• The importance of distances
• Euclidian distance
• some matrix algebra notation
• Distance Exercises: p. 322-323

• High-dimensional data are complex and impossible to visualize in raw form
• Thousands of dimensions, we can only visualize 2-3
• Distances can simplify thousands of dimensions

• Distances can help organize samples and genomic features

• Any clustering or classification of samples and/or genes involves combining or identifying objects that are close or similar.
• Distances or similarities are mathematical representations of what we mean by close or similar.
• The choice of distance is important and requires thought.
  • choice is subject-matter specific

Source: http://master.bioconductor.org/help/course-materials/2002/Summer02Course/Distance/distance.pdf

A metric satisfies the following five properties:

1. non-negativity \(d(a, b) \ge 0\)
2. symmetry \(d(a, b) = d(b, a)\)
3. identification mark \(d(a, a) = 0\)
4. definiteness \(d(a, b) = 0\) if and only if \(a=b\)
5. triangle inequality \(d(a, b) + d(b, c) \ge d(a, c)\)
   • A distance is only required to satisfy 1-3.
   • A similarity function satisfies 1-2, and increases as \(a\) and \(b\) become more similar
   • A dissimilarity function satisfies 1-2, and decreases as \(a\) and \(b\) become more similar

• Remember grade school: Euclidean d = \(\sqrt{ (A_x-B_x)^2 + (A_y-B_y)^2}\).
• Side note: also referred to as \(L_2\) norm

##biocLite("genomicsclass/tissuesGeneExpression")
library(tissuesGeneExpression)
data(tissuesGeneExpression)
dim(e) ##gene expression data

## [1] 22215   189

table(tissue) ##tissue[i] corresponds to e[,i]

## tissue
##  cerebellum       colon endometrium hippocampus      kidney       liver 
##          38          34          15          31          39          26 
##    placenta 
##           6

Interested in identifying similar samples and similar genes

• Points are no longer on the Cartesian plane,
• instead they are in higher dimensions. For example:
  • sample \(i\) is defined by a point in 22,215 dimensional space: \((Y_{1,i},\dots,Y_{22215,i})^\top\).
  • feature \(g\) is defined by a point in 189 dimensions \((Y_{g,189},\dots,Y_{g,189})^\top\)

Euclidean distance as for two dimensions. E.g., the distance between two samples \(i\) and \(j\) is:

\[ \mbox{dist}(i,j) = \sqrt{ \sum_{g=1}^{22215} (Y_{g,i}-Y_{g,j })^2 } \]

and the distance between two features \(h\) and \(g\) is:

\[ \mbox{dist}(h,g) = \sqrt{ \sum_{i=1}^{189} (Y_{h,i}-Y_{g,i})^2 } \]

The distance between samples \(i\) and \(j\) can be written as:

\[ \mbox{dist}(i,j) = \sqrt{ (\mathbf{Y}_i - \mathbf{Y}_j)^\top(\mathbf{Y}_i - \mathbf{Y}_j) }\]

with \(\mathbf{Y}_i\) and \(\mathbf{Y}_j\) columns \(i\) and \(j\).

t(matrix(1:3, ncol=1))

##      [,1] [,2] [,3]
## [1,]    1    2    3

matrix(1:3, ncol=1)

##      [,1]
## [1,]    1
## [2,]    2
## [3,]    3

t(matrix(1:3, ncol=1)) %*% matrix(1:3, ncol=1)

##      [,1]
## [1,]   14

Note: R is very efficient at matrix algebra

kidney1 <- e[, 1]
kidney2 <- e[, 2]
colon1 <- e[, 87]
sqrt(sum((kidney1 - kidney2)^2))

## [1] 85.8546

sqrt(sum((kidney1 - colon1)^2))

## [1] 122.8919

dim(e)

## [1] 22215   189

(d <- dist(t(e[, c(1, 2, 87)])))

##                 GSM11805.CEL.gz GSM11814.CEL.gz
## GSM11814.CEL.gz         85.8546                
## GSM92240.CEL.gz        122.8919        115.4773

class(d)

## [1] "dist"

Excerpt from ?dist:

dist(x, method = "euclidean", diag = FALSE, upper = FALSE, p = 2)

• method: the distance measure to be used.
  • This must be one of "euclidean", "maximum", "manhattan", "canberra", "binary" or "minkowski". Any unambiguous substring can be given.
• dist class output from dist() is used for many clustering algorithms and heatmap functions

Caution: dist(e) creates a 22215 x 22215 matrix that will probably crash your R session.

• In practice, features (e.g. genes) are typically "standardized", i.e. converted to z-score:

\[x_{gi} \leftarrow \frac{(x_{gi} - \bar{x}_g)}{s_g}\]

• This is done because the differences in overall levels between features are often not due to biological effects but technical ones, e.g.:
  • GC bias, PCR amplification efficiency, …
• Euclidian distance and \(1-r\) (Pearson cor) are related:
  • \(\frac{d_E(x, y)^2}{2m} = 1 - r_{xy}\)

• A built html version of my full Chapter 8 lecture is available.
• The source R Markdown is also available from Github.
• Source for this file is also availble from the Book Club Github
  • See folder: Data Analysis for the Life Sciences/Chapter 08 - Distance and Dimension Reduction/