+Energy market liberalization created complex, interconnected trading systems
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+Renewable energy transition introduces uncertainty and volatility from weather-dependent generation
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+Traditional point forecasts are inadequate for modern energy markets with increasing uncertainty
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+Risk inherently is a probabilistic concept
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+Probabilistic forecasting essential for risk management, planning and decision making in volatile energy environments
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+Online learning methods needed for fast-updating models with streaming energy data
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+Image generated by Sora (OpenAI)
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Overview of the Thesis
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+Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2), 105221.
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+Berrisch, J., & Ziel, F. (2024). Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. International Journal of Forecasting, 40(4), 1568–1586.
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+Berrisch, J., & Ziel, F. (2022). Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
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+Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
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+Berrisch, J., Narajewski, M., & Ziel, F. (2023). High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
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+Berrisch, J. (2025). rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
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+Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2), 105221.
+
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+Berrisch, J., & Ziel, F. (2024). Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. International Journal of Forecasting, 40(4), 1568–1586.
+
+Berrisch, J., & Ziel, F. (2022). Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
+
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+Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
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+Berrisch, J., Narajewski, M., & Ziel, F. (2023). High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
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+Berrisch, J. (2025). rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
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+Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2), 105221.
+
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+Berrisch, J., & Ziel, F. (2024). Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. International Journal of Forecasting, 40(4), 1568–1586.
+
+Berrisch, J., & Ziel, F. (2022). Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
+
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+
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+Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
+
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+
+
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+Berrisch, J., Narajewski, M., & Ziel, F. (2023). High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
+
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+Berrisch, J. (2025). rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
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Overview
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Online Distributional Regression
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Distributional Modeling and Forecasting of Natural Gas Prices
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Reduces estimation time by 2-3 orders of magnitude
In stochastic settings, the cumulative Risk should be analyzed Wintenberger (2017): \[\begin{align}
+ &\underbrace{\widetilde{\mathcal{R}}_t = \sum_{i=1}^t \mathbb{E}[\ell(\widetilde{X}_{i},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Forecaster}} \\
+ &\underbrace{\widehat{\mathcal{R}}_{t,k} = \sum_{i=1}^t \mathbb{E}[\ell(\widehat{X}_{i,k},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Experts}}
+ \label{eq_def_cumrisk}
+\end{align}\]
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Optimal Convergence
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The selection problem
+
\[\begin{equation}
+ \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \stackrel{t\to \infty}{\rightarrow} a \quad \text{with} \quad a \leq 0.
+ \label{eq_opt_select}
+\end{equation}\] The forecaster is asymptotically not worse than the best expert \(\widehat{\mathcal{R}}_{t,\min}\).
+
The convex aggregation problem
+
\[\begin{equation}
+ \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right) \stackrel{t\to \infty}{\rightarrow} b \quad \text{with} \quad b \leq 0 .
+ \label{eq_opt_conv}
+\end{equation}\] The forecaster is asymptotically not worse than the best convex combination \(\widehat{X}_{t,\pi}\) in hindsight (oracle).
+
+
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+
Optimal rates with respect to selection \(\eqref{eq_opt_select}\) and convex aggregation \(\eqref{eq_opt_conv}\)Wintenberger (2017):
Bernstein Online Aggregation (BOA) lets us weaken the exp-concavity condition. It satisfies that there exist a \(C>0\) such that for \(x>0\) it holds that
Almost optimal w.r.t. convex aggregation\(\eqref{eq_optp_conv}\)Wintenberger (2017).
+
The same algorithm satisfies that there exist a \(C>0\) such that for \(x>0\) it holds that \[\begin{equation}
+ P\left( \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \leq
+ C\left(\frac{\log(K)+\log(\log(Gt))+ x}{\alpha t}\right)^{\frac{1}{2-\beta}} \right) \geq
+ 1-2e^{-x}
+ \label{eq_boa_opt_select}
+\end{equation}\]
+
if \(Y_t\) is bounded, the considered loss \(\ell\) is convex \(G\)-Lipschitz and weak exp-concave in its first coordinate.
+
Almost optimal w.r.t. selection\(\eqref{eq_optp_select}\)Gaillard & Wintenberger (2018).
+
We show that this holds for QL under feasible conditions.
convexity properties of \(\mathcal{Q}_p\) depend on the conditional distribution \(Y_t|\mathcal{F}_{t-1}\).
+
Proposition 1
+
Let \(Y\) be a univariate random variable with (Radon-Nikodym) \(\nu\)-density \(f\), then for the second subderivative of the quantile risk \(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]\) of \(Y\) it holds for all \(p\in(0,1)\) that \(\mathcal{Q}_p'' = f.\) Additionally, if \(f\) is a continuous Lebesgue-density with \(f\geq\gamma>0\) for some constant \(\gamma>0\) on its support \(\text{spt}(f)\) then is \(\mathcal{Q}_p\) is \(\gamma\)-strongly convex.
A1 and A2 give us almost optimal convergence w.r.t. selection \(\eqref{eq_boa_opt_select}\)
+
Theorem 1
+
The gradient based fully adaptive Bernstein online aggregation (BOAG) applied pointwise for all \(p\in(0,1)\) on \(\text{QL}\) satisfies \(\eqref{eq_boa_opt_conv}\) with minimal CRPS given by
If \(Y_t|\mathcal{F}_{t-1}\) is bounded and has a pdf \(f_t\) satifying \(f_t>\gamma >0\) on its support \(\text{spt}(f_t)\) then \(\ref{eq_boa_opt_select}\) holds with \(\beta=1\) and
CRPS difference to Naive. Negative values correspond to better performance (the best value is bold). Additionally, we show the p-value of the DM-test, testing against Naive. The cells are colored with respect to their values (the greener better).
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Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices
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Berrisch, J., & Ziel, F. (2024). International Journal of Forecasting, 40(4), 1568-1586.
+
+
+
Multivariate CRPS Learning
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+
We extend the B-Smooth and P-Smooth procedures to the multivariate setting:
Let \(\boldsymbol{\psi}^{\text{mv}}=(\psi_1,\ldots, \psi_{D})\) and \(\boldsymbol{\psi}^{\text{pr}}=(\psi_1,\ldots, \psi_{P})\) be two sets of bounded basis functions on \((0,1)\):
Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2), 105221.
+
Berrisch, J., & Ziel, F. (2024). Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. International Journal of Forecasting, 40(4), 1568-1586.
+
+
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+
Modeling Volatility and Dependence of European Carbon and Energy Prices
+
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Finance Research Letters, 52, 103503.
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Motivation
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Understanding European Allowances (EUA) dynamics is important for several fields:
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Portfolio & Risk Management,
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Sustainability Planing
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Political decisions
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EUA prices are obviously connected to the energy market
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How can the dynamics be characterized?
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Several Questions arise:
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Data (Pre)processing
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Modeling Approach
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Evaluation
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Data
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EUA, natural gas, Brent crude oil, coal
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March 15, 2010, until October 14, 2022
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Data was normalized w.r.t. \(\text{CO}_2\) emissions
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Emission-adjusted prices reflects one tonne of \(\text{CO}_2\)
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We adjusted for inflation by Eurostat’s HICP, excluding energy
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Log transformation of the data to stabilize the variance
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ADF Test: All series are stationary in first differences
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Johansen’s likelihood ratio trace test suggests two cointegrating relationships (levels)
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Johansen’s likelihood ratio trace test suggests no cointegrating relationships (logs)
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Data
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Modeling Approach: Overview
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VECM: Vector Error Correction Model
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Modeling the expectaion
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Captures the long-run cointegrating relationship
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Different cointegrating ranks, including rank zero (no cointegration)
\(z_{i,t}\) denotes the \(i\)-th standardized residual from time series \(i\) at time point \(t\)
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\(\Lambda(\cdot)\) is a link function - ensures that \(\Xi_{t}\) is a valid variance covariance matrix - ensures that \(\Xi_{t}\) does not exceed its support space and remains semi-positive definite
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Maximum Likelihood Estimation
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All parameters can be estimated jointly. Using conditional independence: \[\begin{align*}
+ L = f_{X_1} \prod_{i=2}^T f_{X_i|\mathcal{F}_{i-1}},
+\end{align*}\] with multivariate conditional density: \[\begin{align*}
+ f_{\mathbf{X}_t}(\mathbf{x}_t | \mathcal{F}_{t-1}) = c\left[\mathbf{F}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu},
+ \boldsymbol{\lambda});\Xi_t, \Theta\right] \cdot \\ \prod_{i=1}^K f_{X_{i,t}}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu}, \boldsymbol{\lambda})
+\end{align*}\] The copula density \(c\) can be derived analytically.
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Study Design and Evaluation
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Rolling-window forecasting study
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3257 observations total
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Window size: 1000 days (~ four years)
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Forecasting 30-steps-ahead
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=> 2227 potential starting points
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We sample 250 to reduce computational cost
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We draw \(2^{12}= 2048\) trajectories from the joint predictive distribution
where \(\mathbf{x}_t\) is the observed \(K\)-dimensional realization and \(\tilde{\mathbf{X}}_t\), respectively \(\tilde{\mathbf{X}}_t'\) are independent random vectors distributed according to \(F\)
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For univariate cases the Energy Score becomes the Continuous Ranked Probability Score (CRPS)
\(\text{RW}^{\sigma, \rho}\): Random walk with constant volatility and correlation
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Univariate \(\text{ETS}^{\sigma}\) with constant volatility
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Vector ETS \(VES^{\sigma}\) with constant volatility
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Heteroscedasticity is a main driver of ES
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The VECM model without cointegration (essentially a VAR) is the best performing model in terms of ES overall
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For EUA, the ETS Benchmark is the best performing model in terms of ES
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CRPS solely evaluates the marginal distributions
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The cross-sectional dependence is ignored
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VES models deliver poor performance in short horizons
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For Oil prices the RW Benchmark can’t be oupterformed 30 steps ahead
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Both VECM models generally deliver good performance
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Improvement in CRPS of selected models relative to \(\textrm{RW}^{\sigma, \rho}_{}\) in % (higher = better). Colored according to the test statistic of a DM-Test comparing to \(\textrm{RW}^{\sigma, \rho}_{}\) (greener means lower test statistic i.e., better performance compared to \(\textrm{RW}^{\sigma, \rho}_{}\)).
RMSE measures the performance of the forecasts at their mean
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Some models beat the benchmarks at short horizons
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Conclusion: the Improvements seen before must be attributed to other parts of the multivariate probabilistic predictive distribution
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Improvement in RMSE score of selected models relative to \(\textrm{RW}^{\sigma, \rho}_{}\) in % (higher = better). Colored according to the test statistic of a DM-Test comparing to \(\textrm{RW}^{\sigma, \rho}_{}\) (greener means lower test statistic i.e., better performance compared to \(\textrm{RW}^{\sigma, \rho}_{}\)).
Won Western Power Distribution Competition Won Best-Student-Presentation Award
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References
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+Berrisch, J. (2025). Rcpptimer: Rcpp tic-toc timer with OpenMP support. arXiv Preprint arXiv:2501.15856, abs/2501.15856. DOI: 10.48550/arXiv.2501.15856
+
+
+Berrisch, J., Narajewski, M., & Ziel, F. (2023). High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236. DOI: 10.1016/j.egyai.2023.100236
+
+
+Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Modeling volatility and dependence of european carbon and energy prices. Finance Research Letters, 52, 103503. DOI: 10.1016/j.frl.2022.103503
+
+
+Berrisch, J., & Ziel, F. (2022). Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086. DOI: 10.1002/for.2853
+
+
+Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2, Part C), 105221. DOI: 10.1016/j.jeconom.2021.11.008
+
+
+Berrisch, J., & Ziel, F. (2024). Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. International Journal of Forecasting, 40(4), 1568–1586. DOI: 10.1016/j.ijforecast.2024.01.005
+
+
+Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games (pp. I–XII, 1–394). Cambridge university press. DOI: 10.1017/CBO9780511546921
+
+
+Gaillard, P., & Wintenberger, O. (2018). Efficient online algorithms for fast-rate regret bounds under sparsity. In S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, & R. Garnett (Eds.), Proceedings of the 32nd international conference on neural information processing systems (pp. 7026–7036). Cornell University. DOI: 10.48550/arxiv.1805.09174
+
+
+Gneiting, T. (2011). Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494), 746–762. DOI: 10.1198/jasa.2011.r10138
+
+
+Gneiting, T., & Raftery, A.E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477), 359–378. DOI: 10.1198/016214506000001437
+
+Johnson, N.L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John wiley & sons.
+
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+Li, Z., & Cao, J. (2022). General p-splines for non-uniform b-splines. arXiv Preprint. DOI: 10.48550/arXiv.2201.06808
+
+
+Marcjasz, G., Narajewski, M., Weron, R., & Ziel, F. (2023). Distributional neural networks for electricity price forecasting. Energy Economics, 125, 106843. DOI: 10.1016/j.eneco.2023.106843
+
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+Wintenberger, O. (2017). Optimal learning with bernstein online aggregation. Machine Learning, 106(1), 119–141. DOI: 10.1007/s10994-016-5592-6
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