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# Principal Components Analysis (PCA) # The purpose of principal component analysis is to find the best low-dementional representation of the variation in a multivariate data set. url <- “archive.ics.uci.edu/ml/machine-learning-databases/wine/wine.data” wine <- read.table(file = url, sep = “,”)

standardisedconcentration <- as.data.frame(scale(wine)) wine_pca <- prcomp(x = standardisedconcentration)

pcaSummary <- summary(wine_pca)

wine_pca$sdev sum((wine_pca$sdev)^2)

# deciding how many pricipal components to retain # using scree plot screeplot(wine_pca, type = “lines”) # the most obvious change in slope in the screee plot occurs at component 4, which is the ‘elbow’ of the scree plot.

# using Kaiser’s criterion (wine_pca$sdev)^2 # we should only retain principal components for which the variance is above 1 (when principal component analysis was applied to standardised data).

# A thrid way to decide how many principal components to retain is to decide to keep the number of components required to explain at least some minimum amount of the total variance.

# For example, if it is important to explain at least 80% of the variance, we would retain the first five principal components.

# loading for the Principal Components wine_pca$rotation # This means that the first component is a linear combination of the variables: -0.144*Z2 + 0.245*Z3 + 0.002*Z4 + 0.239*Z5 -0.142*Z6 -0.395*Z7 - 0.423*Z8 + 0.299*Z9 - 0.313*Z10 + 0.089*Z11 -0.29Z12 - 0.376*Z13 - 0.287*Z14

sum((wine_pca$rotation[, 1])^2) # Note that the square of the loadings sum to 1

# similarly, we can obtain the loadings for the second principal component wine_pca$rotation # this means that the second principal component is a linear combination of the variables: 0.484*Z2 + 0.225*Z3 + 0.316*Z4 - 0.011*Z5 + 0.300*Z6 + 0.065*Z7 - 0.003*Z8 + 0.029*Z9 + 0.039*Z10 + 0.530*Z11 - 0.279*Z12 - 0.164*Z13 + 0.365*Z14

# ———- Scatterplots of the Principal components plot(x = wine_pca$x[, 1], y = wine_pca$x[, 2],

pch = wine$V1, col = wine$V1, 
xlab = paste('PC1:', pcaSummary$importance[2, 1]*100, '%'),
ylab = paste('PC2:', pcaSummary$importance[2,2]*100, "%"))

legend(‘bottomleft’, legend = levels(as.factor(wine$V1)),

pch = unique(wine$V1), col = unique(wine$V1),
ncol = 1)

abline(h = 0, v = 0, lty = 2, col = ‘black’)

plot(x = wine_pca$x[, 1], y = wine_pca$x,

pch = wine$V1, col = wine$V1, xlab = 'PC1',
ylab = 'PC2', main = 'Principal Components Analysis')

abline(v = 0, h = 0, type = ‘l’)

# ———- Linear Discriminant Analysis # The purpose of principal component analysis is to find # the best low-dimensional representation of the variation # in a multivariate data set.

# 输入的数据相对于PCA来说可以标准化（每个样本的观察值减去# 均值除以标准差）也可以不用标准化

library(MASS) wine_lda <- lda(wine$V1 ~ wine$V2 + wine$V3 + wine$V4 + wine$V5 +

wine$V6 + wine$V7 + wine$V8 + wine$V9 + wine$V10 +
wine$V11 + wine$V12 + wine$V13 + wine$V14  )

wine_lda

# ———- separation achieved by the Discriminant Functions wine_lda_values <- predict(wine_lda, wine) wine_lda_values$x (wine_lda$svd)^2

# ———– A stacked Histogram of the LDA Values ldahist(data = wine_lda_values$x, g = wine$V1)

# ———– Scatterplots of the Discriminant Functions plot(x = wine_lda_values$x, y = wine_lda_values$x,

pch = wine$V1, col = wine$V1, xlab = 'PC1:68.75%', 
ylab = 'PC2:31.25%', 
main = "Linear Discriminant Analysis")

abline(v = 0, h = 0, lwd = 0.5, col = ‘black’,

lty = 2)

################################################################ # # 第二遍学习# ################################################################

# ——- reading multivariate analysis data into R # This data contains 13 different chemicals concentrations in winne # grown in the same region in Italy that are derived from different # cultivars.

# the row represent pre sample, columns represent different chemical’s # concentration.(V1 represent different cultivars)

url <- ‘archive.ics.uci.edu/ml/machine-learning-databases/wine/wine.data’ wine <- read.table(file = url, sep = “,”) head(wine)

library(car) scatterplotMatrix(x = wine)

plot(x = wine$V4, y = wine$V5, pch = wine$V1,

col = wine$V1, xlab = "V4", ylab = 'V5', main = "V5 ~ V4")

legend(‘topleft’, legend = c(“cultivar_1”, “cultivar_2”, ‘cultivar_3’),

pch = unique(wine$V1), col = unique(wine$V1))

cor(wine$V2, wine$V3) minerva::mine(x = wine$V4, y = wine$V5)

# ——— PCA # The purpose of principal component analysis is to find # the best low-dimentional of the variation in a multivariate # data set. wine_standard <- as.data.frame(scale(wine)) str(wine_standard) wine_pca <- prcomp(wine_standard) summary(wine_pca) sum((wine_pca$sdev)^2)

# Deciding to retain how many Principal Components screeplot(x = wine_pca, type = ‘lines’) (wine_pca$sdev)^2 >= 1 # 保留大于等于1的PC

# ——– loading pca plot plot(x = wine_pca$x, y = wine_pca$x,

pch = wine$V1, col = wine$V1,ylim = c(-4,4),
xlab = 'PC1:36.2%', ylab = 'PC2:19.2%',
main = 'Principal Components Analysis')

legend(‘topleft’, legend = c(‘cultivar1’, ‘cultivar2’, ‘cultivar3’),

pch = unique(wine$V1), col = unique(wine$V1), ncol = 3,
cex = 0.7)

abline(v = 0, h = 0, lty = 2)

# ———— analysis rna-seq data 0524 rna <- read.csv(file = ‘/home/yincy/Documents/R/RNA-seq/data/0524/nuron.csv’,

row.names = 1, header = T, stringsAsFactors = T)

rna <- t(rna) rna_standard <- scale(rna) rna_standard rna_pca <- prcomp(x = rna_standard) summary(rna_pca) plot(x = rna_pca$x, y = rna_pca$x,

xlab = 'PC1:40.7%', ylab = 'PC2:34.7%', pch = 16, col = "red")

text(x = rna_pca$x, y = rna_pca$x + 0.12,

labels = rownames(rna), cex = 0.9, xpd = TRUE)

abline(v = 0, h = 0)

#———————————————————–

# ——– Linear Discriminant Analysis # The purpose of principal component analysis is to find the # best low-demensional representation of the variation in a # multivariate data set.

# the purpose of linear discriminant analysis (LDA) is to find # the liear combinations of the original variables that gives # the best possible separation between the groups in our data set.

library(MASS) wine_lda <- lda(wine$V1 ~ wine$V2 + wine$V3 + wine$V4 + wine$V5

+ wine$V6 + wine$V7 + wine$V8 + wine$V9 + wine$V10
+ wine$V11 + wine$V12 + wine$V13 + wine$V14)

wine_lda$scaling wine_lda wine_lda_values <- predict(object = wine_lda, wine)

plot(x = wine_lda_values$x, y = wine_lda_values$x,

xlab = 'PC1:68.8%', ylab = 'PC2:31.2%',
pch = wine$V1, col = wine$V1,
main = 'Linear Discriminate Analysis')

abline(v = 0, h = 0, lty = 2) legend(‘top’, legend = c(‘cultivar1’, ‘cultivar2’, ‘cultivar3’),

pch = unique(wine$V1), col = unique(wine$V1),ncol = 3)