R (≥ 4.0.0)
VignetteBuilder:	knitr
LinkingTo:	Rcpp, RcppArmadillo
Imports:	Rcpp, data.table, dplyr, magrittr, wv, Matrix, httr2, rlang, longmemo
Suggests:	knitr, rmarkdown, knitcitations, raster, rnaturalearth, shape, tibble, tidygeocoder, sf, geodata
URL:	https://smac-group.github.io/gmwmx2/, https://github.com/SMAC-Group/gmwmx2
BugReports:	https://github.com/SMAC-Group/gmwmx2/issues
LazyData:	true
NeedsCompilation:	yes
Packaged:	2026-06-10 09:57:10 UTC; lionel
Author:	Lionel Voirol [aut, cre], Haotian Xu [aut], Yuming Zhang [aut], Luca Insolia [aut], Roberto Molinari [aut], Stéphane Guerrier [aut]
Maintainer:	Lionel Voirol <lionelvoirol@hotmail.com>
Repository:	CRAN
Date/Publication:	2026-06-10 10:20:02 UTC

Add to a sum_model object

Description

Add to a sum_model object

Usage

## S3 method for class 'sum_model'
e1 + e2

Arguments

Value

A sum_model.

Examples

m1 <- wn(sigma2 = 1)
m2 <- ar1(phi = 0.8, sigma2 = 0.5)
m3 <- pl(kappa = 0.3, sigma2 = 2)
model <- (m1 + m2) + m3

Add to a time_series_model object

Description

Combines time_series_model and/or sum_model into a sum_model.

Usage

## S3 method for class 'time_series_model'
e1 + e2

Arguments

Value

A sum_model.

Examples

m1 <- wn(sigma2 = 1)
m2 <- ar1(phi = 0.8, sigma2 = 0.5)
model <- m1 + m2
model

Component name prefix for summed models

Description

Component name prefix for summed models

Usage

.comp_prefix(i)

Arguments

Value

Prefix string like m1_.

Get plot colors for generated time series

Description

Returns a vector of plot colors of length n. If n exceeds the size of gmwmx2_plot_colors, a larger palette is created via interpolation.

Usage

.gmwmx2_get_plot_colors(n)

Arguments

Value

Character vector of hex colors.

See Also

gmwmx2_plot_colors

Matern autocovariance

Description

Computes the Matern correlation at lags x for smoothness alpha. This is used internally to build the full autocovariance vector.

Usage

Ma(x, alpha)

Arguments

Value

Numeric vector of correlations for each lag in x.

AR(1) process (time_series_model)

Description

Constructs a time_series_model for a stationary AR(1) process with parameter phi and innovation variance sigma2. The model is X_t = \phi X_{t-1} + \varepsilon_t, \; \varepsilon_t \stackrel{\text{i.i.d.}}{\sim} N(0, \sigma^2). The autocovariance is \gamma(h) = \mathrm{cov}(X_t, X_{t+h}) = \frac{\sigma^2}{1 - \phi^2}\,\phi^{\lvert h \rvert} .

Usage

ar1(phi = NULL, sigma2 = NULL)

Arguments

Value

A time_series_model object.

Examples

mod <- ar1(phi = 0.8, sigma2 = 1)
mod

Coerce to a list of models

Description

Helper used by + methods to normalize inputs.

Usage

as_model_list(x)

Arguments

Value

List of time_series_model objects.

Estimated northward and eastward velocity and their standard deviation using the GMWMX estimator

Description

Estimated northward and eastward velocity and standard deviation for a subset of 1202 GNSS station with more than 10 years of daily data.

Usage

df_estimated_velocities_gmwmx

Format

A data frame with 1202 rows and 12 variables:

station_name

Name of the GNSS station.

estimated_trend_N

Estimated northward velocity trend (in meters per day).

std_estimated_trend_N

Standard deviation of the estimated northward velocity trend.

estimated_trend_E

Estimated eastward velocity trend (in meters per day).

std_estimated_trend_E

Standard deviation of the estimated eastward velocity trend.

length_signal

Length of the signal (in days).

estimated_trend_N_scaled

Scaled estimated northward velocity trend (multiplying by 365.25 for yearly values).

std_estimated_trend_N_scaled

Scaled standard deviation of the estimated northward velocity trend.

estimated_trend_E_scaled

Scaled estimated eastward velocity trend (multiplying by 365.25 for yearly values).

std_estimated_trend_E_scaled

Scaled standard deviation of the estimated eastward velocity trend.

latitude

Latitude of the GNSS station.

longitude

Longitude of the GNSS station.

Download all stations name and location from the Nevada Geodetic Laboratory

Description

Download all stations name and location from the Nevada Geodetic Laboratory

Usage

download_all_stations_ngl(verbose = FALSE)

Arguments

Value

Return a data.frame with all stations name, latitude, longitude and heights.

Examples

df_all_stations <- download_all_stations_ngl()
head(df_all_stations)

Download estimated velocities using the MIDAS estimator provided by the Nevada Geodetic Laboratory for all stations.

Description

Download estimated velocities using the MIDAS estimator provided by the Nevada Geodetic Laboratory for all stations.

Usage

download_estimated_velocities_ngl(verbose = FALSE)

Arguments

Value

Return a data.frame with all stations name, information about the time series for each station, estimated velocities and estimated standard deviation of the estimated velocities.

Examples

df_estimated_velocities <- download_estimated_velocities_ngl()
head(df_estimated_velocities)

Download GNSS position time series and steps reference from the Nevada Geodetic Laboratory with IGS14 or IGS20 reference frame.

Description

Download GNSS position time series and steps reference from the Nevada Geodetic Laboratory with IGS14 or IGS20 reference frame.

Usage

download_station_ngl(station_name, verbose = FALSE, reference_frame = "IGS20")

Arguments

Value

A list of class gnss_ts_ngl that contains three data.frame: The data.frame df_position which contains the position time series extracted from the .tenv3 file available from the Nevada Geodetic Laboratory, the data.frame df_equipment_software_changes which specify the equipment or software changes for that stations and the data.frame df_earthquakes that specify the earthquakes associated with that station.

Examples

station_1LSU <- download_station_ngl("1LSU")
attributes(station_1LSU)

Fill missing model parameters using initial-parameter functions

Description

Ensures any NULL / missing parameters are populated from the model's get_initial_parameters_function(signal) while preserving any user-provided parameters (assumed to be in the domain).

Usage

fill_missing_parameters(model, signal)

Arguments

Value

Model with all parameters populated.

Flicker noise process (time_series_model)

Description

Constructs a time_series_model for flicker noise with variance sigma2. The process has spectral density S(f) \propto \frac{1}{|f|}. Hence, \kappa = -1 (Bos et al., 2008). The process is non-stationary and its covariance matrix is assumed to be given by

\mathbf C = \sigma^2 \mathbf U^\top \mathbf U,

where \mathbf U \in \mathbb{R}^{N \times N} is an upper-triangular Toeplitz matrix with entries

U_{i,j} = \begin{cases} h_{j-i}, & j \ge i, \\ 0, & j < i, \end{cases} \qquad i,j = 1, \ldots, N.

The coefficients \{h_i\}_{i \ge 0} define a causal linear filter and are given recursively by

h_0 = 1, \qquad h_i = \left(i - \frac{\kappa}{2} - 1\right)\frac{h_{i-1}}{i}, \quad i > 0.

Usage

flicker(sigma2 = NULL)

Arguments

Value

A time_series_model object.

References

Bos MS, Fernandes RMS, Williams SDP, Bastos L (2008). "Fast error analysis of continuous GPS observations." Journal of Geodesy, 82, 157-166.

Examples

mod <- flicker(sigma2 = 1)
mod

Format model text

Description

Internal helper to create a human-readable model label for printing.

Usage

format_model_text(model)

Arguments

Value

A single string describing the model.

Format parameter display

Description

Format parameter display

Usage

format_params(pars, digits = 4)

Arguments

Value

Single string for printing.

Generate a time series from a time_series_model or sum_model object

Description

Generate a time series from a time_series_model or sum_model object

Usage

generate(object, n, seed = NULL, ...)

Arguments

Value

A generated_time_series (single model) or generated_composite_model_time_series (sum model).

Examples

# Single model
m1 <- ar1(phi = 0.8, sigma2 = 1)
y1 <- generate(m1, n = 200, seed = 123)
plot(y1)

# Composite model
m2 <- wn(sigma2 = 1) + pl(kappa = 0.3, sigma2 = 2)
y2 <- generate(m2, n = 200, seed = 123)
plot(y2)

Compute autocovariance for a model (internal)

Description

If theta is NULL, uses domain parameters stored in the model. If theta is provided, it is treated as unconstrained parameters in real space and mapped to the domain via the model's transformation.

Usage

get_autocovariance(object, n, theta = NULL, prep = NULL, ...)

Arguments

Value

Numeric vector of autocovariances of length n.

Theoretical wavelet variance from model parameters

Description

Computes the theoretical WV implied by a model and parameter vector.

Usage

get_theoretical_wv(
  theta,
  model,
  n,
  wv_obj = NULL,
  tau = NULL,
  prep = NULL,
  return_components = FALSE
)

Arguments

Value

Numeric vector of theoretical wavelet variances.

Variance-covariance matrix implied by a model

Description

Constructs the variance-covariance matrix for a model using its get_variance_covariance_matrix_signal method. Supports both single time_series_model objects and composite sum_model objects.

Usage

get_variance_covariance_matrix_model(model, n, theta = NULL, prep = NULL, ...)

Arguments

Value

Variance-covariance matrix of dimension ⁠n x n⁠.

GMWM estimator

Description

Implements the Generalized Method of Wavelet Moments (GMWM) estimator to fit a time_series_model, a sum_model or a numeric vector.

Usage

gmwm2(x, model, omega = NULL, method = "L-BFGS-B", control = list(), ...)

Arguments

Details

The GMWM estimator solves a weighted least-squares criterion of the form

\left\{\hat{\boldsymbol{\nu}} - \boldsymbol{\nu}(\boldsymbol{\theta})\right\}^{\top} \boldsymbol{\Omega} \left\{\hat{\boldsymbol{\nu}} - \boldsymbol{\nu}(\boldsymbol{\theta})\right\}

where \hat{\boldsymbol{\nu}} denotes the empirical wavelet variance and \boldsymbol{\nu}(\boldsymbol{\theta}) the corresponding theoretical wavelet variance implied by the model parameters \boldsymbol{\theta}. The weighting matrix \boldsymbol{\Omega} defaults to a diagonal matrix with entries proportional to the inverse squared width of the empirical WV asymptotic confidence intervals. Provide omega to use a custom weighting (e.g., from a theoretical covariance).

Value

An object of class gmwm2_fit with elements: theta_hat (real space), theta_domain (constrained space), model, empirical_wvar, theoretical_wvar, optim, and n.

References

Guerrier, S., Skaloud, J., Stebler, Y., and Victoria-Feser, M.-P. (2013). Wavelet-variance-based estimation for composite stochastic processes. Journal of the American Statistical Association, 108(503), 1021-1030. doi:10.1080/01621459.2013.799920.

Examples

n = 10000
mod = wn(20) + ar1(phi = .995, sigma2 = .2)
y = generate(mod, n = n, seed = 123)
plot(y)
fit = gmwm2(y, model = wn() + ar1())
fit
plot(fit)

GMWMX estimator

Description

Dispatches either to the generic regression interface (design matrix + response) or to a gnss_ts_ngl workflow.

Convenience wrapper that selects the missing or non-missing implementation based on the presence of NA values in y.

Usage

gmwmx2(X, ...)

## Default S3 method:
gmwmx2(X, y, model, omega = NULL, method = "L-BFGS-B", control = list(), ...)

## S3 method for class 'gnss_ts_ngl'
gmwmx2(
  X,
  n_seasonal = 2,
  vec_earthquakes_relaxation_time = NULL,
  component = NULL,
  model = NULL,
  omega = NULL,
  method = "L-BFGS-B",
  control = list(),
  ...
)

Arguments

Value

A fitted model object.

A fitted model object.

A fitted model object.

GMWMX estimator

Description

GMWMX estimator

Usage

gmwmx2_new_no_missing(
  X = NULL,
  y = NULL,
  model = NULL,
  omega = NULL,
  method = "L-BFGS-B",
  control = list(),
  ...
)

Arguments

Value

A fitted model object (to be defined).

GMWMX estimator with missing

Description

Draft interface for a future gmwmx2() that can be called either with a gnss_ts_ngl object (current workflow) or with a generic design matrix and response vector.

Usage

gmwmx2_new_with_missing(
  X = NULL,
  y = NULL,
  model = NULL,
  omega = NULL,
  method = "L-BFGS-B",
  control = list(),
  ...
)

Arguments

Value

A fitted model object (to be defined).

Plot colors for generated time series

Description

Editable default palette used by the plotting methods for generated time series. When more colors are needed, the palette is expanded with interpolation.

Usage

gmwmx2_plot_colors

Format

A character vector of hex colors.

See Also

.gmwmx2_get_plot_colors

Inverse transform for Matérn alpha

Description

Maps domain values in (1/2, 10) back to the real line.

Usage

inv_trans_alpha_matern(x)

Arguments

Value

Numeric vector on the real line.

Inverse transform for power-law kappa

Description

Maps domain values back to the real line using atanh.

Usage

inv_trans_from_real_to_minus_1_and_1(kappa)

Arguments

Value

Numeric vector on the real line.

Loss function for GMWM optimization (internal)

Description

Computes the weighted squared error between empirical wavelet variance and theoretical wavelet variance implied by a model and parameter vector.

Usage

loss_fn_gmwm(theta, model, n, prep, wv_obj, omega = NULL)

Arguments

Value

Scalar objective value.

Loss function for GMWMX without missing value

Description

Computes the weighted squared error between empirical wavelet variance and theoretical wavelet variance implied by a model and parameter vector.

Usage

loss_fn_gmwmx_no_missing(
  theta,
  model,
  n,
  prep,
  wv_obj,
  quantities_D,
  omega = NULL
)

Arguments

Value

Scalar objective value.

Loss function for GMWMX with missing value

Description

Computes the weighted squared error between empirical wavelet variance and theoretical wavelet variance implied by a model and parameter vector.

Usage

loss_fn_gmwmx_with_missing(
  theta,
  model,
  n,
  prep,
  wv_obj,
  quantities_D,
  vec_autocov_omega,
  pstar_hat,
  omega = NULL
)

Arguments

Value

Scalar objective value.

Markov two-state missingness model (missingness_model)

Description

Constructs a missingness_model representing a two-state Markov process for missing/observed indicators. The process takes values in {0, 1}, where 1 indicates observed and 0 indicates missing.

Usage

markov_two_states(p1 = NULL, p2 = NULL)

Arguments

Value

A missingness_model object.

Examples

mod <- markov_two_states(p1 = 0.05, p2 = 0.95)
mod
z <- generate(mod, n = 200, seed = 123)
plot(z)

Matern process (time_series_model)

Description

Constructs a time_series_model for a Matern covariance process with variance sigma2, range lambda, and smoothness alpha. The autocovariance is \gamma(h) = \mathrm{cov}(X_t, X_{t+h}) = \frac{2 \sigma^2}{\Gamma(\alpha-1 / 2) 2^{\alpha-1 / 2}}|\lambda h|^{\alpha-1 / 2} \mathcal{K}_{|\alpha-1 / 2|}(| \lambda h|) where \mathcal{K}_\omega(x) is the modified Bessel function of the second kind of order \omega.

Usage

matern(sigma2 = NULL, lambda = NULL, alpha = NULL)

Arguments

Value

A time_series_model object.

References

Lilly JM, Sykulski AM, Early JJ, Olhede SC (2017). "Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion." Nonlinear Processes in Geophysics, 24(3), 481-514.

Examples

mod <- matern(sigma2 = 1, lambda = 0.2, alpha = 1.0)
mod

Stationary Power-Law process (time_series_model)

Description

Constructs a time_series_model representing a stationary power-law process with parameters kappa and sigma2. In the frequency domain, a power-law process is often described by a spectrum P(f) = P_0 f^{\kappa} (Bos et al., 2008), where f is the frequency, P_0 is a constant and \kappa is the spectral index. Note that we use the convention that the power spectral density satisfies P(f) \propto |f|^{\kappa}, where \kappa > -1 ensures second-order stationarity. This corresponds to the alternative notation P(f) \propto |f|^{-\alpha} with \alpha = -\kappa. The autocovariance \gamma(h) = \mathrm{cov}(X_t, X_{t+h}) used here (Hosking, 1981) is \gamma(0) = \sigma^{2} \frac{\Gamma(1+\kappa)}{\Gamma\left(1+\kappa/2\right)^2}, and for h > 0 \gamma(h) = \frac{-\kappa/2 + h - 1}{\kappa/2 + h}\,\gamma(h-1).

Usage

pl(kappa = NULL, sigma2 = NULL)

Arguments

Value

A time_series_model object.

References

Bos MS, Fernandes RMS, Williams SDP, Bastos L (2008). "Fast error analysis of continuous GPS observations." Journal of Geodesy, 82, 157-166.

Hosking JRM (1981). "Fractional differencing." Biometrika, 68(1), 165-176.

Examples

mod <- pl(kappa = -0.5, sigma2 = 2)
mod

Plot a generated_composite_model_time_series object

Description

Produces stacked line plots for each component and the sum for a generated_composite_model_time_series object.

Usage

## S3 method for class 'generated_composite_model_time_series'
plot(x, ...)

Arguments

Value

Invisibly returns x.

Examples

m2 <- wn(sigma2 = 1) + ar1(phi = 0.8, sigma2 = 0.5)
y2 <- generate(m2, n = 200, seed = 123)
plot(y2)

Plot a generated_missingness object

Description

Produces a step plot for a generated_missingness object.

Usage

## S3 method for class 'generated_missingness'
plot(x, ...)

Arguments

Value

Invisibly returns x.

Examples

m0 <- markov_two_states(p1 = 0.05, p2 = 0.9)
z0 <- generate(m0, n = 200, seed = 123)
plot(z0)

Plot a generated_time_series object

Description

Produces a single line plot for a generated_time_series object.

Usage

## S3 method for class 'generated_time_series'
plot(x, ...)

Arguments

Value

Invisibly returns x.

Examples

m1 <- wn(sigma2 = 1)
y1 <- generate(m1, n = 200, seed = 123)
plot(y1)

Plot method for a gmwm2_fit object

Description

Plots empirical wavelet variance with the fitted theoretical curve and, for sum models, component-implied theoretical curves.

Usage

## S3 method for class 'gmwm2_fit'
plot(
  x,
  show_ci = TRUE,
  col_emp = "black",
  col_theo = "darkorange",
  col_ci = "#e6f7fb",
  lwd = 2,
  pch_emp = 16,
  pch_theo = 21,
  cex_theo = 1.4,
  legend_pos = "auto",
  ...
)

Arguments

Value

The input object, invisibly.

Examples

n = 10000
mod = wn(20) + ar1(phi = .995, sigma2 = .2)
y = generate(mod, n = n, seed = 123)
plot(y)
fit = gmwm2(y, model = wn() + ar1() )
fit
plot(fit)

Plot a gmwmx2_fit_gnss_ts_ngl object

Description

Plot a gmwmx2_fit_gnss_ts_ngl object

Usage

## S3 method for class 'gmwmx2_fit_gnss_ts_ngl'
plot(x, ...)

Arguments

Value

No return value. Plot a gmwmx2_fit_gnss_ts_ngl object.

Examples

# station_data = gmwmx2::download_station_ngl("1LSU")
# fit station with WN and PL
# fit1 <- gmwmx2(
#   station_data,
#   n_seasonal = 2,
#   model = wn() + pl(), component = "N"
# )
# fit1
# plot(fit1)

Plot a gnss_ts_ngl object

Description

Plot a gnss_ts_ngl object

Usage

## S3 method for class 'gnss_ts_ngl'
plot(x, component = NULL, ...)

Arguments

Value

No return value. Plot a gnss_ts_ngl object.

Examples

station_1LSU <- download_station_ngl("1LSU")
plot(station_1LSU)
plot(station_1LSU, component = "N")
plot(station_1LSU, component = "E")
plot(station_1LSU, component = "V")

Prepare optimization layout for a model

Description

Build a bookkeeping structure that maps a flat optimization vector (theta, in real/unconstrained space) to per-component parameter blocks. This is used throughout the package to:

• Define a consistent parameter order for optimization.

• Provide an initial parameter vector (theta0) in real space.

• Record which subset of theta belongs to each component model.

Usage

prepare_optim_layout(model)

Arguments

Details

The function handles two cases:

• time_series_model: returns a single block of parameters.

• sum_model: concatenates all component blocks and records their indices.

Value

A list with:

• kind: "single" or "sum" depending on the model class.

• theta0: named numeric vector of initial parameters in real space.

• pnames: (single model) parameter names for the block.

• layout: (sum model) list of per-component layouts, each containing: i (component index), idx (indices into theta0), and pnames (parameter names for that component).

Print method for a gmwm2_fit object

Description

Print method for a gmwm2_fit object

Usage

## S3 method for class 'gmwm2_fit'
print(x, digits = 4, show_initial_parameters = FALSE, ...)

Arguments

Value

The input object, invisibly.

Examples

n = 10000
mod = wn(20) + ar1(phi = .995, sigma2 = .2)
y = generate(mod, n = n, seed = 123)
plot(y)
fit = gmwm2(y, model = wn() + ar1() )
fit

Print method for a gmwmx2_fit object

Description

Displays a table of regression coefficients with standard errors and summarizes the fitted stochastic model with estimated parameters.

Usage

## S3 method for class 'gmwmx2_fit'
print(x, digits = 4, ...)

Arguments

Value

The input object, invisibly.

Print method for a gmwmx2_fit_gnss_ts_ngl object

Description

Displays regression coefficients with standard errors and confidence intervals, along with the fitted stochastic and missingness models.

Usage

## S3 method for class 'gmwmx2_fit_gnss_ts_ngl'
print(x, digits = 4, ...)

Arguments

Value

The input object, invisibly.

Random walk process (time_series_model)

Description

Constructs a time_series_model for a random walk with innovation variance sigma2. The autocovariance returned is the mean of the diagonal and super-diagonals of the covariance matrix. The model is X_t = X_{t-1} + \varepsilon_t, \; \varepsilon_t \stackrel{\text{i.i.d.}}{\sim} N(0, \sigma^2).

Usage

rw(sigma2 = NULL)

Arguments

Value

A time_series_model object.

Examples

mod <- rw(sigma2 = 1)
mod

Sum of stochastic models (internal)

Description

Creates a composite model representing a sum of independent time_series_model components. This is an internal constructor used by the + methods.

Usage

sum_model(models)

Arguments

Value

A sum_model object.

Examples

mod <- pl(kappa = 0.5, sigma2 = 2) + wn(sigma2 = 1)
print(mod)

Convert optimization parameters to domain parameters

Description

Maps a real-valued optimization vector theta into the constrained parameter space used by each model component.

Usage

theta_to_domain(model, theta, prep = NULL)

Arguments

Value

Named numeric vector (single model) or a named list (sum model).

Transform Matérn alpha from real line to domain

Description

Ensures Matérn smoothness parameter alpha is in (1/2, 10) by mapping real values with a scaled logistic transform.

Usage

trans_alpha_matern(x)

Arguments

Value

Numeric vector with values > 1/2.

Transform power-law kappa from real line to domain

Description

Maps unconstrained real values to the stationary domain for the power-law parameter kappa using a tanh transform.

Usage

trans_from_real_to_minus_1_and_1(x)

Arguments

Value

Numeric vector of transformed values in (-1, 1).

White noise process (time_series_model)

Description

Constructs a time_series_model for white noise with variance sigma2. The process is defined as X_t \stackrel{\text{i.i.d.}}{\sim} N(0, \sigma^2) with autocovariance \gamma(h) = \mathrm{cov}(X_t, X_{t+h}) = \sigma^2 \mathbf{1}\{h=0\}

Usage

wn(sigma2 = NULL)

Arguments

Value

A time_series_model object.

Examples

mod <- wn(sigma2 = 1)
mod