Synthetic Example {murphydiagram}

Analytical Expressions from the Synthetic Example in Section 3.3 and Appendix B

Description

Functions to compute the analytical expressions in Table 3 of the paper by Ehm et al (2016). These expressions yield the expected score of various forecasters, given the synthetic setup studied in Section 3.3 and Appendix B of the paper. The expressions can be used to replicate Figure 2 in the paper.

Usage

expected_score_mean(theta, forecaster = "P")
expected_score_quantile(theta, alpha, forecaster = "P")

Arguments

theta

Value of the parameter $theta$, indexing the extremal score

alpha

Quantile level, between zero and one

forecaster

ID of the forecaster, string of length one. Either "P" (perfect forecaster), "C" (climatological forecaster), "U" (unfocused forecaster), or "SR" (sign-reversed forecaster).

Value

Expected value of the extremal score, given the synthetic setup described in Section 3.3 of Ehm et al (2016).

Author(s)

Alexander Jordan, Fabian Krueger

References

Ehm, W., Gneiting, T., Jordan, A. and Krueger, F. (2016): Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings. Journal of the Royal Statistical Society (Series B) 78, 1-29. doi: 10.1111/rssb.12154 (open access).

Examples

## Not run: 
# Color palette, obtained from http://www.cookbook-r.com/Graphs/Colors_
cbbPalette <- c("#000000", "#E69F00", "#56B4E9", "#009E73")
cbbPalette <- cbbPalette[c(1, 4, 2, 3)]

# Labeling stuff
forecasters <- c("P", "C", "U", "SR")
names <- c("Perfect", "Climatological", "Unfocused", "Sign-Reversed")
x_label <- expression(paste("Parameter ", theta))

# Figure 2, top left

# Grid for theta
theta_grid1 <- seq(-3, 3, 0.01)
# Expected scores for all forecasters
scores1 <- sapply(forecasters, expected_score_mean, theta = theta_grid1)
# Plot
matplot(x = theta_grid1, y = scores1[, 4:1], type = "l", lty = 1, col = cbbPalette[4:1], 
        lwd = 2, bty = "n", xlab = x_label, ylab = expression("Expected Score"))
legend("topright", names, col = cbbPalette, lwd = 2, bty = "n")

## End(Not run)