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Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. Journal of Econometrics, 237(2), 105221.
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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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Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. arXiv preprint arXiv:2407.08750.
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Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
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Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
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Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
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Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
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## Overview of the Thesis {transition="fade" transition-speed="slow"}
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Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. Journal of Econometrics, 237(2), 105221.
|
|
|
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.
|
|
|
Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. arXiv preprint arXiv:2407.08750.
|
|
|
Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
|
|
|
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
|
|
|
Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
|
|
|
Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
|
## Overview of the Thesis {transition="fade" transition-speed="slow"}
|
|
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. CRPS learning. Journal of Econometrics, 237(2), 105221.
|
|
|
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.
|
|
|
Hirsch, S., Berrisch, J., & Ziel, F. [-@hirsch2024online]. Online Distributional Regression. arXiv preprint arXiv:2407.08750.
|
|
|
Berrisch, J., & Ziel, F. [-@berrisch2022distributional]. Distributional modeling and forecasting of natural gas prices. Journal of Forecasting, 41(6), 1065–1086.
|
|
|
Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. [-@berrisch2023modeling]. Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.
|
|
|
Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]. High-resolution peak demand estimation using generalized additive models and deep neural networks. Energy and AI, 13, 100236.
|
|
|
Berrisch, J. [-@berrisch2025rcpptimer]. rcpptimer: Rcpp Tic-Toc Timer with OpenMP Support. arXiv preprint arXiv:2501.15856.
|
# CRPS Learning
Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. *Journal of Econometrics*, 237(2), 105221.
## Introduction
:::: {.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, cache = TRUE}
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, cache = TRUE}
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 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
- @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 @gneiting2011making:
:::: {.columns}
::: {.column width="48%"}
$\ell_2$ loss:
\begin{equation}
\ell_2(x, y) = | x -y|^2 \label{eq:elltwo}
\end{equation}
Strictly proper for *mean* prediction
:::
::: {.column width="4%"}
:::
::: {.column width="48%"}
$\ell_1$ loss:
\begin{equation}
\ell_1(x, y) = | x -y| \label{eq:ellone}
\end{equation}
Strictly proper for *median* predictions
:::
::::
## Popular Algorithms and the Risk
