Difference between revisions of "TF ErrorAna PropOfErr"

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where
 
where
<math>\sigma^2_{LW} = \left [\frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \right ]^2}{N}</math>
+
<math>\sigma^2_{LW} = \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \right ]^2}{N}</math>
  
  
 
[http://wiki.iac.isu.edu/index.php/Forest_Error_Analysis_for_the_Physical_Sciences] [[Forest_Error_Analysis_for_the_Physical_Sciences]]
 
[http://wiki.iac.isu.edu/index.php/Forest_Error_Analysis_for_the_Physical_Sciences] [[Forest_Error_Analysis_for_the_Physical_Sciences]]

Revision as of 21:38, 9 January 2010

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×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)+f(x)|x=ax1!+f(x)|x=ax22!+...

=inftyn=0f(n)(x)|x=axnn!


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

1+x=10|12(1+x)1/2|x=0x11!+1212(1+x)3/2|x=0x22!

=1+x2x24


The talylor expansion of a function with two variables(x1,x1) about the average of the two variables(¯x1,¯x2) is given by

f(x1,x2)=f(ˉx1,ˉx2)+(x1ˉx1)fx1|(x1=ˉx1,x2=ˉx2)+(x2ˉx2)fx2|(x1=ˉx1,x2=ˉx2)

or

f(x1,x2)f(ˉx1,ˉx2)=(x1ˉx1)fx1|(x1=ˉx1,x2=ˉx2)+(x2ˉx2)fx2|(x1=ˉx1,x2=ˉx2)

The term

f(x1,x2)f(ˉx1,ˉx2)

represents a small fluctuation of the function from its average f(ˉx1,ˉx2) if we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).

Based on the Definition of Variance

σ2=i=Ni=1(xiˉx)2N


We can write the variance of the area

σ2A=i=Ni=1(AiˉA)2N
=i=Ni=1[(LˉL)AL|ˉLˉW+(WˉW)AW|ˉLˉWW]2N


=i=Ni=1[(LˉL)AL|ˉLˉW]2N+i=Ni=1[(WˉW)AW|ˉLˉW]2N
+2i=Ni=1[(LˉL)(WˉW)AL|ˉLˉW+AW|ˉLˉW]2N
=σ2L(AL)2+σ2W(AW)2+σ2LWALAW

where σ2LW=i=Ni=1[(LˉL)(WˉW)]2N


[1] Forest_Error_Analysis_for_the_Physical_Sciences