diff --git a/docs/Users_Guide/difficulty_index.rst b/docs/Users_Guide/difficulty_index.rst index 22b0cd11..8e31cae7 100644 --- a/docs/Users_Guide/difficulty_index.rst +++ b/docs/Users_Guide/difficulty_index.rst @@ -52,4 +52,53 @@ This accounts for the notion that a forecast is more difficult when it is slight than slightly above. The value of :math:`A` then ramps down to zero for large values of :math:`\bar{x}_{i,j}`. +To gain a sense of how the difficulty index performs, consider the interplay between probability of +exceedance, normalized ensemble spread, and the mean forecast value (which sets the value of +:math:`A`) shown in Tables 1-3. Each row is for a different probability of threshold exceedance, +:math:`P(x_{i,j} \geq thresh)`, each column is for a different value of normalized uncertainty, +quantized as small, :math:`(\sigma/\bar{x})/(\sigma/\bar{x})_{ref}=0.01`, medium, +:math:`(\sigma/\bar{x})/(\sigma/\bar{x})_{ref}=0.05`, and large, +:math:`(\sigma/\bar{x})/(\sigma/\bar{x})_{ref}=1.0`. Each box contains the calculation of +:math:`d_{i,j}` for that case. +When :math:`\bar{x}` is very large or very small the difficulty index is dominated by :math:`A`. +Regardless of the spread or the probability of exceedance the difficulty index takes on a value near +zero and the forecast is considered to be easy (see Table 1). + +When :math:`\bar{x}` is near the threshold (e.g. 25kt or 37kt), the situation is a bit more complex +(see Table 2). For small values of spread the only interesting case is when the probability is +equally distributed about the threshold. For large spread, all probability values deserve a look, and +the case where the probability is equally distributed about the threshold is deemed difficult. + +When :math:`\bar{x}` is close to but slightly below the threshold (e.g. between 28kt and 34kt), +almost all combinations of probability of exceedance and spread deserve a look, and all values of the +difficulty index for medium and large spread are difficult or nearly difficult. + +.. list-table:: Table 1: Example of an obviously easy forecast. :math:`\bar{x}` is very large (e.g. 48 kt) or very small (e.g. 7kt), making :math:`A/2=0.1/2=0.05`. + :widths: auto + :header-rows: 1 + + * - + - Small Spread + - Medium Spread + - Large Spread + * - 1 + - 0.05*(0.01+0.5) = 0.026 + - 0.05*(0.5+0.5) = 0.05 + - 0.05*(1+0.5) = 0.075 + * - 0.75 + - 0.05*(0.01+0.75) = 0.038 + - 0.05*(0.5+0.75) = 0.063 + - 0.05*(1+0.75) = 0.088 + * - 0.5 + - 0.05*(0.01+1) = 0.051 + - 0.05*(0.5+1) = 0.075 + - 0.05*(1+1) = 0.1 + * - 0.25 + - 0.05*(0.01+0.75) = 0.038 + - 0.05*(0.5+0.75) = 0.063 + - 0.05*(1+0.75) = 0.088 + * - 0 + - 0.05*(0.01+0.5) = 0.026 + - 0.05*(0.5+0.5) = 0.05 + - 0.05*(1+0.5) = 0.075