A.7.3 Performance Metric Equations

Statistical formulae used to compute performance measures are given below:

Bias (C_IS) = Bias in estimate of CDU = E (C_IS – C_DU)

The estimate CIS of CDU is unbiased when Bias (C_IS) = 0

Relative bias (C_IS) = Abs (E (C_IS – C_DU)) / C_DU = |E (C_IS – C_DU)| / C_DU

FE in mean estimate = FE (C_IS) = Relative Bias (C_IS) = |E (C_IS – C_DU)| / C_DU

From the above, it is noted that reduction in FE and its variance are directly related to reduction in the bias in the estimate of DU mean.

MSE and RMSE associated with the ISM mean estimate CIS are given by

MSE(C_IS) = E(C_IS - C_DU)² = Var(C_IS) + (Bias(C_IS))²;

S²_FE = Var (FE) = MSE / c²_DU ; and

Standard Deviation (FE) = Sd (FE) = sqrt (MSE) / C_DU

The (1 – α) 100% Chebyshev UCL of the mean based on r ISM replicates is given by

Student’s-t-statistic based (1 – α) 100% UCL of mean is given by