CRPS-Based Trajectory Learning

• The paper demonstrates that aggregating CRPS over sequential predictions yields calibrated marginal distributions without enforcing a fixed parametric law.
• Methodologies include direct ensemble generation via neural models, stochastic parametrization of dynamical systems, and online quantile aggregation for practical forecasting applications.
• Empirical evaluations reveal competitive performance, enhanced sampling speed, and improved forecast reliability while addressing calibration and stability challenges.

CRPS-based trajectory learning denotes a class of probabilistic forecasting methods in which the continuous ranked probability score (CRPS) is used as the optimization target over sequential predictions rather than only for ex post evaluation. In the cited literature, this idea appears in at least three operational forms: direct training of ensemble-generating dynamical or neural models over forecast trajectories, marginally decomposed training of regional weather ensembles, and online aggregation of multivariate quantile trajectories for electricity-price forecasting (Ephrati et al., 29 Aug 2025, Larsson et al., 10 Oct 2025, Berrisch et al., 2023). Across these formulations, the common principle is to optimize scalar or marginal predictive distributions at each variable, location, or horizon, and then aggregate the resulting CRPS contributions over time and state. This yields calibrated marginals, and in several settings also accurate and sharp ensembles, without imposing a fixed parametric predictive law (Larsson et al., 10 Oct 2025, Ephrati et al., 29 Aug 2025).

1. Mathematical basis of CRPS in trajectory settings

For a predictive cumulative distribution function $F$ and realization $y$, the univariate CRPS is

$$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$

An equivalent representation is

$$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$

where $X$ and $X'$ are i.i.d. draws from $F$. When forecasts are available as an ensemble $\{x_i\}_{i=1}^M$, the empirical form is

$$\mathrm{CRPS}=\frac{1}{M}\sum_{i=1}^M |x_i-y|-\frac{1}{2M^2}\sum_{i=1}^M\sum_{j=1}^M |x_i-x_j|.$$

In CRPS-LAM, training uses the unbiased “fair CRPS,” which excludes self-pairs in the second term in order to correct small-sample bias:

$$\mathrm{CRPS}_{\text{fair}}(\{x_n\}_{n=1}^N,y)=\frac{1}{N}\sum_{n=1}^{N}|x_n-y|-\frac{1}{2N(N-1)}\sum_{n=1}^{N}\sum_{n^*\ne n}|x_n-x_{n^*}|.$$

The cited works treat CRPS as a strictly proper scoring rule for univariate or marginal distributions: for any true data distribution $y$0, expected CRPS is minimized by $y$1, which is why optimization of CRPS encourages calibrated marginal distributions (Larsson et al., 10 Oct 2025).

In quantile-based aggregation settings, CRPS is approximated on a dense probability grid by pinball losses. The approximation used in the electricity-price studies is

$y$2

This connects CRPS learning to quantile regression while retaining a score tied to the full marginal predictive distribution rather than to a finite set of point functionals (Berrisch et al., 2023, Nitka et al., 2023).

2. Trajectory learning as marginal score aggregation over time and state

The defining operational move in CRPS-based trajectory learning is to compute CRPS at each component of a trajectory and then aggregate those contributions over the forecast horizon. In the Lorenz ’96 case study, CRPS is computed separately for each slow state variable $y$3 at each time $y$4, using the ensemble of coarse forecasts, and the total objective is

$y$5

The paper explicitly uses scalar-by-component CRPS rather than a multivariate CRPS; the per-variable scores are summed and given equal weight (Ephrati et al., 29 Aug 2025).

CRPS-LAM applies the same principle to regional weather forecasting. The model minimizes marginal fair CRPS across all variables, grid points, and forecast time steps. For interior grid point $y$6, variable $y$7, ensemble members $y$8, and observation $y$9, the per-$$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$0 loss is

$$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$1

The full training loss is averaged over spatial dimensions and summed across variables; under two-step autoregressive training it is averaged over the two time steps, and no additional variable or level weighting is applied (Larsson et al., 10 Oct 2025).

In multivariate electricity-price forecasting, the trajectory is the 24-dimensional hourly day-ahead price curve. The multivariate CRPS-learning framework does not optimize a vector-valued score such as the energy score. Instead, it performs horizontal aggregation across quantiles for each marginal and sums CRPS across marginals:

$$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$2

The rationale stated in the paper is that, by Sklar’s theorem, improving marginals via a strictly proper score like CRPS remains meaningful even when the copula is not directly updated (Berrisch et al., 2023).

3. Principal methodological paradigms

One major paradigm is direct generative ensemble modeling. CRPS-LAM repurposes the backbone of Diffusion-LAM but replaces diffusion objectives with fair CRPS. It uses a convolutional U-Net-style architecture with conditional normalization layers and distinct interior and boundary inputs for limited-area modeling. The interior input is $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$3, the boundary input is $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$4, and the model outputs $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$5 at a 3-hour step, rolling out autoregressively up to 57 hours. Diversity is produced by sampling a latent noise vector $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$6 of dimension 32 per ensemble member and injecting it through conditional normalization layers, yielding a stochastic mapping $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$7. Because the same $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$8 conditions all outputs at a step, the method induces coherent spatial and cross-variable structure even though the loss only constrains marginals (Larsson et al., 10 Oct 2025).

A second paradigm is stochastic parametrization learning for dynamical systems. In the Lorenz ’96 study, the coarse model is equipped with a stochastic parametrization $$\mathrm{CRPS}(F,y)=\int_{-\infty}^{\infty}\big(F(z)-\mathbf{1}\{z\ge y\}\big)^2\,dz.$$9 and trained end-to-end by backpropagating CRPS through a differentiable Stratonovich SSPRK3 integrator. Two parametrizations are developed: an additive model using POD modes and a coupled OU process in POD space, and a multiplicative model in which the POD coefficients are scaled by $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$0. In both cases, the ensemble is generated by integrating the stochastic coarse model over short windows and comparing the resulting forecast trajectories to the coarsened truth (Ephrati et al., 29 Aug 2025).

A third paradigm is online quantile aggregation. In multivariate probabilistic CRPS learning, experts provide quantile forecasts $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$1, and the aggregated quantile is a convex combination

$$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$2

Weights vary jointly over marginal index $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$3 and probability level $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$4, which the paper terms horizontal aggregation. To stabilize learning over the $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$5 grid, it introduces either dimensionality reduction using basis matrices or penalized smoothing of the weight surfaces, and updates weights online by Bernstein Online Aggregation (BOA) with Bernstein correction and adaptive learning rates (Berrisch et al., 2023).

4. Optimization, rollout design, and stability

Trajectory length and rollout structure are central design variables. CRPS-LAM is first trained for single-step forecasts $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$6 and then extended to two-step autoregressive rollouts $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$7, averaging the loss across both steps. The paper states that this stabilizes the usage of the latent noise and the trajectory generation. Optimization uses a staged learning-rate schedule $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$8 over $$\mathrm{CRPS}(F,y)=\mathbb{E}[|X-y|]-\tfrac{1}{2}\mathbb{E}[|X-X'|],$$9 epochs, and the $X$0 pairwise term in fair CRPS is reported as tractable for ensemble sizes up to $X$1. The authors also report occasional early-training collapse to near-deterministic behavior, in which the latent is ignored, and identify two-step autoregressive training as the mitigation adopted in the paper (Larsson et al., 10 Oct 2025).

In the Lorenz ’96 study, optimization uses Adam with fixed learning rates $X$2 for additive OU models and $X$3 for multiplicative models, with 400 epochs and 125 batches per epoch. Trajectory lengths $X$4 are explored, with batch size adjusted so that $X$5 remains roughly constant. The paper reports unstable growth for additive models at $X$6, attributing this to chaotic sensitivity, and notes that one multiplicative configuration became unstable in long integrations (Ephrati et al., 29 Aug 2025).

In online aggregation, BOA provides the update mechanism. The multivariate extension carries out learning on the $X$7 grid, optionally in a reduced basis space, and may combine smoothing, forgetting, and shrinkage operators. The papers distinguish between static tuning and dynamic hyperparameter adaptation; in the day-ahead electricity-price application, Bayesian Online and Sampling Online are reported as more robust than Bayesian Fix under structural changes (Berrisch et al., 2023). A related bidding study uses BOA with penalized probabilistic smoothing, no forgetting, and quantile-specific time-varying weights, and reports substantially higher computational cost than equal weighting, though still below 20 seconds for the full test period on a standard laptop (Nitka et al., 2023).

5. Empirical realizations and reported performance

In regional weather forecasting on the MEPS limited-area dataset, CRPS-LAM is evaluated at native 2.5 km resolution with training data subsampled to 10 km and produces 25-member ensembles. The model forecasts 3-hour steps autoregressively to 57 hours. Relative to Diffusion-LAM and Graph-EFM, the paper reports low errors comparable to diffusion models, SSR comparable to Graph-EFM and better than Diffusion-LAM at longer lead times, and energy spectra indicating that CRPS-LAM retains more fine-scale energy than Graph-EFM, while diffusion preserves slightly more at the highest frequencies. Sampling speed is about 0.5 s per ensemble member on a single A100 GPU, and the method achieves up to $X$8 faster sampling than Diffusion-LAM while remaining similar in per-member speed to Graph-EFM or deterministic models (Larsson et al., 10 Oct 2025).

In the Lorenz ’96 case study, CRPS-based trajectory learning is evaluated on short-range forecasts up to 2 MTU and on long-range climate statistics. Additive OU models trained with $X$9 and $X'$0 achieve the lowest CRPS across lead times for both $X'$1 and $X'$2. Multiplicative models achieve the lowest MSE and the smallest spread, but not always the lowest CRPS, which the paper relates to the $X'$3-type nature of CRPS. Reliability diagnostics show additive models are generally overdispersive, whereas multiplicative models are initially underdispersive. For long-term climate, CRPS-trained additive models improve over derivative-fitting baselines in KS and Hellinger distance, although many parametrizations still produce PDFs that are too wide (Ephrati et al., 29 Aug 2025).

In day-ahead electricity prices, the multivariate CRPS-learning framework is applied to 24-dimensional German power-price trajectories over 736 days, with 99 quantiles and eight distributional neural-network experts in the main application. The reported best overall specification is BOA with Bayesian Online and Smooth.Forget using equidistant knots, and the paper states that forgetting contributed most of the gains, indicating structural changes in the data. Quantile crossing is reported as rare with smoothing; in the best BOA Bayesian Online Smooth.Forget configuration, crossings occurred on 67 of 554 test days in at least one marginal (Berrisch et al., 2023).

A separate decision-focused study reaches a more qualified conclusion. In that work, CRPS learning improves probabilistic accuracy over equal-weighted quantile aggregation, and the diversified JSU ensemble with LEAR add-ons and CRPS learning attains the best CRPS. However, the higher computational cost is not offset by higher profits in day-ahead battery bidding, and the paper explicitly argues that minimizing CRPS does not necessarily lead to optimal downstream decisions when the decision rule depends nonlinearly on medians and tail quantiles (Nitka et al., 2023).

6. Limitations, misconceptions, and directions for extension

A recurrent limitation is that CRPS-based trajectory learning usually matches marginals rather than the full joint spatiotemporal distribution. CRPS-LAM states this directly: the loss does not explicitly penalize or shape multivariate dependence across space, time, and variables, and coherence arises indirectly from the shared latent $X'$4, autoregressive rollout, and convolutional inductive bias (Larsson et al., 10 Oct 2025). The electricity-price aggregation papers make an analogous point in a different form: weights are learned for marginal hourly distributions, while cross-hour dependence remains implicit in the experts and is not explicitly modeled by the combination rule (Nitka et al., 2023, Berrisch et al., 2023).

A related misconception is that better CRPS necessarily implies better decision quality. The bidding study provides a direct counterexample: lower CRPS did not consistently yield higher profits, especially when the trading rule was sensitive to extreme lower percentiles. The paper’s diagnosis is that gains concentrated in the center of the predictive distribution can be diluted or reversed by a decision map that depends disproportionately on tail quantiles (Nitka et al., 2023).

Calibration also remains an open issue. In CRPS-LAM, all models show some underdispersion, with SSR below 1. In Lorenz ’96, additive models are overdispersive and multiplicative models underdispersive for part of the forecast range, and no explicit post-training calibration is applied. The Lorenz paper notes several possible remedies: introducing a hyperparameter on the self-spread term in CRPS during training, adding a penalty to align spread with squared error, or applying post-training variance inflation or EMOS-like calibration (Larsson et al., 10 Oct 2025, Ephrati et al., 29 Aug 2025).

The extensions proposed in the cited literature indicate a shared research trajectory. CRPS-LAM identifies multivariate proper scores such as the energy score and variogram score, copula-based assembly of calibrated marginals into a coherent joint, autoregressive latent processes $X'$5, and hybrid or multi-scale losses as routes toward stronger dependence modeling (Larsson et al., 10 Oct 2025). The multivariate electricity-price work similarly points to copula modeling, hybrid CRPS/ES objectives, graph-based penalties, and explicit structural-break detection (Berrisch et al., 2023). This suggests that CRPS-based trajectory learning is evolving less as a single algorithm than as a design pattern: optimize strictly proper marginal scores over trajectories, then supplement them with architectural, dynamical, or post-hoc mechanisms that recover the aspects of joint structure most important for the application.