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• Steep slope implies higher confidence.
• Shallow slope implies lower confidence.

 Fig8.1.4.1.1Fig8191.A: 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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The Probability Density Function (PDF) is the first derivative of the cumulative distribution function CDF). 

Fig8.1.4.1.2A leftFig8191.B(left): Example cumulative distribution function (CDF).

Fig8.1.4.1.2B rightFig8191.B(right): The probability density function (PDF) is defined as the first derivative of the CDF.  The graphs correspond to the example CDF curves in Fig8.1.4.1.2A Fig8191.B with the temperature M-climate (blue) and the forecast distribution (red). 

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The median and any other percentile is given by reading the point on the x-axis where a horizontal probability line intersects the curve.  The most likely values are associated with those where the CDF is steepest.  Similarly, the PDF shows peaks in the curve at the highest probability intervals.  The EFI can be understood and interpreted with both the CDF and PDF in mind; the former relates to the EFI value, the latter clarifies the connection to probabilities.  A steep slope of the CDF, or equivantly a narrow peak of the PDF, implies a high confidence in the forecast.

In the upper frames of Fig8.1.4.1.2 Fig8191.B 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 Fig8.1.4.1.2 Fig8191.B 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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Sometimes the distribution of possible outcomes can have two favoured solutions.  This is called "bimodality".  On a PDF this is clearly shown by two peaks.  On a CDF curve it will be denoted by a step. A scenario in which one can sometimes see bimodal solutions is for the maximum wind gust parameter, close to the track of an active, small scale frontal wave cyclone.  North of the track relatively light winds are favoured whilst south of the track very strong winds are favoured.  Values in between may be less likely overall.

Fig8.1.4.1.3Fig8191.C: This example probability density function (PDF) diagram shows the ensemble members to be widely distributed but fall towards two distinct more likely wind speeds - one set suggests a most probable wind speed centred around the peak at W₁ and a second set suggests a probable wind speed centred around the peak at W₂.  The CDF associated with the example PDF shows the probability of (i.e. the percentage of ensemble members) attaining wind speeds.

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It is also possible to estimate, graphically, the mean value of ensemble forecasts (or the model climate) from a CDF, using the method shown on Fig8Fig8191.1.4.1.4D.

Fig8.1.4.1.4Fig8191.D: 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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