Utility Functions

The FactorGraph provides several utility functions to evaluate and compare obtained results.

The WLS results

The function provides the estimate obtained by the weighted least-squares (WLS) method and root mean square error (RMSE), the mean absolute error (MAE) and the weighted residual sum of squares (WRSS) error metrics evaluated according to the WLS solutions. These results can be used to compare results obtained by the GBP algorithm.

exact = wls(gbp)

The function returns the composite type WeightedLeastSquares with fields estimate, rmse, mae, wrss. Note that results are obtained according to variables ContinuousSystem.coefficient, ContinuousSystem.observation and ContinuousSystem.variance.

The GBP error metrics

The package provides the function to obtain RMSE, MAE, and WRSS error metrics of the GBP algorithm.

evaluation = errorMetric(gbp)

The function returns the composite type ErrorMetric with fields rmse, mae, wrss. Further, passing the composite type WeightedLeastSquares, we obtained additional fields rmseGBPWLS and maeGBPWLS that determine the distance between the GBP estimate and WLS estimate.

evaluation = errorMetric(gbp, exact)

The function returns the composite type ErrorMetricWiden with fields rmse, mae, wrss, rmseGBPWLS, maeGBPWLS.

Error metrics

The root mean square error, the mean absolute error and the weighted residual sum of squares are evaluated according to:

\[ \begin{aligned} \text{rmse} = \sqrt {\cfrac{\sum_{i=1}^m \left[z_i - h_i(\hat{\mathbf x}) \right]^2}{m}}; \quad \text{mae} = \cfrac{\sum_{i=1}^m \left|z_i - h_i(\hat{\mathbf x}) \right|}{m}; \quad \text{wrss} = \sum_{i=1}^m \cfrac{\left[z_i - h_i(\hat{\mathbf x}) \right]^2}{v_i}, \end{aligned}\]

where $m$ denotes the number of observations, $z_i$ is observation value, $v_i$ is observation variance, and corresponding equation $h_i(\hat{\mathbf x})$ is evaluated at the point $\hat{\mathbf x}$ obtained using the GBP or WLS algorithm. Note, wrss is the value of the objective function of the optimization problem we are solving.

Fields rmseGBPWLS and maeGBPWLS determine distance beetwen the GBP estimate $\hat{x}_{\text{gbp},i}$ and WLS estimate $\hat{x}_{\text{wls},i}$, where root mean square error and mean absolute error are obtained using:

\[ \begin{aligned} \text{rmse} = \sqrt {\cfrac{\sum_{i=1}^n \left[\hat{x}_{\text{wls},i} - \hat{x}_{{\text{gbp}},i}) \right]^2}{n}}; \quad \text{mae} = {\cfrac{\sum_{i=1}^n \left|\hat{x}_{\text{wls},i} - \hat{x}_{{\text{gbp}},i}) \right|}{n}}, \end{aligned}\]

where $n$ is the number of state variables.