pit_sample {scoringutils}

Probability Integral Transformation (sample-based version)

Description

Uses a Probability Integral Transformation (PIT) (or a randomised PIT for integer forecasts) to assess the calibration of predictive Monte Carlo samples. Returns a p-values resulting from an Anderson-Darling test for uniformity of the (randomised) PIT as well as a PIT histogram if specified.

Usage

pit_sample(true_values, predictions, n_replicates = 100)

Arguments

true_values

A vector with the true observed values of size n

predictions

nxN matrix of predictive samples, n (number of rows) being the number of data points and N (number of columns) the number of Monte Carlo samples. Alternatively, predictions can just be a vector of size n.

n_replicates

the number of draws for the randomised PIT for integer predictions.

Details

Calibration or reliability of forecasts is the ability of a model to correctly identify its own uncertainty in making predictions. In a model with perfect calibration, the observed data at each time point look as if they came from the predictive probability distribution at that time.

Equivalently, one can inspect the probability integral transform of the predictive distribution at time t,

u_t = F_t (x_t)

where x_t is the observed data point at time t \textrm{ in } t_1, …, t_n, n being the number of forecasts, and F_t is the (continuous) predictive cumulative probability distribution at time t. If the true probability distribution of outcomes at time t is G_t then the forecasts F_t are said to be ideal if F_t = G_t at all times t. In that case, the probabilities u_t are distributed uniformly.

In the case of discrete outcomes such as incidence counts, the PIT is no longer uniform even when forecasts are ideal. In that case a randomised PIT can be used instead:

u_t = P_t(k_t) + v * (P_t(k_t) - P_t(k_t - 1) )

where k_t is the observed count, P_t(x) is the predictive cumulative probability of observing incidence k at time t, P_t (-1) = 0 by definition and v is standard uniform and independent of k. If P_t is the true cumulative probability distribution, then u_t is standard uniform.

The function checks whether integer or continuous forecasts were provided. It then applies the (randomised) probability integral and tests the values u_t for uniformity using the Anderson-Darling test.

As a rule of thumb, there is no evidence to suggest a forecasting model is miscalibrated if the p-value found was greater than a threshold of p >= 0.1, some evidence that it was miscalibrated if 0.01 < p < 0.1, and good evidence that it was miscalibrated if p <= 0.01. However, the AD-p-values may be overly strict and there actual usefulness may be questionable. In this context it should be noted, though, that uniformity of the PIT is a necessary but not sufficient condition of calibration.

Value

A vector with PIT-values. For continuous forecasts, the vector will correspond to the length of true_values. For integer forecasts, a randomised PIT will be returned of length length(true_values) * n_replicates

References

Sebastian Funk, Anton Camacho, Adam J. Kucharski, Rachel Lowe, Rosalind M. Eggo, W. John Edmunds (2019) Assessing the performance of real-time epidemic forecasts: A case study of Ebola in the Western Area region of Sierra Leone, 2014-15, doi:10.1371/journal.pcbi.1006785

See Also

pit()

Examples

## continuous predictions
true_values <- rnorm(20, mean = 1:20)
predictions <- replicate(100, rnorm(n = 20, mean = 1:20))
pit <- pit_sample(true_values, predictions)
plot_pit(pit)

## integer predictions
true_values <- rpois(50, lambda = 1:50)
predictions <- replicate(2000, rpois(n = 50, lambda = 1:50))
pit <- pit_sample(true_values, predictions, n_replicates = 30)
plot_pit(pit)