3:30 PM - 5:00 PM
[SGD01-P12] Model selection for the GNSS-A analysis software GARPOS-MCMC with the widely applicable Bayesian information criterion (WBIC)
Keywords:GNSS-A, GARPOS, MCMC, widely applicable Bayesian information criterion (WBIC), model selection
Precise seafloor positioning by the GNSS-Acoustic ranging combination technique (GNSS-A), as seafloor geodesy, is applied for the observations of the crustal deformation in the plate subduction zones (e.g., Spiess et al., 1998; Fujita et al., 2006; Ishikawa et al., 2020). For the precise positioning with the GNSS-A, it is required to appropriately cancel or correct the effects of sound speed variation on acoustic travel time. Watanabe et al. (2020) introduced a perturbation field model which is naturally tied to the spatiotemporal variations of sound speed and formulated the methods to directly estimate the perturbation field with an assumption that each coefficient of the perturbation field changes smoothly. By modeling the perturbation field with a linear function, they derived a semi-analytical solution of position and perturbation coefficients, under given hyperparameters which control the degree of smoothness and correlations among data errors. Based on the empirical Bayes approach, it was implemented in an open-source software named GARPOS (the latest version is v1.0.1, https://doi.org/10.5281/zenodo.6414642), in which the hyperparameters are selected to minimize the Akaike Bayesian Information Criterion (ABIC; Akaike, 1980).
Watanabe et al. (under review, preprint https://doi.org/10.21203/rs.3.rs-1881756/v1) developed the upgraded version of GARPOS, i.e., GARPOS-MCMC (the latest version is v1.0.0, https://doi.org/10.5281/zenodo.6825238), with a full-Bayes GNSS-A analysis scheme, where the hyperparameters are also expressed as probability density functions and estimated with the Markov chain Monte Carlo (MCMC) method. This enabled us to directly sample from the joint posterior of parameters including any hyperparameters and evaluate the correlations between those parameters. The GARPOS-MCMC has advantages in introducing more flexible perturbation field expressions, because it does not require the assumptions of the Gaussian distribution nor the linearity of the observation equation. Actually, Watanabe et al. (under review) introduced the simple and realistic, but nonlinear, single-gradient layer model, and discussed the differences between the assumed models.
On the other hand, there is also a room for improvements in GARPOS-MCMC. One is calculation speed, due to the requirement to calculate the inverse of the data-error covariance matrix, E (\Sigma_d in the paper), which is N * N matrix (N is the number of acoustic data) with finite non-diagonal components, in each MCMC step. The non-diagonal components, introduced for statistical appropriateness (see Watanabe et al., 2020), are controlled by the estimation parameter m (\mu, in the paper) in GARPOS-MCMC. Another issue is to obtain some method for the statistical evaluation of the models within a given dataset, such as to compare the results of different perturbation field models.
For the former issue, Watanabe et al. (under review) showed the results that m is less correlated with the position and perturbation coefficients. Therefore, if one can select sufficiently appropriate value for m, as m= mk, the distributions of position and perturbation coefficients can be less affected even when putting the selected mk as a fixed value. With fixed mk, the calculation of E-1 is not required in each MCMC step, which reduces the computational costs. This also reduces to the model selection problem to select the proper mk, similar to the latter issue.
In this study, we compare the MCMC models by applying the widely applicable Bayesian information criterion (WBIC; Watanabe, 2013), which has the same asymptotic behavior as the Bayes free energy. The WBIC value can be easily calculated from a single MCMC sample series with an inverse temperature of 1/log(N).
Watanabe et al. (under review, preprint https://doi.org/10.21203/rs.3.rs-1881756/v1) developed the upgraded version of GARPOS, i.e., GARPOS-MCMC (the latest version is v1.0.0, https://doi.org/10.5281/zenodo.6825238), with a full-Bayes GNSS-A analysis scheme, where the hyperparameters are also expressed as probability density functions and estimated with the Markov chain Monte Carlo (MCMC) method. This enabled us to directly sample from the joint posterior of parameters including any hyperparameters and evaluate the correlations between those parameters. The GARPOS-MCMC has advantages in introducing more flexible perturbation field expressions, because it does not require the assumptions of the Gaussian distribution nor the linearity of the observation equation. Actually, Watanabe et al. (under review) introduced the simple and realistic, but nonlinear, single-gradient layer model, and discussed the differences between the assumed models.
On the other hand, there is also a room for improvements in GARPOS-MCMC. One is calculation speed, due to the requirement to calculate the inverse of the data-error covariance matrix, E (\Sigma_d in the paper), which is N * N matrix (N is the number of acoustic data) with finite non-diagonal components, in each MCMC step. The non-diagonal components, introduced for statistical appropriateness (see Watanabe et al., 2020), are controlled by the estimation parameter m (\mu, in the paper) in GARPOS-MCMC. Another issue is to obtain some method for the statistical evaluation of the models within a given dataset, such as to compare the results of different perturbation field models.
For the former issue, Watanabe et al. (under review) showed the results that m is less correlated with the position and perturbation coefficients. Therefore, if one can select sufficiently appropriate value for m, as m= mk, the distributions of position and perturbation coefficients can be less affected even when putting the selected mk as a fixed value. With fixed mk, the calculation of E-1 is not required in each MCMC step, which reduces the computational costs. This also reduces to the model selection problem to select the proper mk, similar to the latter issue.
In this study, we compare the MCMC models by applying the widely applicable Bayesian information criterion (WBIC; Watanabe, 2013), which has the same asymptotic behavior as the Bayes free energy. The WBIC value can be easily calculated from a single MCMC sample series with an inverse temperature of 1/log(N).