Implements the maximum likelihood estimation (MLE) procedure for the fluctuating mean sequence \(\{\boldsymbol{\mu}_t\}_{t=1}^n\) in the proposed model, given the parameters (or their estimates) \(\Sigma_{\boldsymbol{\eta}}\), \(\Sigma_{\boldsymbol{\nu}}\), \(\Phi\), and \(\Gamma_{\boldsymbol{\epsilon}}(0)\).
estimate_mus(data, Sig_eta, Sig_nu, Phi, Sig_e1)A numeric matrix of dimension \(n \times p\), containing the estimated fluctuating mean vectors \(\hat{\boldsymbol{\mu}}_t\).
Numeric matrix of dimension \(n \times p\), representing the observed time series \(\{\mathbf{y}_t\}_{t=1}^n\).
Numeric \(p \times p\) matrix \(\Sigma_{\boldsymbol{\eta}}\), covariance of the random walk innovation.
Numeric \(p \times p\) matrix \(\Sigma_{\boldsymbol{\nu}}\), covariance of the VAR(1) innovation.
Numeric \(p \times p\) autoregressive coefficient matrix \(\Phi\).
Numeric \(p \times p\) initial-state covariance matrix \(\Gamma_{\boldsymbol{\epsilon}}(0)\).
The algorithm performs forward and backward recursions to compute the MLE of \(\boldsymbol{\mu}_t\) under the proposed model with Gaussian noises: $$\mathbf{y}_t = \boldsymbol{\mu}_t + \boldsymbol{\epsilon}_t, \quad \boldsymbol{\mu}_t = \boldsymbol{\mu}_{t-1} + \boldsymbol{\eta}_t, \quad \boldsymbol{\epsilon}_t = \Phi \boldsymbol{\epsilon}_{t-1} + \boldsymbol{\nu}_t.$$
This estimation provides the smoothed mean trajectory that captures gradual fluctuations between change points, conditioned on the given model parameters.