Distances Between Genes
and Samples
Naomi Altman
Oct. 06
Why Do We Care about
Distance?
Genes that are "close" may be in the same
pathway or have similar function.
Samples that are "close" may come from similar
subpopulations, or have an evolutionary
relationship.
Human Vs Chimp Brains
Some ideas from the picture
1. Some genes are much more highly expressed
in some tissue.
2. Brain regions are more similar across species
than species across brain regions (for these
genes).
Some questions based on the
picture
Suppose that we were missing the
expression for a gene in this picture for a
particular sample. How might we "impute"
a value?
Suppose we had a new sample and wanted
to know the species and brain region?
How could we infer this?
Suppose we have a gene of unknown
function in the first cluster. What might
one infer about the function?
Which gene is closest in
expression pattern to the
black gene?
Some common distance measures
'euclidean': Usual square distance between the two vectors (2
norm).
'maximum': Maximum distance between two components of x and
y (supremum norm)
'manhattan': Absolute distance between the two vectors (1 norm).
'canberra': sum(|x_i - y_i| / |x_i + y_i|). Terms with zero numerator
and denominator are omitted from the sum and treated as if the
values were missing.
'minkowski': The p norm, the pth root of the sum of the pth
powers of the differences of the components.
'correlation': 1-r where r is the Pearson or Spearman correlation
'absolute correlation': 1-|r|
Preprocessing
Often we preprocess, either by gene (row) or by
sample (or treatment) (column)
Centering
Y-c(Y) where c(Y) is the sample mean or
sample median
Standardizing (scaling, z-score)
Y-c(Y)
where s(Y) is a measure of spread
s(Y)
such as SD or IQR
Which gene is closest in
expression pattern to the
black gene?
Centered
Centered
and Scaled
Data Imputation
Some classification and clustering methods
require expression values for every sample (or
every gene in every sample)
Imputation is a technical term for clever methods
to make up data.
2 reasonable methods:
K-nearest neighbors (KNN)
Singular Value Decomposition (SVD)
KNN
Based on the distance metric of your choice, and
all the non-missing values of your gene G
compute d(G,x) for all other genes and select
the K closest x1 ... xk
The imputed value is the weighted average
S wixi where closer genes are more heavily
weighted.
Naturally, we use the imputed value only to fill in
the missing value - the other values for G are
kept.
SVD
Any matrix M can be decomposed as
Mg,s=Ug,gDg,sV's,s
where D is zero except on the
diagonal
Then Mvi = divi
and uiM=diui
V captures the main patterns of variation in the rows of
M
U captures the main directions of variation in the
columns of M
SVD
I made up a matrix as follows:
x=1:12
G15,12 has 4 rows which are sin(2p x/12) + noise
3 rows which are sin(2p x/12 + 1.5) +noise
7 rows which are noise
SVD
Any matrix M can be decomposed as
Mg,s=Ug,gDg,sVs,s
where D is zero except on the
diagonal
Use the k columns of V corresponding to the largest values of D
Then
is considered to be a less noisy version of M
M is the gene expression matrix with rows=genes and
columns = samples.
We start by putting the row average in any missing value
and then iteratively estimate V and regress the
observed values of the gene on the selected columns
of V to fill in the missing values
SVD
g.svd=svd(g)
Using SVD
The diagonal elements of D are called the
eigenvalues. The sum of the elements squared
is a measure of the overall variability of the
data.
We generally assume that treatment differences
create more variability than the noise in the
data.
So, the largest eigenvalues should express the
variability due to the treatments.
Using SVD
Scree is the jumble of rock
at the base of a mountain.
The scree plot
(eigenvalues vs rank) is
used to pick out the
informative eigenvalues
from the jumble (noise).
"Denoised Reconstruction"
Mg,s=Ug,gDg,sV's,s
If there are k informative eigenvalues, we pick the
corresponding k columns of U and V
M*=Ug,kDk,kV's,k is a denoised reconstruction of M
To use to fill in missing data: start by filling in M with
some estimate (e.g. row or column mean)
Compute M* and replace missing values in M by
corresponding values in M*
Iterate
The first 9 rows
of the matrix and
the
reconstruction
based on 2
eigenvalues.
Note that rows 8
and 9 are where
the pure noise
begin.
Some terminology for Data
Imputation
• Missing completely at random:
Pr( R| X(obs), X(miss) ) = Pr( R )
does not depend on the values in X .
• Missing at random:
Pr( R| X(obs), X(miss) ) = Pr( R| X(obs) )
depends on the values in X only through the ones we get to
observe.
• Missing NOT at random:
Pr( R| X(obs), X(miss) )
depends also on the values we do not get to observe – most
complicated situation
(These simple imputation methods do not work.)