To set up the probabilistic prediction model for visibility, the characteristics of data from observed visibility and from the corresponding ensemble forecasts had to be considered. The observed visibility was censored at 10 km, while the corresponding ensemble forecasts generated using LENS had positive real numbers. In other words, even if visibility was observed for more than 10 km, the observed visibility was recorded as 10 km; in contrast, the corresponding values from the ensemble forecasts were used as they were generated. Therefore, we needed a probabilistic model that reconciled the results obtained from these different datasets.
3.1. Weighted Model Averaging
We consider WMA, which models the predictive PDF of a weather quantity of interest
, as a mixture of the conditional PDFs. Let
be the
th ensemble member forecast and
a mixture of the conditional PDF given a specific forecast
. The WMA predictive PDF is given by
where
is the weight of the
th ensemble member forecast and refers to its relative skill over the training period. The weights are constrained to be non-negative and sum to 1.
To determine the mixture of conditional PDF , we consider a two-component model. The first part consists of a point mass at 10 km and corresponds to the probability that the recorded visibility is 10 km, which is conditional on the kth forecast in the ensemble. The second component of the model assigns a member-specific PDF to visibility, given that it is less than 10 km. We use a two-sided truncated normal distribution defined in (0, 10).
First, we apply a logistic regression model to estimate the probability that the observed visibility is 10 km, given the forecast of the
kth ensemble member,
, as follows:
where
and
are regression coefficients, and these parameters are estimated using a logistic regression model using the member forecasts in the training period as predictors and a vector of binary indicator of
y = 10 as the response variable.
To predict visibility when the observed visibility is less than 10 km, we consider a two-sided truncated normal distribution. Observed visibility
y has a normal distribution, with mean
and variance
defined for 0 <
y < 10, and the PDF for 0 <
y < 10 is given by
where
is the PDF of the standard normal distribution and
is its cumulative distribution function.
Combining the two components of the model, we build a final conditional PDF for visibility, given the
kth ensemble member forecast, as follows:
where
is a member-specified truncated normal distribution. The final WMA model for the predictive probability density function of visibility
y is given by
where
with mean
and standard deviation
of the truncated normal distribution.
We consider a relative humidity variable for inclusion in each component of the model. The model for binary outcome
is represented by
where
and
are the member-specified visibility and relative humidity forecasts, respectively. Given that
, the mean and standard deviation of the associated member-specific two-sided truncated normal distribution is specified as
and
By inserting these two components into Equation (6), we consider the predictive PDF that takes into account the relative humidity variable.
For the given observation of visibility less than 10 km, parameters and , and standard deviation are estimated using the method of maximum likelihood.
Parameters
are estimated as follows: After estimating parameters
, standard deviation
, and the probability that the observed visibility is 10 km given the forecast of the
kth ensemble member
over the training period, the median value is derived from the truncated normal distribution based on Equation (6) to estimate the visibility for less than 10 km. The corresponding estimates of observed visibility
during the training period are obtained as
where
,
;
K is the total number of ensemble members; and
n is the total number of observations.
We used the mean absolute error (MAE) and non-negative least squares to determine weights
The weight based on MAE is used to assign the largest weight to the ensemble member forecast with the smallest prediction error in the training period. In contrast, the weights based on non-negative least squares are determined by minimizing the following weighted combinations:
To select one of the two estimated weights, each prediction error is calculated for the training period; then, the weights () that provide the smallest prediction error are finally selected. WMA is applied to each station individually.
3.2. Scoring Rules
Prior to setting up a statistical model of ensemble member forecasts, we evaluated the prediction skills of ensembles. The measures used for comparing the prediction skills were the Brier score [
26,
27] for the binary events (
y = 10), the continuous ranked probability score (CRPS), the mean absolute error (MAE), and the root mean square error (RMSE). The Brier score (BS) is defined as the mean squared error of the forecast probability for
y = 10, as follows:
where
n is the number of observations;
is the forecasted probability of
; and
is 1 if
y = 10 and 0 otherwise. The
BS takes a value in the range between 0 and 1, and the perfect
BS has a value of zero. The CRPS [
22,
28] is an accurate scoring rule, which is defined as
where
is the cumulative distribution function of the forecast,
y is the observation, and
is the indicator function. CRPS is a generalization of the MAE and is a more general measure of model fit than the
BS.