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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).  The calculations are made so that more weight is given to differences in the tails of the (climatological) distribution.

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Fig8.1.9.2-1: Schematic CDF diagram showing positive EFI as the area between the M-climate curve and the ENS curve.  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.

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Fig8.1.9.2-2: Schematic CDF diagram showing negative EFI as the area between the M-climate curve and the ENS curve.  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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Fig8.1.9.2-3: Schematic CDF diagram illustrating the impact on the EFI when there are both positive and negative contributions to the integral.  This arises whenever the ENS (red) and M-climate (black) curves cross.

Fig8.1.9.2-4: Schematic CDF diagram for rainfall showing positive EFI as the area between the M-climate curve and the ENS curve.  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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Fig8.1.9.2-5: Schematic CDF diagram for rainfall showing negative EFI as the area between the M-climate curve and the ENS curve.  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 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. 

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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 does show the risk of anomalously dry weather.

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 Fig8.1.9.2-6: 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.  The CDFs and PDFs both give, in different ways, a visual indication of mean, spread and asymmetry.

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