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Add overview slides and cleanup bib

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Jonathan Berrisch
2025-05-31 12:08:20 +02:00
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@@ -98,11 +98,197 @@ col_orange <- "#ffa600"
col_yellow <- "#FCE135"
```
## Motivation
## Overview of the Thesis {transition="fade" transition-speed="slow"}
<table style="width: 100%; border-collapse: separate; border-spacing: 0 1em; border: none;">
<tr style="border: none;">
<td style="vertical-align: top; width: 2em; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. <em>Journal of Econometrics</em>, 237(2), 105221.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH20241568]. Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. <em>International Journal of Forecasting</em>, 40(4), 1568–1586.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. <em>arXiv preprint</em> arXiv:2407.08750.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. <em>Journal of Forecasting</em>, 41(6), 1065–1086.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. <em>Finance Research Letters</em>, 52, 103503.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. <em>Energy and AI</em>, 13, 100236.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. <em>arXiv preprint</em> arXiv:2501.15856.
</td>
</tr>
</table>
## Overview of the Thesis {transition="fade" transition-speed="slow"}
<table style="width: 100%; border-collapse: separate; border-spacing: 0 1em; border: none;">
<tr style="border: none;">
<td style="vertical-align: top; width: 2em; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. <em>Journal of Econometrics</em>, 237(2), 105221.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH20241568]. Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. <em>International Journal of Forecasting</em>, 40(4), 1568–1586.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. <em>arXiv preprint</em> arXiv:2407.08750.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. <em>Journal of Forecasting</em>, 41(6), 1065–1086.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. <em>Finance Research Letters</em>, 52, 103503.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. <em>Energy and AI</em>, 13, 100236.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. <em>arXiv preprint</em> arXiv:2501.15856.
</td>
</tr>
</table>
## Overview of the Thesis {transition="fade" transition-speed="slow"}
<table style="width: 100%; border-collapse: separate; border-spacing: 0 1em; border: none;">
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; width: 2em; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. <em>Journal of Econometrics</em>, 237(2), 105221.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@BERRISCH20241568]. Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. <em>International Journal of Forecasting</em>, 40(4), 1568–1586.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. <em>arXiv preprint</em> arXiv:2407.08750.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. <em>Journal of Forecasting</em>, 41(6), 1065–1086.
</td>
</tr>
<tr class = "greyed-out" style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. <em>Finance Research Letters</em>, 52, 103503.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. <em>Energy and AI</em>, 13, 100236.
</td>
</tr>
<tr style="border: none;">
<td style="vertical-align: top; border: none;">
<i class="fa fa-fw fa-newspaper"></i>
</td>
<td style="border: none;">
Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. <em>arXiv preprint</em> arXiv:2501.15856.
</td>
</tr>
</table>
# CRPS Learning
Berrisch, J., & Ziel, F. (2023). *Journal of Econometrics*, 237(2), 105221.
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. *Journal of Econometrics*, 237(2), 105221.
## Motivation
## Introduction
:::: {.columns}
@@ -482,7 +668,7 @@ EWA satisfies optimal selection convergence \eqref{eq_optp_select} in a determin
- Loss $\ell$ is exp-concave
- Learning-rate $\eta$ is chosen correctly
Those results can be converted to stochastic iid settings @kakade2008generalization, @gaillard2014second.
Those results can be converted to any stochastic setting @wintenberger2017optimal.
Optimal convex aggregation convergence \eqref{eq_optp_conv} can be satisfied by applying the kernel-trick:
@@ -532,8 +718,11 @@ Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what
## Almost Optimal Convergence
:::: {style="font-size: 90%;"}
`r fontawesome::fa("exclamation", fill = col_orange)` QL is convex, but not exp-concave `r fontawesome::fa("arrow-right", fill ="#000000")` 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
`r fontawesome::fa("exclamation", fill = col_orange)` QL is convex, but not exp-concave
`r fontawesome::fa("arrow-right", fill ="#000000")` 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
\begin{equation}
P\left( \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right) \leq C \log(\log(t)) \left(\sqrt{\frac{\log(K)}{t}} + \frac{\log(K)+x}{t}\right) \right) \geq
@@ -557,6 +746,8 @@ if $Y_t$ is bounded, the considered loss $\ell$ is convex $G$-Lipschitz and weak
`r fontawesome::fa("arrow-right", fill ="#000000")` We show that this holds for QL under feasible conditions.
:::
## Conditions + Lemma
@@ -564,34 +755,6 @@ if $Y_t$ is bounded, the considered loss $\ell$ is convex $G$-Lipschitz and weak
::: {.column width="48%"}
**A1**
For some $G>0$ it holds
for all $x_1,x_2\in \mathbb{R}$ and $t>0$ that
$$ | \ell(x_1, Y_t)-\ell(x_2, Y_t) | \leq G |x_1-x_2|$$
**A2** For some $\alpha>0$, $\beta\in[0,1]$ it holds
for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that
\begin{align*}
\mathbb{E}[
& \ell(x_1, Y_t)-\ell(x_2, Y_t) | \mathcal{F}_{t-1}] \leq \\
& \mathbb{E}[ \ell'(x_1, Y_t)(x_1 - x_2) |\mathcal{F}_{t-1}] \\
& +
\mathbb{E}\left[ \left. \left( \alpha(\ell'(x_1, Y_t)(x_1 - x_2))^{2}\right)^{1/\beta} \right|\mathcal{F}_{t-1}\right]
\end{align*}
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient.
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
**Lemma 1**
\begin{align}
@@ -617,6 +780,34 @@ QL is Lipschitz continuous:
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
**A1**
For some $G>0$ it holds
for all $x_1,x_2\in \mathbb{R}$ and $t>0$ that
$$ | \ell(x_1, Y_t)-\ell(x_2, Y_t) | \leq G |x_1-x_2|$$
**A2** For some $\alpha>0$, $\beta\in[0,1]$ it holds
for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that
\begin{align*}
\mathbb{E}[
& \ell(x_1, Y_t)-\ell(x_2, Y_t) | \mathcal{F}_{t-1}] \leq \\
& \mathbb{E}[ \ell'(x_1, Y_t)(x_1 - x_2) |\mathcal{F}_{t-1}] \\
& +
\mathbb{E}\left[ \left. \left( \alpha(\ell'(x_1, Y_t)(x_1 - x_2))^{2}\right)^{1/\beta} \right|\mathcal{F}_{t-1}\right]
\end{align*}
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient.
:::
::::
## Proposition + Theorem
@@ -639,7 +830,7 @@ $\mathcal{Q}_p'' = f.$
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.
Strong convexity with $\beta=1$ implies **A2** `r fontawesome::fa("check", fill ="#ffa600")` @gaillard2018efficient
Strong convexity with $\beta=1$ implies weak exp-concavity **A2** `r fontawesome::fa("check", fill ="#ffa600")` @gaillard2018efficient
:::
@@ -1092,30 +1283,6 @@ weights_preprocessed %>%
::::
## Possible Extensions
**Forgetting**
- Only taking part of the old cumulative regret into account
- Exponential forgetting of past regret
\begin{align*}
R_{t,k} & = R_{t-1,k}(1-\xi) + \ell(\widetilde{F}_{t},Y_i) - \ell(\widehat{F}_{t,k},Y_i) \label{eq_regret_forget}
\end{align*}
**Fixed Shares** @herbster1998tracking
- Adding fixed shares to the weights
- Shrinkage towards a constant solution
\begin{align*}
\widetilde{w}_{t,k} = \rho \frac{1}{K} + (1-\rho) w_{t,k}
\label{fixed_share_simple}.
\end{align*}
TODO: Move these to the multivariate slides
## Application Study
::: {.panel-tabset}