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Probabilistic forecasts assign a probability to every possible future. Yet, all probabilistic forecasts are not equally accurate, and metrics are needed to assess the respective accuracy of distinct probabilistic forecasts. Simple accuracy metrics such as MAE (Mean Absolute Error) or MAPE (Mean Absolute Percentage Error) are not directly applicable to probabilistic forecasts. The

This metric notably differs from simpler metrics such as MAE because of its asymmetric expression: while the forecasts are probabilistic, the observations are deterministic. Unlike the pinball loss function, the CPRS does not focus on any specific point of the probability distribution, but considers the distribution of the forecasts as a whole.

Let $F$ be the cumulative distribution function (CDF) of $X$, such as $F(y)=\mathbf{P}\left[X \leq y\right]$.

Let $x$ be the observation, and $F$ the CDF associated with an empirical probabilistic forecast.

The CRPS between $x$ and $F$ is defined as: $$CRPS(F, x) = \int_{-\infty}^{\infty}\Big(F(y)- 𝟙(y - x)\Big)^2dy$$ where $𝟙$ is the Heaviside step function and denotes a step function along the real line that attains:

- the value of 1 if the real argument is positive or zero,
- the value of 0 otherwise.

The CRPS is expressed in the same unit as the observed variable. The CRPS generalizes the mean absolute error; in fact, it reduces to the mean absolute error (MAE) if the forecast is deterministic.

`crps()`

function:
Accuracy = crps(Z, X)where

`Z`

is expected to be a distribution intended to represent the probabilistic forecast, and `X`

is expected to be a number intended to represent the observed values.- $X$ and $X^*$ are independent copies of a linear random variable,
- $X$ is the random variable associated with the cumulative distribution function $F$,
- $\mathbf{E}[X]$ is the expected value of $X$.

- Gneiting, T. and Raftery, A. E. (2004). Strictly proper scoring rules, prediction, and estimation. Technical Report no. 463, Department of Statistics, University of Washington, Seattle, Washington, USA.

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