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

The Extreme Forecast Index is calculated according to the formula:

 where Qf(Q) denotes the proportion of ENS members lying below the Q quantile of the M-climate record.  This is shown diagramatically in Fig8.1.4.2.1 where : Q-Qf(Q) is represented by the blue line and the green shaded area. The term Q(1-Q) gives more weight towards the extremes of M-climate.

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Fig8.1.4.2.1: Schematic CDF diagram showing positive EFI as the area between the M-climate and  curve and the ENS curvescurve.  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 (green dashed line) are not used in evaluating EFI and so how extreme these actually are is not accounted for. The Shift of Tails (SOT) concept was developed in part to address this disadvantage.

Fig8.1.4.2.2: Schematic CDF diagram showing negative EFI as the area between the M-climate curve and the ENS curvescurve.  The area is negative where the ENS curve (red line) is to the left (i.e. values less than) M-climate (black line).  Note forecast values beyond the limits of the M-climate (green dashed line) are not used in evaluating EFI and so how extreme these actually are is not accounted for. The Shift of Tails (SOT) concept was developed in part to address this disadvantage.

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Fig8.1.4.2.3: Schematic CDF diagram illustrating the impact on the final EFI value of having EFI when there are both positive and negative contributions to the integral, which .  This arises whenever the ENS (red) and M-climate (black) curves cross.

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Fig8.1.4.2.4: Schematic CDF diagram for rainfall showing positive EFI as the area between the the M-climate and  curve and the ENS curvescurve.  The area is positive where the ENS curve (red line) is to the right (i.e. values greater than) M-climate (black line).  Note forecast values beyond the limits of the M-climate (green dashed line) are not used in evaluating EFI and so how extreme these actually are is not accounted for. The Shift of Tails (SOT) concept was developed in part to address this disadvantage.

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Fig8.1.4.2.5: Schematic CDF diagram for rainfall showing negative EFI as the area between the the M-climate and  curve and the ENS curvescurve.  The area is negative where the ENS curve (red line) is to the left (i.e. values less than) M-climate (black line).  The lower bound of the ENS forecast precipitation can be no lower than 0mm.  The lower bound of the M-climate for the vast majority of locations is also 0mm.  Thus negative values of the EFI for 24hr total precipitation do not provide sensible information and should not be used.  In most places a dry day is not considered extreme or severe anyway. 

Since generally there can be no ENS members below the lower limit of the M-climate the Shift of Tails (SOT) concept generally does not have a meaning for the dry ENS extremities of the CDF.  Theoretically it is possible at some very rare locations, where there has locations where there has never been any dry days, it is possible for the minimum of M-climate is to be above 0mm.   In these cases the Shift of Tails (SOT) concept can apply when no rain is forecast but it is not very informative.

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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 they will be incorporated when creating the model climate (M-climate).  However, negative EFI does make sense for accumulations over longer periods as few places consistently experience completely dry conditions during these longer intervals, and that should be reflected in a long-period model climate.  Negative precipitation EFI in this case shows does show the risk of anomalously dry weather.

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 Fig8.1.4.2.6: 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. Note how the CDFs and PDFs both give, in different ways, a visual indication of mean, spread and asymmetry.

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A convenient and powerful way to depict the temporal evolution of successive ENS forecasts for a given day (akin also to a "lagged ensemble") is to overlay the CDFs corresponding to each of those runs, as in Figs 8.1.4.2.6 (for e.g. rainfall) and 8.1.4.2.7 (for e.g. 2m temperature).  In the first example, for rainfall (Fig 8.1.4.2.6) 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 very unusual rainfall compared to climatology.  The steepness of the CDF and hence the peaked shape of the PDF charts at T+48 indicate that many of the ENS members are showing similar results and thus an extreme event of the magnitude indicated (on the x-axis) can be considered quite likely (assuming of course that the forecasts are not systematically biased).

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