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Jonathan Berrisch
2025-06-01 10:56:07 +02:00
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@@ -614,7 +614,7 @@ The Idea:
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@@ -784,7 +784,7 @@ Each day, $t = 1, 2, ... T$
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@@ -884,7 +884,7 @@ Du kannst dann auch easy darauf verweisen: \ref{eq:exp_combination}.
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@@ -928,7 +928,7 @@ The forecaster is asymptotically not worse than the best convex combination $\wi
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@@ -1005,7 +1005,7 @@ It's strictly proper @gneiting2007strictly.
Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what if the experts' performance varies in parts of the distribution? Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what if the experts' performance varies in parts of the distribution?
`r fontawesome::fa("lightbulb", fill = col_yellow)` Utilize this relation: <i class="fa fa-fw fa-lightbulb" style="color:var(--col_yellow_9);"></i> Utilize this relation:
\begin{equation*} \begin{equation*}
\text{CRPS}(F, y) = 2 \int_0^{1} \text{QL}_p(F^{-1}(p), y) dp.\label{eq_crps_qs} \text{CRPS}(F, y) = 2 \int_0^{1} \text{QL}_p(F^{-1}(p), y) dp.\label{eq_crps_qs}
@@ -1025,9 +1025,9 @@ Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what
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`r fontawesome::fa("exclamation", fill = col_orange)` QL is convex, but not exp-concave <i class="fa fa-fw fa-exclamation" style="color:var(--col_orange_10);"></i> 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 <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> 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} \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 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
@@ -1035,7 +1035,7 @@ Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what
\label{eq_boa_opt_conv} \label{eq_boa_opt_conv}
\end{equation} \end{equation}
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *convex aggregation* \eqref{eq_optp_conv} @wintenberger2017optimal. <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Almost optimal w.r.t. *convex aggregation* \eqref{eq_optp_conv} @wintenberger2017optimal.
The same algorithm satisfies that there exist a $C>0$ such that for $x>0$ it holds that The same algorithm satisfies that there exist a $C>0$ such that for $x>0$ it holds that
\begin{equation} \begin{equation}
@@ -1047,9 +1047,9 @@ The same algorithm satisfies that there exist a $C>0$ such that for $x>0$ it hol
if $Y_t$ is bounded, the considered loss $\ell$ is convex $G$-Lipschitz and weak exp-concave in its first coordinate. if $Y_t$ is bounded, the considered loss $\ell$ is convex $G$-Lipschitz and weak exp-concave in its first coordinate.
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient. <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient.
`r fontawesome::fa("arrow-right", fill ="#000000")` We show that this holds for QL under feasible conditions. <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> We show that this holds for QL under feasible conditions.
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@@ -1075,17 +1075,17 @@ Pointwise can outperform constant procedures
QL is convex but not exp-concave: QL is convex but not exp-concave:
`r fontawesome::fa("arrow-right")` Almost optimal convergence w.r.t. *convex aggregation* \eqref{eq_boa_opt_conv} `r fontawesome::fa("check", fill ="#00b02f")` </br> <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Almost optimal convergence w.r.t. *convex aggregation* \eqref{eq_boa_opt_conv} <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> </br>
For almost optimal congerence w.r.t. *selection* \eqref{eq_boa_opt_select} we need to check **A1** and **A2**: For almost optimal congerence w.r.t. *selection* \eqref{eq_boa_opt_select} we need to check **A1** and **A2**:
QL is Lipschitz continuous: QL is Lipschitz continuous:
`r fontawesome::fa("arrow-right")` **A1** holds `r fontawesome::fa("check", fill ="#ffa600")` </br> <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> **A1** holds <i class="fa fa-fw fa-check" style="color:var(--col_orange_9);"></i>
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@@ -1109,7 +1109,7 @@ for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that
\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] \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*} \end{align*}
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient. <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient.
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@@ -1123,7 +1123,7 @@ for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that
Conditional quantile risk: $\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]$. Conditional quantile risk: $\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]$.
`r fontawesome::fa("arrow-right")` convexity properties of $\mathcal{Q}_p$ depend on the <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> convexity properties of $\mathcal{Q}_p$ depend on the
conditional distribution $Y_t|\mathcal{F}_{t-1}$. conditional distribution $Y_t|\mathcal{F}_{t-1}$.
**Proposition 1** **Proposition 1**
@@ -1135,17 +1135,17 @@ $\mathcal{Q}_p'' = f.$
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 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. is $\mathcal{Q}_p$ is $\gamma$-strongly convex.
Strong convexity with $\beta=1$ implies weak exp-concavity **A2** `r fontawesome::fa("check", fill ="#ffa600")` @gaillard2018efficient Strong convexity with $\beta=1$ implies weak exp-concavity **A2** <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> @gaillard2018efficient
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`r fontawesome::fa("arrow-right")` **A1** and **A2** give us almost optimal convergence w.r.t. selection \eqref{eq_boa_opt_select} `r fontawesome::fa("check", fill ="#00b02f")` </br> <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> **A1** and **A2** give us almost optimal convergence w.r.t. selection \eqref{eq_boa_opt_select} <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> </br>
**Theorem 1** **Theorem 1**
@@ -1198,7 +1198,7 @@ Simple Example:
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@@ -1361,14 +1361,14 @@ $\boldsymbol \beta_{t,k}$ is calculated using a reduced regret matrix:
\underbrace{\boldsymbol r_{t}}_{\text{LxK}} = \frac{L}{P} \underbrace{\boldsymbol B'}_{\text{LxP}} \underbrace{\left({\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widetilde{\boldsymbol X}_{t},Y_t)- {\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widehat{\boldsymbol X}_{t},Y_t)\right)}_{\text{PxK}} \underbrace{\boldsymbol r_{t}}_{\text{LxK}} = \frac{L}{P} \underbrace{\boldsymbol B'}_{\text{LxP}} \underbrace{\left({\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widetilde{\boldsymbol X}_{t},Y_t)- {\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widehat{\boldsymbol X}_{t},Y_t)\right)}_{\text{PxK}}
\end{equation} \end{equation}
`r fontawesome::fa("arrow-right", fill ="#000000")` $\boldsymbol r_{t}$ is transformed from PxK to LxK <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> $\boldsymbol r_{t}$ is transformed from PxK to LxK
If $L = P$ it holds that $\boldsymbol \varphi = \boldsymbol{I}$ If $L = P$ it holds that $\boldsymbol \varphi = \boldsymbol{I}$
For $L = 1$ we receive constant weights For $L = 1$ we receive constant weights
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@@ -1481,7 +1481,7 @@ Data Generating Process of the [simple probabilistic example](#simple_example):
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@@ -1535,11 +1535,11 @@ The same simulation carried out for different algorithms (1000 runs):
\widehat{X}_{t,2} &\sim \widehat{F}_{2} = \mathcal{N}(3,\,4) \widehat{X}_{t,2} &\sim \widehat{F}_{2} = \mathcal{N}(3,\,4)
\end{align*} \end{align*}
`r fontawesome::fa("arrow-right", fill ="#000000")` Changing optimal weights <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Changing optimal weights
`r fontawesome::fa("arrow-right", fill ="#000000")` Single run example depicted aside <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> Single run example depicted aside
`r fontawesome::fa("arrow-right", fill ="#000000")` No forgetting leads to long-term constant weights <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> No forgetting leads to long-term constant weights
<center> <center>
<img src="assets/crps_learning/forget.png"> <img src="assets/crps_learning/forget.png">
@@ -2082,7 +2082,7 @@ knitr::include_graphics("assets/mcrps_learning/algorithm.svg")
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@@ -2651,7 +2651,7 @@ TODO: Add actual algorithm to backup slides
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The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Package: The [<i class="fa-brands fa-fw fa-github" style="color:var(--col_grey_10);"></i> profoc](https://profoc.berrisch.biz/) R Package:
- Implements all algorithms discussed above - Implements all algorithms discussed above
- Is written using RcppArmadillo `r fontawesome::fa("arrow-right", fill ="#000000")` its fast - Is written using RcppArmadillo <i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> its fast
- Accepts vectors for most parameters - Accepts vectors for most parameters
- The best parameter combination is chosen online - The best parameter combination is chosen online
- Implements - Implements
@@ -2708,9 +2708,9 @@ The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Packa
Pubications: Pubications:
[{{< fa newspaper >}}]{style="color:var(--col_grey_7);"} Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. *Journal of Econometrics*, 237(2), 105221. <i class="fa fa-fw fa-newspaper" style="color:var(--col_grey_10);"></i> Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. *Journal of Econometrics*, 237(2), 105221.
[{{< fa newspaper >}}]{style="color:var(--col_grey_7);"} Berrisch, J., & Ziel, F. [-@BERRISCH20241568]. Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. *International Journal of Forecasting*, 40(4), 1568-1586. <i class="fa fa-fw fa-newspaper" style="color:var(--col_grey_10);"></i> Berrisch, J., & Ziel, F. [-@BERRISCH20241568]. Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices. *International Journal of Forecasting*, 40(4), 1568-1586.
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@@ -2751,7 +2751,7 @@ Several Questions arise:
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@@ -2935,7 +2935,7 @@ $$\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}$$
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@@ -3476,10 +3476,12 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
<img src="assets/voldep/frame.png"> <img src="assets/voldep/frame.png">
</center> </center>
`r fontawesome::fa("newspaper")` @berrisch2023modeling <i class="fa fa-fw fa-newspaper" style="color:var(--col_grey_10);"></i> @berrisch2023modeling
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## Conclusion
## References ## References