Toby A. Patterson, Bernie J. McConnell, Mike A. Fedak, Mark V. Bravington, and Mark A. Hindell. 2010. Using GPS data to evaluate the accuracy of state–space methods for correction of Argos satellite telemetry error. Ecology 91:273–285.

Appendix D. Diagnostics for the Kalman filter.

In this appendix we examine the errors (KF relative to GPS) and residual diagnostics (KF relative to Argos locations). We thus assess model fit and see if diagnostics without reference to the GPS data can aid in determining an appropriate maximum speed threshold. To illustrate the effects of a range of SF thresholds, we chose three examples starting with lenient application of the SF retaining data implying speeds less than 200 km/h.  Filtering with such impossibly high speeds removes only the most erroneous points in the Argos data. The other cases show are an example of harsh application of the SF, retaining only data implying speeds <2km/h and the case of the SF yielding the lowest RMSE (5km/h) for S2. Due to the different error variances for each location class, residuals were standardized by dividing by the standard deviation of residuals within location class.

Results

The QQ-plots (Fig. D1) showed departure from normality and that long tailed residuals and outliers persisted despite the speed filtering. The QQ-plots also indicated concentration of errors around zero. There was no indication of temporal dependence in residuals although errors were autocorrelated (Fig. D2). For S1 the plot of residuals through time indicates that the smallest spread of residuals coincided with the 0.55ms^-1 (2kmh^-1) speed threshold (Fig. D1) that also yielded the worst RMSE. The minimal SF (55.5 ms^-1/ 200kmh^-1) gave heavy-tailed residuals in the QQ-plot in both the longitude and latitude components for both seals. However, for S1 the RMSE was close to that obtained when using data with a 5km/h SF applied. At 5km/h SF, the QQ-plot indicated less mass in the tails and fewer outliers. This was also the case for the 2km/h SF data. For S2 the 5km/h SF resulted in the least variable residuals (Fig. D1). However there was little to differentiate between the residual QQ-plots with 2km/h and 5km/h speed thresholds despite a factor of 2 increase in RMSE for 2km/h SF relative to the 5 km/h SF.

The very small spread of residuals for the most stringent application SF may be useful in diagnosing when the SF has been applied too harshly. However, the heavy-tailed residuals resulting from minimal speed filtering indicated different things for each seal’s data. For S1 the RMSE and errors through time (Fig. D2) only changed slightly despite the inclusion of many apparent outliers in the data set indicating that the KF reliably gave inaccurate points little weight when interpolating positions. Consequently there was little to differentiate the error QQ-plots (Fig. D1). However, the RMSE increased by a factor of ~10 when minimal SF was used on S2’s data suggesting that outliers did influence the predictions when ideally, they should have carried less weight. This was reflected in heavy tailed error distributions (Fig. D1).

FIG. D1. Residuals through time (left column) and QQ-plots for the residuals for (A–D) S1 and (E–H) S2. For each seal, northing residuals are shown in the upper row and easting residuals in the lower.

FIG. D2. (A–B) Easting errors at three different speed thresholds for S1. (C–D)  The same for S2.