Supported models
The gmwmx package allows to estimate linear model with
correlated residuals that are described by a functional model and a
stochastic noise model.
Functional model
More precisely, for the functional model, we consider a linear model
which can be expressed as:
\[\begin{equation}
\mathbf{Y} = \mathbf{A} {{\bf x}}_0 + \boldsymbol{\varepsilon},
\end{equation}\]
where \(\mathbf{Y} \in {\rm
I\!R}^n\) denotes the response variable of interest (i.e., vector
of GNSS observations), \(\mathbf{A} \in {\rm
I\!R}^{n \times p}\) a fixed design matrix, \({{\bf x}}_0 \in \mathcal{X} \subset {\rm
I\!R}^p\) a vector of unknown constants and \(\boldsymbol{\varepsilon} \in {\rm I\!R}^n\)
a vector of (zero mean) residuals.
The gmwmx package allows to estimate functional models
for which the \(i\)-th component of the
vector \(\mathbf{A} {{\bf x}}_0\) can
be described as follows:
\[\begin{equation}
\mathbb{E}[\mathbf{Y}_i] = \mathbf{A}_i^T {{\bf x}}_0 =
a+b\left(t_{i}-t_{0}\right)+\sum_{h=1}^{2}\left[c_{h} \sin \left(2 \pi
f_{h} t_{i}\right)+d_{h} \cos \left(2 \pi f_{h} t_{i}\right)\right] +
\sum_{k=1}^{n_{g}} g_{k} H\left(t_{i}-t_{k}\right),
\end{equation}\]
where \(a\) is the initial position
at the reference epoch \(t_0\), \(b\) is the velocity parameter, \(c_k\) and \(d_k\) are the periodic motion parameters
(\(h = 1\) and \(h = 2\) represent the annual and
semi-annual seasonal terms, respectively). The offset terms models
earthquakes, equipment changes or human intervention in which \(g_k\) is the magnitude of the change at
epochs \(t_k\), \(n_g\) is the total number of offsets, and
\(H\) is the Heaviside step
function.
Stochastic model
Regarding the stochastic model, we assume that \(\boldsymbol{\varepsilon}_i=\mathbf{Y}_i-\mathbb{E}[\mathbf{Y}_i]\)
is a strictly (intrinsically) stationary process and that
\[\begin{equation}
\boldsymbol{\varepsilon} \sim \mathcal{F} \left\{\mathbf{0},
\boldsymbol{\Sigma}(\boldsymbol{\gamma}_0)\right\} ,
\label{eq:model:noise}
\end{equation}\]
where \(\mathcal{F}\) denotes some
probability distribution in \({\rm
I\!R}^n\) with mean \({\bf 0}\)
and covariance \(\boldsymbol{\Sigma}(\boldsymbol{\gamma}_0)\).
We assume that \(\boldsymbol{\Sigma}(\boldsymbol{\gamma}_0) >
0\) and that it depends on the unknown parameter vector \(\boldsymbol{\gamma}_0 \in \boldsymbol{\Gamma}
\subset {\rm I\!R}^q\). This parameter vector specifies the
covariance of the observations and is often referred to as the
stochastic parameters.
Hence, we let \(\boldsymbol{\theta}_0 =
\left[\boldsymbol{{\bf x}}_0^{\rm T} \;\; \boldsymbol{\gamma}_0^{\rm
T}\right]^{\rm T} \in \boldsymbol{\Theta} = \mathcal{X} \times
\boldsymbol{\Gamma} \subset {\rm I\!R}^{p + k}\) denote the
unknown parameter vector of the model described above.
The gmwmx allows to estimate parameters of a specified
functional model as well as parameters of a stochastic model (i.e. \(\hat{\boldsymbol{\theta}} =
\left[\boldsymbol{\hat{\boldsymbol{x}}}^{T} \;\;
\hat{\boldsymbol{\gamma}}^{T}\right]\)) defined by a combinations
of
- White noise (
wn)
- Matérn process (
matern)
- Fractional Gaussian noise (
fgn) and
- Power Law process (
powerlaw).
Estimate a model with the GMWMX
You can estimate a model using the GMWMX estimator with the function
estimate_gmwmx().
The stochastic model considered is specified by a string provided to
the argument model_string which is a combination of the
strings wn, powerlaw, matern and
fgn separated by the character +.
You specify the initialization values for solving the optimization
problem at the GMWM estimation step that estimate the stochastic model
by providing a numeric vector of the correct length (the total number of
parameters of the stochastic model specified in
model_string) to the argument theta_0.
You can compute confidence intervals for estimated functional
parameters of an estimated model by setting the argument ci
to TRUE.
Let us consider a single sinusoidal signal with the jumps specified
in the gnssts object and a combination of a White noise and
a Power Law process for the stochastic model.
fit_dobs_wn_plp_gmwmx = estimate_gmwmx(x = data_dobs, theta_0 = c(0.1, 0.1, 0.1),
model_string = "wn+powerlaw",
n_seasonal = 1, ci = T)
Estimated models are of class gnsstsmodel
class(fit_dobs_wn_plp_gmwmx)
## [1] "gnsstsmodel"
We can print the estimated model or extract estimated parameters
(functional and stochastic) as such:
print(fit_dobs_wn_plp_gmwmx)
## GNSS time series model
##
## * Model: wn + powerlaw
##
## * Functional parameters:
## bias : +0.012013 +/- 0.020
## trend : +0.000008 +/- 0.000
## A*cos(U) : -0.000655 +/- 0.000
## A*sin(U) : +0.000016 +/- 0.000
## jump : -0.003758 +/- 0.001
## jump : +0.001568 +/- 0.001
##
## * Stochastc parameters:
## wn_sigma2 : +0.00000079
## powerlaw_sigma2 : +0.00000025
## powerlaw_d : +0.49990000
##
## * Estimation time: 1.66 s
fit_dobs_wn_plp_gmwmx$beta_hat
## bias trend A*cos(U) A*sin(U) jump
## 1.201297e-02 8.351512e-06 -6.553985e-04 1.578605e-05 -3.758074e-03
## jump
## 1.567881e-03
fit_dobs_wn_plp_gmwmx$theta_hat
## wn_sigma2 powerlaw_sigma2 powerlaw_d
## 7.916985e-07 2.525867e-07 4.999000e-01
We can also plot graphically the estimated functional model on the
time series and the Wavelet variance of residuals by calling the
plot.gnsstsmodel method on a gnsstsmodel
object.
plot(fit_dobs_wn_plp_gmwmx)
We can specify the number of iterations of the GMWMX to compute
respectively the GMWMX-1 and GMWMX-2 or other iteration of the GMWMX
with the argument k_iter. For example we can compute the
GMWMX-2 as such:
fit_dobs_wn_plp_gmwmx_2 = estimate_gmwmx(x = data_dobs,
theta_0 = c(0.1, 0.1, 0.1),
model_string = "wn+powerlaw",
n_seasonal = 1,
k_iter = 2)
Estimate a model with the MLE implemented in Hector
Assuming that you have Hector available on the PATH, an
estimation of the model can the be performed using the Maximum
Likelihood Estimation (MLE) method implemented in Hector as such:
fit_dobs_wn_plp_mle = estimate_hector(x = data_dobs,
model_string = "wn+powerlaw",
n_seasonal = 1)
Similarly we can plot and extract the model parameters of the
estimated model:
plot(fit_dobs_wn_plp_mle)
fit_dobs_wn_plp_mle$beta_hat
## bias trend A*cos(U) A*sin(U) jump
## 1.200000e-02 8.451006e-06 -6.672690e-04 1.359080e-06 -4.194170e-03
## jump
## 9.071300e-04
fit_dobs_wn_plp_mle$theta_hat
## wn_sigma2 powerlaw_sigma2 powerlaw_d
## 7.779727e-07 2.700504e-07 4.854970e-01
