Outlines

Several popular dimension reduction approaches

• Linear methods
  • Principal component analysis (PCA)
  • Singular value decomposition (SVD)
  • Non-negative matrix factorization (NMF)
• Non-linear methods
  • Multi dimension scaling (MDS)
  • T-Distributed Stochastic Neighbor Embedding (tSNE)

Principal component analysis

• Can effectively perform dimension reduction
• A toy example of projecting 2-dimensional data onto 1-dimensional data.

Which direction to project?

• The information regarding the original data should be retained as much as possible.
• Or equivalently, the variance along the projected direction is maximized.

Projection results

An example

• Usually use the first principle component and the second principle component direction to visualize the data
• Motivating example: iris data, 150 observations, 4 variables (features)

head(iris)

##   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1          5.1         3.5          1.4         0.2  setosa
## 2          4.9         3.0          1.4         0.2  setosa
## 3          4.7         3.2          1.3         0.2  setosa
## 4          4.6         3.1          1.5         0.2  setosa
## 5          5.0         3.6          1.4         0.2  setosa
## 6          5.4         3.9          1.7         0.4  setosa

dim(iris)

## [1] 150   5

• Perform PCA
  • project these 150 observations from 4 dimensional space onto 2 dimensional space
  • Usually, we need to standardize each feature to mean 0 and sd 1.
• visualize the data.

iris.data <- iris[,1:4]
ir.pca <- prcomp(iris.data,
                 center = TRUE,
                 scale = TRUE) 
PC1 <- ir.pca$x[,"PC1"]
PC2 <- ir.pca$x[,"PC2"]
variance <- ir.pca$sdev^2 / sum(ir.pca$sdev^2)
v1 <- paste0("variance: ",signif(variance[1] * 100,3), "%")
v2 <- paste0("variance: ",signif(variance[2] * 100,3), "%")
plot(PC1, PC2, col=as.numeric(iris$Species),pch=19, xlab=v1, ylab=v2)
legend("topright", legend = levels(iris$Species), col =  unique(iris$Species), pch = 19)

Biplot

• Data points: projection of the 150 observations onto the first two PCs in a Biplot.
• Project the observations onto arrow gives original (standardized) value of that variable.
• Arrow length: Variance of a variable.
• Angle between Arrows: Correlation = cos(angle).

biplot(ir.pca)

• Perform PCA (using ggplot)
  • project these 150 observations from 4 dimentional space onto 2 dimensional space
• visualize the data.

suppressWarnings(suppressMessages(library(tidyverse)))
iris.data <- iris[,1:4]
ir.pca <- prcomp(iris.data,
                 center = TRUE,
                 scale = TRUE) 
PC1 <- ir.pca$x[,"PC1"]
PC2 <- ir.pca$x[,"PC2"]
data_PCA <- tibble(PC1=PC1, PC2=PC2, Species = iris$Species)
variance <- ir.pca$sdev^2 / sum(ir.pca$sdev^2)
v1 <- paste0("variance: ",signif(variance[1] * 100,3), "%")
v2 <- paste0("variance: ",signif(variance[2] * 100,3), "%")
black.bold.text <- element_text(face = "bold", color = "black", size=20)

data_PCA %>% 
  ggplot() +
  aes(x=PC1,y=PC2,color=Species) + 
  geom_point() + 
  labs(x = v1, y=v2) +    
  theme_bw() + 
  theme(text = black.bold.text) 

Geometrical motivation for PCA (population version)

• \({\bf x} \in \mathbb{R}^{G}\) is one sample with \(G\) features.
• Without loss of generosity,
  • assume \(\mathbb{E}({\bf x}) = {\bf 0}\)
  • \(Var({\bf x}) = \mathbb{E}({\bf x} {\bf x}^\top) = \Sigma \in \mathbb{R}^{G\times G}\)
• We want to a direction (in unit length) \(\alpha \in \mathbb{R}^{G}\) such that the variance of the projected value (\(\alpha^\top {\bf x}\)) is maximized. \[\max_\alpha Var(\alpha^\top {\bf x}) = \max_\alpha \alpha^\top \Sigma \alpha, s.t. \|\alpha\|_2 = 1\]
• Claim the solution: \(\alpha = {\bf v}_1\) which is the first eigen-value of \(\Sigma\)

Proof (page 1):

• Since \(\Sigma \in \mathbb{R}^{G\times G}\) is a symmetric and positive - definite matrix, there exists matrix \(V \in \mathbb{R}^{G\times G}\) and diagonal matrix \(D \in \mathbb{R}^{G\times G}\) such that \[\Sigma = V D V^\top,\]

where \(D = \begin{pmatrix} d_1 & 0 & 0 & \\ 0 & d_2 & & \\ 0 & & \ddots & 0\\ & & 0 & d_G \end{pmatrix},\)

\(d_1 \ge d_2 \ge \ldots \ge d_G \ge 0\) are eigenvalues. \({\bf V} = ({\bf v}_1, {\bf v}_2, \ldots, {\bf v}_G)\) are eigen-vectors.

• \({\bf v}\)’s are orthonormal
  • \({\bf v}_i^\top {\bf v}_j = 0\) if \(i \ne j\)
  • \(\|{\bf v}_g\|_2^2 = {\bf v}_g^\top {\bf v}_g = 1\), \(\forall 1 \le g \le G\)
• \(\Sigma {\bf v}_g = d_g {\bf v}_g\)

Proof (page 2):

For any vector \(\alpha \in \mathbb{R}^{G}\), it can be spanned as linear combination of \({\bf v}_i\)’s

• \(\alpha = \sum_{g=1}^G a_g {\bf v}_g\).
  • \(\sum_{g=1}^G a_g^2 = 1\) if \(\|\alpha\|_2^2 = 1\)
• When \(\|\alpha\|_2^2 = 1\),

\(\begin{aligned} Var(\alpha^\top {\bf x}) &= \alpha^\top \Sigma \alpha \\ &= (\sum_{g=1}^G a_g {\bf v}_g)^\top \Sigma (\sum_{g=1}^G a_g {\bf v}_g) \\ &= (\sum_{g=1}^G a_g {\bf v}_g)^\top(\sum_{g=1}^G a_g d_g {\bf v}_g) \\ &= \sum_{g=1}^G d_g a_g^2 \|{\bf v}_g\|_2^2 = \sum_{g=1}^G d_g a_g^2 \\ &\le \sum_{g=1}^G d_1 a_g^2 \\ &\le d_1 \\ \end{aligned}\)

Proof (page 3):

• If \(\alpha = {\bf v}_1\),

\(\begin{aligned} Var(\alpha^\top {\bf x}) &= \alpha^\top \Sigma \alpha \\ &= {\bf v}_1^\top \Sigma {\bf v}_1 \\ &= d_1 \\ \end{aligned}\)

• Therefore, \(\alpha = {\bf v}_1\) maximize \(\max_\alpha Var(\alpha^\top {\bf x})\).

• And the projected random variable \(L_1 = {\bf v}_1^\top {\bf x}\) has the largest variance \(d_1\)

• \({\bf v}_1 = (v_{11}, \ldots, v_{1G})\) are called loadings of PC1.

• Similarly, \(L_2 = {\bf v}_2^\top {\bf x}\) has the largest variance among all possible projections orthogonal to \(L_1\).

• Similarly, \(L_g = {\bf v}_g^\top {\bf x}\) has the largest variance among all possible projections orthogonal to \(L_1, L_2, \ldots, L_{g-1}\) for \(1 \le g \le G\).

PCA 2D example

Variance explained

• \(trace(\Sigma)\) is sum of variance of all features, is invariant of change of basis.
  • \(trace(A) = trace(V\Sigma V^\top) = trace(\Sigma V^\top V) = trace(\Sigma)\)
  • \(trace(\Sigma) = \sum_g d_g = \sum_{g} Var(L_g)\).
• Define \(r_g = \frac{d_g}{\sum_{g'} d_{g'}}\) where \(r_g\) represents the proportion of variance explained by the \(g^{th}\) PC.
• \(R_g = \sum_{g'=1}^g r_{g'}\) is the cummulative proportion of variance explained by the first \(g\) PCs.

PCA summary

• Project the original data into orthonormal space (\(\bf v_1, v_2, \ldots\)) where \(\bf v_g\) is the \(g^{th}\) eigen vector of the covariance matrix \(\Sigma\), such that the variance of the projected value along \(\bf v_1\) is maximized, then along \(\bf v_2\) is maximized, …
• \(L_1 = {\bf v_1}^\top {\bf x}\) is the projected value of \(x\) on the first principal component
  • \({\bf v_1}\) is referred as the loading of the first PC
  • \(L_1\) is referred as the score of the first PC.
  • The new score is the linear combination of original data
• \(L_2 = {\bf v_2}^\top {\bf x}\) is the projected value of \(x\) on the second principal component
• These projection directions are the eigenvectors by applying eigenvalue decomposition to the covariance matrix \(\Sigma\).
• Eigen value \(d_1\) is the variance explained by \(L_1\)
• The proportion of variance explained by the first \(g\) PCs is \(r_g = \frac{d_g}{\sum_{g'} d_{g'}}\).

Example 1, HAPMAP data

• The International HapMap Project was an organization that aimed to develop a haplotype map (HapMap) of the human genome, to describe the common patterns of human genetic variation.
• A subset of HAPMAP data (n = 210):
  • CEU Utah residents with Northern and Western European ancestry from the CEPH collection
  • CHB Han Chinese in Beijing, China
  • JPT Japanese in Tokyo, Japan
  • YRI Yoruba in Ibadan, Nigeria
• 712,940 SNPs:
  • AA: 0
  • AB: 0.5
  • BB: 1
• Project 210 samples on 712,940 dimentional space onto a 2 dimentional space.
  • \(210 \times 712,940\) matrix becomes \(210 \times 2\).

HAPMAP data visualization

In HW (Visualize HAPMAP data using chromosome 1)

• Human being have 23 pairs of chromosomes.

• Chromosome 1 of HAPMAP data is avaliable on /blue/phc6068/share/data/HAPMAP/chr1Data.Rdata

• Annotation is on /blue/phc6068/share/data/HAPMAP/relationships_w_pops_121708.txt

• These data are avaliable on hpg.rc.ufl.edu

• You can either perform the analysis on HiperGator, or downloaded the data to your local computer and perform the PCA analysis.

PCA steps (sample version)

• Center the data to mean 0 for each feature.
• Also standardize the varaince of each feature to be 1 after centering, otherwise feature with large variance will dominate.
• Calculate the sample covariance matrix.
• Perform eigen-value decomposition and obtain eigen-values and eigen-vectors.
• Project the original data onto the first eigen-vector,
  • the resulting projected value on the first eigen-vector is called first principal component.
  • the first eigen-value is the total variance explained by the first principal component.
• Repeat the projection step for the \(k^{th}\) eigen-value and eigen-vectors, \((1 \le k \le K)\)
  • the direction of \(k^{th}\) eigen-vector is perpendicular to all previous eigen-vectors.

The PCA has been implemented in many R packages. You don’t have to implement these individual steps.

How many principal component to use?

• Select \(K\) by eyeballing the inflection point of the scree plot, or total variance is greater than certain threshold (\(>90\%\))
• Scree plot

Example 2, yeast cycle data

• The yeast cell cycle analysis project’s goal is to identify all genes whose mRNA levels are regulated by the cell cycle. http://genome-www.stanford.edu/cellcycle/

https://le.ac.uk/-/media/uol/images/facilities/vgec/topics/cell-cycle-sc1.gif

Yeast cell cycle data

• Contains the expression profile of 800 genes \(\times\) 77 samples
• 800 genes can be categorized as 5 types by their peak time.
  1. G1
  2. S
  3. S/G2
  4. G2/M
  5. M/G1
• 77 yeast cells were first synchronized to the same G0 stage by four different chemicals:
  • alpha
  • cdc15
  • cdc28
  • elu

Cell cycle data (will be on HW)

library(impute)
raw = read.table("http://Caleb-Huo.github.io/teaching/data/cellCycle/cellCycle.txt",header=TRUE,as.is=TRUE)
cellCycle = raw
cellCycle[,2:78]<- impute.knn(as.matrix(raw[,2:78]))$data ## missing value imputation
dim(cellCycle)

Yeast cell cycle data PCA for genes

• Project 800 genes onto the first two PCs (samples are features).
• Each dot is a gene

Yeast cycle data PCA for samples

• Project all samples onto the first two PCs (genes are features).
• Each dot is a sample

Singular value decomposition (SVD)

• Singular value decomposition of \(M \in \mathbb{R}^{m \times n}\) can be factorized as \(U E V^\top\)
  • If \(m > n\)
    • \(U \in \mathbb{R}^{m \times n}\) such that \(U^\top U = I_n\).
    • \(E \in \mathbb{R}^{n \times n}\) diagonal matrix.
    • \(V \in \mathbb{R}^{n \times n}\) such that \(V^\top V = I_n\).
  • If \(m \le n\), \(M \in \mathbb{R}^{m \times n}\), \(M = UE V^\top\)
    • \(U \in \mathbb{R}^{m \times m}\) such that \(U^\top U= I_m\).
    • \(E \in \mathbb{R}^{m \times m}\) rectangular diagonal matrix.
    • \(V \in \mathbb{R}^{n \times m}\) such that \(V^\top V= I_m\).

Equivalence between PCA and SVD

• PCA

  • Covariance matrix \(\Sigma = Var(x)\) \[\Sigma = V D V^\top, or \mbox{ } V^\top \Sigma V= D\]

  • \(v_1\) is the first principal component
  • \(d_1\) is the variance explained by \(v_1\)

  • \(v_1^\top x\) is the score of x on the first principal component

• SVD
  • Suppose \(X = U E V^\top\), where \(X\) has been standardized (mean 0 and sd 1).

\(\begin{aligned} V^\top X^\top X V &= V^\top (U E V^\top)^\top U E V^\top V \\ &= V^\top V E U^\top U E V^\top V \\ &= E^2 \end{aligned}\)

• Equivalence of PCA and SVD
  • \(X_0 \in \mathbb{R}^{n \times p}\) is the observed data matrix.
  • \(\hat{\Sigma} = \frac{1}{n - 1} X_0^\top X_0\)
• Performing eigen-value decomposition on \(X^\top X\) is equivalent to perform SVD on \(X\)
  • \(V\) are eigen-vectors (projected directions)
  • \(E^2\) are eigen-values (explained variances)
  • Projected value \(XV = U E V^\top V = U E\)

Validate the SVD results using iris data

iris.data <- iris[,1:4]
iris.data.center <- scale(iris.data) ## mean 0 and std 1
asvd <- svd(iris.data.center)
UE <- asvd$u %*% diag(asvd$d)
PC1 <- UE[,1]
PC2 <- UE[,2]
variance <- asvd$d^2 / sum(asvd$d^2)
v1 <- paste0("variance: ",signif(variance[1] * 100,3), "%")
v2 <- paste0("variance: ",signif(variance[2] * 100,3), "%")
plot(PC1, PC2, col=as.numeric(iris$Species),pch=19, xlab=v1, ylab=v2)
legend("topright", legend = levels(iris$Species), col =  unique(iris$Species), pch = 19)

Important notes for PCA and SVD

• It is the best practice to scale each feature to
  • mean 0
  • sd 1

Non negetive matrix factorization (NMF)

• Non-negative matrix factorization (NMF) is a dimension reduction algorithm where a non-negative matrix, X, is factorized into two non-negative matrices
  • W and H, all elements must be equal to or greater than zero.
  • The method has been applied in image recognition, text mining and bioinformatics.

\[\min_{W\in \mathbb{R}^{p\times r}, H \in \mathbb{R}^{r \times n}} \|X - WH\|_F,\]
where \(X \in \mathbb{R}^{p \times n}\), \(\|X\|_F = \sqrt{\sum_{j=1}^p \sum_{i=1}^n X_{ji}^2}\)

• W: score matrix
• H: loading matrix

NMF example

suppressMessages(library(NMF))

iris.data <- iris[,1:4]
anmf <- nmf(iris.data, rank = 2, method = "lee")

W <- anmf@fit@W
H <- anmf@fit@H

plot(W[,1], W[,2], col=as.numeric(iris$Species),pch=19)
legend("topright", legend = levels(iris$Species), col =  unique(iris$Species), pch = 19)

NMF example

basismap(anmf) ## W

coefmap(anmf) ## H

Multi dimensional Scaling (MDS)

• Multi-dimensional scaling (MDS) aims to map data (distance or dissimilarity) structure to low dimensional space (usually 2D or 3D Euclidian space).
  • The input is dissimilarity matrix, not high dimentional data matrix (which is for PCA and SVD)
• Flight mileage of ten cities obtained from http://www.webyer.com/travel/mileage_calculator/.

z <- read.delim("https://Caleb-Huo.github.io/teaching/data/UScityDistance/Mileage.txt", row.names=1)
knitr::kable(z, caption = "distance between cities (miles)")

Multi dimensional Scaling (MDS)

• after apply MDS

• Will be on HW
  • Perform MDS on the following data, with both classical method and Sammon’s stress.

Math behind MDS

• objective function for classical MDS \[ L = \min \sum_{i<j} (d_{ij} - \delta_{ij})^2\] Assume the underlying data \(X \in \mathbb{n\times q}\) (Usually \(q=2\)) and \(\|X_i - X_j\|_2 = d_{ij}\).

• parameters:
  • \(i, j\): sample index
  • \(\delta_{ij}\): the dissimilarity measure between object i and j in the original data space.
  • \(d_{ij}\): the distance between the two objects after mapping to the targeted low-dimensional space.
• In the loss function above, large distances can dominate the optimization and ignore the local structure for pairs of short distances.
  • A possible modification may be to minimize the percent of squared loss. \[ L = \min \sum_{i<j} (\frac{d_{ij} - \delta_{ij}}{\delta_{ij}})^2 \]

MDS with Sammon’s stress

• This modification, however, may over-emphasize the local structure and easily distort the global structure.
• A better balance between the two is the Sammon’s stress. \[ L = \min \sum_{i<j} (\frac{d_{ij} - \delta_{ij}}{\delta_{ij}})^2 \times w_{ij},\] \(w_{ij} = \delta_{ij} / (\sum_{i<j} \delta_{ij})\)

Note:

• Reflection, translation and/or rotation (i.e. isometry) of an MDS solution is also an MDS solution since MDS only considers the preservation of the dissimilarity structure.

Classical mulditimensional scaling

• \(\delta_{ij}\) is the observed distance between sample \(i\) and \(j\), (\(1\le i \le n\), \(1\le j \le n\)).
• \(D = \{d_{ij}\}_{1\le i,j \le n}\) is derived from Euclidean distance of an unkown \(n \times q\) data matrix \(X \in \mathbb{R}^{n\times q}\) (Usually \(q=2\)).
  • \(\|X_i - X_j\|_2 = d_{ij}\).
  • Such a solution is not unique, because if \(X\) is the solution, \(X^* = X + c\), \(c \in \mathbb{R}^{q}\) is also solution, since \(\|X_i^* - X_j^*\|_2 = \|(X_i + c) - (X_j + c)\|_2 = \|X_i - X_j\|_2\)
  • So suppose \(X_i\) is centered, (i.e. \(\sum_{i=1}^n X_{iq} = 0\), for all \(q\)).
  • \(B = XX^{\top}\), \(B_{ij} = b_{ij}\)
• The following relationship can be derived
  • \(d_{ij}^2 = b_{ii} + b_{jj} - 2b_{ij}\)
  • \(\sum_{i=1}^n b_{ij} = 0\)
  • \(T = trace(B) = \sum_{i=1}^n b_{ii}\)
  • \(\sum_{i=1}^n d_{ij}^2 = T + nb_{jj}\)
  • \(\sum_{j=1}^n d_{ij}^2 = T + nb_{ii}\)
  • \(\sum_{i=1}^n \sum_{j=1}^n d_{ij}^2 = 2 n T\)

Classical mulditimensional scaling continue

• We can solve for: \[b_{ij} = -1/2 (d_{ij}^2 - d{\cdot j}^2 - d{i\cdot}^2 + d_{\cdot\cdot}^2)\]
  • \(d_{.j} = \frac{1}{n} \sum_{i = 1}^n d_{ij}\)
  • \(d_{i.} = \frac{1}{n} \sum_{j = 1}^n d_{ij}\)
  • \(d_{..} = \frac{1}{n^2} \sum_{i = 1}^n \sum_{j = 1}^n d_{ij}\)
• Use \(\delta_{ij}\) to approximate \(d_{ij}\).
• A solution \(X\) is then given by the eigen-decomposition of \(B\).
  • \(B = V\Lambda V^{\top}\)
  • \(X = \Lambda^{1/2} V^{\top}\)

MDS in R

• classical: cmdscale()
• MDS with Sammon’s stress: MASS::sammon()

Another MDS example - Genetic dissimilarity between races

z0 <- read.csv("https://Caleb-Huo.github.io/teaching/data/SNP_MDS/SNP_PCA.csv", row.names=1)
knitr::kable(z0, caption = "Genetic dissimilarity between races")

Another MDS example - Genetic dissimilarity between races

library(ggplot2)
library(ggrepel)

z <- as.dist(cbind(z0,NA))
z_mds = - cmdscale(z,k=2)

mdsDataFrame <- data.frame(Race=rownames(z_mds),x = z_mds[,1], y = z_mds[,2])

ggplot(data=mdsDataFrame, aes(x=x, y=y, label=Race)) +  
  geom_point(aes(color=Race)) + 
  geom_text_repel(data=mdsDataFrame, aes(label=Race)) +
  theme(legend.position = "none")

T-Distributed Stochastic Neighbor Embedding (tSNE)

• tSNE is a non linear dimension reduction method
• well suited for visualizing high-dimensional data in low-dimensional space
• try to perserve the dissimilarity structure after dimention reduction
• very popular for single cell RNA-Seq data visualization

applications with tSNE

library(tsne)

data_tsne0 <- tsne(iris[,1:4])

## Warning in if (class(X) == "dist") {: the condition has length > 1 and only the
## first element will be used

## sigma summary: Min. : 0.486505661043274 |1st Qu. : 0.587913800179832 |Median : 0.614872437640536 |Mean : 0.623051089344394 |3rd Qu. : 0.654914112723525 |Max. : 0.796707932771489 |

## Epoch: Iteration #100 error is: 12.990063489305

## Epoch: Iteration #200 error is: 0.277712292102509

## Epoch: Iteration #300 error is: 0.257101580790775

## Epoch: Iteration #400 error is: 0.246665630776044

## Epoch: Iteration #500 error is: 0.224003366859357

## Epoch: Iteration #600 error is: 0.223580775226078

## Epoch: Iteration #700 error is: 0.223572977457498

## Epoch: Iteration #800 error is: 0.223572819069482

## Epoch: Iteration #900 error is: 0.223572816097557

## Epoch: Iteration #1000 error is: 0.223572816039835

data_tsne <- data.frame(tsne1 = data_tsne0[,1], tsne2 = data_tsne0[,2], species = iris$Species)

ggplot(data_tsne) +
aes(x = tsne1, y = tsne2, color = species) +
geom_point()

Math behind tSNE

• High dimentional data (original data):
  • \(x_i \in \mathbb{R}^G\)
  • Distance between sample \(i\) and \(j\): \(p_{ij} = D_x(x_i, x_j)\)
• Low dimentional data (projected data):
  • \(y_i \in \mathbb{R}^2\) (2 dimension, can be 1 or 3 or more)
  • Distance between sample \(i\) and \(j\): \(q_{ij} =D_y(y_i, y_j)\)
• Objective function of tSNE
  • \(min_{y_i} \sum_{i\ne j} dist(p_{ij}, q_{ij})\)
  • \(p_{i,j}\)
    • \(p_{j|i} = \frac{\exp(-\|x_i - x_j\|^2/2\sigma_i)}{\sum_{k \ne i}\exp(-\|x_i - x_k\|^2/2\sigma_i)}\)
    • \(p_{j|i}\) is standardized Gaussian density function.
    • \(p_{i,j} = \frac{p_{j|i} + p_{i|j}}{2}\)
  • \(q_{ij}\)
    • \(q_{ij} = \frac{(1 + \|y_i - y_j\|^2)^{-1}}{\sum_{l \ne k}(1 + \|y_l - y_k\|^2)^{-1}}\)
    • \(q_{ij}\) is standardized t distribution with degree of freedom 1 (Cauchy distribution)
  • \(dist\)
    • \(dist(p_{ij}, q_{ij}) = KL(P||Q) = \sum_{i,j} p_{ij} \log \frac{p_{ij}}{q_{ij}}\)
    • KL represents KL divergence, which measures the distance between probabilities.