Determines Bias of Quantile Forecasts based on the range of the prediction intervals

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

Value

scalar with the quantile bias for a single quantile prediction

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.

Author

Nikos Bosse nikosbosse@gmail.com

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
)
#> [1] -0.8