Distance Metrics
Clustering
Clustering, like the name implies, is a way to group data points to find patterns. Clustering is type of unsupervised learning - where patterns are discovered without labeled data.
Distance Metrics
Before generating clusters we need to calculate the distance between observations:
To do so we have a few different options for distance metrics:
Euclidean Distance
\[d_{euc}(x,y) = \sqrt{\sum_{i=1}^n{(x_i - y_i)^2}}\]
Explanation of Terms
- \(x\) variable x
- \(y\) variable y
- \(n\) number of observations
Manhattan Distance
\[d_{man}(x,y) = \sum_{i=1}^n{|(x_i - y_i)|}\]
Explanation of Terms
- \(x\) variable x
- \(y\) variable y
- \(n\) number of observations
Eisen Cosine Correlation Distance
\[d_{eis}(x,y) = 1 - \frac{|\sum_{i=1}^n{x_iy_i}|}{\sqrt{\sum_{i=1}^n {x_i^2}\sum_{i=1}^n {y_i^2}}}\]
Explanation of Terms
- \(x\) variable x
- \(y\) variable y
- \(n\) number of observations
Pearson Correlation Distance
\[d_{pearson}(x,y) = 1 - \frac{\sum_{i=1}^n{(x - \mu_x)(y - \mu_y)}}{\sqrt{\sum_{i=1}^n{(x - \mu_x)^2} \sum_{i=1}^n{(y - \mu_y)^2}}} \]
Explanation of Terms
- \(x\) variable x
- \(y\) variable y
- \(\mu_x\) mean of variable x
- \(\mu_y\) mean of variable y
- \(n\) number of observations
Spearman Correlation Distance
\[d_{spearman}(x,y) = 1 - \frac{\sum{(x\prime - \mu_{x\prime} )(y\prime - \mu_{y\prime} )}}{\sqrt{\sum{(x\prime - \mu_{x\prime} )^2} \sum{(y\prime - \mu_{y\prime} )^2}}}\]
Explanation of Terms
- \(x\) variable x
- \(y\) variable y
- \(\mu_x\) mean of variable x
- \(\mu_y\) mean of variable y
- \(n\) number of observations
Distance Metrics In R
Let's try creating a distance matrix!
# load the libraries
.libPaths(c("/cluster/tufts/hpc/tools/R/4.0.0"))
library(tidyverse)
library(factoextra)
# load our counts data
counts <- read.csv(
file="data/gbm_cptac_2021/data_mrna_seq_fpkm.txt",
header = T,
sep = "\t")
# make the genes our rownames
rownames(counts) <- make.names(counts$Hugo_Symbol,unique = TRUE)
# remove the gene symbol column
counts <- counts %>%
select(-c(Hugo_Symbol))
# log2 transform our data
# transpose our data so that our patients are rows
counts <- t(log2(counts + 1))
# Change NA counts to 0
counts[!is.finite(counts)] <- 0
# generate correlation distance matrix
dist <- get_dist(counts,method = "pearson")
# plot correlation distance matrix
fviz_dist(dist) +
theme(axis.text = element_text(size = 3)) +
labs(
title = "Pearson Correlation Distances Between Samples",
fill = "Pearson Correlation"
)
