Total uncertainty

Total uncertainty

Consider the result of measurement Y expressed by a function of other measurements X₁, X₂, ..., X_n

Y = f(X₁, X₂, ..., X_n)

Total uncertainty of independent measurements

The total uncertainty is a combination of all measurements, and the measurement results can be independent or correlate with each other. For independent measurements, the total variance (u_c²(y)) is determined by the formula:

u_c²(y) = Σⁿ_i=1[df/dx_i]²u²(x_i)
Where f is the measurement model, u_i is the uncertainty of type A or B.

If the nonlinearity of the function f is critical, the Taylor series for the derivative df/dx_i should include senior degrees:
Σⁿ_i=1Σⁿ_j=1[½[d²f/(dx_idx_j)]² + df/dx_i d³f/dx_idx_j²]u²(x_i)u²(x_j)

Partial derivatives of the measurement model calculated at the point μ(x_i) are called coefficients sensitivity and describe the change in the mathematical expectation y depending on the mathematical expectation independent measurement values. In particular, the change in y caused by a small change in Δx_i, expressed as: (Δy)_i = (df/dx_i)(Δx_i). If the reason for this the change is the uncertainty of the mathematical expectation x_i, the change y is expressed as (df/dx_i)u(x_i). The total variance u_c²(y) can be expressed as the sum of the variances of each of x_i, hence:

u_c² = Σ_i=1ⁿ[c_iu(x_i)]² = Σ u_i²(y)
Where c_i = df/dx_i and u_i(y) =|c_i|u(x_i)

The total uncertainty can be calculated by replacing c_iu(x_i) with the following expression:

Z_i = ½{f[x₁, ..., x_i + u(x_i), ..., x_n] - f[x₁, ..., x_i - u(x_i), ..., x_n]}

Thus, we calculate the changes of y as a result of the change of x_i in the interval between +u(x_i) and -u(x_i). The value u_i(y) can be taken |Z_i|, the corresponding coefficient sensitivity c_i is equal to Z_i/u(x_i).

The sensitivity coefficient c_i can also be obtained as a result of measuring y at fixed by changing x_i, the true nature of the value of the function f will be lost, since the value c_i will be obtained empirically.

Total uncertainty of dependent measurements

In the case when the values of x_i have a correlation dependence, it is necessary to change the formula total variance:

u²_c(y) = Σⁿ_i=1Σⁿ_j=1df/dx_i • df/dx_j u(x_i,x_j) = Σ_i=1ⁿ [df/dx_i]²u²(x_i) + 2 Σ^n-1_i=1Σⁿ_j=i+1 df/dx_i• df/dx_j u(x_i,x_j)
Where x_i, x_j are the expected values of X_i and X_j, u(x_i,x_j) - covariance of the values of x_i and x_j.
Correlation coefficient:
r(x_i, x_j) = r(x_j, x_i) = u(x_i, x_j)/u(x_i)u(x_j)

r∈ [-1,1], if the mathematical expectations x_i and x_j are independent, then r=0.

u_c² = Σ_i=1ⁿ[c_iu(x_i)]² + 2 Σ^n-1_i=1Σⁿ_j=i+1 c_ic_j u(x_i) u(x_j) r(x_i,x_j)
So, if all values have a direct relationship (r =1), the equation will take the form:
u_c²(y) = [Σⁿ_i=1c_iu(x_i)]² = [Σⁿ_i=1df/dx_iu(x_i)]²

Extended uncertainty

The extended uncertainty (U) is determined by the confidence interval of the total uncertainty: U = ku_c(y). In practice, when the number of degrees of freedom u_c(y) can be neglected, the values are used k=2 for 95% confidence interval and k=3 for 99% confidence interval. When the number of degrees of freedom is known, and the uncertainties obey the law of normal distribution, the Student's criterion is used as the criterion k.

General uncertainty calculation algorithm

1. First of all, it is necessary to make a measurement model Y = f(X₁, X₂, ..., X_n). The measurement model should include all values and all correction values that may affect the result measurements.

2. Determine the statistical estimate of the average value of X_i using statistical analysis or other methods.

3. Express the uncertainty value u(x_i) of each statistical estimate of the average value of x_i. If the average value was obtained by statistical analysis, then uncertainty of type A is used, in the rest in cases of type B uncertainty .

4. Express the covariance values for all measured quantities having a correlation dependence.

5. Calculate the measurement result: statistical estimation of the measured value Y based on the measurement model f, using as statistical estimates X_i the values of x_i obtained at the second stage.

6. Determine the total uncertainty u_c(y) of the measurement result, y, based on the uncertainties and covariance of statistical estimates of averages.

7. If necessary, calculate the extended uncertainty U.