14 WGN, the Journal of the IMO 39:1 (2011) A Note on Poisson inference and extrapolations under low raw data and short interval observation conditions Peter V. Bias 1 To obtain ZHRs under very short observation periods and/or poor observing conditions, meteor counts must be significantly magnified. However, using a large correction factor can lead to substantially uncertain ZHRs. This paper examines the statistical uncertainty that results when large correction factors are applied to poor data. The Poisson distribution used in ZHR calculations is reviewed, concentrating on its significant skew under low raw data conditions. Real structural, asymmetric differences in probability densities between high certainty/high ZHRs and low certainty/high projected ZHRs are shown to exist for the same reported ZHR. Received 2010 October 10 1 Introduction On 2007 September 1, a far-comet outburst of Alpha- Aurigids, which had been predicted earlier by Peter Jenniskens (Jenniskens, 2007) and J´ er´ emie Vaubaillon (Vaubaillon, 2007), was seen by several favorably lo- cated observers on the US west coast. As has become custom, a request was made for observers to immedi- ately send in data so that the IMO could put together an, admittedly informal, on-the-fly ZHR activity pro- file, which has been reproduced in Figure 1. Included in the caption is a short quote from the website listing some of the correction processes that were used to gen- erate the graph. The raw data for the periods used to generate the graph are listed in Table 1. The error amounts and the associated error bars in the graphic are immediately interesting when compared to the raw data. How is it that no Alpha-Aurigids are seen in the first several time intervals and yet there are positive, even high, ZHRs listed for each? In look- ing at previous issues of WGN going back several years or more recent on-the-fly ZHR profiles reported on the IMO website, we see the same thing: some observing intervals have no meteors and yet still report a positive ZHR with error bars reaching zero as the minimum of a symmetrical range around the ZHR. As an example, Arlt and Barentsen’s (Arlt & Barentsen, 2006) presen- tation of the ZHR activity profile for the 2006 Leonids shows the same interesting result. Quoting from their paper, “The last row is a typical effect of small-number statistics as 0 Leonids produce a ZHR of 1.1 which looks odd at first glance. However, the fact that zero mete- ors were seen, can be the result of a true rate (measured over an infinitely long time) larger than 0.” The authors state later in the paper that “In statistical terms, the ZHR is the expectation value of all possible true rates which may have caused the observer to see 0 Leonids. It results from an integration over a Poisson-like func- tion.” An interpretation of this methodology is pre- sented below. 1 111 Lake Hollingsworth Drive Lakeland, Florida 33801, USA Email: pbias@flsouthern.edu IMO bibcode WGN-391-bias-poisson NASA-ADS bibcode 2011JIMO...39...14B Table 1 – Numerical data of the activity profile for the 2007 Alpha-Aurigids. “For each estimation interval: [. . . ] nINT is the number of observing periods and nAUR is the number of Alpha-Aurigids involved. ZHR = (1 + nAUR)/ P n INT i=1 (T eff,i /Ci ) where T eff,i is the effective observ- ing time of observing period i and Ci is the total correc- tion for limiting magnitude, clouds and zenith correction for observing period i.” (International Meteor Organiza- tion, 2007). n INT n AUR ZHR error 2 0 2 ± 2 1 0 4 ± 4 3 0 14 ±14 3 4 30 ±13 4 7 12 ± 4 3 7 15 ± 5 2 1 3 ± 2 3 1 4 ± 3 4 4 6 ± 3 2 0 2 ± 2 16 44 52 ± 8 7 51 140 ±19 11 60 216 ±28 14 40 69 ±11 11 8 14 ± 5 3 4 14 ± 6 1 1 24 ±17 2 1 5 ± 4 1 3 17 ± 8 1 0 5 ± 5 4 3 8 ± 4 Table 2 – A comparison of Poisson’s theoretical expectations (“predicted”) with actual occurrence (“actual”) for the 2001 November 18 Leonid shower (well before peak) using a mean of 1.6 meteors per minute (Bias, 2005). Number of Leonids seen Number of minutes in an in a one-minute interval hour that number is seen Predicted Actual 0 12.75 12 1 19.74 19 2 15.27 19 3 7.89 5 4 3.06 3 5 0.99 1 6 0.24 1 7 0.06 0