bias_range {scoringutils} | R Documentation |
Determines Bias of Quantile Forecasts based on the range of the prediction intervals
Description
Determines bias from quantile forecasts based on the range of the prediction intervals. For an increasing number of quantiles this measure converges against the sample based bias version for integer and continuous forecasts.
Usage
bias_range(lower, upper, range, true_value)
Arguments
lower |
vector of length corresponding to the number of central prediction intervals that holds predictions for the lower bounds of a prediction interval |
upper |
vector of length corresponding to the number of central prediction intervals that holds predictions for the upper bounds of a prediction interval |
range |
vector of corresponding size with information about the width of the central prediction interval |
true_value |
a single true value |
Details
For quantile forecasts, bias is measured as
B_t = (1 - 2 \cdot \max \{i | q_{t,i} \in Q_t \land q_{t,i} \leq x_t\})
\mathbf{1}( x_t \leq q_{t, 0.5}) \\
+ (1 - 2 \cdot \min \{i | q_{t,i} \in Q_t \land q_{t,i} \geq x_t\})
\mathbf{1}( x_t \geq q_{t, 0.5}),
where Q_t
is the set of quantiles that form the predictive
distribution at time t
. They represent our
belief about what the true value x_t
will be. For consistency, we
define
Q_t
such that it always includes the element
q_{t, 0} = - \infty
and q_{t,1} = \infty
.
\mathbf{1}()
is the indicator function that is 1
if the
condition is satisfied and $0$ otherwise. In clearer terms, B_t
is
defined as the maximum percentile rank for which the corresponding quantile
is still below the true value, if the true value is smaller than the
median of the predictive distribution. If the true value is above the
median of the predictive distribution, then $B_t$ is the minimum percentile
rank for which the corresponding quantile is still larger than the true
value. If the true value is exactly the median, both terms cancel out and
B_t
is zero. For a large enough number of quantiles, the
percentile rank will equal the proportion of predictive samples below the
observed true value, and this metric coincides with the one for
continuous forecasts.
Bias can assume values between -1 and 1 and is 0 ideally.
Value
scalar with the quantile bias for a single quantile prediction
Author(s)
Nikos Bosse nikosbosse@gmail.com
See Also
bias_quantile bias_sample
Examples
lower <- c(
6341.000, 6329.500, 6087.014, 5703.500,
5451.000, 5340.500, 4821.996, 4709.000,
4341.500, 4006.250, 1127.000, 705.500
)
upper <- c(
6341.000, 6352.500, 6594.986, 6978.500,
7231.000, 7341.500, 7860.004, 7973.000,
8340.500, 8675.750, 11555.000, 11976.500
)
range <- c(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 98)
true_value <- 8062
bias_range(
lower = lower, upper = upper,
range = range, true_value = true_value
)