An often used measure of central tendency with some desirable properties is the arithmetic mean (AM). AM is a statistical concept that minimizes the error in sample-based estimates of the expected value of quantities of interest. As such, AM is widely used as an ensemble forecast-based estimate of future weather conditions. In traditional applications, AM is applied variable by variable, disregarding covariance present in multivariate systems like the atmosphere. As a result, AM distorts and reduces the amplitude of any feature that may be coherent across the sample data. The lack of realistic variance has been noted as a challenge in the use of AM in weather forecasting.
In this study, we generalize the notion of expected value for multivariate systems with coherent structures across a sample, like features present in a set of ensemble forecasts. The feature-oriented mean (FM) proposed here first co-locates coherent features across all the members to the mean of their position in individual members, before the mean of all members is taken. Thus, FM provides an estimate of the expected state of a multivariate system both in terms of the location and structure of coherent features, instead of the expected value of independently considered single point variables. Though FM is still not a dynamically realizable state, preliminary results suggest the variance and amplitude of features in FM are more realistic than in AM, with no appreciable loss in forecast skill.
