4_clustering: update

4_clustering/clustering.Rmd

@@ -19,16 +19,16 @@ classoption: aspectratio=169
 
 # Introduction
 
-## Programme
+## Program
 
 1. Single-cell RNASeq data from 10X Sequencing (Friday 3 June 2022 - 14:00)
 2. Normalization and spurious effects (Wednesday 8 June 2022 - 14:00)
 3. Dimension reduction and data visualization (Monday 13 June 2022 - 15:00)
 4. Clustering and annotation (Thursday 30 June 2022 2022 - 14:00)
 5. Pseudo-time and velocity inference (Friday 8 July 2022 - 14:00)
-6. Differental expression analysis (Monday 11 July 2022 - 15:30)
+6. Differential expression analysis (Monday 11 July 2022 - 15:30)
 
-## Programme
+## Program
 
 1. Single-cell RNASeq data from 10X Sequencing (Friday 3 June 2022 - 14:00)
 2. Normalization and spurious effects (Wednesday 8 June 2022 - 14:00)
@@ -38,9 +38,9 @@ classoption: aspectratio=169
  - Clustering
  - Classification
 5. Pseudo-time and velocity inference (Friday 8 July 2022 - 14:00)
-6. Differental expression analysis (Monday 11 July 2022 - 15:30)
+6. Differential expression analysis (Monday 11 July 2022 - 15:30)
 
-## Different kind of clustering
+## Different kinds of clustering
 
 \includegraphics[width=\textwidth]{img/learning_type.png}
 
@@ -103,7 +103,7 @@ We have up to $10^6$ rows and $10^6$ columns (cells)
 ### Hermann-Minkowski ${\left(\sum_{i=1}^n (x_i-y_i)^p\right)^{\frac{1}{p}}}$
 
 \begin{center}
-\includegraphics[width=0.7\textwidth]{img/hermann-minkowski.png}
+\includegraphics[width=0.6\textwidth]{img/hermann-minkowski.png}
 \end{center}
 
 ## Statistical divergence
@@ -273,8 +273,8 @@ ggsave("img/cor_kendall.pdf", width = 4, height = 4)
 
 Where:
 \begin{itemize}
-\item $A$ is number of concordant pairs
-\item $B$ is number of discordant pairs
+\item $A$ is the number of concordant pairs
+\item $B$ is the number of discordant pairs
 \item ${n \choose 2} = {n (n-1) \over 2}$ is the Binomial coefficient for the number of ways to choose $2$ items from $n$ items.
 \end{itemize}
 \end{columns}
@@ -403,7 +403,7 @@ The algorithm assigns each point to the closest centroid to get initial clusters
 \column{0.5\textwidth}
 For every cluster, the algorithm recomputes the centroid by taking the average of all points in the cluster. 
 
-Since the centroids change, the algorithm then re-assigns the points to the closest centroid.
+Since the centroids change, the algorithm then reassigns the points to the closest centroid.
 \end{columns}
 \end{center}
 
@@ -439,6 +439,7 @@ When clustering large datasets, you stop the algorithm before reaching convergen
 }
 \column{0.5\textwidth}
 Compute the Within-Cluster-Sum of Squared Errors (WSS) (each point vs the centroid) for different values of $k$
+\[\sum_{i=1}^n (x_i - c_i)^2\]
 \end{columns}
 \end{center}
 
@@ -461,8 +462,8 @@ The silhouette value measures how similar a point is to its own cluster (cohesio
 
 with 
 \begin{itemize}
-  \item $a(i)$ the mean distance between $i$ and aussi cells in the same cluster
-  \item $b(i)$ the mean distance between $i$ and aussi cells in different clusters
+  \item $a(i)$ the mean distance between $i$ and cells in the same cluster
+  \item $b(i)$ the mean distance between $i$ and cells in different clusters
 \end{itemize}
 
 We plot $\frac{1}{n}\sum_{i=1}^n s(i)$
@@ -572,16 +573,15 @@ At each step we merge clusters with their closest neighbor
 }
 \end{center}
 
-# Neighberhood graph
+# Neighborhood graph
 
 ## $k$-NN graph
 
-### Instead of considering every the relations between every cells we can focus on $k$ neighbors
+### Instead of considering every the relations between every cell we can focus on $k$ neighbors
 \begin{center}
 \href{http://www.biomedcentral.com/1471-2164/13/S7/S27}{
   \includegraphics[width=\textwidth]{img/knn_k2.png}
 }
-k = 2
 \end{center}
 
 ## SNN graph
@@ -592,12 +592,27 @@ k = 2
 \href{http://www.biomedcentral.com/1471-2164/13/S7/S27}{
   \includegraphics[width=\textwidth]{img/knn_k2.png}
 }
-k = 2
+\end{center}
+The weights on the edges are the intersection of the kNN neighborhood of two points
+
+## Louvain algorithm
+
+### Move nodes to optimize the modularity
 
+\[Q = \frac{1}{2m}\sum\limits_{ij}\bigg[A_{ij} - \frac{k_i k_j}{2m}\bigg]\delta (c_i,c_j)\]
+
+where:
+\begin{itemize}
+  \item $A_{ij}$ represents the edge weight between nodes $i$ and $j$;
+  \item $k_i$ and $k_j$ are the sum of the weights of the edges attached to nodes $i$ and $j$, respectively;
+  \item $m$ is the sum of all of the edge weights in the graph;
+  \item $c_i$ and $c_j$ are the communities of the nodes; and 
+  \item $\delta (x,y)= 1$ if $x=y$, $0$ otherwise.
+\end{itemize}
 
 ## Louvain algorithm
 
-### Move nodes to optimized the modularity
+### Move nodes to optimize the modularity
 
 \begin{columns}
 \column{0.5\textwidth}
@@ -639,7 +654,7 @@ Measures the density of links inside communities compared to links between commu
 
 ## Louvain algorithm
 
-### Move nodes to optimized the modularity
+### Move nodes to optimize the modularity
 
 \begin{columns}
 \column{0.5\textwidth}
@@ -659,6 +674,14 @@ Measures the density of links inside communities compared to links between commu
 Measures the density of links inside communities compared to links between communities
 
 
+## Louvain algorithm
+
+\begin{center}
+\href{https://www.nature.com/articles/s41467-021-27464-5}{
+  \includegraphics[width=0.7\textwidth]{img/paper_cluster_louvain.png}
+}
+\end{center}
+
 ## Validation methods
 
 \begin{center}
@@ -671,14 +694,14 @@ Partition data in 2 and compare the two clustering
 
 ### Adjusted mutual information (AMI)
 
-**AMI** takes a value of $1$ when the two clusterings are identical and $0$ when the **MI** between two partitions equals the value expected due to chance alone.
+**AMI** takes a value of $1$ when the two clustering are identical and $0$ when the **MI** between two partitions equals the value expected due to chance alone.
 
 ### V-measure
 
-Geometric mean between the **homogeneity** (how much the sample in a cluster are similar) and the **Completeness** (how much similar samples are put together by the clustering algorithm)
+Geometric mean between the **homogeneity** (how much the sample in a cluster is similar) and the **Completeness** (how much similar samples are put together by the clustering algorithm)
 
 \begin{center}
-Cell which flip assignement must be labeled as ambiguous
+Cell which flip assignment must be labeled as ambiguous
 \end{center}
 
 ## SC3
@@ -691,7 +714,7 @@ Cell which flip assignement must be labeled as ambiguous
 
 ## Mixture model
 
-### We model the probability of belonging to one groupe
+### We model the probability of belonging to one group
 
 \begin{columns}
 \column{0.5\textwidth}
@@ -706,7 +729,7 @@ We can fit
 
 with
 \begin{itemize}
-  \item $K$ the number of cluster
+  \item $K$ the number of clusters
   \item $\alpha_i$ the proportion of cluster $i$
   \item $\theta$ the model parameters
 \end{itemize}
@@ -715,7 +738,7 @@ with
 
 ## Mixture model
 
-### We cannot direcly cluster the multidimentional distribution of gene expression
+### We cannot directly cluster the multidimensional distribution of gene expression
 
 \begin{center}
 \href{https://doi.org/10.1093/bioinformatics/btac136}{
@@ -725,7 +748,7 @@ with
 
 ## Mixture model
 
-### We cannot direcly cluster the multidimentional distribution of gene expression
+### We cannot directly cluster the multidimensional distribution of gene expression
 
 \begin{center}
 \href{https://doi.org/10.1093/bioinformatics/btac136}{
@@ -749,7 +772,7 @@ with
 
 \begin{center}
 \href{https://www.nature.com/articles/s42256-019-0037-0/}{
-  \includegraphics[width=\textwidth]{img/scDeepCluster.png}
+  \includegraphics[width=0.8\textwidth]{img/scDeepCluster.png}
 }
 \end{center}
 
@@ -771,6 +794,14 @@ with
 }
 \end{center}
 
+## clusterProfiler
+
+\begin{center}
+\href{https://www.nature.com/articles/s41467-021-27464-5}{
+  \includegraphics[width=0.7\textwidth]{img/paper_cluster_id.png}
+}
+\end{center}
+
 ## Harmony
 
 \begin{center}
@@ -787,6 +818,15 @@ with
 }
 \end{center}
 
+## Harmony
+
+\begin{center}
+\href{https://www.nature.com/articles/s41467-021-27464-5}{
+  \includegraphics[width=0.7\textwidth]{img/paper_harmony.png}
+}
+\end{center}
+
+
 ## GNN
 
 \begin{center}