Mixed moment validation metric for models with multivariate output

Considering the relations among multi-outputs and the mean of single output, the mathematical expectation of single dimensional variable and covariance metric of multi-dimensional variables were introduced into the validation metrics for models. The new metrics of LA-3M and LR-3M were proposed for validating multi-responses at a single validation site, while the metrics of GA-3M and GR-3M were proposed to collect data of multiple responses observed at multiple validation sites. These metrics were examined through a numerical test case and an engineering example to illustrate their feasibility and effectiveness. Results show that the proposed metrics are efficient and they can easily measure the differential degree of multiple responses between calculation model and physical experiment.