# Estimate a model

# Model and estimator

The `gmwmx2` `R` package allows to estimate linear model with correlated residuals in presence of missing data.

More precisely, we assume the following model: \[\begin{equation} \boldsymbol{Y} = \boldsymbol{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}, \quad \boldsymbol{\varepsilon} \sim \mathcal{F}\left\{\boldsymbol{\Sigma}\left(\boldsymbol{\gamma}\right)\right\}, \end{equation}\]

where \(\boldsymbol{X} \in \mathbb{R}^{n \times p}\) is a design matrix of observed predictors, \(\boldsymbol{\beta} \in \mathbb{R}^p\) is the regression parameter vector and \(\boldsymbol{\varepsilon} = \{\varepsilon_{i}\}_{i=1,\ldots,n}\) is a zero-mean process following an unspecified joint distribution \(\mathcal{F}\) with positive-definite covariance function \(\boldsymbol{\Sigma}(\boldsymbol{\gamma}) \in \mathbb{R}^{n\times n}\) characterizing the second-order dependence structure of the process and parameterized by the vector \(\boldsymbol{\gamma} \in \boldsymbol{\Gamma} \subset \mathbb{R}^q\).

We then define the a random variable \(\boldsymbol{Z} =\{Z_{i}\}_{i=1,\ldots,n}\) which describes the missing observation mechanism. More specifically, the vector \(\boldsymbol{Z} \in \{0, 1\}^n\) is a binary-valued stationary process independent of \(\boldsymbol{Y}\) with expectation \(\mu(\boldsymbol{\vartheta}) = \mathbb{E}[Z_i] \in [0, \, 1)\), \(\forall \, i\), and covariance matrix \(\boldsymbol{\Lambda}(\boldsymbol{\vartheta}) = \mathbb{V}[\boldsymbol{Z}] \in \mathbb{R}^{n\times n}\) whose structure is assumed known up to the parameter vector \(\boldsymbol{\vartheta} \in \boldsymbol{\Upsilon} \subset \mathbb{R}^k\)

We then assume that we only observe the stochastic process \(\tilde{\boldsymbol{Y}} = \boldsymbol{Z} \odot \boldsymbol{Y}\), where \(\odot\) denotes the Hadamard product. Hence, \(\tilde{\boldsymbol{Y}} \in \mathbb{R}^n = \boldsymbol{Y} \odot \boldsymbol{Z}\) represents the observed process vector with null elements in the positions where observations are missing.

Using \(\otimes\) to denote the Kronecker product, we then define \(\tilde{\boldsymbol{X}} = \boldsymbol{Z} \otimes \boldsymbol{1}^T \odot \boldsymbol{X} \in \mathbb{R}^{n \times p}\) as the design matrix \(\boldsymbol{X}\) with zero-valued vectors for the rows where observations are missing in \(\boldsymbol{Y}\) (where \(\boldsymbol{1}\) represents a vector of ones of dimension \(p\)).

We estimate the parameters \(\boldsymbol{\beta}\) with the least square estimator:

\[\begin{equation} \hat{\boldsymbol{\beta}} = \left(\tilde{\boldsymbol{X}}^T \tilde{\boldsymbol{X}}\right)^{-1} \tilde{\boldsymbol{X}}^T \tilde{\boldsymbol{Y}}. \end{equation}\]

We compute the estimated residuals as \(\hat{\boldsymbol{\varepsilon}} = {\tilde{\boldsymbol{Y}}} - \tilde{{\boldsymbol{X}}} \hat{\boldsymbol{\beta}}\).

We then estimate with the Maximum Likelihood Estimator the parameters \(\boldsymbol{\vartheta}\) of the missingness process \(\boldsymbol{Z}\) assuming that \(\boldsymbol{Z}\) is generated from a Markov model with the following transition probabilities:

\[\begin{equation} \label{eq:markov_model_def} \begin{aligned} & P\left( Z_2=1 \mid Z_1=1\right)=1 - p_1 \\ & P\left(Z_2=1 \mid Z_1=0\right) = p_2 \\ & P\left(Z_2=0 \mid Z_1=1\right)=p_1 \\ & P\left(Z_2=0 \mid Z_1=0\right)=1-p_2. \end{aligned} \end{equation}\]

We then estimate the parameters \(\boldsymbol{\gamma}\) using a Generalized method of Wavelet Moments approach (Guerrier et al. 2013) and using the fact that the variance-covariance matrix of \(\hat{\boldsymbol{\varepsilon}}\) is given by:

\[\boldsymbol{\Sigma}_{\hat{\boldsymbol{\varepsilon}}}(\boldsymbol{\gamma}) = [\boldsymbol{\Lambda}(\hat{\boldsymbol{\vartheta}}) + \mu(\hat{\boldsymbol{\vartheta}})^2 \mathbf{1} \mathbf{1}^T ] \odot ( \boldsymbol{I} - \boldsymbol{P})\boldsymbol{\Sigma}(\boldsymbol{\gamma}) (\boldsymbol{I} - \boldsymbol{P})\]

where \(\boldsymbol{P} = \boldsymbol{X}(\boldsymbol{X}^T\boldsymbol{X})^{-1}\boldsymbol{X}^T\) and \(\boldsymbol{I}\) is the identity matrix of dimension \(n\times n\).

More precisely, we rely on the result of Xu et al. (2017) that provide a computationally efficient form of the theoretical Allan variance (equivalent to the Haar wavelet variance up to a constant) for zero-mean stochastic processes such as \(\boldsymbol{\varepsilon}\) to avoid computing these large matrices multiplication in the objective function. Indeed in Xu et al. (2017) they generalize the results in Zhang (2008) to zero-mean non-stationary processes by using averages of the diagonals and super-diagonals of the covariance matrix of \(\boldsymbol{\varepsilon}\). What this implies is that the GMWM, which uses this form, does not require the storage of the \(n \times n\) covariance matrix of \(\boldsymbol{\varepsilon}\), but only of a vector of dimension \(n\) which is then plugged into an explicit formula consisting in a linear combination of the elements of this vector (these elements being averages of the diagonal and super-diagonals of the covariance matrix).

# Estimating tectonic velocities and crustal uplift

While the GMWMX as described above and in more details in Voirol et al. (2024), is a general method for estimating large linear model with complex dependence structure in presence of missing data, the `gmwmx2` `R` package is currently developed specifically to estimate tectonic velocities from position times series in graticule distance coordinates system (GD) provided by the Nevada geodetic Laboratory (Blewitt 2024; Blewitt, Hammond, and Kreemer 2018).

## Trajectory model

To estimate the trajectory model (see e.g., Bevis and Brown (2014) for more details), we construct the design matrix \(\boldsymbol{X}\) such that \(i\)-th component of the vector \(\mathbf{X} {{\boldsymbol{\beta}}}\) can be described as follows with \(t_i\) representing the \(i^{th}\) ordered time point (epoch) indicated in Modified Julian Date and \(t_0\) representing the reference epoch located exactly in the middle of start and end point of the time series:

\[\begin{split} \mathbb{E}[\mathbf{Y}_i] &= \mathbf{X}_i^T {{\boldsymbol{\beta}}} \\ &= a + b \left(t_{i} - t_0\right) + \sum_{h=1}^{m}\left[c_{h} \sin \left(2 \pi f_{h} t_{i}\right) + d_h \cos \left(2 \pi f_h t_i\right)\right] + \\& \sum_{j=1}^{r}e_j H\left(t_i - t_j\right) + \sum_{k = 1 }^{s} l_k \left[1- \exp\left(\frac{-(t_i-t_k)}{\tau_k}\right)\right]H\left(t_{i}-\tau_k\right) \end{split}\]

where \(a\) is the initial position at the reference epoch \(t_0\), \(b\) is the velocity parameter, \(c_h\) and \(d_h\) are the periodic motion parameters (\(h = 1\) and \(h = 2\) represent the annual and semi-annual seasonal terms, respectively with \(f_1 = \frac{1}{365.25}\) and \(f_2 = \frac{2}{365.25}\)). The offset terms models earthquakes, equipment changes or human intervention in which \(e_j\) is the magnitude of the step at epochs \(t_j\), \(r\) is the total number of offsets, \(H\) is the Heaviside step function defined as \(H(x):= \begin{cases}1, & x \geq 0 \\ 0, & x<0\end{cases}\) and the last term allow to model post-seismic deformation (see e.g., Sobrero et al. (2020)) where \(s\) is the number of post seismic relaxation time specified, \(t_k\) is the time when the relaxation \(k\) starts in Modified Julian Date (MJD), \(\tau_k\) is the relaxation period in days for the post-seismic relaxation \(k\) and \(l_k\) is the amplitude of the transient. Note that by default the estimates of the functional parameters are provided in unit/day.

When loading data from a specific station using the function `gmwmx2::download_station_ngl()`, we extract from the Nevada Geodetic Laboratory the position time series in GD coordinates, the time steps associated with a equipment or software change and the time steps associated with an earthquake near the station. All these objects are stored in a object of class `gnss_ts_ngl`.

When applying the function `gmwmx2::gmwmx2()` to an object of class `gnss_ts_ngl`, we construct the design matrix \(\boldsymbol{X}\) by considering an offset term for all equipment or software changes steps and all earthquakes indicated by the NGL. We also specify a post-seismic relaxation term for all earthquakes indicated by the NGL. If no relaxation time is specified in the argument `vec_earthquakes_relaxation_time`, we consider a default relaxation time of \(365.25\) days.

## Stochastic model

It is generally recognized that noise in GNSS time series is best described by a combination of colored noise plus white noise (He et al. 2017; Langbein 2008; Williams et al. 2004; Bos et al. 2013) where the white noise generally model noise at high frequencies and the colored noise model the lower frequencies of the spectrum. In a large study on the noise properties of GNSS signals, concluded that the optimal noise models for 80–90% of GNSS time series signals are the power law and white noise model or white noise and flicker/pink noise with model. The package currently support both stochastic model specification.

More precisely, the power spectrum of a power-law noise has the following form: \[ P(f)=P_0\left(f / f_s\right)^\kappa \] where \(f\) is the frequency, \(P_0\) is a constant, \(f_s\) the sampling frequency and the exponent \(\kappa\) is called the spectral index.

Many stochastic noise can be expressed as such, for example, a spectral index \(\kappa=0\) produces a white noise, a spectral index \(\kappa=-2\) produces a red noise or random walk and a spectral index \(\kappa=-1\) produce a flicker noise, also called pink noise.

Granger (1980) and Hosking (1981) showed that power-law noise with a spectral index between \(-1\) and \(1\) can be obtained by using fractional differencing of Gaussian noise:

\[ (1-B)^{-\kappa / 2} \mathbf{v} \]

where \(B\) is the backward-shift operator \(\left(B v_i=v_{i-1}\right)\) and \(\mathbf{v}\) a vector with independent and identically distributed (IID) Gaussian noise.

Following from Hosking’s definition of the fractional differencing, we obtain

\[ \begin{aligned} (1-B)^{-\kappa / 2} & =\sum_{i=0}^{\infty}\binom{-\kappa / 2}{i}(-B)^i \\ & =1-\frac{\kappa}{2} B-\frac{1}{2} \frac{\kappa}{2}\left(1-\frac{\kappa}{2}\right) B^2+\ldots \\ & =\sum_{i=0}^{\infty} h_i \end{aligned} \] with the coefficient \(h_i\) that can be computed using the following recurrence relation (Kasdin and Walter 1992):

\[ \begin{aligned} h_0 & =1 \\ h_i & =\left(i-\frac{\kappa}{2}-1\right) \frac{h_{i-1}}{i} \quad \text { for } i>0 \end{aligned} \]

Assuming an infinite sequence of zero-mean white noise \(\mathbf{v}\), with variance \(\sigma_{P L}^2\), and a spectral index \(kappa > -1\) then the autocovariance \(\gamma(\tau)=\operatorname{Cov}\left(X_t, X_{t+\tau}\right)=\mathbb{E}\left[\left(X_t-\mu\right)\left(X_{t+\tau}-\mu\right)\right]\) is (Bos et al. 2008):

\[ \begin{aligned} & \gamma(0)=\sigma_{p l}^2 \frac{\Gamma(1-\alpha)}{\left(\Gamma\left(1-\frac{\alpha}{2}\right)\right)^2} \\ & \gamma(\tau)=\frac{\frac{\alpha}{2}+\tau-1}{-\frac{\alpha}{\alpha}+\tau} \gamma(\tau-1) \text { for } \tau>0 \end{aligned} \] When the argument `stochastic_model` is set to `"wn + pl"`, the stochastic model considered includes both white noise and power-law with the specified above stationary autocovariance structure. The parameters estimated are: \(\sigma^2_{W N}\), \(\kappa\) (constrained to be greater than \(-1\)) and \(\sigma^2_{P L}\).

When the argument `stochastic_model` is set to `"wn + fl"`, the stochastic model considered includes both white noise and flicker noise (not stationary power-law noise with a spectral index \(\kappa=-1\)) where the variance covariance of the flicker noise \(\omega\) is obtained as follows (see e.g., (Bos et al. 2008)):

\[ \operatorname{Cov}(\omega) = \sigma^2_{F L}\mathbf{U}^T \mathbf{U} \]

where \[ \mathbf{U}=\left(\begin{array}{cccc} h_0 & h_1 & \ldots & h_n \\ 0 & h_0 & & h_{n-1} \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \ldots & h_0 \end{array}\right) \] with the coefficients \(h_i\) computed considering a spectral index \(\kappa=-1\).

The stochastic parameters estimated are: \(\sigma^2_{W N}\), \(\sigma^2_{F L}\) and \(\kappa\) is fixed to \(-1\).

# Example of estimation

Let us showcase how to estimate the tectonic velocity in for one specific component (North, East or Vertical) of one station.

Let us first load the `gmwmx2` package.

    library(gmwmx2)

# Download a station from Nevada Geodetic Laboratory

    station_data <- download_station_ngl("CHML")

# Plot Station

    plot(station_data)

# Plot Northing component

    plot(station_data, component = "N")

# Estimate model on station data

    fit1 <- gmwmx2(station_data, n_seasonal = 2, component = "N", stochastic_model = "wn + pl")

# Extract estimated parameters

    summary(fit1)

    ## Summary of Estimated Model
    ## -------------------------------------------------------------
    ## Functional parameters
    ## -------------------------------------------------------------
    ## Parameter                  Estimate  Std_Deviation  95% CI Lower  95% CI Upper
    ## -------------------------------------------------------------
    ## Intercept               -0.70909375   0.18916301  -1.07984643  -0.33834107
    ## Trend                    0.00007383   0.00003653   0.00000224   0.00014542
    ## Sin (Annual)            -0.00066336   0.00152324  -0.00364885   0.00232212
    ## Cos (Annual)             0.00077064   0.00164693  -0.00245729   0.00399857
    ## Sin (Semi-Annual)        0.00073582   0.00135324  -0.00191648   0.00338812
    ## Cos (Semi-Annual)       -0.00030933   0.00137844  -0.00301101   0.00239236
    ## Jump: MJD 56248          0.01697696   0.00821201   0.00088172   0.03307221
    ## Jump: MJD 55391          0.00754302   0.00830207  -0.00872873   0.02381477
    ## Jump: MJD 55563          0.00163803   0.01285810  -0.02356339   0.02683945
    ## Jump: MJD 55603          0.00276825   0.01824830  -0.03299776   0.03853425
    ## Jump: MJD 55606          0.00077484   0.02364216  -0.04556294   0.04711263
    ## Jump: MJD 56011         -0.00118520   0.01018758  -0.02115250   0.01878209
    ## Jump: MJD 56085          0.00127046   0.01038474  -0.01908325   0.02162417
    ## Jump: MJD 56748         -0.00068122   0.00545275  -0.01136842   0.01000597
    ## Jump: MJD 57281         -0.00035896   0.00810240  -0.01623938   0.01552145
    ## Earthquake: MJD 55391    0.01819421   0.03580356  -0.05197948   0.08836790
    ## Earthquake: MJD 55563    0.00092715   0.19515335  -0.38156640   0.38342069
    ## Earthquake: MJD 55603   -0.37283777   3.36503347  -6.96818219   6.22250664
    ## Earthquake: MJD 55606    0.36328081   3.31283649  -6.12975939   6.85632101
    ## Earthquake: MJD 56011    0.00451098   0.08625675  -0.16454915   0.17357111
    ## Earthquake: MJD 56085   -0.01897612   0.07590773  -0.16775254   0.12980029
    ## Earthquake: MJD 56748   -0.00957194   0.02160005  -0.05190726   0.03276339
    ## Earthquake: MJD 57281   -0.02302075   0.03105817  -0.08389364   0.03785214
    ## -------------------------------------------------------------
    ## Stochastic parameters
    ## -------------------------------------------------------------
    ##  White Noise Variance  :     0.00000058
    ##  Stationary powerlaw Spectral index:    -0.77567027
    ##  Stationary powerlaw Variance:     0.00000343
    ## -------------------------------------------------------------
    ## Missingness parameters
    ## -------------------------------------------------------------
    ##  P(Z_{i+1} = 0 | Z_{i} = 1): 0.00279460
    ##  P(Z_{i+1} = 1 | Z_{i} = 0): 0.31578947
    ##  \hat{E[Z]}: 0.99122807
    ## -------------------------------------------------------------
    ## Running time: 1.14 seconds
    ## -------------------------------------------------------------

By default, the estimated parameters are provided in m/day, we can optionally scale the estimated functional parameters so that they are returned in m/year with the argument `scale_parameters`.

    summary(fit1, scale_parameters = TRUE)

    ## Summary of Estimated Model
    ## -------------------------------------------------------------
    ## Functional parameters
    ## -------------------------------------------------------------
    ## Parameter                  Estimate  Std_Deviation  95% CI Lower  95% CI Upper
    ## -------------------------------------------------------------
    ## Intercept              -258.99649344  69.09178842 -394.41391036 -123.57907651
    ## Trend                    0.02696692   0.01334112   0.00081881   0.05311503
    ## Sin (Annual)            -0.24229327   0.55636168  -1.33274213   0.84815560
    ## Cos (Annual)             0.28147693   0.60154229  -0.89752430   1.46047815
    ## Sin (Semi-Annual)        0.26875991   0.49427057  -0.69999261   1.23751243
    ## Cos (Semi-Annual)       -0.11298102   0.50347365  -1.09977124   0.87380920
    ## Jump: MJD 56248          6.20083621   2.99943624   0.32204921  12.07962322
    ## Jump: MJD 55391          2.75508821   3.03233021  -3.18816978   8.69834621
    ## Jump: MJD 55563          0.59829213   4.69642231  -8.60652645   9.80311071
    ## Jump: MJD 55603          1.01110216   6.66519069 -12.05243155  14.07463586
    ## Jump: MJD 55606          0.28301205   8.63529914 -16.64186325  17.20788735
    ## Jump: MJD 56011         -0.43289581   3.72101510  -7.72595140   6.86015978
    ## Jump: MJD 56085          0.46403594   3.79302525  -6.97015695   7.89822882
    ## Jump: MJD 56748         -0.24881736   1.99161707  -4.15231510   3.65468037
    ## Jump: MJD 57281         -0.13111171   2.95940189  -5.93143283   5.66920941
    ## Earthquake: MJD 55391    6.64543546  13.07725059 -18.98550473  32.27637564
    ## Earthquake: MJD 55563    0.33864093  71.27976220 -139.36712581 140.04440768
    ## Earthquake: MJD 55603  -136.17899727 1229.07847654 -2545.12854547 2272.77055093
    ## Earthquake: MJD 55606  132.68831606 1210.01352704 -2238.89461773 2504.27124986
    ## Earthquake: MJD 56011    1.64763593  31.50527896 -60.10157615  63.39684801
    ## Earthquake: MJD 56085   -6.93102876  27.72529823 -61.27161475  47.40955724
    ## Earthquake: MJD 56748   -3.49614936   7.88941855 -18.95912558  11.96682685
    ## Earthquake: MJD 57281   -8.40832800  11.34399617 -30.64215194  13.82549593
    ## -------------------------------------------------------------
    ## Stochastic parameters
    ## -------------------------------------------------------------
    ##  White Noise Variance  :     0.00000058
    ##  Stationary powerlaw Spectral index:    -0.77567027
    ##  Stationary powerlaw Variance:     0.00000343
    ## -------------------------------------------------------------
    ## Missingness parameters
    ## -------------------------------------------------------------
    ##  P(Z_{i+1} = 0 | Z_{i} = 1): 0.00279460
    ##  P(Z_{i+1} = 1 | Z_{i} = 0): 0.31578947
    ##  \hat{E[Z]}: 0.99122807
    ## -------------------------------------------------------------
    ## Running time: 1.14 seconds
    ## -------------------------------------------------------------

# Plot estimated model

    plot(fit1)

    fit2 <- gmwmx2(station_data, n_seasonal = 2, component = "N", stochastic_model = "wn + fl")

    summary(fit2)

    ## Summary of Estimated Model
    ## -------------------------------------------------------------
    ## Functional parameters
    ## -------------------------------------------------------------
    ## Parameter                  Estimate  Std_Deviation  95% CI Lower  95% CI Upper
    ## -------------------------------------------------------------
    ## Intercept               -0.70909375   0.01066519  -0.72999715  -0.68819036
    ## Trend                    0.00007383   0.00001016   0.00005391   0.00009375
    ## Sin (Annual)            -0.00066336   0.00040724  -0.00146153   0.00013481
    ## Cos (Annual)             0.00077064   0.00042793  -0.00006809   0.00160938
    ## Sin (Semi-Annual)        0.00073582   0.00026087   0.00022454   0.00124711
    ## Cos (Semi-Annual)       -0.00030933   0.00027063  -0.00083974   0.00022109
    ## Jump: MJD 56248          0.01697696   0.00179981   0.01344939   0.02050453
    ## Jump: MJD 55391          0.00754302   0.00171565   0.00418040   0.01090564
    ## Jump: MJD 55563          0.00163803   0.00178150  -0.00185363   0.00512970
    ## Jump: MJD 55603          0.00276825   0.00219231  -0.00152859   0.00706509
    ## Jump: MJD 55606          0.00077484   0.00332771  -0.00574734   0.00729703
    ## Jump: MJD 56011         -0.00118520   0.00180644  -0.00472576   0.00235535
    ## Jump: MJD 56085          0.00127046   0.00187483  -0.00240414   0.00494506
    ## Jump: MJD 56748         -0.00068122   0.00176286  -0.00413637   0.00277393
    ## Jump: MJD 57281         -0.00035896   0.00176295  -0.00381428   0.00309635
    ## Earthquake: MJD 55391    0.01819421   0.00873109   0.00108158   0.03530684
    ## Earthquake: MJD 55563    0.00092715   0.02591243  -0.04986029   0.05171458
    ## Earthquake: MJD 55603   -0.37283777   0.48867274  -1.33061874   0.58494319
    ## Earthquake: MJD 55606    0.36328081   0.48325352  -0.58387868   1.31044030
    ## Earthquake: MJD 56011    0.00451098   0.01414518  -0.02321306   0.03223502
    ## Earthquake: MJD 56085   -0.01897612   0.01363705  -0.04570426   0.00775201
    ## Earthquake: MJD 56748   -0.00957194   0.00680936  -0.02291803   0.00377416
    ## Earthquake: MJD 57281   -0.02302075   0.00761179  -0.03793957  -0.00810192
    ## -------------------------------------------------------------
    ## Stochastic parameters
    ## -------------------------------------------------------------
    ##  White Noise Variance  :     0.00000159
    ##  Flicker Noise Variance:     0.00000224
    ## -------------------------------------------------------------
    ## Missingness parameters
    ## -------------------------------------------------------------
    ##  P(Z_{i+1} = 0 | Z_{i} = 1): 0.00279460
    ##  P(Z_{i+1} = 1 | Z_{i} = 0): 0.31578947
    ##  \hat{E[Z]}: 0.99122807
    ## -------------------------------------------------------------
    ## Running time: 2.01 seconds
    ## -------------------------------------------------------------

    plot(fit2)

# References

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