Error propagation. Alexander Khanov. October 4, PHYS6260: Experimental Methods is HEP Oklahoma State University

Transcription

1 Error propagation Alexander Khanov PHYS660: Experimental Methods is HEP Oklahoma State University October 4, 017

2 Why error propagation? In many cases we measure one thing and want to know something else Example: we want to determine the volume of a ball. We measure its diameter to be 0 ± 1 cm. What is the volume of the ball? Solution: V = πd 3 6, V = πd d. Answer: V = 400 ± 600 cm 3 For a function y = f (x) of a single variable x, σ y = df dx σ x exact meaning of this formula: if x is normally distributed with mean x 0 and standard deviation σ x, and y = f (x) is (approximately) linear near x 0 ( near = within the range of the order of σ x ), then y is (approximately) normally( distributed ) with mean y 0 = f (x 0 ) and df standard deviation σ y = dx x=x 0 σ x A. Khanov (PHYS660, OSU) Particles 10/4/17 / 10

3 What if the result depends on more than one measurement? If z = f (x, y) depends on two independent variables x, y, then Examples: σ z = ( ) f ( ) f σx + σy x y z = x + y: σ z = σx + σy (same for z = x y) exactly true for normally distributed variables ( z = xy: σz = y σx + x σy σz ) ( σx ) ( ) σy, or = + z x y approximately true (xy is not normally distributed) Problem: if two sides of a rectangle are 5 ± 1 cm and 4 ± 1 cm, what is its area? Solution: A = 0 ± 6 cm A. Khanov (PHYS660, OSU) Particles 10/4/17 3 / 10

4 Correlated variables If variables are not independent, their correlations must be taken into account Problem: two persons measured the size of a rectangle. Person A measured one side of the rectangle and found it to be (4 ± 1) cm. Person B measured rectangle s perimeter and found it to be (0 ± ) cm. What is the area of the rectangle? Solution 1: the other side of the rectangle is b = P a = (10 ± 1) (4 ± 1) = (6.0 ± 1.4) cm. The area A = ab = (4 ± 1) ( (6.0 ± ) 1.4) = 4 ± 8 cm. P Solution : A = a a = ap a (P ) ( a ) σ A = a σa + σ P = 4 cm, so A = 4 ± 4 cm. Why the answers are different? (hint: solution 1 is wrong) A. Khanov (PHYS660, OSU) Particles 10/4/17 4 / 10

5 General formula for error propagation If two variables have covariance cov(x, y) = ρ(x, y)σ x σ y, then σ z = ( ) ( ) ( ) ( ) f f f f σx + σy + cov(x, y) x y x y For n variables x 1,..., x n : σf = ( ) ( ) f f cov(x i, x j ) x i x j i,j ( ) f In matrix form, denoting = f, cov(x i, x j ) = V x, x i σ f = f V x f T Finally, if we have m functions f 1,..., f m of n variables x 1,..., x n, then ( ) V f = GV x G T fk, where G ki = x i A. Khanov (PHYS660, OSU) Particles 10/4/17 5 / 10

6 Statistical and systematic uncertainties Uncertainty of a measurement is traditionally divided into two parts statistical and systematic Statistical uncertainties originate from inevitable fluctuations in the measured value Systematic uncertainties introduce a bias in the measurement which only gets more pronounced as the number of measurements increases Statistical uncertainties affect precision, systematic uncertainties affect accuracy A. Khanov (PHYS660, OSU) Particles 10/4/17 6 / 10

7 Statistical uncertainties Suppose we measure the mean lifetime τ of a particle. We do it by counting the rate of particles in a beam crossing two detectors separated by some distance L: N = N 0 exp( L/vτ) If we repeat the measurement many times we will get different results because the number of decays is a random (Poisson distributed) value. The more measurements we do, the better we know the parameter λ of the Poisson distribution The precision of the result scales as the square root of the number of measurements A. Khanov (PHYS660, OSU) Particles 10/4/17 7 / 10

8 Statistical uncertainties () For a Poisson distributed variable (the number of observed decays) average value is N = λ = np, standard deviation N = N this is absolute statistical uncertainty on the number of observed decays relative statistical uncertainty N N = 1 goes down as the N number of measurements increases in theory, by repeating the measurements sufficient number of times, the statistical uncertainty can be made arbitrarily small If the error propagation is involved the dependence may get more complicated N d = N 0 L vτ N d = N 0L vτ τ A. Khanov (PHYS660, OSU) Particles 10/4/17 8 / 10

9 Origin of systematic uncertainties While we convert the measured number of particles crossing the detectors to the lifetime, we rely on several parameters which are not known exactly: the distance between the two detectors is not perfectly known the detectors have finite efficiency (probability to register a particle). This efficiency can be derived separately, or it can be determined simultaneously with the lifetime measurement. E.g. the beam may consist of two types of particles, one which is stable and another one which lifetime we are trying to measure. We can use the stable particles to estimate the detector efficiency. In this case the uncertainty related to the detector efficiency will improve with the number of measurements, just like the statistical uncertainty the mathematical models involved in the measurement have their own uncertainties. E.g., in our experiment some particles are lost due to interactions while they move between the detectors. If we try to take it into account by using the Bethe formula, we need to account for its uncertainties The exact separation between statistical and systematic uncertainty is not well defined but usually clear from common sense A. Khanov (PHYS660, OSU) Particles 10/4/17 9 / 10

10 How to deal with systematic uncertainties Method 1: use the error propagation easy if the uncertainty is on a parameter in the formula often doesn t work well because unlike statistical uncertainties, systematic uncertainties are usually not Gaussian Method : vary the parameter characterizing the uncertainty, see what happens to the result as an option, one can generate a number of pseudo experiments and look at the distribution of the result Method 3: add systematic uncertainties to the list of parameters determined in the experiment A. Khanov (PHYS660, OSU) Particles 10/4/17 10 / 10