Fix typos throughout the document
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28
index.qmd
28
index.qmd
@@ -400,7 +400,7 @@ col_yellow <- "#FCE135"
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Reduces estimation time by 2-3 orders of magnitude
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Maintainins competitive forecasting accuracy
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Maintains competitive forecasting accuracy
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Real-World Validation in Energy Markets
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@@ -1059,7 +1059,7 @@ chart = {
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Each day, $t = 1, 2, ... T$
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- The **forecaster** receives predictions $\widehat{X}_{t,k}$ from $K$ **experts**
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- The **forecaster** assings weights $w_{t,k}$ to each **expert**
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- The **forecaster** assigns weights $w_{t,k}$ to each **expert**
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- The **forecaster** calculates her prediction:
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\begin{equation}
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\widetilde{X}_{t} = \sum_{k=1}^K w_{t,k} \widehat{X}_{t,k}.
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@@ -1230,7 +1230,7 @@ Optimal rates with respect to selection \eqref{eq_opt_select} and convex aggrega
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\label{eq_optp_conv}
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\end{align}
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Algorithms can statisfy both \eqref{eq_optp_select} and \eqref{eq_optp_conv} depending on:
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Algorithms can satisfy both \eqref{eq_optp_select} and \eqref{eq_optp_conv} depending on:
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- The loss function
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- Regularity conditions on $Y_t$ and $\widehat{X}_{t,k}$
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@@ -1312,7 +1312,7 @@ Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what
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<i class="fa fa-fw fa-triangle-exclamation" style="color:var(--col_amber_9);"></i> QL is convex, but not exp-concave
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<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
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<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 exists a $C>0$ such that for $x>0$ it holds that
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\begin{equation}
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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
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@@ -1324,7 +1324,7 @@ if the loss function is convex.
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<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.
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The same algorithm satisfies that there exist a $C>0$ such that for $x>0$ it holds that
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The same algorithm satisfies that there exists a $C>0$ such that for $x>0$ it holds that
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\begin{equation}
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P\left( \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \leq
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C\left(\frac{\log(K)+\log(\log(Gt))+ x}{\alpha t}\right)^{\frac{1}{2-\beta}} \right) \geq
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@@ -1362,7 +1362,7 @@ Pointwise can outperform constant procedures
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<i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> $\text{QL}$ is convex: 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>
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For almost optimal congerence w.r.t. *selection* \eqref{eq_boa_opt_select} we need:
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For almost optimal convergence w.r.t. *selection* \eqref{eq_boa_opt_select} we need:
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**A1: Lipschitz Continuity**
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@@ -1443,7 +1443,7 @@ The gradient based fully adaptive Bernstein online aggregation (BOAG) applied po
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$$\widehat{\mathcal{R}}_{t,\pi} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\pi}.$$
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If $Y_t|\mathcal{F}_{t-1}$ is bounded
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and has a pdf $f_t$ satifying $f_t>\gamma >0$ on its
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and has a pdf $f_t$ satisfying $f_t>\gamma >0$ on its
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support $\text{spt}(f_t)$ then \eqref{eq_boa_opt_select} holds with $\beta=1$ and
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$$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\min}$$
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@@ -1684,7 +1684,7 @@ Weights converge to the constant solution if $L\rightarrow 1$
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### Initialization:
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Array of expert predicitons: $\widehat{X}_{t,p,k}$
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Array of expert predictions: $\widehat{X}_{t,p,k}$
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Vector of Prediction targets: $Y_t$
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@@ -1908,7 +1908,7 @@ Combination methods:
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::: {.column width="69%"}
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Tuning paramter grids:
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Tuning parameter grids:
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- Smoothing Penalty: $\Lambda= \{0\}\cup \{2^x|x\in \{-4,-3.5,\ldots,12\}\}$
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- Learning Rates: $\mathcal{E}= \{2^x|x\in \{-1,-0.5,\ldots,9\}\}$
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@@ -2301,7 +2301,7 @@ Let $\boldsymbol{\psi}^{\text{mv}}=(\psi_1,\ldots, \psi_{D})$ and $\boldsymbol{\
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\boldsymbol w_{t,k} = \boldsymbol{\psi}^{\text{mv}} \boldsymbol{b}_{t,k} {\boldsymbol{\psi}^{pr}}'
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\end{equation*}
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with parameter matix $\boldsymbol b_{t,k}$. The latter is estimated to penalize $L_2$-smoothing which minimizes
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with parameter matrix $\boldsymbol b_{t,k}$. The latter is estimated to penalize $L_2$-smoothing which minimizes
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\begin{align}
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& \| \boldsymbol{\beta}_{t,d, k}' \boldsymbol{\varphi}^{\text{pr}} - \boldsymbol b_{t, d, k}' \boldsymbol{\psi}^{\text{pr}} \|^2_2 + \lambda^{\text{pr}} \| \mathcal{D}_{q} (\boldsymbol b_{t, d, k}' \boldsymbol{\psi}^{\text{pr}}) \|^2_2 + \nonumber \\
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@@ -3168,7 +3168,7 @@ It holds that:
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with: $\boldsymbol{u}_t =(u_{1,t},\ldots, u_{K,t})^\intercal$, $u_{k,t} = F_{X_{k,t}|\mathcal{F}_{t-1}}(x_{k,t})$
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For brewity we drop the conditioning on $\mathcal{F}_{t-1}$.
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For brevity we drop the conditioning on $\mathcal{F}_{t-1}$.
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The model can be specified as follows
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@@ -3267,7 +3267,7 @@ $\Lambda(\cdot)$ is a link function:
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### Estimation
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Joint maximum lieklihood estimation:
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Joint maximum likelihood estimation:
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\begin{align*}
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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},
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@@ -3417,7 +3417,7 @@ table_energy %>%
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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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- For Oil prices the RW Benchmark can't be outperformed 30 steps ahead
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- Both VECM models generally deliver good performance
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:::
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@@ -3742,7 +3742,7 @@ plot_quant_data %>% ggplot(aes(x = date, y = value)) +
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Accounting for heteroscedasticity or stabilizing the variance via log transformation is crucial for good performance in terms of ES
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- Price dynamics emerged way before the russian invaion into ukraine
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- Price dynamics emerged way before the Russian invasion into Ukraine
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- Linear dependence between the series reacted only right after the invasion
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- Improvements in forecasting performance is mainly attributed to:
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- the tails
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