---
title: "Data Science Methods for Forecasting in Energy and Economics"
date: 2025-07-10
author: 
    - name: Jonathan Berrisch
      affiliations:
        - ref: hemf
affiliations: 
    - id: hemf
      name: University of Duisburg-Essen, House of Energy Markets and Finance
format: 
    revealjs:
        embed-resources: true
        footer: ""
        logo: logos_combined.png
        theme: [default, clean.scss]
        smaller: true
        fig-format: svg
execute:
  daemon: false
highlight-style: github
---

## Outline

::: {.hidden}
$$
\newcommand{\A}{{\mathbb A}}
$$
:::

<br>

::: {style="font-size: 150%;"}

[{{< fa bars-staggered >}}]{style="color: #404040;"} &ensp; Introduction & Research Motivation

[{{< fa bars-staggered >}}]{style="color: #404040;"} &ensp; Overview of the Thesis

[{{< fa table >}}]{style="color: #404040;"} &ensp; Online Learning

[{{< fa circle-nodes >}}]{style="color: #404040;"} &ensp; Probabilistic Forecasting of European Carbon and Energy Prices

[{{< fa lightbulb >}}]{style="color: #404040;"} &ensp; Limitations

[{{< fa binoculars >}}]{style="color: #404040;"} &ensp; Contributions & Outlook

:::

## PHD DeFence

```{r, setup, include=FALSE}
# Compile with: rmarkdown::render("crps_learning.Rmd")
library(latex2exp)
library(ggplot2)
library(dplyr)
library(tidyr)
library(purrr)
library(kableExtra)
knitr::opts_chunk$set(
  dev = "svglite" # Use svg figures
)
library(RefManageR)
BibOptions(
  check.entries = TRUE,
  bib.style = "authoryear",
  cite.style = "authoryear",
  style = "html",
  hyperlink = TRUE,
  dashed = FALSE
)
my_bib <- ReadBib("assets/library.bib", check = FALSE)
col_lightgray <- "#e7e7e7"
col_blue <- "#000088"
col_smooth_expost <- "#a7008b"
col_smooth <- "#187a00"
col_pointwise <- "#008790"
col_constant <- "#dd9002"
col_optimum <- "#666666"
```

```{r xaringan-panelset, echo=FALSE}
xaringanExtra::use_panelset()
```

```{r xaringanExtra-freezeframe, echo=FALSE}
xaringanExtra::use_freezeframe(responsive = TRUE)
```

# Outline

- [Motivation](#motivation)
- [The Framework of Prediction under Expert Advice](#pred_under_exp_advice)
- [The Continious Ranked Probability Scrore](#crps)
- [Optimality of (Pointwise) CRPS-Learning](#crps_optim)
- [A Simple Probabilistic Example](#simple_example)
- [The Proposed CRPS-Learning Algorithm](#proposed_algorithm)
- [Simulation Results](#simulation)
- [Possible Extensions](#extensions)
- [Application Study](#application)
- [Wrap-Up](#conclusion)
- [References](#references)

---

class: center, middle, sydney-blue

# Motivation

name: motivation

## Motivation

:::: {.columns}

::: {.column width="48%"}

The Idea:

- Combine multiple forecasts instead of choosing one

- Combination weights may vary over **time**, over the **distribution** or **both**

2 Popular options for combining distributions:

- Combining across quantiles (this paper)
  - Horizontal aggregation, vincentization
- Combining across probabilities
  - Vertical aggregation

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

::: {.panel-tabset}

## Time

```{r, echo = FALSE, fig.height=6}
par(mfrow = c(3, 3), mar = c(2, 2, 2, 2))
set.seed(1)
# Data
X <- matrix(ncol = 3, nrow = 15)
X[, 1] <- seq(from = 8, to = 12, length.out = 15) + 0.25 * rnorm(15)
X[, 2] <- 10 + 0.25 * rnorm(15)
X[, 3] <- seq(from = 12, to = 8, length.out = 15) + 0.25 * rnorm(15)
# Weights
w <- matrix(ncol = 3, nrow = 15)
w[, 1] <- sin(0.1 * 1:15)
w[, 2] <- cos(0.1 * 1:15)
w[, 3] <- seq(from = -2, 0.25, length.out = 15)^2
w <- (w / rowSums(w))
# Vis
plot(X[, 1],
  lwd = 4,
  type = "l",
  ylim = c(8, 12),
  xlab = "",
  ylab = "",
  xaxt = "n",
  yaxt = "n",
  bty = "n",
  col = "#2050f0"
)
plot(w[, 1],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#2050f0"
)
text(6, 0.5, TeX("$w_1(t)$"), cex = 2, col = "#2050f0")
arrows(13, 0.25, 15, 0.0, , lwd = 4, bty = "n")
plot.new()
plot(X[, 2],
  lwd = 4,
  type = "l", ylim = c(8, 12),
  xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "purple"
)
plot(w[, 2],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "purple"
)
text(6, 0.6, TeX("$w_2(t)$"), cex = 2, col = "purple")
arrows(13, 0.5, 15, 0.5, , lwd = 4, bty = "n")
plot(rowSums(X * w), lwd = 4, type = "l", xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#298829")
plot(X[, 3],
  lwd = 4,
  type = "l", ylim = c(8, 12),
  xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#e423b4"
)
plot(w[, 3],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#e423b4"
)
text(6, 0.25, TeX("$w_3(t)$"), cex = 2, col = "#e423b4")
arrows(13, 0.75, 15, 1, , lwd = 4, bty = "n")
```

## Distribution

```{r, echo = FALSE, fig.height=6}
par(mfrow = c(3, 3), mar = c(2, 2, 2, 2))
set.seed(1)
# Data
X <- matrix(ncol = 3, nrow = 31)

X[, 1] <- dchisq(0:30, df = 10)
X[, 2] <- dnorm(0:30, mean = 15, sd = 5)
X[, 3] <- dexp(0:30, 0.2)
# Weights
w <- matrix(ncol = 3, nrow = 31)
w[, 1] <- sin(0.05 * 0:30)
w[, 2] <- cos(0.05 * 0:30)
w[, 3] <- seq(from = -2, 0.25, length.out = 31)^2
w <- (w / rowSums(w))
# Vis
plot(X[, 1],
  lwd = 4,
  type = "l",
  xlab = "",
  ylab = "",
  xaxt = "n",
  yaxt = "n",
  bty = "n",
  col = "#2050f0"
)
plot(X[, 2],
  lwd = 4,
  type = "l",
  xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "purple"
)
plot(X[, 3],
  lwd = 4,
  type = "l",
  xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#e423b4"
)
plot(w[, 1],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#2050f0"
)
text(12, 0.5, TeX("$w_1(x)$"), cex = 2, col = "#2050f0")
arrows(26, 0.25, 31, 0.0, , lwd = 4, bty = "n")
plot(w[, 2],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "purple"
)
text(15, 0.5, TeX("$w_2(x)$"), cex = 2, col = "purple")
arrows(15, 0.25, 15, 0, , lwd = 4, bty = "n")
plot(w[, 3],
  lwd = 4, type = "l",
  ylim = c(0, 1),
  xlab = "",
  ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#e423b4"
)
text(20, 0.5, TeX("$w_3(x)$"), cex = 2, col = "#e423b4")
arrows(5, 0.25, 0, 0, , lwd = 4, bty = "n")
plot.new()
plot(rowSums(X * w), lwd = 4, type = "l", xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#298829")
```

:::

:::

::::

# The Framework of Prediction under Expert Advice

## 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:
\begin{equation}
    \widetilde{X}_{t} = \sum_{k=1}^K w_{t,k} \widehat{X}_{t,k}.
    \label{eq_forecast_def}
\end{equation}
- The realization for $t$ is observedilities
  - Vertical aggregation

:::

::: {.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
- `r Citet(my_bib, "cesa2006prediction")`

:::

::::

---

## The Regret

Weights are updated sequentially according to the past performance of the $K$ experts.

That is, a loss function $\ell$ is needed. This is used 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 the expert $k$ until time $t$.
- Measures how much the forecaster *regrets* not having followed the expert's advice

Popular loss functions for point forecasting `r Citet(my_bib, "gneiting2011making")`:
.pull-left[
- $\ell_2$-loss $\ell_2(x, y) = | x -y|^2$
  - optimal for mean prediction 
]
.pull-right[
- $\ell_1$-loss $\ell_1(x, y) = | x -y|$ 
  - optimal for median predictions 
]


:::: {.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 

:::

::::


## Popular Aggregation Algorithms

#### The naive combination

\begin{equation}
    w_{t,k}^{\text{Naive}} = \frac{1}{K}
\end{equation}

#### The exponentially weighted average forecaster (EWA)

\begin{align}
    w_{t,k}^{\text{EWA}} & = \frac{e^{\eta R_{t,k}} }{\sum_{k = 1}^K e^{\eta R_{t,k}}}
    =
    \frac{e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }{\sum_{k = 1}^K e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }
    \label{eq_ewa_general}
\end{align}

#### The polynomial weighted aggregation (PWA)

\begin{align}
    w_{t,k}^{\text{PWA}} & = \frac{ 2(R_{t,k})^{q-1}_{+} }{ \|(R_t)_{+}\|^{q-2}_q}
    \label{eq_pwa_general}
\end{align}

with $q\geq 2$ and $x_{+}$ the (vector) of positive parts of $x$.

## Optimality

In stochastic settings, the cumulative Risk should be analyezed `r Citet(my_bib, "wintenberger2017optimal")`:

\begin{align}
    \underbrace{\widetilde{\mathcal{R}}_t = \sum_{i=1}^t \mathbb{E}[\ell(\widetilde{X}_{i},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Forecaster}} \qquad\qquad\qquad \text{ and } \qquad\qquad\qquad
    \underbrace{\widehat{\mathcal{R}}_{t,k} = \sum_{i=1}^t \mathbb{E}[\ell(\widehat{X}_{i,k},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Experts}}
    \label{eq_def_cumrisk}
\end{align}

There are two problems that an algorithm should solve in iid settings:

:::: {.columns}

::: {.column width="48%"}

### The selection problem
\begin{equation}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \stackrel{t\to \infty}{\rightarrow} a \quad \text{with} \quad a \leq 0.
    \label{eq_opt_select}
\end{equation}
The forecaster is asymptotically not worse than the best expert $\widehat{\mathcal{R}}_{t,\min}$.

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

### The convex aggregation problem

\begin{equation}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right) \stackrel{t\to \infty}{\rightarrow} b \quad \text{with} \quad b \leq 0 .
    \label{eq_opt_conv}
\end{equation}
The forecaster is asymptotically not worse than the best convex combination $\widehat{X}_{t,\pi}$ in hindsight (**oracle**).

:::

::::

## Optimality

Satisfying the convexity property \eqref{eq_opt_conv} comes at the cost of slower possible convergence.

According to `r Citet(my_bib, "wintenberger2017optimal")`, an algorithm has optimal rates with respect to selection \eqref{eq_opt_select} and convex aggregation \eqref{eq_opt_conv} if

\begin{align}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) & =
    \mathcal{O}\left(\frac{\log(K)}{t}\right)\label{eq_optp_select}
\end{align}

\begin{align}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right)  & =
    \mathcal{O}\left(\sqrt{\frac{\log(K)}{t}}\right)
    \label{eq_optp_conv}
\end{align}

Algorithms can statisfy both \eqref{eq_optp_select} and \eqref{eq_optp_conv} depending on:

- The loss function
- Regularity conditions on $Y_t$ and $\widehat{X}_{t,k}$
- The weighting scheme

## Optimality

According to `r Citet(my_bib, "cesa2006prediction")` EWA \eqref{eq_ewa_general} satisfies the optimal selection convergence \eqref{eq_optp_select} in a deterministic setting if the:
- Loss $\ell$ is exp-concave
- Learning-rate $\eta$ is chosen correctly

Those results can be converted to stochastic iid settings `r Citet(my_bib, "kakade2008generalization")` `r Citet(my_bib, "gaillard2014second")`.

The optimal convex aggregation convergence \eqref{eq_optp_conv} can be satisfied by applying the kernel-trick. Thereby, the loss is linearized:
\begin{align}
\ell^{\nabla}(x,y) = \ell'(\widetilde{X},y) x
\end{align}
$\ell'$ is the subgradient of $\ell$ in its first coordinate evaluated at forecast combination $\widetilde{X}$.

Combining probabilistic forecasts calls for a probabilistic loss function

:::: {.notes}

We apply Bernstein Online Aggregation (BOA). It lets us weaken the exp-concavity condition while almost keeping the optimalities \ref{eq_optp_select} and \ref{eq_optp_conv}.

::::

## The Continuous Ranked Probability Score

:::: {.columns}

::: {.column width="48%"}

**An appropriate choice:**

\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 `r Citet(my_bib, "gneiting2007strictly")`.

Using the CRPS, we can calculate time-adaptive weight $w_{t,k}$. However, what if the experts' performance is not uniform over all parts of the distribution? 

The idea: 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*}

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

to combine quantiles of the probabilistic forecasts individually using the quantile-loss (QL):
\begin{align*}
    \text{QL}_p(q, y) & = (\mathbb{1}\{y < q\} -p)(q - y)
\end{align*}

</br>

**But is it optimal?**

CRPS is exp-concave `r fontawesome::fa("check", fill ="#00b02f")`

`r fontawesome::fa("arrow-right", fill ="#000000")` EWA \eqref{eq_ewa_general} with CRPS satisfies \eqref{eq_optp_select} and \eqref{eq_optp_conv}

QL is convex, but not exp-concave `r fontawesome::fa("exclamation", fill ="#ffa600")`

`r fontawesome::fa("arrow-right", fill ="#000000")` Bernstein Online Aggregation (BOA) lets us weaken the exp-concavity condition while almost keeping optimal convergence

:::

::::

## CRPS-Learning Optimality

For convex losses, BOAG 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
    1-e^{x}
    \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} `r Citet(my_bib, "wintenberger2017optimal")` .

The same algorithm 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,\min} \right) \leq
    C\left(\frac{\log(K)+\log(\log(Gt))+ x}{\alpha t}\right)^{\frac{1}{2-\beta}} \right) \geq
    1-e^{x}
    \label{eq_boa_opt_select}
\end{equation}

if $Y_t$ is bounded, the considered loss $\ell$ is convex $G$-Lipschitz and weak exp-concave in its first coordinate. 

This is for losses that satisfy **A1** and **A2**.

## CRPS-Learning Optimality

:::: {.columns}

::: {.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} `r Citet(my_bib, "gaillard2018efficient")`.

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

**Lemma 1**

\begin{align}
    2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\min}
      & \leq \widehat{\mathcal{R}}^{\text{CRPS}}_{t,\min}
    \label{eq_risk_ql_crps_expert}                        \\
    2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\pi}
      & \leq \widehat{\mathcal{R}}^{\text{CRPS}}_{t,\pi} .
    \label{eq_risk_ql_crps_convex}
\end{align}

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")` </br>

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")` </br>

:::

::::


## CRPS-Learning Optimality

:::: {.columns}

::: {.column width="48%"}

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
conditional distribution $Y_t|\mathcal{F}_{t-1}$.

**Proposition 1**

Let $Y$ be a univariate random variable with (Radon-Nikodym) $\nu$-density $f$, then for the second subderivative of the quantile risk
$\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]$
of $Y$ it holds for all $p\in(0,1)$ that
$\mathcal{Q}_p'' = f.$
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")` `r Citet(my_bib, "gaillard2018efficient")`

:::

::: {.column width="2%"}

:::

::: {.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")` </br>

**Theorem 1**

The gradient based fully adaptive Bernstein online aggregation (BOAG) applied pointwise for all $p\in(0,1)$ on $\text{QL}$ satisfies
\eqref{eq_boa_opt_conv} with  minimal CRPS given by

$$\widehat{\mathcal{R}}_{t,\pi} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\pi}.$$

If $Y_t|\mathcal{F}_{t-1}$ is bounded 
and has a pdf $f_t$ satifying $f_t>\gamma >0$ on its 
support $\text{spt}(f_t)$ then \ref{eq_boa_opt_select} holds with $\beta=1$ and

$$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}_{t,\min}$$.

:::

::::


## A Probabilistic Example



:::: {.columns}

::: {.column width="48%"}

Simple Example:


\begin{align}
    Y_t               & \sim \mathcal{N}(0,\,1)                     \\
    \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_sim1}
\end{align}

- True weights vary over $p$
- Figures show the ECDF and calculated weights using $T=25$ realizations
- Pointwise solution creates rough estimates
- Pointwise is better than constant
- Smooth solution is better than pointwise

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

foo

:::

::::






## Columns Template

:::: {.columns}

::: {.column width="48%"}

Baz

:::

::: {.column width="2%"}

:::

::: {.column width="48%"}

foo

:::

::::


# References

```{r refs1, echo=FALSE, results="asis"}
PrintBibliography(my_bib, .opts = list(style = "text"))
```

