TF ErrorAna PropOfErr

Instrumental and Statistical Uncertainties

Example of cosmic counting experiments. Is the variation statistical?

[math]\sigma^2_{Poisson} = \mu =[/math] Mean Coinc/Hr

Date	Time (hrs)	[math]\theta[/math]	Coincidence Counts	Mean Coinc/Hr	[math]\sqrt{Mean Counts/Hr}[/math]	[math]\left | \sigma \right |[/math] 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

[math]\frac{\sum CPM_i}{8} = 97.44[/math]

[math]\frac{\sum Counts}{\sum time} = \frac{16316}{167.5}=97.41[/math]

The above 2 expressions are equivalent because all the denominators are essentially the value 21

[math]\frac{sum (x_i-\mu)^2}{8-1} = 10.8[/math]

P=68% = Probability that a measurement of a Gaussian variant will lie within 1 [math]\sigma[/math] of the mean

[math]= 0.68 * 8 = 5[/math]

Looks like we have 7/8 events within 1[math] \sigma[/math] = 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

[math]A= L \times W[/math]

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

[math]A = f(L,W)[/math]

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

[math]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!} + ...[/math]

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

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

[math]\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!}[/math]

[math]=1 + \frac{x}{2} - \frac{x^2}{4}[/math]

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

[math]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)}[/math]

or

[math]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)}[/math]

The term

[math]f(x_1, x_2)-f(\bar {x}_1, \bar{x}_2)[/math]

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

Based on the Definition of Variance

[math]\sigma^2 = \frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N}[/math]

We can write the variance of the area

[math]\sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - \bar{A})^2}{N}[/math]

[math]= \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}[/math]

[math]= \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}[/math]

[math]+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} [/math]

[math]= \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} [/math]

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

Weighted Mean and variance

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

[math]\bar{x} = \frac{ \sum_{i=1}^{i=n} \frac{x_i}{\delta x_i}}{\sum_{i=1}^{i=n} \frac{1}{\delta x_i}}[/math]

The variance of the distribution is defined as

[math]\bar{x} = \sum_{i=1}^{i=n} \frac{1}{\delta x_i}[/math]

Date	Time (hrs)	[math]\theta[/math]	Coincidence Counts	Coinc/Hour	[math]\sqrt{N}[/math]
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