Burst Statistics

Jim Linnemann

Michigan State University

Mar 7, 2004

A Few Updates• Graduate student Aws left Milagro—personal issues

– Talking with a possible replacement, but beginning student(s)

• Workshop at MSU on statistical software• Feigelson, Beers, R (Luc Tierney), HEP folk

– Feigelson & Babu, Astrostatistics• Relationship of Root and R language (can send summary)

– R is used by research statisticians• Future: software repository for Astronomy/HEP

– Can now read some Root trees into R– R has:

• Plot types Root doesn’t (1-Dim and n-Dim)– But: missing histograms with error bars

• Many statistical tests Root doesn’t• Elegant programming language (interactive + macro)• Large repository of procedures CRAN

– Very slick download mechanism• Documentation published in books!

Burst Statistics

• Brenda asked me to look at circulars and check the probability calculations

• Lots of help from Liz, Andy, David

Short Bursts• Purely Poisson calculation by table interpolation

– Completely ignores uncertainty in background

• Ignore sky motion 48 sec max = .2 degree• b = n(T) N(t*) / N(T) t/t* (no bin exclusion)

n(T) = in bin during map period (T = ½h, 2h?)

N(T) = all sky counts in mapt = search period t* = max(10s, t)

• Define effective α = (δb)2 / b (like Ton/Toff)• α (t*/T) + n(T)/N(T) t*/T + b/Rt + H.O.

• Assume Poisson; R = trigger rate is roughly constant• t*/T > .005-.025; b/Rt small correction

– α is small, and neglecting it a good approximation

Milagro Bursts x b pre-trials annual =b/ p_circular = 0Date DurationDeclinationsignal bkg est signif rate Y to Z Z17 Z17

7-Oct-03 25.1 s 16 132 67 1.57E-12 4 0.0156 4285 6.97 6.93 7.0010-Oct-03 10.0 s 15 42 11 4.04E-13 2 0.0062 1723 7.16 7.18 7.2211-Oct-03 39.8 s 51 238 146 2.36E-12 5 0.0245 5985 6.91 6.84 6.9413-Oct-03 25.1 s 60 103 48 3.58E-12 48 0.0152 3163 6.85 6.82 6.8913-Oct-03 15.8 s 60 80 31 9.15E-14 0 0.0100 3066 7.36 7.35 7.4016-Oct-03 25.1 s 37 210 122 4.35E-13 1 0.0170 7187 7.15 7.10 7.1717-Oct-03 39.8 s 35 261 164 1.55E-12 3 0.0247 6617 6.97 6.89 6.9921-Oct-03 15.8 s 69 56 18 1.27E-12 5 0.0095 1930 7.00 7.00 7.0521-Oct-03 39.8 s 43 243 150 1.63E-12 3 0.0245 6107 6.97 6.89 6.9922-Oct-03 25.2 s 29 153 80 3.05E-13 1 0.0160 5017 7.20 7.15 7.2330-Oct-03 .63 s 12 17 1 1.74E-13 8 0.0069 198 7.27 7.28 7.375-Nov-03 39.8 s 59 175 96 2.58E-13 1 0.0236 4050 7.22 7.14 7.251-Dec-03 15.8 s 25 93 41 2.29E-12 9 0.0104 3936 6.92 6.91 6.95

13-Feb-04 2.51 s 14 31 6 2.86E-13 5 0.0070 836 7.21 7.22 7.27

Short Burst StatisticsOct 03 to mid-Feb 04

I performed many checks in Excel and mathematica

Spreadsheet to Brenda: histogram declination…

Li Ma 17• Li Ma 17 includes background uncertainty

– But in a Gaussian approximation

• Can rewrite as:

½ Z217= x Ln(x/b) + (x+y) Ln r, wherex = Non, b = α y y = Noff

r = (1 + α) / (1 + αm) = (1 + α) / (1 + x/y)

Can with care take limit α 0

Z217(α=0) = x Ln(x/b) + b – x

Find: ZPoisson Z17(α=0) to pretty good accuracy|Z error| < .1 for x > 20

Long Bursts

• α = 2.1o/cos(δ) / (30o - 2.1o/cos(δ) ) (T=2h)– Standard calculation

• What happens if < 2h?

• Not using Li Ma 17 for significance calculation– If I understand meaning of variables:

• Seems to over-report significance by .3 - .5– Probability too high by 3-10?

• ZBinomial Z17 for long bursts—no strong need to change

Long Burst Statistics Oct 03 to mid-Feb 04

Milagro Bursts x b pre-trials prob annual p_circular alpha=0Date DurationDeclinationsignal bkg est signif Z_17 rate to Z Z17 Zbino Z1722-Oct-03 1 d 80 506 369 1.59E-08 1.78E-07 23 0.664 5.53 5.09 5.07 6.7229-Oct-03 1 d 2 617 486 5.05E-09 2.13E-08 6 0.075 5.73 5.48 5.47 5.7031-Oct-03 2 h 75 255 168 4.39E-09 8.95E-08 89 0.367 5.75 5.22 5.19 6.241-Nov-03 2 h 26 233 157 3.09E-09 3.52E-08 86 0.085 5.81 5.39 5.37 5.652-Nov-03 2 h 54 554 424 1.39E-09 9.01E-09 6 0.137 5.94 5.63 5.61 6.04

16-Nov-03 2 h 69 264 171 1.01E-10 3.52E-09 2 0.246 6.36 5.79 5.76 6.5723-Nov-03 2 h 69 394 284 2.12E-09 2.25E-08 14 0.240 5.88 5.47 5.45 6.1623-Nov-03 1 d 14 1824 1585 3.60E-09 9.55E-09 7 0.078 5.79 5.62 5.60 5.853-Dec-03 1 d 14 1865 1631 1.12E-08 2.66E-08 17 0.078 5.59 5.44 5.41 5.668-Dec-03 1 w 46 15717 15049 1.15E-07 1.53E-07 19 0.111 5.17 5.12 5.12 5.40

10-Dec-03 2 h 72 303 208 2.75E-09 4.40E-08 24 0.293 5.83 5.35 5.32 6.1810-Dec-03 1 d 14 1917 1679 1.03E-08 2.38E-08 12 0.078 5.61 5.46 5.45 5.6815-Dec-03 1 w 46 25539 24599 6.35E-09 8.50E-09 1 0.112 5.69 5.64 5.69 5.9516-Dec-03 1 d 23 1895 1659 1.19E-08 2.51E-08 20 0.083 5.58 5.45 5.42 5.6824-Dec-03 2 h 42 390 287 3.73E-09 2.66E-08 20 0.105 5.78 5.44 5.43 5.7511-Jan-04 2 h 75 216 136 2.61E-09 7.60E-09 32 0.384 5.84 5.25 5.22 6.3315-Jan-04 2 h 76 176 97 1.20E-09 2.94E-09 12 0.415 5.97 5.82 5.78 7.1622-Jan-04 2 h 0 118 67 1.52E-09 5.48E-08 17 0.075 5.93 5.31 5.27 5.5522-Jan-04 2 h 78 136 73 1.01E-09 1.70E-07 47 0.527 6.00 5.10 5.05 6.5726-Jan-04 2 h 26 228 152 1.39E-09 1.90E-08 17 0.085 5.94 5.50 5.48 5.7627-Jan-04 1 d 62 1814 1561 1.50E-09 4.46E-09 5 0.173 5.93 5.75 5.73 6.2529-Jan-04 2 h 59 265 182 4.35E-09 5.19E-08 47 0.156 5.75 5.32 5.30 5.7831-Jan-04 2 h 66 529 401 3.22E-09 2.13E-08 45 0.204 5.80 5.48 5.47 6.078-Feb-04 1 d 65 1411 1201 1.54E-08 4.40E-08 44 0.202 5.54 5.35 5.35 5.909-Feb-04 1 d 38.3 3752 3412 1.34E-08 2.38E-08 25 0.098 5.56 5.46 5.46 5.73

Why need to “recalibrate”?• Rate per year: needs

– Correct probability for individual bursts– Correct trials

• Live time needed to calculate them correctly• Trigger Mix dependent?• Oversampling

– Is this optimized?

– Correct interaction between time scales• Circular Issue Criteria Needed• Probability given short burst passes longer burst

– Should be able to calculate burst by burst (x,b, then x’-x, b’-b)– Or maybe MC to calculate effective trials; or measure?

• Will need time staring at David’s web pages (and Liz…)

– I don’t see how the trigger rate enters very strongly• Enters in size of background estimate• Taken into account in probability calculation

Queries• Is the P distribution flat?

• Is its log-slope = -1?– If not, then probability calculation is wrong– Or complications with trials/oversampling…

Comments• I still suspect that FDR would be a useful• Untriggered searches--some interesting papers: Jeff

Scargle (NASA) at PHYSTAT 2003Bruce Knuteson (HEP) at PHYSTAT 2003 & earlier

• Both search for “best” regions (Varonni Tesselations)• Then calculate probability of such a fluctuation• An alternative to pre-determined trials choices• As usual, should study with MC the efficiency tradeoff

– You can do better if you have a model

• I will provide references

• Minimize <limit>/typical signal? Billor paper?

In Progress

• Write this down, and circular criteria• Be sure understanding Liz’s “Off” variable• Understand Trials and Multiple time scales• Dump Long Burst database to Excel• Brenda asks: why flat in declination?

– What drives it?• Solid angle?• Trials? (more 2h after all…)