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#### Reliability Diagrams

Reliability diagrams (Hartmann et al. 2002) are simply graphs of the Observed frequency of an event plotted against the Forecast probability of an event. This effectively tells the user how often (as a percentage) a forecast probability actually occurred.

In theory, a perfect forecast system will result in forecasts with a probability of X% being consistent with the eventual outcome X% of the time. Hence when plotting a reliability diagram comparisons are made against the diagonal (see below figure).

With climate forecasting, it is also often the case that probabilities will be clustered together, generally near the 0.5 probability, as chaos dictates that situations are rarely perfectly predictable/historically identical, for similar initial conditions. Hence it is important to also plot some form of frequency distribution to indicate how the forecasts are spread over time. Are they all clustered together, or are they well spread? The below figure shows a probability function plotted for each curve, in order to demonstrate that the majority of forecasts were indeed clustered in the central probability bins.

The technique for constructing the reliability diagram is similar to that for calculating the ROC score, only instead of plotting the hit rate against the false alarm rate, the hit rate is calculated only from the sets of forecasts for each probability separately. It is then plotted against the corresponding forecast probabilities.

The hit rate for each probability bin is defined as:

where O = number of Observed instances,
NO = number of non-observed instances

Frequency histograms are constructed from the same contingency tables as those used to produce the reliability diagrams. Frequency histograms show the frequency of forecasts as a function of the probability bin. The frequency of forecasts (Fn) for probability bin n is defined as:

where T is the total number of forecasts.

An example of a reliability diagram for Level 1 of the SVSLRF is given below:

Reliability diagrams and frequency histograms are mandatory for level 1.

References:

Hartmann, H.C., Pagano, T.C., Sorooshiam, S. and Bales, R. 2002. Confidence builder: evaluating seasonal climate forecasts from user perspectives. Bull Amer. Met. Soc., 84, 683-698

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