Init presentation boilerplate

.gitignore

@@ -82,3 +82,7 @@ data/*
 
 # Whitelist selected files
 !data/example_witelisted.csv
+
+# Ignore html files for now
+# TODO: Remove later
+*.html

25_07_phd_defense/_extensions/quarto-ext/fontawesome/_extension.yml

@@ -0,0 +1,7 @@
+title: Font Awesome support
+author: Carlos Scheidegger
+version: 1.2.0
+quarto-required: ">=1.2.269"
+contributes:
+  shortcodes:
+    - fontawesome.lua

25_07_phd_defense/_extensions/quarto-ext/fontawesome/assets/css/latex-fontsize.css

@@ -0,0 +1,30 @@
+.fa-tiny {
+  font-size: 0.5em;
+}
+.fa-scriptsize {
+  font-size: 0.7em;
+}
+.fa-footnotesize {
+  font-size: 0.8em;
+}
+.fa-small {
+  font-size: 0.9em;
+}
+.fa-normalsize {
+  font-size: 1em;
+}
+.fa-large {
+  font-size: 1.2em;
+}
+.fa-Large {
+  font-size: 1.5em;
+}
+.fa-LARGE {
+  font-size: 1.75em;
+}
+.fa-huge {
+  font-size: 2em;
+}
+.fa-Huge {
+  font-size: 2.5em;
+}

25_07_phd_defense/_extensions/quarto-ext/fontawesome/fontawesome.lua

@@ -0,0 +1,84 @@
+local function ensureLatexDeps()
+  quarto.doc.use_latex_package("fontawesome5")
+end
+
+local function ensureHtmlDeps()
+  quarto.doc.add_html_dependency({
+    name = 'fontawesome6',
+    version = '1.2.0',
+    stylesheets = {'assets/css/all.min.css', 'assets/css/latex-fontsize.css'}
+  })
+end
+
+local function isEmpty(s)
+  return s == nil or s == ''
+end
+
+local function isValidSize(size)
+  local validSizes = {
+    "tiny",
+    "scriptsize",
+    "footnotesize",
+    "small",
+    "normalsize",
+    "large",
+    "Large",
+    "LARGE",
+    "huge",
+    "Huge"
+  }
+  for _, v in ipairs(validSizes) do
+    if v == size then
+      return size
+    end
+  end
+  return ""
+end
+
+return {
+  ["fa"] = function(args, kwargs)
+
+    local group = "solid"
+    local icon = pandoc.utils.stringify(args[1])
+    if #args > 1 then
+      group = icon
+      icon = pandoc.utils.stringify(args[2])
+    end
+    
+    local title = pandoc.utils.stringify(kwargs["title"])
+    if not isEmpty(title) then
+      title = " title=\"" .. title  .. "\""
+    end
+
+    local label = pandoc.utils.stringify(kwargs["label"])
+    if isEmpty(label) then
+      label = " aria-label=\"" .. icon  .. "\""
+    else
+      label = " aria-label=\"" .. label  .. "\""
+    end
+
+    local size = pandoc.utils.stringify(kwargs["size"])
+    
+    -- detect html (excluding epub which won't handle fa)
+    if quarto.doc.is_format("html:js") then
+      ensureHtmlDeps()
+      if not isEmpty(size) then
+        size = " fa-" .. size
+      end
+      return pandoc.RawInline(
+        'html',
+        "<i class=\"fa-" .. group .. " fa-" .. icon .. size .. "\"" .. title .. label .. "></i>"
+      )
+    -- detect pdf / beamer / latex / etc
+    elseif quarto.doc.is_format("pdf") then
+      ensureLatexDeps()
+      if isEmpty(isValidSize(size)) then
+        return pandoc.RawInline('tex', "\\faIcon{" .. icon .. "}")
+      else
+        return pandoc.RawInline('tex', "{\\" .. size .. "\\faIcon{" .. icon .. "}}")
+      end
+    else
+      return pandoc.Null()
+    end
+  end
+}

25_07_phd_defense/clean.scss

@@ -0,0 +1,261 @@
+/*-- scss:defaults --*/
+
+// Custom colours and variables
+
+$jet: #131516;
+$accent: #107895;
+$accent2: #9a2515;
+// $accent2: #e64173;
+$right-arrow: "\2192"; // Unicode character for right arrow
+
+// fonts
+
+/*
+Note: This theme uses the Roboto font family, which it imports from Google
+  Fonts to ensure consistent weighting in addition to availability. While
+  you can use a local installation of Roboto, this is generally not 
+  recommended since the weighting will likely be wrong (probably too
+  light). OTOH, importing from Google Fonts can cause some issues in
+  certain secure environments due the external CDN (see:
+  https://github.com/grantmcdermott/quarto-revealjs-clean/issues/7). If
+  that's the case for you, simply comment out the `@import url(...)` line
+  below and it will default for the default Sans Serif font on your system
+  (e.g., Helvetica on a Mac). Circling back to the earlier point about
+  preserving consistent font weights, you may also wish to remove "Roboto"
+  from the choice set if the family is installed locally.
+*/
+@import url(https://fonts.googleapis.com/css?family=Roboto:200,200i,300,300i,350,350i,400,400i);
+$font-family-sans-serif: "Roboto", sans-serif !default;
+$presentation-heading-font: "Roboto", sans-serif !default;
+
+$presentation-heading-color: $jet !default;
+$presentation-heading-font-weight: lighter;
+//$presentation-heading-line-height: 2;
+//$presentation-block-margin: 28px;
+$presentation-font-size-root: 32px;
+
+// colors
+//$body-bg: #f0f1eb !default;
+$body-color: $jet !default;
+$link-color: $accent !default;
+$selection-bg: #26351c !default;
+
+
+/*-- scss:rules --*/
+
+.reveal a {
+  line-height: 1.5em;
+}
+
+.reveal p {
+  // font-weight: 300;
+  font-weight: lighter;
+  margin-top: 1.25em;
+}
+
+// title and headings
+
+#title-slide {
+  text-align: left;
+
+  .title {
+    color: $body-color;
+    font-size: 1.4em;
+    // font-weight: 350;
+    font-weight: lighter;
+  }
+
+  .subtitle {
+    color: $accent;
+    font-style: italic;
+    margin-top: 0em;
+    font-weight: lighter;
+  }
+
+  .institute,
+  .quarto-title-affiliation,
+  .quarto-title-author-email {
+    font-style: italic;
+    // font-size: 80%;
+    // color: #7F7F7F;
+  }
+
+  .author,
+  .quarto-title-author-name {
+    color: $body-color;
+  }
+
+  .quarto-title-authors {
+    display: flex;
+    justify-content: left;
+
+    .quarto-title-author {
+      padding-left: 0em;
+      padding-right: 0em;
+      width: 100%;
+    }
+  }
+
+}
+
+
+.reveal h2 {
+  // font-weight: 350;
+  font-weight: lighter;
+  font-size: 1.4em;
+}
+
+.reveal h3 {
+  color: $accent;
+  font-style: italic;
+  // font-weight: 350;
+  font-weight: lighter;
+  font-size: 0.95em;
+}
+
+.reveal h4 {
+  color: $accent2;
+  // font-weight: 350;
+  font-weight: normal;
+  margin-top: 1.25em;
+}
+
+// alerts etc.
+
+.alert {
+  color: $accent2;
+}
+
+.fg {
+  color: var(--col, $jet);
+}
+
+.bg {
+  background-color: var(--col, #fff);
+  padding: 0.1em;
+  border-radius: 5px;
+  display: inline-block;
+}
+
+// lists
+
+// Unordered lists
+
+.reveal ul {
+  // font-weight: 300;
+  font-weight: lighter;
+  padding-left: 16px;
+
+  li::marker {
+    color: mix($accent, white, 70%);
+  }
+}
+
+.reveal ul ul {
+  list-style: none;
+
+  li:before {
+    content: $right-arrow;
+    color: mix($accent, white, 60%);
+    display: inline-block;
+    width: 1em;
+    margin-left: -1em;
+    margin-right: 0.5em;
+  }
+}
+
+// Ordered lists
+
+.reveal ol {
+  // font-weight: 300;
+  font-weight: lighter;
+  padding-left: 16px;
+
+  li::marker {
+    color: $accent;
+  }
+}
+
+// Move "hamburger" menu button to top right
+
+.reveal .slide-menu-button {
+  position: fixed;
+  top: 6px;
+  right: 0;
+  display: flex;
+  justify-content: flex-end;
+  align-items: flex-start;
+  pointer-events: none;
+}
+
+.reveal .slide-menu-button > * {
+  pointer-events: auto;
+}
+
+
+// Same for chalkboard buttons (with an offset)
+
+.reveal .slide-chalkboard-buttons {
+  position: fixed;
+  top: 12px;
+  right: 24px;
+  display: flex;
+  justify-content: flex-end;
+  align-items: flex-start;
+  pointer-events: none;
+}
+
+.reveal .slide-chalkboard-buttons > * {
+  pointer-events: auto;
+}
+
+// Beamer-style button link environment
+
+.button {
+  display: inline-block;
+  padding: 6px 12px;
+  margin-bottom: 0;
+  font-size: 14px;
+  font-weight: 400;
+  line-height: 1.42857143;
+  text-align: center;
+  white-space: nowrap;
+  vertical-align: middle;
+  cursor: pointer;
+  background-color: $accent;
+  border: 1px solid $accent;
+  color: #fff !important;
+  text-decoration: none;
+  border-radius: 4px;
+  transition: all 0.2s ease-in-out;
+}
+
+.button:hover {
+  background-color: #0056b3;
+  border-color: #0056b3;
+}
+
+.button::before {
+  content: "▶";
+  margin-right: 5px;
+}
+
+// Special catch for etable environment to ensure these table images
+// don't overflow the slide.
+// See: https://lrberge.github.io/fixest/articles/etable_new_features.html
+
+.etable {
+  width: 100%;
+  height: calc(100% - 3em); /* Adjust 3em based on the height of your header, if necessary */
+  display: flex;
+  align-items: center;
+  justify-content: center;
+}
+
+.etable img {
+  max-width: 100%;
+  max-height: 100%;
+  width: auto;
+  height: auto;
+  object-fit: contain;
+}

25_07_phd_defense/custom.scss

@@ -0,0 +1,11 @@
+/*-- scss:defaults --*/
+
+$body-bg: #ffffff;
+$body-color: #7a6f69;
+$link-color: #005088;
+$selection-color: #00b0dc;
+
+$presentation-heading-color: $selection-color;
+$tabset-border-color: #bbb3b0;
+
+/*-- scss:rules --*/

25_07_phd_defense/index.qmd

@@ -0,0 +1,696 @@
+---
+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
+
+:::
+
+## EfeMOD
+
+**Empirisch fundierte Elektrizitätsmarkt-Modellierung mit Open Data**
+
+:::: {.columns}
+
+::: {.column width="65%"}
+
+[{{< fa users-gear >}}]{style="color: #404040;"} **Project Entities:**
+
+Chair of Prof. Dr. Christoph Weber (Management Sciences and Energy Economics)
+
+Chair of Prof. Dr. Florian Ziel (Data Science in Energy and Environment)
+
+[{{< fa bullseye >}}]{style="color: #404040;"} &ensp; **Project Goal:**
+
+Use publicly available data (particularly ENTSO-E Transparency Platform) to estimate parameters for energy system and energy market models.
+
+:::
+
+::: {.column width="5%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="30%"}
+![](figures/BMWK.webp)
+
+:::
+
+::::
+
+## EfeMOD
+
+![](figures/power_plant_list.jpg)
+
+## Motivation and Objective
+
+**Identification of Power Plant Operation States Using Clustering**
+
+[{{< fa earth-europe >}}]{style="color: #404040;"} Gain Knowledge about the Power Plant Characteristics
+
+- Operation Points,
+- Efficiency
+- Capacity, etc.
+
+[{{< fa display >}}]{style="color: #404040;"} This Presentation: 
+
+Identify Operation States:
+
+- Stable Operation
+- Startup
+- Minimum-Stable Operation, etc.
+
+Provide these characteristics to other researchers
+
+[{{< fa right-long >}}]{style="color: #404040;"} e.g. to estimate efficiency
+
+## Data
+
+[{{< fa database >}}]{style="color:#404040;"} Entsoe Data: 
+
+- ActualGenerationOutputPerGenerationUnit_16.1.A
+- UnavailabilityOfGenerationUnits_15.1.A_B
+
+[{{< fa fire-flame-simple >}}]{style="color:rgb(0, 200, 255);"} We focus on natural gas units:
+
+- 63 units in `DE_LU` bidding zone
+- 299 units across all bidding zones
+
+[{{< fa calendar-days >}}]{style="color:#404040;"} We use recent data:
+
+- 2020-01-01 until "now"
+
+## Data
+
+![](figures/Block%20AGuD/0_data1.jpg)
+
+## Data
+
+![](figures/Block%20AGuD/0_data2.jpg)
+
+## Data
+
+
+::: {.panel-tabset}
+
+## Lausward
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+**Heizkraftwerk Lausward **
+
+Location: Düsseldorf
+
+Block Anton (*Block AGuD*)
+
+Combined cycle gas turbine (CCGT)
+
+Electrical output: 103 MW [{{< fa bolt >}}]{style="color: #ffc400;"}
+
+75 MW of district heating can be decoupled
+
+Efficiency: 54%
+
+Fuel Utilization Rate: 87% (with district heating)
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+![](figures/Block%20AGuD/0_data3.jpg)
+
+:::
+
+::::
+
+## Emsland
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+**Erdgaskraftwerk Emsland**
+
+Location: Lingen (Ems)
+
+*Block C*
+
+Combined cycle gas turbine (CCGT)
+
+Electrical output: 475 MW [{{< fa bolt >}}]{style="color: #ffc400;"}
+
+Efficiency: 46%
+
+Black start enabled.
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+![](figures/Emsland%20C/0_data3.jpg)
+
+:::
+
+::::
+
+:::
+
+
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Overview
+
+Empirical identification of states
+
+3-Step Approach:
+
+- Prior Partitioning
+  - We create preliminary clusters
+  - They will be used to initialize the main clustering
+- Main Clustering
+  - Gaussian Model Based Clustering
+- Label Assignment 
+  - We assign meaningful labels to the final clusters
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+![](figures/Block%20AGuD/0_data3.jpg)
+
+:::
+
+::::
+
+:::
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Prior Partitioning
+
+[{{< fa arrow-up-right-dots >}}]{style="color: #202020FF;"} Divide the space in meaningful partitions: 
+
+Define the Capacity: $\zeta = max(t0)$
+
+Define a threshold: $\gamma = \frac{\zeta}{50}$
+
+[{{< fa circle >}}]{style="color: #2D7D32FF;"} $\pm \gamma$ around the diagonal: Stable <br>
+[{{< fa circle >}}]{style="color: #202020FF;"} $t0 < 1$ & $t1 < 1$: Zero <br>
+[{{< fa circle >}}]{style="color: #FA8C00FF;"} $t0 < \gamma$ & $t1 > 1$: Startup <br>
+[{{< fa circle >}}]{style="color: #D81A5FFF;"} $t0 > 1$ & $t1 < \gamma$: Shutdown <br>
+[{{< fa circle >}}]{style="color: #FDD834FF;"} $t1 > t0$: Ramp-Up <br>
+[{{< fa circle >}}]{style="color: #8D24AAFF;"} $t1 < t0$: Ramp-Down
+
+We project <b style="color: #2D7D32FF;">Stable</b> observations onto the diagonal, <font style = "opacity: 0.4;"> <b style="color: #FA8C00FF;">Startup</b>  on $t1$ and <b style="color: #D81A5FFF;">Shutdown</b> on $t0$ for the next step. </font>
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+![](figures/Block%20AGuD/1_pre-partition.jpg)
+
+:::
+
+::::
+
+:::
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Prior Partitioning
+
+Model-Based Clustering of the Regions using `mclust::Mclust` in `R`.
+
+- <b style="color: #2D7D32FF;">Stable</b>: 2-5 Clusters
+- <b style="color: #FDD834FF;">Ramp Up</b>: 2-4 Clusters
+- <b style="color: #8D24AAFF;">Ramp Down</b>: 2-4 Clusters
+
+[{{< fa lightbulb >}}]{style="color:rgb(255, 166, 0);"} Obtain finite mixture distribution:
+
+$$\sum_{k=1}^{G}{\pi_k f_k (\mathbf{x}; \mathbf{\theta}_k)}$$
+
+$f_k$ Density of k's component<br>
+$\pi_k$ Mixture weights<br>
+$\theta_k$ parameters of k's density component
+
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+![](figures/Block%20AGuD/1_pre-partition.jpg)
+
+:::
+
+::::
+
+## Empirical Approach
+
+### Prior Partitioning
+:::: {.columns}
+
+::: {.column width="49%"}
+
+$$f(\mathbf{x}; \mathbf{\Psi}) = \sum_{k=1}^{G}{\pi_k \phi (\mathbf{x}; \mathbf{\mu}_k; \mathbf{\Sigma}_k)}$$
+
+$\phi(\cdot)$ Multivariate Gaussian density<br>
+
+Maximum Likelihood Estimation via Expectation Maximization (EM) algorithm
+
+Likelihood for Gaussian Mixture Models (GMMs):
+
+\begin{align}
+  \ell(\Psi) = \sum_{i=1}^n \log \left\{ \sum_{k=1}^G \pi_k \phi(x_i; \mu_k, \Sigma_k) \right\}
+\end{align}
+
+[{{< fa retweet >}}]{style="color: #404040;"} We Re-Formulate this likelihood to a complete-data likelihood to utilize the EM algorithm
+
+
+:::
+
+::: {.column width="2%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="49%"}
+
+\begin{align}
+  \ell_{\mathcal{C}}(\Psi) = \sum_{i=1}^n \sum_{k=1}^G z_{ik} \left\{ \log \pi_k + \log \phi(x_i; \mu_k, \Sigma_k) \right\}
+\end{align}
+
+\begin{align}
+  z_{ik} = 
+  \begin{cases} 
+  1 & \text{if } x_i \text{ belongs to component }k \\
+  0 & \text{otherwise.}
+  \end{cases}
+\end{align}
+
+E-Step:
+
+\begin{align}
+  \hat{z}_{ik} = \frac{\hat{\pi}_k \phi(x_i; \hat{\mu}_k, \hat{\Sigma}_k)}{\sum_{g=1}^{G} \hat{\pi}_g \phi(x_i; \hat{\mu}_g, \hat{\Sigma}_g)},
+\end{align}
+
+M-Step:
+
+\begin{align}
+\quad \hat{\mu}_k = \frac{\sum_{i=1}^{n} \hat{z}_{ik} x_i}{n_k}, \quad \text{where} \quad n_k = \sum_{i=1}^{n} \hat{z}_{ik}.
+\end{align}
+
+
+:::
+
+::::
+
+
+
+::: {.notes}
+
+- log-likelihood in (2.2) is hard to maximize directly
+-  even numerically 
+
+- As a consequence, mixture models are usually fitted by reformulating the mixture
+problem as an incomplete-data problem within the EM framework.
+
+General EM Steps:
+
+- Init
+- Estimate latent component memberships
+- M-Step obtain the updated parameter estimates
+- Check convergence criteria
+
+::: 
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Prior Partitioning
+
+**Initialization**
+
+We initialize the EM algorithm (E-Step) using the partitions
+obtained from model-based agglomerative hierarchical clustering (MBAHC)
+
+**Estimation**
+
+The Bayesian information criterion (BIC) is used for model selection
+
+**Prior Partitioning Results**
+
+Right graph shows prior clusters.
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+::: {.panel-tabset}
+
+## Lausward
+
+![](figures/Block%20AGuD/2_partition.jpg)
+
+## Emsland
+
+![](figures/Emsland%20C/2_partition.jpg)
+
+:::
+
+:::
+
+::::
+
+::: {.notes}
+
+recursively merging the two clusters that yield the maximum
+likelihood of a probability model over all possible merges
+
+:::
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Main Clustering 
+
+**MBAHC**
+
+Prior Clusters are used in MBAHC
+
+The results of the MBAHC are used to initialize the EM Algorithm in the main Gaussian Model Based Clustering
+
+**Main Clustering Results**
+
+Right graph shows *Maximum A Posteriori (MAP) Classification*
+
+Colour indicates cumulated log(density) of all components.
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"}
+
+::: {.panel-tabset}
+
+## Lausward
+
+![](figures/Block%20AGuD/3_cluster.jpg)
+
+## Emsland
+
+![](figures/Emsland%20C/3_cluster.jpg)
+
+:::
+
+:::
+
+::::
+
+::: {.notes}
+
+recursively merging the two clusters that yield the maximum
+likelihood of a probability model over all possible merges
+
+:::
+
+
+
+## Empirical Approach
+
+### Label Assignment
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+We assign labels to the clusters using their mean $\mu$ and correlation $\rho$
+
+Multiple clusters may describe one Generation State (e.g., along the diagonal)
+
+:::
+
+::: {.column width="4%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="48%"}
+
+```{r}
+library(dplyr)
+load("figures/Block AGuD/clusters.RDS")
+clusters %>%
+  select(classification, mu_t0, mu_t1, cor) %>%
+  head()
+```
+
+:::
+
+::::
+
+\begin{align}
+\text{State} =
+  \begin{cases}
+    \color{#202020FF}{\text{Zero}} & (\mu_{t0} < 1) \land (\mu_{t1} < 1), \\
+    \text{MSO} & \left[ (\mu_{t0} > \zeta/10) \land (\mu_{t1} > \zeta / 10)  \land (\right| \mu_{t0} - \mu_{t1} \left| > \zeta / 10) \right]\\ &   \rightarrow \operatorname{argmin}(\mu_{t0} + \mu_{t1}), \\
+    \text{Max Capacity} & \rightarrow \operatorname{argmax}(\mu_{t0} + \mu_{t1}), \\
+    \text{Startup} & (\mu_{t1} \geq \zeta / 10) \land (\mu_{t0} < \gamma) \land (\rho < 0.3), \\
+    \text{Shutdown} & (\mu_{t0} \geq \zeta / 10) \land (\mu_{t1} < \gamma) \land (\rho < 0.3), \\
+    \text{Stable Operation} & \text{Remaining clusters with cor} > 0.8, \\
+    \text{Ramp Up} & \text{Remaining clusters: } \mu_{t1} > \mu_{t0}, \\
+    \text{Ramp Down} & \text{Remaining clusters: } \mu_{t1} < \mu_{t0}.
+  \end{cases}
+\end{align}
+
+::: {.notes}
+
+recursively merging the two clusters that yield the maximum
+likelihood of a probability model over all possible merges
+
+:::
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="39%"}
+
+### Label Assignment
+
+Right graphs show *assigned states*
+
+The points are coloured according to
+
+- MAP
+- Probability (each pure colour reflects a probability of 1)
+
+Some points below /above the diagonal are assigned to Ramp Up / Ramp Down
+
+- Can be easily fixed for MAP
+- Fixing probabilistic predictions not that easy
+
+:::
+
+::: {.column width="2%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="59%"}
+
+::: {.panel-tabset}
+
+## LSW
+
+![](figures/Block%20AGuD/4_assignments.jpg)
+
+## LSW Pr
+
+![](figures/Block%20AGuD/4_assignments_prob.jpg)
+
+## LSW Pr
+
+![](figures/Block%20AGuD/4_probability.jpg)
+
+## EMS
+
+![](figures/Emsland%20C/4_assignments.jpg)
+
+## EMS Pr
+
+![](figures/Emsland%20C/4_assignments_prob.jpg)
+
+## EMS Pr
+
+![](figures/Emsland%20C/4_probability.jpg)
+
+:::
+
+:::
+
+::::
+
+::: {.notes}
+
+recursively merging the two clusters that yield the maximum
+likelihood of a probability model over all possible merges
+
+:::
+
+## Empirical Approach
+
+:::: {.columns}
+
+::: {.column width="42%"}
+
+### Label Assignment
+
+*Fixing assignments*
+
+Relabeling Ramp Up and Ramp Down MAP predictions is trivial:
+
+\begin{align}
+\text{State} =
+  \begin{cases}
+    \text{Ramp Up} & x_{t1} > x_{t0}, \\
+    \text{Ramp Down} & x_{t1} < x_{t0}.
+  \end{cases}
+\end{align}
+
+Fixing the probability array is more involved:
+
+Find observations $x_{t1} < x_{t0}$ that can not be "Ramp Up":
+
+Set probability of all Ramp Up clusters to $0$.
+
+Normalize the probabilities.
+
+:::
+
+::: {.column width="3%"}
+<!-- empty column to create gap -->
+:::
+
+::: {.column width="55%"} 
+
+::: {.panel-tabset}
+
+
+## LSW Pr
+
+![](figures/Block%20AGuD/4_assignments_prob_fixed.jpg)
+
+## LSW Pr
+
+![](figures/Block%20AGuD/4_probability_fixed.jpg)
+
+## EMS Pr
+
+![](figures/Emsland%20C/4_assignments_prob_fixed.jpg)
+
+## EMS Pr
+
+![](figures/Emsland%20C/4_probability_fixed.jpg)
+
+:::
+
+:::
+
+::::
+
+## Outlook
+
+<br>
+<br>
+
+- The approach works in general
+- Conceptually simple
+- Label assignment needs some more work
+- Probabilistic statements may need adjustments for Ramp-Up Ramp-Down predictions
+- Some kind of validation would be desirable
+- Results will be used party on another research project in the EFEMOD project