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Finalize CRPS + MCRS begin working on VolDep

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Jonathan Berrisch
2025-05-25 17:24:06 +02:00
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@@ -97,6 +97,8 @@ col_yellow <- "#FCE135"
# CRPS Learning
Berrisch, J., & Ziel, F. (2023). *Journal of Econometrics*, 237(2), 105221.
## Motivation
:::: {.columns}
@@ -1022,21 +1024,17 @@ The same simulation carried out for different algorithms (1000 runs):
## Study Forget
::::
## Simulation Study
:::: {.columns}
::: {.column width="48%"}
::: {.column width="38%"}
**New DGP:**
#### New DGP:
\begin{align}
Y_t & \sim \mathcal{N}\left(\frac{\sin(0.005 \pi t )}{2},\,1\right) \\
\widehat{X}_{t,1} & \sim \widehat{F}_{1} = \mathcal{N}(-1,\,1) \\
\widehat{X}_{t,2} & \sim \widehat{F}_{2} = \mathcal{N}(3,\,4) \label{eq_dgp_sim2}
\end{align}
\begin{align*}
Y_t &\sim \mathcal{N}\left(\frac{\sin(0.005 \pi t )}{2},\,1\right) \\
\widehat{X}_{t,1} &\sim \widehat{F}_{1} = \mathcal{N}(-1,\,1) \\
\widehat{X}_{t,2} &\sim \widehat{F}_{2} = \mathcal{N}(3,\,4)
\end{align*}
`r fontawesome::fa("arrow-right", fill ="#000000")` Changing optimal weights
@@ -1044,20 +1042,25 @@ The same simulation carried out for different algorithms (1000 runs):
`r fontawesome::fa("arrow-right", fill ="#000000")` No forgetting leads to long-term constant weights
:::
::: {.column width="2%"}
<center>
<img src="assets/crps_learning/forget.png">
</center>
:::
::: {.column width="48%"}
::: {.column width="4%"}
:::
::: {.column width="58%"}
### &nbsp;
**Weights of expert 2**
```{r, echo = FALSE, fig.width=7, fig.height=5, fig.align='center', cache = TRUE}
load("assets/crps_learning/changing_weights.rds")
mod_labs <- c("Optimum", "Pointwise", "Smooth", "Constant")
names(mod_labs) <- c("TOptimum", "Pointwise", "Smooth", "Constant")
load("assets/crps_learning/weights_preprocessed.rda")
mod_labs <- c("Optimum", "No Forget\nPointwise", "No Forget\nP-Smooth", "Forget\nPointwise", "Forget\nP-Smooth")
names(mod_labs) <- c("Optimum", "nf_ptw", "nf_psmth", "f_ptw", "f_psmth")
colseq <- c(grey(.99), "orange", "red", "purple", "blue", "darkblue", "black")
weights_preprocessed %>%
mutate(w = 1 - w) %>%
@@ -1084,19 +1087,10 @@ weights_preprocessed %>%
::::
## Simulation Results
The simulation using the new DGP carried out for different algorithms (1000 runs):
<center>
<img src="assets/crps_learning/algos_changing.gif">
</center>
::::
## Possible Extensions
:::: {.columns}
::: {.column width="48%"}
**Forgetting**
@@ -1117,34 +1111,20 @@ The simulation using the new DGP carried out for different algorithms (1000 runs
\label{fixed_share_simple}.
\end{align*}
:::
TODO: Move these to the multivariate slides
::: {.column width="2%"}
## Application Study
:::
::: {.panel-tabset}
::: {.column width="48%"}
**Non-Equidistant Knots**
- Non-equidistant spline-basis could be used
- Potentially improves the tail-behavior
- Destroys shrinkage towards constant
<center>
<img src="assets/crps_learning/uneven_grid.gif">
</center>
:::
::::
## Application Study: Overview
## Overview
:::: {.columns}
::: {.column width="29%"}
::: {style="font-size: 85%;"}
Data:
- Forecasting European emission allowances (EUA)
@@ -1160,6 +1140,8 @@ Tuning paramter grids:
- Smoothing Penalty: $\Lambda= \{0\}\cup \{2^x|x\in \{-4,-3.5,\ldots,12\}\}$
- Learning Rates: $\mathcal{E}= \{2^x|x\in \{-1,-0.5,\ldots,9\}\}$
::::
:::
::: {.column width="2%"}
@@ -1203,7 +1185,9 @@ overview
::::
## Application Study: Experts
## Experts
::: {style="font-size: 90%;"}
Simple exponential smoothing with additive errors (**ETS-ANN**):
@@ -1235,6 +1219,7 @@ ARIMA(0,1,0)-GARCH(1,1) with student-t errors (**I-GARCHt**):
Y_{t} = \mu + Y_{t-1} + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \sigma_t Z, \quad \sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \quad \text{and} \quad Z_t \sim t(0,1, \nu)
\end{align*}
::::
## Results
@@ -1242,6 +1227,8 @@ Y_{t} = \mu + Y_{t-1} + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \
## Significance
<br/>
```{r, echo = FALSE, fig.width=7, fig.height=5.5, fig.align='center', cache = TRUE, results='asis'}
load("assets/crps_learning/bernstein_application_study_estimations+learnings_rev1.RData")
@@ -1494,247 +1481,47 @@ weights %>%
::::
## Wrap-Up
:::: {.columns}
::: {.column width="48%"}
Potential Downsides:
- Pointwise optimization can induce quantile crossing
- Can be solved by sorting the predictions
Upsides:
- Pointwise learning outperforms the Naive solution significantly
- Online learning is much faster than batch methods
- Smoothing further improves the predictive performance
- Asymptotically not worse than the best convex combination
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
Important:
- The choice of the learning rate is crucial
- The loss function has to meet certain criteria
The [`r fontawesome::fa("github")` 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
- Accepts vectors for most parameters
- The best parameter combination is chosen online
- Implements
- Forgetting, Fixed Share
- Different loss functions + gradients
:::
::::
<!-- :::: {.notes}
Execution Times:
T = 5000
Opera:
Ml-Poly > 157 ms
Boa > 212 ms
Profoc:
Ml-Poly > 17
BOA > 16 -->
# Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices
---
Berrisch, J., & Ziel, F. (2024). *International Journal of Forecasting*, 40(4), 1568-1586.
## Outline
</br>
**Multivariate CRPS Learning**
- Introduction
- Smoothing procedures
- Application to multivariate electricity price forecasts
**The `profoc` R package**
- Package overview
- Implementation details
- Illustrative examples
## The Framework of Prediction under Expert Advice
### The sequential framework
:::: {.columns}
::: {.column width="48%"}
Each day, $t = 1, 2, ... T$
- The **forecaster** receives predictions $\widehat{X}_{t,k}$ from $K$ **experts**
- The **forecaster** assings weights $w_{t,k}$ to each **expert**
- The **forecaster** calculates her prediction:
$$\widetilde{X}_{t}=\sum_{k=1}^K w_{t,k}\widehat{X}_{t,k}$$
- The realization for $t$ is observed
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
- The experts can be institutions, persons, or models
- The forecasts can be point-forecasts (i.e., mean or median) or full predictive distributions
- We do not need any assumptions concerning the underlying data
- @cesa2006prediction
:::
::::
## The Regret
Weights are updated sequentially according to the past performance of the $K$ experts.
`r fontawesome::fa("arrow-right", fill ="#000000")` A loss function $\ell$ is needed (to compute the **cumulative regret** $R_{t,k}$)
\begin{equation}
R_{t,k} = \widetilde{L}_{t} - \widehat{L}_{t,k} = \sum_{i = 1}^t \ell(\widetilde{X}_{i},Y_i) - \ell(\widehat{X}_{i,k},Y_i)
\label{eq_regret}
\end{equation}
The cumulative regret:
- Indicates the predictive accuracy of expert $k$ until time $t$.
- Measures how much the forecaster *regrets* not having followed the expert's advice
Popular loss functions for point forecasting @gneiting2011making:
:::: {.columns}
::: {.column width="48%"}
- $\ell_2$-loss $\ell_2(x, y) = | x -y|^2$
- optimal for mean prediction
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
- $\ell_1$-loss $\ell_1(x, y) = | x -y|$
- optimal for median predictions
:::
::::
---
:::: {.columns}
::: {.column width="48%"}
### Probabilistic Setting
An appropriate loss:
\begin{align*}
\text{CRPS}(F, y) & = \int_{\mathbb{R}} {(F(x) - \mathbb{1}\{ x > y \})}^2 dx
\label{eq_crps}
\end{align*}
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:
\begin{align*}
\text{CRPS}(F, y) = 2 \int_0^{1} \text{QL}_p(F^{-1}(p), y) \, d p.
\label{eq_crps_qs}
\end{align*}
... to combine quantiles of the probabilistic forecasts individually using the quantile-loss QL.
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
### Optimal Convergence
</br>
`r fontawesome::fa("exclamation", fill = col_orange)` exp-concavity of the loss is required for *selection* and *convex aggregation* properties
`r fontawesome::fa("exclamation", fill = col_orange)` QL is convex, but not exp-concave
`r fontawesome::fa("arrow-right", fill ="#000000")` The Bernstein Online Aggregation (BOA) lets us weaken the exp-concavity condition.
Convergence rates of BOA are:
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t *selection* @gaillard2018efficient.
`r fontawesome::fa("arrow-right", fill ="#000000")` Almost optimal w.r.t *convex aggregation* @wintenberger2017optimal.
:::
::::
## Multivariate CRPS Learning
:::: {.columns}
::: {.column width="48%"}
Additionally, we extend the **B-Smooth** and **P-Smooth** procedures to the multivariate setting:
- Basis matrices for reducing
- - the probabilistic dimension from $P$ to $\widetilde P$
- - the multivariate dimension from $D$ to $\widetilde D$
::: {.column width="45%"}
- Hat matrices
- - penalized smoothing across P and D dimensions
We extend the **B-Smooth** and **P-Smooth** procedures to the multivariate setting:
We utilize the mean Pinball Score over the entire space for hyperparameter optimization (e.g, $\lambda$)
::: {.panel-tabset}
:::
## Penalized Smoothing
::: {.column width="2%"}
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 $(0,1)$:
:::
\begin{equation*}
\boldsymbol w_{t,k} = \boldsymbol{\psi}^{\text{mv}} \boldsymbol{b}_{t,k} {\boldsymbol{\psi}^{pr}}'
\end{equation*}
::: {.column width="48%"}
with parameter matix $\boldsymbol b_{t,k}$. The latter is estimated to penalize $L_2$-smoothing which minimizes
*Basis Smoothing*
\begin{align}
& \| \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 \\
& \| \boldsymbol{\beta}_{t, p, k}' \boldsymbol{\varphi}^{\text{mv}} - \boldsymbol b_{t, p, k}' \boldsymbol{\psi}^{\text{mv}} \|^2_2 + \lambda^{\text{mv}} \| \mathcal{D}_{q} (\boldsymbol b_{t, p, k}' \boldsymbol{\psi}^{\text{mv}}) \|^2_2 \nonumber
\end{align}
Represent weights as linear combinations of bounded basis functions:
with differential operator $\mathcal{D}_q$ of order $q$
[{{< fa calculator >}}]{style="color:var(--col_green_10);"} We have an analytical solution.
## Basis Smoothing
Linear combinations of bounded basis functions:
\begin{equation}
\underbrace{\boldsymbol w_{t,k}}_{D \text{ x } P} = \sum_{j=1}^{\widetilde D} \sum_{l=1}^{\widetilde P} \beta_{t,j,l,k} \varphi^{\text{mv}}_{j} \varphi^{\text{pr}}_{l} = \underbrace{\boldsymbol \varphi^{\text{mv}}}_{D\text{ x }\widetilde D} \boldsymbol \beta_{t,k} \underbrace{{\boldsymbol\varphi^{\text{pr}}}'}_{\widetilde P \text{ x }P} \nonumber
@@ -1750,42 +1537,15 @@ If $\widetilde P = P$ it holds that $\boldsymbol \varphi^{pr} = \boldsymbol{I}$
For $\widetilde P = 1$ we receive constant weights
:::
::::
## Multivariate CRPS Learning
:::: {.columns}
::: {.column width="48%"}
**Penalized smoothing:**
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 $(0,1)$:
\begin{equation}
\boldsymbol w_{t,k} = \boldsymbol{\psi}^{\text{mv}} \boldsymbol{b}_{t,k} {\boldsymbol{\psi}^{pr}}'
\end{equation}
with parameter matix $\boldsymbol b_{t,k}$. The latter is estimated to penalize $L_2$-smoothing which minimizes
\begin{align}
& \| \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 \\
& \| \boldsymbol{\beta}_{t, p, k}' \boldsymbol{\varphi}^{\text{mv}} - \boldsymbol b_{t, p, k}' \boldsymbol{\psi}^{\text{mv}} \|^2_2 + \lambda^{\text{mv}} \| \mathcal{D}_{q} (\boldsymbol b_{t, p, k}' \boldsymbol{\psi}^{\text{mv}}) \|^2_2 \nonumber
\end{align}
with differential operator $\mathcal{D}_q$ of order $q$
Computation is easy since we have an analytical solution.
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
::: {.column width="53%"}
```{r, fig.align="center", echo=FALSE, out.width = "1000px", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/algorithm.svg")
@@ -1841,63 +1601,6 @@ Computation Time: ~30 Minutes
::::
## Special Cases
:::: {.columns}
::: {.column width="48%"}
::: {.panel-tabset}
## Constant
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant.svg")
```
## Constant PR
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant_pr.svg")
```
## Constant MV
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant_mv.svg")
```
::::
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
::: {.panel-tabset}
## Pointwise
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/pointwise.svg")
```
## Smooth
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/smooth_best.svg")
```
::::
:::
::::
## Results
:::: {.columns}
@@ -2040,7 +1743,41 @@ table_performance %>%
::: {.column width = "45%"}
Foo
<br/>
::: {.panel-tabset}
## Constant
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant.svg")
```
## Pointwise
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/pointwise.svg")
```
## B Constant PR
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant_pr.svg")
```
## B Constant MV
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/constant_mv.svg")
```
## Smooth.Forget
```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
knitr::include_graphics("assets/mcrps_learning/smooth_best.svg")
```
::::
:::
@@ -2048,7 +1785,11 @@ Foo
## Results
```{r, warning=FALSE, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
::: {.panel-tabset}
## Chosen Parameters
```{r, warning=FALSE, fig.align="center", echo=FALSE, fig.width=12, fig.height=5.5, cache = TRUE}
load("assets/mcrps_learning/pars_data.rds")
pars_data %>%
ggplot(aes(x = dates, y = value)) +
@@ -2085,9 +1826,9 @@ pars_data %>%
)
```
## Results: Hour 16:00-17:00
## Weights: Hour 16:00-17:00
```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=5.5, cache = TRUE}
load("assets/mcrps_learning/weights_h.rds")
weights_h %>%
ggplot(aes(date, q, fill = weight)) +
@@ -2125,9 +1866,9 @@ weights_h %>%
scale_y_continuous(breaks = c(0.1, 0.5, 0.9))
```
## Results: Median
## Weights: Median
```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=5.5, cache = TRUE}
load("assets/mcrps_learning/weights_q.rds")
weights_q %>%
mutate(hour = as.numeric(hour) - 1) %>%
@@ -2166,51 +1907,9 @@ weights_q %>%
scale_y_continuous(breaks = c(0, 8, 16, 24))
```
## Profoc R Package
:::: {.columns}
::: {.column width="48%"}
### Probabilistic Forecast Combination - profoc
Available on [Github](https://github.com/BerriJ/profoc) and [CRAN](https://CRAN.R-project.org/package=profoc)
Main Function: `online()` for online learning.
- Works with multivariate and/or probabilistic data
- Implements BOA, ML-POLY, EWA (and the gradient versions)
- Implements many extensions like smoothing, forgetting, thresholding, etc.
- Various loss functions are available
- Various methods (`predict`, `update`, `plot`, etc.)
:::
::: {.column width="2%"}
:::
::: {.column width="48%"}
### Speed
Large parts of profoc are implemented in C++.
<center>
<img src="assets/mcrps_learning/profoc_langs.png">
</center>
We use `Rcpp`, `RcppArmadillo`, and OpenMP.
We use `Rcpp` modules to expose a class to R
- Offers great flexibility for the end-user
- Requires very little knowledge of C++ code
- High-Level interface is easy to use
:::
::::
## Profoc - B-Spline Basis
## Non-Equidistant Knots
::: {.panel-tabset}
@@ -2315,8 +2014,8 @@ chart = {
});
// Build SVG
const width = 800;
const height = 400;
const width = 1200;
const height = 450;
const margin = {top: 40, right: 20, bottom: 40, left: 40};
const innerWidth = width - margin.left - margin.right;
const innerHeight = height - margin.top - margin.bottom;
@@ -2347,15 +2046,6 @@ chart = {
.attr("preserveAspectRatio", "xMidYMid meet")
.attr("style", "max-width: 100%; height: auto;");
// Add chart title
// svg.append("text")
// .attr("class", "chart-title")
// .attr("x", width / 2)
// .attr("y", 20)
// .attr("text-anchor", "middle")
// .attr("font-size", "20px")
// .attr("font-weight", "bold");
// Create the chart group
const g = svg.append("g")
.attr("transform", `translate(${margin.left},${margin.top})`);
@@ -2372,20 +2062,6 @@ chart = {
.call(d3.axisLeft(y).ticks(5))
.style("font-size", "20px");
// Add axis labels
// g.append("text")
// .attr("x", innerWidth / 2)
// .attr("y", innerHeight + 35)
// .attr("text-anchor", "middle")
// .text("x");
// g.append("text")
// .attr("transform", "rotate(-90)")
// .attr("x", -innerHeight / 2)
// .attr("y", -30)
// .attr("text-anchor", "middle")
// .text("y");
// Add a horizontal line at y = 0
g.append("line")
.attr("x1", 0)
@@ -2482,23 +2158,27 @@ TODO: Add actual algorithm to backup slides
## Wrap-Up
:::: {.columns}
::: {.column width="48%"}
The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Package:
[{{< fa triangle-exclamation >}}]{style="color:var(--col_red_9);"} Potential Downsides:
Profoc is a flexible framework for online learning.
- Pointwise optimization can induce quantile crossing
- Can be solved by sorting the predictions
- It implements several algorithms
- It implements several loss functions
- It implements several extensions
- Its high- and low-level interfaces offer great flexibility
[{{< fa magnifying-glass >}}]{style="color:var(--col_orange_9);"} Important:
Profoc is fast.
- The choice of the learning rate is crucial
- The loss function has to meet certain criteria
- The core components are written in C++
- The core components utilize OpenMP for parallelization
[{{< fa rocket >}}]{style="color:var(--col_green_9);"} Upsides:
- Pointwise learning outperforms the Naive solution significantly
- Online learning is much faster than batch methods
- Smoothing further improves the predictive performance
- Asymptotically not worse than the best convex combination
:::
@@ -2508,17 +2188,21 @@ Profoc is fast.
::: {.column width="48%"}
Multivariate Extension:
The [`r fontawesome::fa("github")` profoc](https://profoc.berrisch.biz/) R Package:
- Code is available now
- [Pre-Print](https://arxiv.org/abs/2303.10019) is available now
- Implements all algorithms discussed above
- Is written using RcppArmadillo `r fontawesome::fa("arrow-right", fill ="#000000")` its fast
- Accepts vectors for most parameters
- The best parameter combination is chosen online
- Implements
- Forgetting, Fixed Share
- Different loss functions + gradients
Get these slides:
Pubications:
<center>
<img src="assets/mcrps_learning/web_pres.png">
</center>
[https://berrisch.biz/slides/23_06_ecmi/](https://berrisch.biz/slides/23_06_ecmi/)
[{{< fa newspaper >}}]{style="color:var(--col_grey_7);"} 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.
:::
@@ -2526,7 +2210,7 @@ Get these slides:
# Modeling Volatility and Dependence of European Carbon and Energy Prices
TODO: Add Reference
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). *Finance Research Letters*, 52, 103503.
---