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Cumulative distribution function and Probability density function

Cumulative distribution function (CDF)

The cumulative distribution function (CDF) is the probability that a continuous random variable has a value less than or equal to a given value.  Each member of the ensemble gives a different forecast value (e.g. of temperature) for a given time and location, and consequently these results may be used to define a CDF where the x-axis is the forecast variable (e.g. temperature) and the y-axis the number of ensemble members (expressed as a proportion of the total number of ensemble members) forecasting a value less than  a given threshold.  The median value will be where the CDF is 50%.

Cumulative distribution function plot design.

The CDF for the ensemble values is constructed from the temperature or precipitation forecast by each ensemble member (red line in the examples below) together with the CDF of the temperature or rainfall M-climate (black line) for that location for the date in question.

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 Fig8.1.4.1.1: 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.  

Probability Density Function (PDF)

The Probability Density Function (PDF) is the first derivative of the cumulative distribution function CDF). 

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In the lower frames of Fig8.1.4.1.2 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.

Bi-modality

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.

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Bi-modal patterns can occur, for example, when there is uncertainty whether a depression will pass one side or the other of the location in question.  The diagrams say nothing about the direction of the winds (e.g. they may be moderate easterlies to the north of the location or strong westerlies to the south (N Hem)) nor about timing of the depression (e.g. it may be slower or faster).  The diagrams only give information on the variation among the ensemble member solutions. 

Estimation of the mean value from a cumulative distribution function

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 Fig8.1.4.1.4

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