diff --git a/.gitignore b/.gitignore
index fa2357b..d70b31b 100644
--- a/.gitignore
+++ b/.gitignore
@@ -86,3 +86,4 @@ data/*
 # Ignore html files for now
 # TODO: Remove later
 *.html
+25_07_phd_defense/index_cache/*
diff --git a/25_07_phd_defense/index.qmd b/25_07_phd_defense/index.qmd
index 2fdd145..97c29f0 100644
--- a/25_07_phd_defense/index.qmd
+++ b/25_07_phd_defense/index.qmd
@@ -22,23 +22,25 @@ execute:
 highlight-style: github
 ---
 
+## Outline
+
 <!--
 Render with: quarto preview /home/jonathan/git/PHD-Presentation/25_07_phd_defense/index.qmd --no-browser --port 6074
 -->
 
-## Outline
-
 ::: {.hidden}
 $$
 \newcommand{\A}{{\mathbb A}}
 $$
 :::
 
-<br>
-
 ::: {style="font-size: 150%;"}
 
-[{{< fa bars-staggered >}}]{style="color: #404040;"} &ensp; Introduction & Research Motivation
+:::
+
+<br>
+
+[{{< fa bars-staggered >}}]{style="color: #404040;"} &ensp;  [Introduction & Research Motivation](#motivation)
 
 [{{< fa bars-staggered >}}]{style="color: #404040;"} &ensp; Overview of the Thesis
 
@@ -50,10 +52,6 @@ $$
 
 [{{< fa binoculars >}}]{style="color: #404040;"} &ensp; Contributions & Outlook
 
-:::
-
-## PHD DeFence
-
 ```{r, setup, include=FALSE}
 # Compile with: rmarkdown::render("crps_learning.Rmd")
 library(latex2exp)
@@ -62,10 +60,10 @@ library(dplyr)
 library(tidyr)
 library(purrr)
 library(kableExtra)
+library(RefManageR)
 knitr::opts_chunk$set(
   dev = "svglite" # Use svg figures
 )
-library(RefManageR)
 BibOptions(
   check.entries = TRUE,
   bib.style = "authoryear",
@@ -84,31 +82,7 @@ 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)
-
----
-
-# Motivation
+# CRPS Learning
 
 ## Motivation
 
@@ -141,7 +115,7 @@ The Idea:
 
 ## Time
 
-```{r, echo = FALSE, fig.height=6}
+```{r, echo = FALSE, fig.height=6, cache = TRUE}
 par(mfrow = c(3, 3), mar = c(2, 2, 2, 2))
 set.seed(1)
 # Data
@@ -207,7 +181,7 @@ arrows(13, 0.75, 15, 1, , lwd = 4, bty = "n")
 
 ## Distribution
 
-```{r, echo = FALSE, fig.height=6}
+```{r, echo = FALSE, fig.height=6, cache = TRUE}
 par(mfrow = c(3, 3), mar = c(2, 2, 2, 2))
 set.seed(1)
 # Data
@@ -277,8 +251,6 @@ plot(rowSums(X * w), lwd = 4, type = "l", xlab = "", ylab = "", xaxt = "n", yaxt
 
 ::::
 
-# The Framework of Prediction under Expert Advice
-
 ## The Framework of Prediction under Expert Advice
 
 ### The sequential framework
@@ -323,32 +295,26 @@ Weights are updated sequentially according to the past performance of the $K$ ex
 
 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}
+$$
+R_{t,k}  = \widetilde{L}_{t} - \widehat{L}_{t,k} =  \sum_{i = 1}^t \ell(\widetilde{X}_{i},Y_i) - \ell(\widehat{X}_{i,k},Y_i)
+$${#eq-regret}
 
 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
+$\ell_2$ loss:
+
+$$\ell_2(x, y) = | x -y|^2$${#eq-elltwo}
+
+Strictly proper for *mean* prediction
 
 :::
 
@@ -358,8 +324,11 @@ Popular loss functions for point forecasting `r Citet(my_bib, "gneiting2011makin
 
 ::: {.column width="48%"}
 
-- $\ell_1$-loss $\ell_1(x, y) = | x -y|$
-  - optimal for median predictions
+$\ell_1$ loss:
+
+$$\ell_1(x, y) = | x -y|$${#eq-ellone}
+
+Strictly proper for *median* predictions
 
 :::
 
@@ -370,9 +339,9 @@ Popular loss functions for point forecasting `r Citet(my_bib, "gneiting2011makin
 
 #### The naive combination
 
-\begin{equation}
-    w_{t,k}^{\text{Naive}} = \frac{1}{K}
-\end{equation}
+$$
+w_{t,k}^{\text{Naive}} = \frac{1}{K}
+$${#eq-wtk_naive}
 
 #### The exponentially weighted average forecaster (EWA)
 
@@ -703,7 +672,7 @@ Simple Example:
 
 ## CDFs
 
-```{r, echo = FALSE, fig.width=7, fig.height=6, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=7, fig.height=6, fig.align='center', cache = TRUE}
 source("assets/01_common.R")
 load("assets/crps_learning/01_motivation_01.RData")
 ggplot(df, aes(x = x, y = y, xend = xend, yend = yend)) +
@@ -751,7 +720,7 @@ ggplot(df, aes(x = x, y = y, xend = xend, yend = yend)) +
 
 ## Weights
 
-```{r, echo = FALSE, fig.width=7, fig.height=6, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=7, fig.height=6, fig.align='center', cache = TRUE}
 source("assets/01_common.R")
 load("assets/crps_learning/01_motivation_02.RData")
 ggplot() +
@@ -832,9 +801,6 @@ We receive the constant solution for high values of $\lambda$ when setting $d=1$
 
 ::::
 
-
-# The Proposed CRPS-Learning Algorithm
-
 ---
 
 ## The Proposed CRPS-Learning Algorithm
@@ -980,7 +946,7 @@ The same simulation carried out for different algorithms (1000 runs):
 
 **Weights of expert 2**
 
-```{r, echo = FALSE, fig.width=7, fig.height=5, fig.align='center', cache = FALSE}
+```{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")
@@ -1094,7 +1060,7 @@ Tuning paramter grids:
 
 ::: {.column width="69%"}
 
-```{r, echo = FALSE, fig.width=7, fig.height=5, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=7, fig.height=5, fig.align='center', cache = TRUE}
 load("assets/crps_learning/overview_data.rds")
 
 data %>%
@@ -1168,7 +1134,7 @@ 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 = FALSE, results='asis'}
+```{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")
 
 quantile_loss <- function(X, y, tau) {
@@ -1243,7 +1209,7 @@ for (j in 1:ncol(table)) {
 table_out
 ```
 
-```{r, echo = FALSE, fig.width=7, fig.height=5.5, fig.align='center', cache = FALSE, results='asis'}
+```{r, echo = FALSE, fig.width=7, fig.height=5.5, fig.align='center', cache = TRUE, results='asis'}
 MO <- 6
 OUT <- Mout
 OUT.num <- OUT
@@ -1287,7 +1253,7 @@ table_out2 %>%
 
 ## QL
 
-```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = TRUE}
 
 ##### Performance across probabilities
 M <- length(mnames)
@@ -1335,7 +1301,7 @@ t(RQL) %>%
 
 ## Cumulative Loss Difference
 
-```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = TRUE}
 DQL <- t(apply(apply(QL[1:KK, -c(1:TTinit), ], c(1, 2), mean), 1, cumsum))
 
 rownames(DQL) <- mnames
@@ -1378,7 +1344,7 @@ data %>%
 
 ## Weights (BOAG P-Smooth)
 
-```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = TRUE}
 load("assets/crps_learning/weights_data.RData")
 weights_data %>%
   ggplot(aes(Date, p, fill = w)) +
@@ -1401,7 +1367,7 @@ weights_data %>%
 
 ## Weights (Last)
 
-```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = FALSE}
+```{r, echo = FALSE, fig.width=13, fig.height=5.5, fig.align='center', cache = TRUE}
 load("assets/crps_learning/weights_example.RData")
 weights %>%
   ggplot(aes(x = p, y = weights, col = Model)) +
@@ -1726,7 +1692,7 @@ Computation is easy since we have an analytical solution.
 
 ::: {.column width="48%"}
 
-```{r, fig.align="center", echo=FALSE, out.width = "1000px"}
+```{r, fig.align="center", echo=FALSE, out.width = "1000px", cache = TRUE}
 knitr::include_graphics("assets/mcrps_learning/algorithm.svg")
 ```
 
@@ -1791,19 +1757,19 @@ Computation Time: ~30 Minutes
 
 ## Constant
 
-```{r, fig.align="center", echo=FALSE, out.width = "400"}
+```{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"}
+```{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"}
+```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
 knitr::include_graphics("assets/mcrps_learning/constant_mv.svg")
 ```
 
@@ -1821,13 +1787,13 @@ knitr::include_graphics("assets/mcrps_learning/constant_mv.svg")
 
 ## Pointwise
 
-```{r, fig.align="center", echo=FALSE, out.width = "400"}
+```{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"}
+```{r, fig.align="center", echo=FALSE, out.width = "400", cache = TRUE}
 knitr::include_graphics("assets/mcrps_learning/smooth_best.svg")
 ```
 
@@ -1843,7 +1809,7 @@ knitr::include_graphics("assets/mcrps_learning/smooth_best.svg")
 
 ::: {.column width="55%"}
 
-```{r}
+```{r, cache = TRUE}
 load("assets/mcrps_learning/naive_table_df.rds")
 
 table_naive <- naive_table_df %>%
@@ -1987,7 +1953,7 @@ Foo
 
 ## Results
 
-```{r, warning=FALSE, fig.align="center", echo=FALSE, fig.width=12, fig.height=6}
+```{r, warning=FALSE, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
 load("assets/mcrps_learning/pars_data.rds")
 pars_data %>%
     ggplot(aes(x = dates, y = value)) +
@@ -2026,7 +1992,7 @@ pars_data %>%
 
 ## Results: Hour 16:00-17:00
 
-```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6}
+```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
 load("assets/mcrps_learning/weights_h.rds")
 weights_h %>%
         ggplot(aes(date, q, fill = weight)) +
@@ -2066,7 +2032,7 @@ weights_h %>%
 
 ## Results: Median
 
-```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6}
+```{r, fig.align="center", echo=FALSE, fig.width=12, fig.height=6, cache = TRUE}
 load("assets/mcrps_learning/weights_q.rds")
 weights_q %>%
     mutate(hour = as.numeric(hour) - 1) %>%
@@ -2157,7 +2123,7 @@ We use `Rcpp` modules to expose a class to R
 
 Basis specification `b_smooth_pr` is internally passed to `make_basis_mats()`:
 
-```{r, echo = TRUE, eval = FALSE, cache = FALSE}
+```{r, echo = TRUE, eval = FALSE, cache = TRUE}
 mod <- online(
   y = Y,
   experts = experts,
@@ -2237,6 +2203,8 @@ Get these slides:
 
 # Modeling Volatility and Dependence of European Carbon and Energy Prices
 
+TODO: Add Reference
+
 ---
 
 ## Motivation
@@ -2296,7 +2264,7 @@ How can the dynamics be characterized?
 
 ## Data
 
-```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center", cache = TRUE}
 readr::read_csv("assets/voldep/2022_10_14_eur_ref_co2_adj_hvpi_ex_nrg.csv") %>%
   select(-EUR_USD, -hvpi_x_nrg) %>%
   pivot_longer(-Date) %>%
@@ -2573,7 +2541,7 @@ Relative improvement in ES compared to $\text{RW}^{\sigma, \rho}$
 
 Cellcolor: w.r.t test statistic of Diebold-Mariano test (testing wether the model outperformes the benchmark, greener = better).
 
-```{r, echo=FALSE, results='asis'}
+```{r, echo=FALSE, results='asis', cache = TRUE}
 load("assets/voldep/energy_df.Rdata")
 table_energy <- energy %>%
   kbl(
@@ -2673,7 +2641,7 @@ table_energy %>%
 
 Improvement in CRPS of selected models relative to $\textrm{RW}^{\sigma, \rho}_{}$ in % (higher = better). Colored according to the test statistic of a DM-Test comparing to $\textrm{RW}^{\sigma, \rho}_{}$  (greener means lower test statistic i.e., better performance compared to $\textrm{RW}^{\sigma, \rho}_{}$).
 
-```{r, echo=FALSE, results = 'asis'}
+```{r, echo=FALSE, results = 'asis', cache = TRUE}
 load("assets/voldep/crps_df.Rdata")
 
 table_crps <- crps %>%
@@ -2752,7 +2720,7 @@ Conclusion: the Improvements seen before must be attributed to other parts of th
 
 Improvement in RMSE score of selected models relative to $\textrm{RW}^{\sigma, \rho}_{}$ in % (higher = better). Colored according to the test statistic of a DM-Test comparing to $\textrm{RW}^{\sigma, \rho}_{}$  (greener means lower test statistic i.e., better performance compared to $\textrm{RW}^{\sigma, \rho}_{}$).
 
-```{r, echo=FALSE, results = 'asis'}
+```{r, echo=FALSE, results = 'asis', cache = TRUE}
 load("assets/voldep/rmsq_df.Rdata")
 
 table_rmsq <- rmsq %>%
@@ -2802,7 +2770,7 @@ table_rmsq %>%
 
 ## Evolution of Linear Dependence $\Xi$
 
-```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center", cache = TRUE}
 load("assets/voldep/plot_rho_df.Rdata")
 ggplot() +
   geom_line(
@@ -2887,7 +2855,7 @@ ggplot() +
 
 ## Predictive Quantiles (Russian Invasion)
 
-```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center", cache = TRUE}
 load("assets/voldep/plot_quant_df.Rdata")
 
 plot_quant_data %>% ggplot(aes(x = date, y = value)) +
@@ -2981,43 +2949,7 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
 
 ::::
 
-## Columns Template
-
-:::: {.columns}
-
-::: {.column width="48%"}
-
-Baz
-
-:::
-
-::: {.column width="2%"}
-
-:::
-
-::: {.column width="48%"}
-
-foo
-
-:::
-
-::::
-
-## Paneltabset Template
-
-::: {.panel-tabset}
-
-## Baz
-
-Bar
-
-## Bam
-
-Foo
-
-::::
-
-# References
+## References
 
 ```{r refs1, echo=FALSE, results="asis"}
 PrintBibliography(my_bib, .opts = list(style = "text"))
diff --git a/25_07_phd_defense/sydney.scss b/25_07_phd_defense/sydney.scss
index c2b4078..a87e65a 100644
--- a/25_07_phd_defense/sydney.scss
+++ b/25_07_phd_defense/sydney.scss
@@ -1,7 +1,7 @@
 // See https://quarto.org/docs/presentations/revealjs/themes.html#saas-variables
 
 $brand-red: #e64626;
-$brand-blue: #0148A4;
+$brand-blue: #fcfcfc;
 $brand-yellow: #FFB800;
 $brand-charcoal: #424242;
 $brand-gray: #F1F1F1;