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The Extreme Forecast Index is computed from the difference between Cumulative Distribution Function (CDF) curves of the M-climate and the current ENS forecast distribution.  The calculations are made so that more weight is given to differences in the tails of the distribution.  The  Extreme Forecast Index is calculated according to the formula


 where Ff(p) denotes the proportion of ENS members lying below the p quantile of the M-climate record.  This is shown diagramatically in Fig8.1.4.4 where: p-Ff(p) is resented by pink lines; p(1-p) gives more weight to the extremes of M-climate.

One can visually estimate the EFI by assessing the area between the M-climate (black) and ENS forecast (red) curves, and dividing this by what the area would be if all the ENS members predicted the M-climate extreme (i.e. a vertical red line that meets the black curve at y=100%). Whilst the answer derived by this method is only approximate it can nonetheless be useful aid to understanding.


 Fig8.1.4.4: Schematic CDF diagram showing positive EFI as the area between the M-climate and ENS curves.  The area is positive where the ENS curve (red line) is to the right of (i.e. values greater than) M-climate (black line).  Note forecast values beyond the limits of the M-climate are not used in evaluating EFI and the additional weight towards an extreme is missed.  Shift of Tails (SOT) has been developed to offset this disadvantage.

Fig8.1.4.5: Schematic CDF diagram showing negative EFI as the area between the M-climate and ENS curves.  The area is negative where the ENS curve (red line) is to the left (i.e. values less than) M-climate (black line).

Fig8.1.4.6: Schematic CDF diagram illustrating the impact on the final EFI value of having both positive and negative contributions to the integral, which arises whenever the ENS (red) and M-climate (black) curves cross.


The forecast CDF or the associated PDF (mean, spread and asymmetry) curve normally does not agree precisely with the M-climate CDF or PDF and the EFI therefore normally takes non-zero values.

  • EFI = 0 where the ENS forecast probability distribution agrees precisely with the M-climate distribution, or when the overall total of positive and negative area contributions is zero.
  • EFI = +1 where all ENS members forecast values to be above the absolute maximum of the M-climate.
  • EFI = -1 where all ENS members forecast values to be below the absolute minimum of the M-climate

The significant values of EFI may be taken as:

  • EFI values of 0.5 to 0.8 (irrespective of sign) generally signify an unusual” event.
  • EFI values above 0.8 usually signify a very unusual or extreme event.

Negative EFI for precipitation for 24-hour accumulations does not make sense because the model climate (M-climate) precipitation curve is bounded by 0.  This is because completely dry days occur in almost all places and this will be incorporated when creating the model climate (M-climate).  However, negative EFI does make sense for accumulations over longer periods as few places experience consistently completely dry conditions during the longer intervals used in creating the model climate.  Negative precipitation EFI in this case shows the risk of dry weather.

Similarly, negative wave height EFI indicates relatively calm seas.

 Fig8.1.4.7: A schematic illustration of the CDF (left) and PDF (right) for forecasts of 12hr accumulated precipitation showing the ENS T+48hr forecast (light green),  ENS T+96hr forecast (dark green) and ENS T+144hr (blue), together with the M-climate (black) verifying at the same time in the future. 

A convenient way to depict the current forecast together with previous runs verifying at the same time (a “lagged ensemble”) is to depict the CDF from previous runs as in Fig8.1.4.7.  The area between the CDF lines and M-climate, and hence the EFI, is becoming greater as the verifying time approaches suggesting increasing probability of an unusual rainfall event.   Indeed the EFI is approaching +1 on the T+48 forecast suggesting a very unusual rainfall compared to climatology.  The steepness of the CDF and hence the peaked shape of the PDF charts suggest many of the ENS members show similar results and thus an extreme event is more likely than usual.

Fig8.1.4.8: Schematic set of idealised CDFs from a series of ENS runs (cyan earliest, red latest). 

Ideally each ENS run should show the following:

  • At long lead-times, the CDFs may be fairly similar to the M-climate because of a broad variation within the ENS forecast results.
  • As forecast lead-time shortens, successive CDFs should show less variations relative to one another, as the ENS forecast results become increasingly similar.
  • As forecast lead-time shortens, the CDFs should become steeper as the members within each successive ENS forecast become increasingly similar, implying higher confidence.

In practice there is often a non-uniformity of progress through this idealised sequence of events, particularly for maximum wind gust and rainfall.  The forecaster needs to identify patterns in the series of CDFs as forecast lead-time shortens and investigate any departures from the expected changes.

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