TF ErrorAna PropOfErr

Instrumental and Statistical Uncertainties

Example of cosmic counting experiments. Is the variation statistical?

$\sigma^2_{Poisson} = \mu =$ Mean Coinc/Hr

Date	Time (hrs)	$\theta$	Coincidence Counts	Mean Coinc/Hr	$\sqrt{Mean Counts/Hr}$	$\left | \sigma \right |$ from Mean
9/12/07	20.5	330	2233	109	10.4	1
9/14/07	21	330	1582	75	8.7	2
10/3/07	21	330	2282	100	10.4	1
10/4/07	21	330	2029	97	9.8	0.1
10/15/07	21	330	2180	100	10	0.6
10/18/07	21	330	2064	99	9.9	0.1
10/23/07	21	330	2003	95	9.7	0.2
10/26/07	21	330	1943	93	9.6	0.5

$\frac{\sum CPM_i}{8} = 97.44$

$\frac{\sum Counts}{\sum time} = \frac{16316}{167.5}=97.41$

$\frac{sum (x_i-\mu)^2}{8-1} = 10.8$

P=68% = Probability that a measurement of a Gaussian variant will lie within 1 $\sigma$ of the mean

$= 0.68 * 8 = 5$

Looks like we have 7/8 events within 1$\sigma$ = 87.5%

Taylor Expansion

A quantity which is calculated using quantities with known uncertainties will have an uncertainty based upon the uncertainty of the quantities used in the calculation.

To determine the uncertainty in a quantity which is a function of other quantities, you can consider the dependence of these quantities in terms of a tayler expansion

Consider a calculation of a Table's Area

$A= L \times W$

The mean that the Area (A) is a function of the Length (L) and the Width (W) of the table.

$A = f(L,W)$

The Taylor series expansion of a function f(x) about the point a is given as

$f(x) = f(a) + \left . f^{\prime}(x)\right |_{x=a} \frac{x}{1!} + \left . f^{\prime \prime}(x)\right |_{x=a} \frac{x^2}{2!} + ...$

$= \left . \sum_{n=0}^{\infty} f^{(n)}(x)\right |_{x=a} \frac{x^n}{n!}$

For small values of x (x << 1) we can expand the function about 0 such that

$\sqrt{1+x} = \left . \sqrt{1-0} \frac{1}{2}(1+x)^{-1/2}\right |_{x=0} \frac{x^1}{1!}+ \left . \frac{1}{2}\frac{-1}{2}(1+x)^{-3/2} \right |_{x=0} \frac{x^2}{2!}$

$=1 + \frac{x}{2} - \frac{x^2}{4}$

The talylor expansion of a function with two variables$(x_1 , x_2)$ about the average of the two variables$(\bar {x_1} , \bar{x_2} )$ is given by

$f(x_1, x_2)=f(\bar {x}_1, \bar{x}_2)+(x_1-\bar {x}_1) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = \bar {x}_1, x_2 = \bar{x}_2)} +(x_2-\bar{x}_2) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = \bar {x}_1, x_2 = \bar{x}_2)}$

or

$f(x_1, x_2)-f(\bar {x}_1, \bar{x}_2)=(x_1-\bar {x}_1) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = \bar {x}_1, x_2 = \bar{x}_2)} +(x_2-\bar{x}_2) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = \bar {x}_1, x_2 = \bar{x}_2)}$

The term

$f(x_1, x_2)-f(\bar {x}_1, \bar{x}_2)$

represents a small fluctuation of the function from its average $f(\bar {x}_1, \bar{x}_2)$ if we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).

Based on the Definition of Variance

$\sigma^2 = \frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N}$

We can write the variance of the area

$\sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - \bar{A})^2}{N}$

$= \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} + (W-\bar W) \frac{\partial A}{\partial W} \bigg |_{\bar L \bar WW} \right] ^2}{N}$

$= \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} \right ] ^2}{N} + \frac{\sum_{i=1}^{i=N} \left [ (W-\bar W) \frac{\partial A}{\partial W} \bigg |_{\bar L \bar W} \right] ^2 }{N}$

$+2 \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} \frac{\partial A}{\partial W} \bigg |_{\bar L \bar W} \right]^2}{N}$

$= \sigma^2_L \left ( \frac{\partial A}{\partial L} \right )^2 +\sigma^2_W \left ( \frac{\partial A}{\partial W} \right )^2 + 2 \sigma^2_{LW} \frac{\partial A}{\partial L} \frac{\partial A}{\partial W}$

where $\sigma^2_{LW} = \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \right ]^2}{N}$ is defined as the Covariance between $L$ and $W$.

Weighted Mean and variance

If each observable ($x_i$) is accompanied by an estimate of the uncertainty in that observable ($\delta x_i$) then weighted mean is defined as

$\bar{x} = \frac{ \sum_{i=1}^{i=n} \frac{x_i}{\delta x_i}}{\sum_{i=1}^{i=n} \frac{1}{\delta x_i}}$

The variance of the distribution is defined as

$\bar{x} = \sum_{i=1}^{i=n} \frac{1}{\delta x_i}$

Date	Time (hrs)	$\theta$	Coincidence Counts	Coinc/Hour	$\sqrt{N}$
9/6/07	18	45	1065	59.2
9/7/07	14.66	45	881	60.1
9/9/07	43	60	1558	36.23
9/12/07	20.5	330	2233	108.93
9/13/07	21	315	2261	107.67
9/14/07	21	330	1582	75.33
9/18/07	21	300	1108	52.8
9/19/07	21	300	1210	57.62
9/20/07	21	300	1111	52.69
9/21/07	21	300	1012	57.62
9/26/07	21	315	1669	79.48
9/27/07	21	315	1756	83.29
9/29/07	24.5	315	2334	95.27
10/3/07	21	330	2282	108.67
10/4/07	21	330	2029	96.62
10/10/07	21	315	1947	92.71
10/15/07	69	330	2180	31.59
10/18/07	21	330	2064	98.52
10/23/07	21	330	2003	95.38
10/26/07	21	330	1943	92.52
11/2/07	21	330	2784
11/5/07	69	330	10251	148.57
11/16/07	21	30	3581	170.52

[1] Forest_Error_Analysis_for_the_Physical_Sciences