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Per #1853, work in progress getting the formatting correct.
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JohnHalleyGotway committed Jul 22, 2021
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Expand Up @@ -948,10 +948,10 @@ While the above probabilistic verification measures utilize dichotomous observat

Let :math:`\text{J}` be the number of categories, then both the forecast, :math:`\text{f} = (f_1,…,f_J)`, and observation, :math:`\text{o} = (o_1,…,o_J)`, are length-:math:`\text{J}` vectors, where the components of :math:`\text{f}` include the probabilities forecast for each category :math:`\text{1,…,J}` and :math:`\text{o}` contains 1 in the category that is realized and zero everywhere else. The cumulative forecasts, :math:`F_m`, and observations, :math:`O_m`, are defined to be:

Then :math:`\text{F_m} = \sum_{j=1}^m (f_j)` and math:`\text{O_m} = \sum_{j=1}^m (o_j), m = 1,…,J`.
:math:`F_m = \sum_{j=1}^m (f_j)` and math:`O_m = \sum_{j=1}^m (o_j), m = 1,…,J`.


To clarify, :math:`\text{F_1 = f_1}` is the first component of :math:`\text{F_m}`, :math:`\text{F_2 = f_1+f_2}`, etc., and :math:`\text{F_J = 1}`. Similarly, if :math:`\text{o_j = 1} and :math:`\text{i < j}`, then :math:`\text{O_i = 0}` and when :math:`\text{i≥j}, :math:`\text{O_i = 1}`, and of course, :math:`\text{O_J = 1}`. Finally, the RPS is defined to be:
To clarify, :math:`F_1 = f_1` is the first component of :math:`F_m`, :math:`F_2 = f_1+f_2`, etc., and :math:`F_J = 1`. Similarly, if :math:`o_j = 1` and :math:`i < j`, then :math:`O_i = 0` and when :math:`i≥j`, :math:`O_i = 1`, and of course, :math:`O_J = 1`. Finally, the RPS is defined to be:

"RPS"=∑_(m=1)^J▒(F_m-O_m )^2 =∑_(m=1)^J▒〖BS_m 〗,
where BS_m is the Brier score for the m-th category (:ref:`Tödter and Ahrens, 2012<Todter-2012>`). Subsequently, the RPS lends itself to a decomposition into reliability, resolution and uncertainty components, noting that each component is aggregated over the different categories; these are the line types RPS_REL, RPS_RES and RPS_UNC.
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