|Event occurs||Event does not occur|
|Action not taken||L||0|
For a forecast based on climatological information alone, one would either always take action or never take action, whichever is cheaper, yielding an optimal expected mean expense of min(C, PclimL).
For a perfect forecast with no misses or false alarms the mean expense would be Pclim*C.
The mean expense for a particular forecast system over a sample of N forecasts is obtained by multiplying the expense matrix by the 2x2 contingency table to yield 1/N (hits*C + false_alarms*C + misses*L).
It can be shown that the maximum value of V occurs where the cost/loss ratio equals the climatological probability, and that the value at this point is equal to the Hanssen and Kuipers discriminant.
A nice primer on forecast value and the cost-loss model is available from the WMO CLIPS curriculum.(follow links to Tools, talk by David Richardson).
When applied to probabilistic forecasts the envelope of relative value curves represents the potential value since all decision thresholds are possible. In practice a user would choose an optimal decision threshold for his/her cost/loss ratio based on past performance, and apply this threshold to future (independent) forecasts. This actual value may be lower than the potential value if the conditions vary in time, or if the size of the sample used to estimate the optimal decision threshold was too small (Atger, 2001). Also, the value of a probabilistic forecast system will not be truly optimal unless the forecasts are reliable (unbiassed), which may require some calibration.
Atger, F., 2001: Verification of intense precipitation forecasts from single models and ensemble prediction systems. Nonlin. Proc. Geophys., 8, 401-417. Click here to see the abstract and get the PDF (295 Kb).
Richardson, D.S., 2000: Skill and relative economic value of the ECMWF ensemble prediction system. Quart. J. Royal Meteorol. Soc., 126, 649-667.
Wilks, D.S., 2001: A skill score based on economic value for probability forecasts. Meteorol. Appl., 8, 209-219.