# Continuous Ranked Probability Score - CRPS

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Category: { Time Series }
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Summary: The Continuous Ranked Probability Score, known as CRPS, is a score to measure how a proposed distribution approximates the data, without knowledge about the true distributions of the data. Definition CRPS is defined as1 $$CRPS(P, x_a) = \int_{-\infty}^\infty \lVert P(x) - H(x - x_a) \rVert_2 dx,$$ where $x_a$ is the true value of $x$, P(x) is our proposed cumulative distribution for $x$, $H(x)$ is the Heaviside step function $$H(x) = \begin{cases} 1, &\qquad x=0\\ 0, &\qquad x\leq 0\\ \end{cases}$$ $\lVert \cdot \rVert_2$ is the L2 norm. Explain it The formula looks abstract on first sight, but it becomes crystal clear once we understand it.
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# Data Generating Processes for Time Series Data

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Category: { Time Series }
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# Root Relative Squared Error (RSE)

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Category: { Time Series }
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Summary: The Root Relative Squared Error (RSE) is an evaluation metric in time series forecasting,1 $$\mathrm{RSE} = \frac{ \sqrt{ \sum_{i, t} ( y^{(i)}_t - \hat y^{(i)}_t )^2 } }{ \sqrt{ \sum_{i, t} ( y^{(i)}_t - \bar y )^2 } }$$ where $y^{(i)}$ is the $i$th time series, ${} _ t$ denotes the time step $t$, and $\bar y$ is the mean of the forecasted series, i.e., $\bar y = \operatorname{mean}(y^{(i\in\{0, 1, \cdots, N\})} _ { t\in \{T _ f, T _ {f+1}, \cdots T _ {f+H}\} })$. Lai2017 Lai G, Chang W-C, Yang Y, Liu H.
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# Empirical Correlation Coefficient (CORR)

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Category: { Time Series }
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Summary: The Empirical Correlation Coefficient (CORR) is an evaluation metric in time series forecasting,1 $$\mathrm{CORR} = \frac{1}{N} \sum_{i=1}^N \frac{ \sum_t (y^{(i)}_t - \bar y^{(i)} ) ( \hat y^{(i)}_t -\bar{ \hat y}^{(i)} ) }{ \sqrt{ \sum_t (y^{(i)}_t - \bar y^{(i)} )^2 ( \hat y^{(i)}_t -\bar{\hat y}^{(i)} )^2 } }$$ where $y^{(i)}$ is the $i$th time series, ${} _ t$ denotes the time step $t$, and $\bar y^{(i)}$ is the mean of the $i$th forecasted series, i.e., $\bar y^{(i)} = \operatorname{mean}( y^{(i)} _ { t \in \{T _ f, T _ {f+1}, \cdots T _ {f+H}\} } )$. Lai2017 Lai G, Chang W-C, Yang Y, Liu H.
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