In vguillemot/ReMUSE: The package for the e-MUSE R course
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
echo = TRUE
)
library(ggplot2)
First things first
We will need the packages corrplot, FactoMineR and factoextra:
- Check that
corrplot, FactoMineR and factoextra are installed
- If not, install them, then load them
library(corrplot)
library(FactoMineR)
library(factoextra)
We also need to load dplyr, tidyr and the "fruits" data, :
library(dplyr)
library(tidyr)
data("fruits", package = "ReMUSE")
Unsupervised/Exploratory vs Supervised Analysis
- In the supervised methods, we observe both a set of features/variables (e.g. gene expression) for each object, as well as a response or outcome variable (e.g. metastasis information or survival information of the patients).
- The goal is then to predict the response using the variables (e.g. which genes predict best or are associated with the survival of the patients).
- Here we instead focus on exploratory analysis, we are not interested (yet) in prediction.
The two exploratory methods we are interested in...
-
Principal Component Analysis, whose goal is to extract the most "important" sources of variability in the quantitative data
-
Hierachical Agglomerative Clustering, whose goal is to define clusters of samples
Program for this morning
Covariance and correlation :
- reminders, and/or not
- visualization
Principal Component Analysis:
- principle (vocabulary!), method, a little bit of theory
- PCA with
FactoMineR and visualization with factoextra
Covariance and correlation
A little bit of history {.columns-2}
{width="100%"}
{width="100%"}
On fruits {.columns-2}
library(ggplot2)
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point() + theme_bw()
\newcolumn
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
Exercise {.columns-2}
Make a graph as similar as the one the right!
Try to focus on the essential.
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point(color = "white", shape = 2) + theme_dark() + scale_y_log10() + scale_x_log10() + theme(panel.grid = element_line(color = "lightgrey"))
Barycenter (reminder)
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point(alpha = 0.5) +
annotate("point",
x = mean(fruits$Phosphore),
y = mean(fruits$Potassium),
shape = 17, size = 5, col = 2) +
annotate("label",
x = mean(fruits$Phosphore) + 10,
y = mean(fruits$Potassium) + 50,
label = "Barycenter",
size = 5, col = 2) +
annotate("text",
x = mean(fruits$Phosphore) + 6,
y = 50,
label = "Mean of x",
col = 2) +
annotate("text",
x = 6.5,
y = mean(fruits$Potassium) + 50,
label = "Mean of y",
col = 2) +
annotate("segment",
x = mean(fruits$Phosphore),
xend = 0,
y = mean(fruits$Potassium),
yend = mean(fruits$Potassium),
col = 2 , linetype = "dashed") +
annotate("segment",
x = mean(fruits$Phosphore),
xend = mean(fruits$Phosphore),
y = mean(fruits$Potassium),
yend = 0,
col = 2 , linetype = "dashed") +
scale_x_log10() + scale_y_log10() +
theme_bw()
Covariance (reminder)
How far away is a "dot" from the barycenter ? Individual rectangle area :
[
\left(x_i - \bar x\right) \times \left(y_i - \bar y\right)
]
The covariance is (almost) the mean area:
[
\operatorname{cov} (x, y) = \frac{1}{n-1} \sum_{i=1}^n\left(x_i - \bar x\right) \times \left(y_i - \bar y\right)
]
Correlation coefficient
Covariance can vary between $-\infty$ and $+\infty$.
Correlation is, by definition, a measure of linear relationship between -1 and +1:
- -1 represents a perfect negative linear relationship,
- 0 represents no linear relationship,
- +1 represents a perfect positive linear relationship.
Pearson's correlation coefficient
[
\operatorname{cor}(x, y) = \frac{\displaystyle\sum_{i = 1}^n (x_i - \bar x)(y_i - \bar y)}{\displaystyle\sqrt{\sum_{i = 1}^n (x_i - \bar x)^2}\sqrt{\sum_{i = 1}^n (y_i - \bar y)^2}}
]
... in short...
[
\operatorname{cor}(x, y) = \frac{\operatorname{cov}(x, y)}{\operatorname{sd}(x) \operatorname{sd}(y) }
]
"Exercises" {.columns-2}
Covariance between two variables with R:
cov(fruits$Potassium,
fruits$Phosphore)
Correlation between two variables with R:
cor(fruits$Potassium,
fruits$Phosphore)
Exercise / challenge !
Compute the correlation between fruits$Potassium and fruits$Phosphore.
Constraints!
Use only the following functions/operations:
length to compute $n$,* to multiply, / to divide, - to substract,mean to compute the mean,sd to compute the standard-deviation,sum for the "$\sum_{i=1}^n$"- as many brackets as you need
Spearman's correlation coefficient
Often noted $\rho$. Same as Pearson's, but on the ranks! Let:
- $r_x$ be the ranks of $x$,
- $r_y$ be the ranks of $y$.
[
\rho(x, y) = \operatorname{cor}(r_x, r_y)
]
Key properties:
- not sensitive to extreme values,
- invariant by monotonous (increasing) transformation (e.g. log, square-root),
- not adapted when there are ex-aequos
"Exercises" {.columns-2}
With the argument method set to "spearman:
cor(fruits$Potassium,
fruits$Phosphore,
method = "spearman")
Same, but with the rank function:
cor(rank(fruits$Potassium),
rank(fruits$Phosphore))
Kendall's correlation coefficient
Pairs of points: select two samples $i$ and $j$.
- Concordant Pair : $\left(x_{i}
x_{j} \text { et } y_{i}>y_{j}\right)$
- Discordant Pair : $\left(x_{i}
y_{j}\right)$ OU $\left(x_{i}>x_{j} \text { et } y_{i}<y_{j}\right)$
[
\tau(x, y) = \displaystyle \frac{n_C - n_D}{n_0},
]
where $n_C$ is the number of concordant pairs, $n_D$ the number of discordant pairs and $n_0$ total number of pairs.
"Exercise": compare and comment {.columns-2}
Pearson: nice linear relationship.
cor(fruits$Potassium,
fruits$Phosphore,
method = "pearson")
Spearman: relationship is non-linear but monotonous.
cor(fruits$Potassium,
fruits$Phosphore,
method = "spearman")
Kendall: ex-aequos.
cor(fruits$Potassium,
fruits$Phosphore,
method = "kendall")
Beware of naked numbers!
Plot the correlation {.columns-2}
To compute all correlations:
cormat <- cor(fruits[, -(1:2)])
To make a "correlogram":
corrplot(cormat)
corrplot(cormat)
Exercise
Make a correlogram on the fruit data and change the color of the labels.
PCA on a two-dimensional data-set
Recall the first graph {.columns-2}
The real data:
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
To make my life easier
xy <- data.frame(
scale(
cbind(
x = log10(fruits$Phosphore),
y = log10(fruits$Potassium))))
ggplot(xy, aes(x)) +
geom_point(aes(y = y)) +
theme_bw() + coord_equal()
Exercise / discussion: What did I do? {.columns-2}
ggplot(fruits, aes(Phosphore, Potassium)) +
geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
xy <- data.frame(
scale(
cbind(
x = log10(fruits$Phosphore),
y = log10(fruits$Potassium))))
ggplot(xy, aes(x)) +
geom_point(aes(y = y)) +
theme_bw() + coord_equal()
Linear Regression, aka Linear Model {.columns-2}
- Model: $y = a x + b$
- $a = \frac{\operatorname{cov}(x, y)}{\operatorname{var}(x)}$ is the slope of the line,
- $b = \bar y - a \bar x$ is the intercept.
res.lm <- lm(y ~ x, data = xy)
ab <- coef(res.lm)
xy <- data.frame(
xy,
yhat = predict(res.lm))
ggplot(xy, aes(x)) +
geom_point(aes(y = y)) +
geom_abline(intercept = ab[1], slope = ab[2], color = 3) +
geom_point(aes(y = yhat), color = 3, shape = 4) +
geom_segment(aes(y = y, xend = x, yend = yhat), color = 3) +
theme_bw() + coord_equal()
First principal component {.columns-2}
- $\text{PC}_1 = a_1 x + a_2y$
- $a_1$ and $a_2$ are the loadings
- $\text{PC}_1$ is the first principal component
- $a_1$ and $a_2$ are computed such that $\text{PC}_1$ has maximum variance AND $a_1^2 + a_2^2 = 1$
res.pc <- prcomp(xy[, 1:2], scale. = FALSE)
pcslope <- res.pc$rotation[2, 1] / res.pc$rotation[1, 1]
pcintercept <- 0
ROT1 <- res.pc$rotation[, 1] %*% t(res.pc$rotation[, 1])
pchatx <- (as.matrix(xy[, 1:2]) %*% ROT1)[, 1]
pchaty <- (as.matrix(xy[, 1:2]) %*% ROT1)[, 2]
xy <- data.frame(
xy,
pchatx = pchatx,
pchaty = pchaty)
dat.ell <- data.frame(ellipse::ellipse(cov(xy[, 1:2])))
ggplot(xy) +
geom_point(aes(x = x, y = y)) +
geom_abline(intercept = pcintercept, slope = pcslope, color = 2) +
geom_point(aes(x = pchatx, y = pchaty), color = 2, shape = 3) +
geom_segment(aes(x = x, y = y, xend = pchatx, yend = pchaty), color = 2) +
geom_path(mapping = aes(x, y), data = dat.ell, color = 2, alpha = 0.5) +
theme_bw() + coord_equal()
Both on the same plot
ggplot(xy) +
geom_point(aes(x = x, y = y)) +
geom_abline(intercept = pcintercept, slope = pcslope, color = 2) +
geom_abline(intercept = ab[1], slope = ab[2], color = 3) +
theme_bw() + coord_equal()
Switching to another set of slides...
vguillemot/ReMUSE documentation built on Dec. 23, 2021, 3:09 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( collapse = TRUE, comment = "#>", echo = TRUE ) library(ggplot2)
First things first
We will need the packages corrplot, FactoMineR and factoextra:
- Check that
corrplot,FactoMineRandfactoextraare installed - If not, install them, then load them
library(corrplot) library(FactoMineR) library(factoextra)
We also need to load dplyr, tidyr and the "fruits" data, :
library(dplyr) library(tidyr) data("fruits", package = "ReMUSE")
Unsupervised/Exploratory vs Supervised Analysis
- In the supervised methods, we observe both a set of features/variables (e.g. gene expression) for each object, as well as a response or outcome variable (e.g. metastasis information or survival information of the patients).
- The goal is then to predict the response using the variables (e.g. which genes predict best or are associated with the survival of the patients).
- Here we instead focus on exploratory analysis, we are not interested (yet) in prediction.
The two exploratory methods we are interested in...
-
Principal Component Analysis, whose goal is to extract the most "important" sources of variability in the quantitative data
-
Hierachical Agglomerative Clustering, whose goal is to define clusters of samples
Program for this morning
Covariance and correlation :
- reminders, and/or not
- visualization
Principal Component Analysis:
- principle (vocabulary!), method, a little bit of theory
- PCA with
FactoMineRand visualization withfactoextra
Covariance and correlation
A little bit of history {.columns-2}
{width="100%"}
{width="100%"}
On fruits {.columns-2}
library(ggplot2) ggplot(fruits, aes(Phosphore, Potassium)) + geom_point() + theme_bw()
\newcolumn
ggplot(fruits, aes(Phosphore, Potassium)) + geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
Exercise {.columns-2}
Make a graph as similar as the one the right!
Try to focus on the essential.
ggplot(fruits, aes(Phosphore, Potassium)) + geom_point(color = "white", shape = 2) + theme_dark() + scale_y_log10() + scale_x_log10() + theme(panel.grid = element_line(color = "lightgrey"))
Barycenter (reminder)
ggplot(fruits, aes(Phosphore, Potassium)) + geom_point(alpha = 0.5) + annotate("point", x = mean(fruits$Phosphore), y = mean(fruits$Potassium), shape = 17, size = 5, col = 2) + annotate("label", x = mean(fruits$Phosphore) + 10, y = mean(fruits$Potassium) + 50, label = "Barycenter", size = 5, col = 2) + annotate("text", x = mean(fruits$Phosphore) + 6, y = 50, label = "Mean of x", col = 2) + annotate("text", x = 6.5, y = mean(fruits$Potassium) + 50, label = "Mean of y", col = 2) + annotate("segment", x = mean(fruits$Phosphore), xend = 0, y = mean(fruits$Potassium), yend = mean(fruits$Potassium), col = 2 , linetype = "dashed") + annotate("segment", x = mean(fruits$Phosphore), xend = mean(fruits$Phosphore), y = mean(fruits$Potassium), yend = 0, col = 2 , linetype = "dashed") + scale_x_log10() + scale_y_log10() + theme_bw()
Covariance (reminder)
How far away is a "dot" from the barycenter ? Individual rectangle area :
[ \left(x_i - \bar x\right) \times \left(y_i - \bar y\right) ]
The covariance is (almost) the mean area:
[ \operatorname{cov} (x, y) = \frac{1}{n-1} \sum_{i=1}^n\left(x_i - \bar x\right) \times \left(y_i - \bar y\right) ]
Correlation coefficient
Covariance can vary between $-\infty$ and $+\infty$.
Correlation is, by definition, a measure of linear relationship between -1 and +1:
- -1 represents a perfect negative linear relationship,
- 0 represents no linear relationship,
- +1 represents a perfect positive linear relationship.
Pearson's correlation coefficient
[ \operatorname{cor}(x, y) = \frac{\displaystyle\sum_{i = 1}^n (x_i - \bar x)(y_i - \bar y)}{\displaystyle\sqrt{\sum_{i = 1}^n (x_i - \bar x)^2}\sqrt{\sum_{i = 1}^n (y_i - \bar y)^2}} ]
... in short...
[ \operatorname{cor}(x, y) = \frac{\operatorname{cov}(x, y)}{\operatorname{sd}(x) \operatorname{sd}(y) } ]
"Exercises" {.columns-2}
Covariance between two variables with R:
cov(fruits$Potassium, fruits$Phosphore)
Correlation between two variables with R:
cor(fruits$Potassium, fruits$Phosphore)
Exercise / challenge !
Compute the correlation between fruits$Potassium and fruits$Phosphore.
Constraints!
Use only the following functions/operations:
lengthto compute $n$,*to multiply,/to divide,-to substract,meanto compute the mean,sdto compute the standard-deviation,sumfor the "$\sum_{i=1}^n$"- as many brackets as you need
Spearman's correlation coefficient
Often noted $\rho$. Same as Pearson's, but on the ranks! Let:
- $r_x$ be the ranks of $x$,
- $r_y$ be the ranks of $y$.
[ \rho(x, y) = \operatorname{cor}(r_x, r_y) ]
Key properties:
- not sensitive to extreme values,
- invariant by monotonous (increasing) transformation (e.g. log, square-root),
- not adapted when there are ex-aequos
"Exercises" {.columns-2}
With the argument method set to "spearman:
cor(fruits$Potassium, fruits$Phosphore, method = "spearman")
Same, but with the rank function:
cor(rank(fruits$Potassium), rank(fruits$Phosphore))
Kendall's correlation coefficient
Pairs of points: select two samples $i$ and $j$.
- Concordant Pair : $\left(x_{i}
- Discordant Pair : $\left(x_{i}
[ \tau(x, y) = \displaystyle \frac{n_C - n_D}{n_0}, ] where $n_C$ is the number of concordant pairs, $n_D$ the number of discordant pairs and $n_0$ total number of pairs.
"Exercise": compare and comment {.columns-2}
Pearson: nice linear relationship.
cor(fruits$Potassium, fruits$Phosphore, method = "pearson")
Spearman: relationship is non-linear but monotonous.
cor(fruits$Potassium, fruits$Phosphore, method = "spearman")
Kendall: ex-aequos.
cor(fruits$Potassium, fruits$Phosphore, method = "kendall")
Beware of naked numbers!
Plot the correlation {.columns-2}
To compute all correlations:
cormat <- cor(fruits[, -(1:2)])
To make a "correlogram":
corrplot(cormat)
corrplot(cormat)
Exercise
Make a correlogram on the fruit data and change the color of the labels.
PCA on a two-dimensional data-set
Recall the first graph {.columns-2}
The real data:
ggplot(fruits, aes(Phosphore, Potassium)) + geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
To make my life easier
xy <- data.frame( scale( cbind( x = log10(fruits$Phosphore), y = log10(fruits$Potassium)))) ggplot(xy, aes(x)) + geom_point(aes(y = y)) + theme_bw() + coord_equal()
Exercise / discussion: What did I do? {.columns-2}
ggplot(fruits, aes(Phosphore, Potassium)) + geom_point() + theme_bw() + scale_y_log10() + scale_x_log10()
xy <- data.frame( scale( cbind( x = log10(fruits$Phosphore), y = log10(fruits$Potassium)))) ggplot(xy, aes(x)) + geom_point(aes(y = y)) + theme_bw() + coord_equal()
Linear Regression, aka Linear Model {.columns-2}
- Model: $y = a x + b$
- $a = \frac{\operatorname{cov}(x, y)}{\operatorname{var}(x)}$ is the slope of the line,
- $b = \bar y - a \bar x$ is the intercept.
res.lm <- lm(y ~ x, data = xy) ab <- coef(res.lm) xy <- data.frame( xy, yhat = predict(res.lm)) ggplot(xy, aes(x)) + geom_point(aes(y = y)) + geom_abline(intercept = ab[1], slope = ab[2], color = 3) + geom_point(aes(y = yhat), color = 3, shape = 4) + geom_segment(aes(y = y, xend = x, yend = yhat), color = 3) + theme_bw() + coord_equal()
First principal component {.columns-2}
- $\text{PC}_1 = a_1 x + a_2y$
- $a_1$ and $a_2$ are the loadings
- $\text{PC}_1$ is the first principal component
- $a_1$ and $a_2$ are computed such that $\text{PC}_1$ has maximum variance AND $a_1^2 + a_2^2 = 1$
res.pc <- prcomp(xy[, 1:2], scale. = FALSE) pcslope <- res.pc$rotation[2, 1] / res.pc$rotation[1, 1] pcintercept <- 0 ROT1 <- res.pc$rotation[, 1] %*% t(res.pc$rotation[, 1]) pchatx <- (as.matrix(xy[, 1:2]) %*% ROT1)[, 1] pchaty <- (as.matrix(xy[, 1:2]) %*% ROT1)[, 2] xy <- data.frame( xy, pchatx = pchatx, pchaty = pchaty) dat.ell <- data.frame(ellipse::ellipse(cov(xy[, 1:2]))) ggplot(xy) + geom_point(aes(x = x, y = y)) + geom_abline(intercept = pcintercept, slope = pcslope, color = 2) + geom_point(aes(x = pchatx, y = pchaty), color = 2, shape = 3) + geom_segment(aes(x = x, y = y, xend = pchatx, yend = pchaty), color = 2) + geom_path(mapping = aes(x, y), data = dat.ell, color = 2, alpha = 0.5) + theme_bw() + coord_equal()
Both on the same plot
ggplot(xy) + geom_point(aes(x = x, y = y)) + geom_abline(intercept = pcintercept, slope = pcslope, color = 2) + geom_abline(intercept = ab[1], slope = ab[2], color = 3) + theme_bw() + coord_equal()
Switching to another set of slides...
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