diff --git a/assets/crps_learning/forget.png b/assets/crps_learning/forget.png new file mode 100644 index 0000000..56f654c Binary files /dev/null and b/assets/crps_learning/forget.png differ diff --git a/assets/library.bib b/assets/library.bib index 64db732..371ac6e 100644 --- a/assets/library.bib +++ b/assets/library.bib @@ -555,4 +555,36 @@ pages = {115600}, year = {2020}, publisher = {Elsevier} +} +@article{BERRISCH2023105221, + title = {CRPS learning}, + author = {Jonathan Berrisch and Florian Ziel}, + year = {2023}, + month = {12}, + journal = {Journal of Econometrics}, + publisher = {Elsevier BV}, + volume = {237}, + number = {2, Part C}, + pages = {105221}, + doi = {10.1016/j.jeconom.2021.11.008}, + issn = {0304-4076}, + url = {https://arxiv.org/pdf/2102.00968}, + abstract = {Combination and aggregation techniques can significantly improve forecast accuracy. This also holds for probabilistic forecasting methods where predictive distributions are combined. There are several time-varying and adaptive weighting schemes such as Bayesian model averaging (BMA). However, the quality of different forecasts may vary not only over time but also within the distribution. For example, some distribution forecasts may be more accurate in the center of the distributions, while others are better at predicting the tails. Therefore, we introduce a new weighting method that considers the differences in performance over time and within the distribution. We discuss pointwise combination algorithms based on aggregation across quantiles that optimize with respect to the continuous ranked probability score (CRPS). After analyzing the theoretical properties of pointwise CRPS learning, we discuss B- and P-Spline-based estimation techniques for batch and online learning, based on quantile regression and prediction with expert advice. We prove that the proposed fully adaptive Bernstein online aggregation (BOA) method for pointwise CRPS online learning has optimal convergence properties. They are confirmed in simulations and a probabilistic forecasting study for European emission allowance (EUA) prices.}, + keywords = {Combination, Aggregation, Online, Probabilistic, Forecasting, Quantile, Time series, Distribution, Density, Prediction, Splines} +} + +@article{BERRISCH20241568, + title = {Multivariate probabilistic CRPS learning with an application to day-ahead electricity prices}, + author = {Jonathan Berrisch and Florian Ziel}, + year = {2024}, + month = {10}, + journal = {International Journal of Forecasting}, + publisher = {Elsevier BV}, + volume = {40}, + number = {4}, + pages = {1568--1586}, + doi = {10.1016/j.ijforecast.2024.01.005}, + issn = {0169-2070}, + abstract = {This paper presents a new method for combining (or aggregating or ensembling) multivariate probabilistic forecasts, considering dependencies between quantiles and marginals through a smoothing procedure that allows for online learning. We discuss two smoothing methods: dimensionality reduction using Basis matrices and penalized smoothing. The new online learning algorithm generalizes the standard CRPS learning framework into multivariate dimensions. It is based on Bernstein Online Aggregation (BOA) and yields optimal asymptotic learning properties. The procedure uses horizontal aggregation, i.e., aggregation across quantiles. We provide an in-depth discussion on possible extensions of the algorithm and several nested cases related to the existing literature on online forecast combination. We apply the proposed methodology to forecasting day-ahead electricity prices, which are 24-dimensional distributional forecasts. The proposed method yields significant improvements over uniform combination in terms of continuous ranked probability score (CRPS). We discuss the temporal evolution of the weights and hyperparameters and present the results of reduced versions of the preferred model. A fast C＋＋implementation of the proposed algorithm is provided in the open-source R-Package profoc on CRAN.}, + keywords = {Combination, Aggregation, Ensembling, Online, Multivariate, Probabilistic, Forecasting, Quantile, Time series, Distribution, Density, Prediction, Splines} } \ No newline at end of file diff --git a/index.qmd b/index.qmd index 0e4d11e..b1d8390 100644 --- a/index.qmd +++ b/index.qmd @@ -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%"} +
+ +
::: -::: {.column width="48%"} +::: {.column width="4%"} + +::: + +::: {.column width="58%"} + +###   -**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): - -
- -
+:::: ## 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 - -
- -
- -::: - -:::: - -## 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 +
+ ```{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 - -::: - :::: - - # 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 - -
- -**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 - -
- -`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 +
+ +::: {.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++. - -
- -
- -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: -
- -
-[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. ---