Type B methods for evaluating correlation
As we saw on the first page of this recipe, Type B evaluations of error correlation are based not on statistics, but instead on other forms of knowledge. On this page, we’ll examine Type B error correlation evaluation by returning to the example of the rolling average that we looked at in the previous recipe. There, we used our knowledge of how the rolling average was being calculated to determine the correlation structure, which were saw to be triangular. On this page, we will again use our knowledge of the rolling average to explore a mathematical approach to the evaluation of error correlation. In order to do so, let’s reintroduce our rolling average example from the previous recipe, as shown below.
Let’s look at two neighbouring averages from the diagram above:
{\bar x_a} = \frac{{{x_0} + {x_1} + {x_2}}}{3}
{\bar x_b} = \frac{{{x_1} + {x_2} + {x_3}}}{3}
As we saw in the previous recipe, these two averages share the common elements {x_1} and {x_2} , and errors in these values are common to both averages. The covariance is from the common error, in this case, the error in the common elements {x_1} and {x_2}.\; We can use this information to write an expression for the covariance, u\left( {{{\bar x}_a},{{\bar x}_b}} \right) :
u\left( {{{\bar x}_a},{{\bar x}_b}} \right) = \frac{{\partial {{\bar x}_a}}}{{\partial {x_1}}}\frac{{\partial {{\bar x}_b}}}{{\partial {x_1}}}{u^2}\left( {{x_1}} \right) + \frac{{\partial {{\bar x}_a}}}{{\partial {x_2}}}\frac{{\partial {{\bar x}_b}}}{{\partial {x_2}}}{u^2}\left( {{x_2}} \right)
where u\left( {{x_1}} \right) is the standard uncertainty in {x_1} and u\left( {{x_2}} \right) is the standard uncertainty in {x_2} .
Since the partial derivatives in the equation above are all equal to
\;\frac{1}{3}
, we can simplify our expression for the covariance as follows:
u\left( {{{\bar x}_a},{{\bar x}_b}} \right) = {\left( {\frac{1}{3}} \right)^2}{u^2}\left( {{x_1}} \right) + {\left( {\frac{1}{3}} \right)^2}{u^2}\left( {{x_2}} \right)
A similar process can be used to evaluate the covariance associated with other pairs of averages giving the triangular correlation structure that we examined in the previous recipe.
Note that once we have determined a value for the covariance, we can use it to determine the correlation coefficient, r\left( {{{\bar x}_a},{{\bar x}_b}} \right), as follows:
r\left( {{{\bar x}_a},{{\bar x}_b}} \right) = \frac{{u\left( {{{\bar x}_a},{{\bar x}_b}} \right)}}{{u\left( {{{\bar x}_a}} \right)u\left( {{{\bar x}_b}} \right)}}
where, again, u\left( {{x_1}} \right) is the standard uncertainty in {x_1} and u\left( {{x_2}} \right) is the standard uncertainty in {x_2} .
Generalising our approach
The approach to evaluating error correlation outlined above can be generalised. For instance, for the following two functions:
{y_1} = f\left( {{N_{11}},{N_{12}} \ldots {N_{1i}};{C_1},{C_2} \ldots {C_j} \ldots } \right)
{y_2} = g\left( {{N_{21}},{N_{22}} \ldots {N_{2i}};{C_1},{C_2} \ldots {C_j} \ldots } \right)
where:
- C represents a common value that is used in the calculation of both {y_1} and {y_2}
- N represents a non-common value
we can determine the covariance, u\left( {{y_1},{y_2}} \right) , as follows:
u\left( {{y_1},{y_2}} \right) = \sum \limits_j \frac{{\partial f}}{{\partial {c_j}}}\frac{{\partial g}}{{\partial {c_j}}}{u^2}\left( {{c_j}} \right).
The correlation coefficient, r({y_1},{y_2} ), can then be calculated from:
r\left( {{y_1},{y_2}} \right) = \frac{{u\left( {{y_1},{y_2}} \right)}}{{u\left( {{y_1}} \right)u\left( {{y_2}} \right)}}