Current Results:

This describes work done since the 12/1/2006 analysis posted previously, which contains history and details of the analysis methodology not repeated here. This summary includes 225 AORs from the two standard calibrators HD 163588 and HD 180711 from MC3 through MC47, analyzed using S14 and S16 pipeline data. (S14 and S16 are equivalent for 70µ data.)

Run   Vers    Data          #AORs      Star     
 A1     S14  (MC3-MC29)        82       HD 163588
 A2     S16  (MC30-MC46)       38       HD 163588
 A           (MC3-MC46)       120       HD 163588

 B1     S14  (MC4-MC29)        67       HD 180711
 B2     S16  (MC30-MC47)       38       HD 180711
 B           (MC4-MC47)       105       HD 180711

For all the data described here, the filtered BCD pixel data were used. The unfiltered uncertainties were input, but not used (except to remove bad pixels) as unweighted linear least squares (LLSQ) fitting was employed for each DCE, in order to avoid a small bias previously found in the fits, believed due to a correlation between the data and the weights, as described earlier.

Summary of Results:

Following are the fluxes, uncertainties, and repeatabilities for both runs, when weighted in the four different combinations described previously.

Run       UD-UA              UD-WA              WD-UA              WD-WA
     Flux  Unc   Rpt    Flux  Unc   Rpt   Flux   Unc   Rpt    Flux  Unc   Rpt
 A. 318.20(76)  2.61%  317.82(69)  2.39%  318.17(79)  2.71%  317.85(70)  2.40%
 B. 413.06(89)  2.21%  412.31(84)  2.08%  413.09(89)  2.21%  412.33(84)  2.08%

Here Flux and uncertainty Unc are in mJy, based on a calibration factor of 702. The headings UD-UA, UD-WA, WD-UA, and WD-WA refer to the weighting (W), or not (U), of the DCEs (D) in an AOR, and of the AORs (A) in the entire run, giving four combinations in all. However, since each DCE has been analyzed here by unweighted LLSQ fitting to the pixel data in order to avoid correlation bias (see below), the DCEs in an AOR have nearly equal uncertainties, so that the UD and WD cases are almost the same.

For the unweighted cases, the repeatability (Rpt) is the standard deviation of the fluxes, expressed as a percent of the mean, and the standard uncertainty (Unc) is the Std.Dev/sqrt(N), where N is the number of samples in the average. Since this takes no account of the degradation due to data points with large error bars, it should be conservative.

For the weighted cases, the uncertainties are computed from the weighted error formula, derated by the factor sqrt(chisq/nu), where chisq/nu is the reduced chisquare for the data on the weighted mean hypothesis for the true flux. These weighted uncertainty estimates, Unc, would be exactly correct for either the DCE or AOR case if the data uncertainties were all in error by single common factors, for the DCEs in an AOR, or for the AORs in the entire data set, respectively. The equivalent repeatabilities for the weighted case (where the data values of course have different uncertainties) have been estimated by derating by the factor from the reduced chisquares, sqrt(chisq/nu):

 Rpt(%)  = 100*sqrt(chisq/nu)*sqrt(N)*wsigbar/wfluxbar

where again chisq/nu is the reduced chisquare, wfluxbar is the weighted average flux, and wsigbar is the uncertainty of wfluxbar, by the usual weighted average formula.

Figure 1 shows the time series of normalized fluxes for each AOR for HD 163588 with the WD-WA weighting.

Figure 1: HD163588 normalized fluxes with WD-WA weighting.
Data are here.

Figure 2 shows the time series of normalized fluxes for each AOR for HD 180711 with the WD-WA weighting.

Figure 2: HD180711 normalized fluxes with WD-WA weighting.
Data are here.

Discussion

The use of unweighted LLSQ was found to be necessary in order to remove a bias, apparently due to some correlation between the residuals (data-model) and the estimated pixel sigmas in the input data. Although weighted LLSQ is theoretically optimal in the sense of minimum least squares error in the estimated parameters, any correlation between the pixel data and the weights (ie, the sigmas) can introduce a bias that overwhelms the advantage of optimality. Earlier experience showed that the filtered BCD (fbcd) data gave better results than the plain (bcd) data, at the cost of losing all information about the true background, due to the column filtering done for the fbcd. Without this background information, there seems no hope of estimating the pixel sigmas in an uncorrelated way (eg, iteratively from the fitted model), but this defect may be reparable later by returning to the plain BCD data. It was found by experment that the bad effect of unweighted LLSQ on the repeatability is small compared to the bias introduced by the correlation.