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 Fig8.1.9.1-1: The cumulative distribution function (CDF) shows the probability not exceeding a threshold value (e.g. say, not exceeding 20°C).  The figure is a schematic explanation of the principle behind the Extreme Forecast Index (EFI).  The blue line shows the cumulative probability of temperatures evaluated by M-climate for a given location, time of year and forecast lead time.  The red line shows the corresponding cumulative probability of temperatures evaluated by the ensemble.  EFI is measured by the area between the CDFs of the M-Climate (blue) and the CDFs of the ensemble members (red).   Almost all the ensemble forecast temperatures are above the M-climate median and about 15% are above the M-climate maximum.  In this case, the EFI is positive (the red line to the right of the blue line), indicating higher than normal probabilities of warm anomalies.  

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In the upper frames of Fig8.1.9.1-2 the peak of the forecast PDF (red) is to the right of the peak of the M-climate PDF (blue), indicating that the forecast predicts warmer than normal conditions and the sharpness of the peak indicates fairly high probability.

In the lower frames of Fig8.1.9.1-2 the peak of the forecast PDF (red) is to the left of the peak of the M-climate PDF (blue), indicating that the forecast predicts colder than normal conditions and the sharpness of the peak indicates high probability.

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The third diagram is ensemble forecast and M-Climate CDFs for maximum wind gusts at 45.9°N 45.28°W during the period Saturday 24 March 2018 00 UTC to Sunday 25 March 2018 00 UTC.  The trace show CDFs at this location from a series of recent ensemble forecasts for this period and the black line is the CDF from the M-climate.  The red (most recent) trace shows a flat interval (at about 57% probability of not exceeding 20m/s gusts) indicating bi-modal structure of the PDF.   

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Fig8.1.9.1-4: Estimating the mean value for a CDF graphically, using a 2m temperature example.  The mean value of a set of ensemble forecast results may be obtained by adjusting a vertical line V laterally until the area A above the CDF curve equals the area B below the curve. In this example the mean V for the black (M-Climate) profile is slightly above the median (where the y-axis probability = 50%), implying some skew to the distribution (related to the longer positive tail).  The same approach could be used to estimate the mean for any of the coloured (forecast) CDFs shown.

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