Photospheric Magnetic Field Properties of Flaring versus Flare-quiet Active Regions. II. Discriminant Analysis

We apply statistical tests based on discriminant analysis to the wide range of photospheric magnetic parameters described in a companion paper by Leka & Barnes, with the goal of identifying those properties that are important for the production of energetic events such as solar flares. The photospheric vector magnetic field data from the University of Hawai'i Imaging Vector Magnetograph are well sampled both temporally and spatially, and we include here data covering 24 flare-event and flare-quiet epochs taken from seven active regions. The mean value and rate of change of each magnetic parameter are treated as separate variables, thus evaluating both the parameter's state and its evolution, to determine which properties are associated with flaring. Considering single variables first, Hotelling's T²-tests show small statistical differences between flare-producing and flare-quiet epochs. Even pairs of variables considered simultaneously, which do show a statistical difference for a number of properties, have high error rates, implying a large degree of overlap of the samples. To better distinguish between flare-producing and flare-quiet populations, larger numbers of variables are simultaneously considered; lower error rates result, but no unique combination of variables is clearly the best discriminator. The sample size is too small to directly compare the predictive power of large numbers of variables simultaneously. Instead, we rank all possible four-variable permutations based on Hotelling's T²-test and look for the most frequently appearing variables in the best permutations, with the interpretation that they are most likely to be associated with flaring. These variables include an increasing kurtosis of the twist parameter and a larger standard deviation of the twist parameter, but a smaller standard deviation of the distribution of the horizontal shear angle and a horizontal field that has a smaller standard deviation but a larger kurtosis. To support the "sorting all permutations" method of selecting the most frequently occurring variables, we show that the results of a single 10-variable discriminant analysis are consistent with the ranking. We demonstrate that individually, the variables considered here have little ability to differentiate between flaring and flare-quiet populations, but with multivariable combinations, the populations may be distinguished.