## Binary Categorical Skill Scores

Skill scores can be a more meaningful way of describing a forecast’s quality. By definition, skill scores compare the performance of the forecasts to some standard or “reference forecast” (e.g., climatology, persistence, perfect forecasts, random forecasts). They often combine aspects of the previously-listed scalar statistics and can serve as a starting point for creating your own skill score that is better suited to your forecasts’ properties. Skill scores create a summary view of the contingency table, which is in contrast to the scalar statistics’ focus on one attribute at a time.

Three of the most popular skill statistics for categorical variables are the Heidke Skill Score (HSS), the Hanssen-Kuipers Discriminant (HK), and the Gilbert Skill Score (GSS). These measures are described here.

### Heidke Skill Score (HSS)

The HSS measures the proportion correct relative to the expected proportion correct that would be achieved by a “reference” forecast, denoted by C2 in the equation. In this instance, the reference forecast denotes a forecast that is completely independent of the observation dataset. In practice, the reference forecast often is based on a random, climatology, or persistence forecast. By combining the probability of a correct “yes” forecast (i.e., a hit) with the probability of a correct “no” forecast (i.e. a correct rejection) the resulting equation is

HSS can range from -1 to 1, with a perfect forecast receiving a score of 1. The equation presented above is a compact version which uses a sample climatology, C2 based on the counts in the contingency table. The C2 term expands to

This is a basic “traditional” version of HSS. METplus also calculates a modified HSS, that allows users to control how the C2 term is defined. This additional control allows users to apply an alternative standard of comparison, such as another forecast or a basic standard such as a persistence forecast or climatology. See how to use these skill scores in METplus!

### Hanssen-Kuipers Discriminant (HK)

HK is known by several names, including the Peirce Skill Score and the True Skill Statistic. This score is similar to HSS (ranges from -1 to 1, perfect forecast is 1, etc.). HK is formulated relative to a random forecast that is constrained to be unbiased. In general, the focus of the HK is on how well the forecast discriminates between observed “yes” events and observed “no” events. The equation for HK is

which is equivalent to “POD minus POFD”. Because of its dependence on POD, HK can be similarly affected by infrequent events and is suggested as a more useful skill score for frequent events. See how to use this skill score in METplus!

### Gilbert Skill Score (GSS)

Finally, GSS measures the correspondence between forecasted and observed “yes” events. Sometimes called the Equitable Threat Score (ETS), GSS is a good option for those forecasted events where the observed “yes” event is rare. In particular, the number of correct negatives (which for a rare event would be large) are not considered in the GSS equation and thus do not influence the GSS values. The GSS is given as

GSS ranges from -1 to 1, with a perfect forecast receiving a score of 1. Similar to HSS, a compact version of GSS is presented using the C1 term. This term expands to