The cumulative distribution function

The EFI value is computed from the difference between two cumulative distribution function (CDF) curves: one for the M-Climate, and the other for the current EPS distribution. The calculations are made so that more weight is given to differences in the tails of the distribution (see Figure 50).

Figure 50:  A schematic explanation of the principle behind the Extreme Forecast Index, measured by the area between the cumulative distribution functions (CDFs) of the M-Climate and the 50 EPS members. The steeper the slope of the CDF in an interval, the higher the probability in that interval.The EFI is, in this case, positive (red line to the right of the blue), indicating higher than normal probabilities of  warm anomalies.

From a CDF curve it is also easy to determine the median and any other percentile as the point on the x-axis where a horizontal line intersects the curve. The most likely values are associated with those where the CDF is steepest. Another way to assess it is by the probability density function (pdf), which is a derivative of the CDF (i.e. the gradient of the curve). The highest probability intervals are easily recognised as the peaks in a pdf (see Figure 51).

Figure 51:  The temperature climatology (blue curve) and the EPS forecast distribution (red curve) presented as probability density functions corresponding to the CDF curves in Figure 50. The pdf is essentially the derivative of the CDF. The EPS pdf is to the right (red curve) of the M-climate pdf (blue curve), indicating that the EPS has higher than normal probabilities of warmer anomalies, consistent with the conclusions on positive EFI from Figure 50.

The EFI can be understood and interpreted with both the CDF and pdf; the former relates to the EFI value, the latter clarifies the connection to probabilities.