Uncertainty of peak position via Monte-Carlo of simulated peak scans, bootstrap analysis of both simulated and real scans, and comparison of cal on/cal off, LCP/RCP data are significantly different

• Simulated data are configured to be similar to TPTCSPNT_070330 peak scans, with additive Gaussian noise. These scans used the X-Band reciever, and the nominal FWHM of the peak is 82".
• Simulated and real data are processed with a nonlinear optimization (same configuration) to minimize the squared error between the scan and Gaussian peak with 3rd order polynomial baseline.
• Data (both real and simulated) are bootstrapped to estimate the uncertainty in position estimate
• Simulated data are both bootstrapped and Monte-Carlo'd to verify the validity of the bootstrap.
• Simulated data have Gaussian peaks and additive IID Gaussian noise. SNR (signal to noise ratio) is (peak-baseline)/standard devition of noise. The same definition of SNR is used for real data.
• Using estimated peak amplitude and baseline residuals, the position error is predicted using the formulation of Kaper, et al.

• Agreement of Kaper, bootstrap, and Monte-Carlo. The figure below compares the results for simulated data (error units are arc-sec), which are in good agreement.

• Real data show excess error. The absolute difference between LCP and RCP position estimates aggregated across all forward and backward scans and for both cal on and off are shown below. The sample deviation from the bootstrap and the sample deviation of the mean of each peak bootstrap are marked for the two cases which were single -source tracks on two different sources, one with nominal SNR of 75 and the other with nomical SNR of 350. The real data thus have substantial excess error. The standard deviation of the error is more closely approximated by estimates from the bootstrap of the real data. Note that the markers are placed at the sample estimate divided by root-2 since the underlying data are a difference of two random variates. The bootstrap appears to be a reasonable estimate of the actual uncertainty, while in both cases the Kaper estimate substantially under-estimates the peak uncertainty. Note that estimates of "squint" from these data show that LCP and RCP mean positions differ by less than 0".1, so aggregation of elevation and azimuth scans should not introduce any biases.

• Histograms of the LCP-RCP difference show some skewness. It's not clear why this should (or could) be, although some subtle effect caused by structure vibration might be the culprit. See Astronomical Data Analysis, DTD 4/16/2007 pp 17.

• Relevance to astronomical data collection for pointing model generation.
  • Bootstrap analysis could be used to replace the robust fitting method now used with an explicit weighted least squares fit.
  • X-band data collection can yield 1-sigma position estimates of less than ~ 0".4 if SNR is above 100. This should be adequate for fitting models as good as the current pointing objectives require.
  • What is the source of this excess uncertainty? The Kaper prediction under-estimates by at least a factor of two for SNRs > 75. Candidates are things like DCR timing errors (although this is not very plausible, requiring an unreasonably large timing error), or quantization error (the 10 Hz sampling rate and 40"/sec source relative rate yeilds quanta of 4"), etc.