diff --git a/index.qmd b/index.qmd index e6a0dad..9a48607 100644 --- a/index.qmd +++ b/index.qmd @@ -614,7 +614,7 @@ The Idea: ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -784,7 +784,7 @@ Each day, $t = 1, 2, ... T$ ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -884,7 +884,7 @@ Du kannst dann auch easy darauf verweisen: \ref{eq:exp_combination}. ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -928,7 +928,7 @@ The forecaster is asymptotically not worse than the best convex combination $\wi ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -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? -`r fontawesome::fa("lightbulb", fill = col_yellow)` Utilize this relation: + Utilize this relation: \begin{equation*} \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 :::: {style="font-size: 90%;"} -`r fontawesome::fa("exclamation", fill = col_orange)` QL is convex, but not exp-concave + 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 + 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 @@ -1035,7 +1035,7 @@ Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what \label{eq_boa_opt_conv} \end{equation} -`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *convex aggregation* \eqref{eq_optp_conv} @wintenberger2017optimal. + 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 \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. -`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient. + 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. + We show that this holds for QL under feasible conditions. ::: @@ -1075,17 +1075,17 @@ Pointwise can outperform constant procedures 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")`
+ Almost optimal convergence w.r.t. *convex aggregation* \eqref{eq_boa_opt_conv}
For almost optimal congerence w.r.t. *selection* \eqref{eq_boa_opt_select} we need to check **A1** and **A2**: QL is Lipschitz continuous: -`r fontawesome::fa("arrow-right")` **A1** holds `r fontawesome::fa("check", fill ="#ffa600")`
+ **A1** holds ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -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] \end{align*} -`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient. + Almost optimal w.r.t. *selection* \eqref{eq_optp_select} @gaillard2018efficient. ::: @@ -1123,7 +1123,7 @@ for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that Conditional quantile risk: $\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]$. -`r fontawesome::fa("arrow-right")` convexity properties of $\mathcal{Q}_p$ depend on the + convexity properties of $\mathcal{Q}_p$ depend on the conditional distribution $Y_t|\mathcal{F}_{t-1}$. **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 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** @gaillard2018efficient ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: ::: {.column width="48%"} -`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")`
+ **A1** and **A2** give us almost optimal convergence w.r.t. selection \eqref{eq_boa_opt_select}
**Theorem 1** @@ -1198,7 +1198,7 @@ Simple Example: ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -1325,7 +1325,7 @@ Computation is easy, since we have an analytical solution ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -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}} \end{equation} -`r fontawesome::fa("arrow-right", fill ="#000000")` $\boldsymbol r_{t}$ is transformed from PxK to LxK + $\boldsymbol r_{t}$ is transformed from PxK to LxK If $L = P$ it holds that $\boldsymbol \varphi = \boldsymbol{I}$ For $L = 1$ we receive constant weights ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -1481,7 +1481,7 @@ Data Generating Process of the [simple probabilistic example](#simple_example): ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -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) \end{align*} -`r fontawesome::fa("arrow-right", fill ="#000000")` Changing optimal weights + Changing optimal weights -`r fontawesome::fa("arrow-right", fill ="#000000")` Single run example depicted aside + Single run example depicted aside -`r fontawesome::fa("arrow-right", fill ="#000000")` No forgetting leads to long-term constant weights + No forgetting leads to long-term constant weights
@@ -2082,7 +2082,7 @@ knitr::include_graphics("assets/mcrps_learning/algorithm.svg") ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -2651,7 +2651,7 @@ TODO: Add actual algorithm to backup slides ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -2690,16 +2690,16 @@ TODO: Add actual algorithm to backup slides ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: ::: {.column width="48%"} -The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Package: +The [ profoc](https://profoc.berrisch.biz/) R Package: - Implements all algorithms discussed above -- Is written using RcppArmadillo `r fontawesome::fa("arrow-right", fill ="#000000")` its fast +- Is written using RcppArmadillo its fast - Accepts vectors for most parameters - The best parameter combination is chosen online - Implements @@ -2708,9 +2708,9 @@ The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Packa Pubications: -[{{< fa newspaper >}}]{style="color:var(--col_grey_7);"} Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. *Journal of Econometrics*, 237(2), 105221. + 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. + 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. ::: @@ -2751,7 +2751,7 @@ Several Questions arise: ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -2879,7 +2879,7 @@ Sklars theorem: decompose target into ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -2935,7 +2935,7 @@ $$\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}$$ ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -2991,7 +2991,7 @@ $\Lambda(\cdot)$ is a link function ::: -::: {.column width="2%"} +::: {.column width="4%"} ::: @@ -3476,10 +3476,12 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
-`r fontawesome::fa("newspaper")` @berrisch2023modeling + @berrisch2023modeling ::: :::: +## Conclusion + ## References \ No newline at end of file