1. 880.P20 Winter 2006 Richard Kass Propagation of Errors Suppose we measure the branching fraction BR(Higgs  +  - ) using the number of produced Higgs Bosons (N produced ), the number of Higgs  +  - decays found (N found ), and the efficiency for finding a Higgs  +  – decay (  ). BR(Higgs  +  - )=N found /(  N produced ), If we know the uncertainties (  ’s) of N produced, N found, and  what is the uncertainty on BR(Higgs  +  - ) ? More formally we could ask, given that we have a functional relationship between several measured variables (x, y, z), i.e. Q = f(x, y, z) What is the uncertainty in Q if the uncertainties in x, y, and z are known? Usually when we talk about uncertainties in a measured variable such as x we assume that the value of x represents the mean of a Gaussian distribution and the uncertainty in x is the standard deviation (  ) of the Guassian distribution. A word of caution here, not all measurements can be represented by Gaussian distributions, but more on that later! To answer this question we use a technique called Propagation of Errors.

2. 880.P20 Winter 2006 Richard Kass Propagation of Errors Note: To first order the average of a function is the function evaluated at its average value(s): =Q(  )

3. 880.P20 Winter 2006 Richard Kass Propagation of Errors Example: Error in BR(Higgs  +  – ). Assume: N produced =100  10, N found =10  3,  = 0.2  0.02 BR(Higgs  +  – ) =0.5  0.2

4. 880.P20 Winter 2006 Richard Kass Propagation of Errors Example: The error in the average. The average of several measurements each with the same uncertainty (  ) is given by: “error in the mean” This is a very important result! It says that we can determine the mean better by combining measurements. Unfortunately, the precision only increases as the square root of the number of measurements. Do not confuse   with  !  is related to the width of the pdf (e.g. gaussian) that the measurements come from.  does not get smaller as we combine measurements. A slightly more complicated problem is the case of the weighted average or unequal  ’s: Using same procedure as above we obtain: “error in the weighted mean”

5. 880.P20 Winter 2006 Richard Kass Propagation of Errors Problems with Propagation of Errors: In calculating the variance using propagation of errors we usually assume that we are dealing with Gaussian errors for the measured variable (e.g. x). Unfortunately, just because x is described by a Gaussian distribution does not mean that f(x) will be described by a Gaussian distribution. 0 20 40 60 80 100 020 40 dN/dy 10 30 y = 2x with x = 10   xy y Start with a gaussian with  =10,  =2. Get another gaussian with  =20,  = 4

6. 880.P20 Winter 2006 Richard Kass Error of Propagation of Errors Example when the new distribution is non-Gaussian: Let y = 2/x The transformed probability distribution function for y does not have the form of a Gaussian pdf. 0 20 40 60 80 100 0.10.20.3 dN/dy 0.40.50.6 y = 2/x with x = 10  2  xy y x 2 Start with a gaussian with  =10,  =2. DO NOT get another gaussian ! Get a pdf with  = 0.2,  = 0.04. This new pdf has longer tails than a gaussian pdf. Prob(y>  y +5  y ) =5x10 -3, for gaussian  3x10 -7 Unphysical situations can arise if we use the propagation of errors results blindly! Example: Suppose we measure the volume of a cylinder: V =  R 2 L. Let R = 1 cm exact, and L = 1.0 ± 0.5 cm. Using propagation of errors we have:  V =  R 2  L =  /2 cm 3. and V =  ±  /2 cm 3 However, if the error on V (  V ) is to be interpreted in the Gaussian sense then the above result says that there’s a finite probability (≈ 3%) that the volume (V) is < 0 since V is only two standard deviations away from than 0! Clearly this is unphysical ! Care must be taken in interpreting the meaning of  V.  G

7. 880.P20 Winter 2006 Richard Kass Generalization of Propagation of Errors We can generalize the propagation of errors formula: In matrix notation: We can generalize to any number of variables:  2 =d T Vd with d an N-dimensional vector of derivatives and V an NxN matrix of variances and covariances V is often called the “error matrix” or “covariance matrix”. V is a symmetric matrix (NxN and V=V T ) Example: Error in BR(Higgs  +  – ). Assume: BR=N fd /(  N pr )

8. 880.P20 Winter 2006 Richard Kass 8 A Real Life Example We want to measure the branching fraction for B -  D 0 K *-. We can measure it using three different decay modes of the D0: D 0      D 0        and D 0         D 0  K  0 D0KD0K D 0  K3  Statistical uncertainties only    , efficiency (%) 13.304.608.82 B(D 0  X) (%) 3.8012.847.46 N, Yield (events) 144.4  13.2185.4  18.6195.0  18.2 B(B -  D 0 K *- )x10 -4 5.15  0.455.65  0.545.34  0.48 Also have to take into account systematic errors How should we combine the 3 measurements?

9. 880.P20 Winter 2006 Richard Kass 9 A Real Life Example Also have to take into account systematic errors Summary of Systematic Errors uncorrelated 3.1% 6.6% 4.7% uncorrelated 5.2% 6.2% 7.3% correlated standard recipes data (B -  D 0  - ) Vs MC study data & MC, vary cuts PDG BF uncertainties finite MC samples lumi script study data & MC, vary cuts Some sources of systematic errors are correlated, some are not. Correlated errors wind up in the off-diagonal elements of the error matrix.

10. 880.P20 Winter 2006 Richard Kass 10 A Real Life Example Since the 3 measurements have different precision we will do a weighted average. But, a bit tricky because we have statistical and systematic errors and some of the systematic errors are correlated. B(B -  D 0 K *- )=w 1 B(B -  D K *- )+w 2 B(B -    D K *- )+w 3 B(B -  D K *- ) Follow the procedure outlined in Lyons et al., NIMA 270, 110 (1988) We want to find the weights (w i ) that minimizes:  2 =w T Vw subject to:  w i =1 (Here the derivative vector is just the weights) Can solve this problem using Lagrange multiplier technique:   2 =w T Vw+ (w T I-1) here I is a vector of 1’s. Need to find the multiplier,. From constraint equation we get: Can now solve the problem: For the uncorrelated case the weights are the same as using: V is a 3x3 symmetric matrix

11. 880.P20 Winter 2006 Richard Kass 11 Branching Fraction Averaging Procedure The weights (w 1, w 2, w 3 ) are calculated using the error matrix, V: V= V statistics + V systematics Calculate weights: st=statistics syTOT=total systematics syC=correlated systematics Calculate variances: Note: an off diagonal element is a “dot” product of the errors of the 3 modes, e.g.: where 1=tracking eff, 2=particle ID, etc. published in PRD 773, 111104(R) (2006)