Adjacency matrix based on polynomial regression
adjacency.polyReg calculates a network adjacency matrix by fitting polynomial regression models to pairs of variables (i.e. pairs of columns from
datExpr). Each polynomial fit results in a model fitting index R.squared.
Thus, the n columns of datExpr result in an n x n dimensional matrix whose entries contain R.squared
measures. This matrix is typically non-symmetric. To arrive at a (symmetric) adjacency matrix, one can
specify different symmetrization methods with symmetrizationMethod.
adjacency.polyReg(datExpr, degree=3, symmetrizationMethod = "mean")
datExpr |
data frame containing numeric variables. Example: Columns may correspond to genes and rows to observations (samples). |
degree |
the degree of the polynomial. Must be less than the number of unique points. |
symmetrizationMethod |
character string (eg "none", "min","max","mean") that specifies the method used to symmetrize the pairwise model fitting index matrix (see details). |
A network adjacency matrix is a symmetric matrix whose entries lie between 0 and 1. It is a special case of a similarity matrix.
Each variable (column of datExpr) is regressed on every other variable, with each model fitting index recorded in a square matrix. Note that the model fitting index of regressing variable x and variable y is usually different from that of regressing y on x. From the polynomial regression model glm(y ~ poly(x,degree)) one can calculate the model fitting index R.squared(y,x).
R.squared(y,x) is a number between 0 and 1. The closer it is to 1, the better the polynomial describes the relationship between x and y and the more significant is the pairwise relationship between the 2 variables. One can also reverse the roles of x and y to arrive at a model fitting index R.squared(x,y). If degree>1 then R.squared(x,y) is typically different from R.squared(y,x). Assume a set of n variables x1,...,xn (corresponding to the columns of datExpr then one can define R.squared(xi,xj). The model fitting indices for the elements of an n x n dimensional matrix (R.squared(ij)).
symmetrizationMethod implements the following symmetrization methods:
A.min(ij)=min(R.squared(ij),R.squared(ji)),
A.ave(ij)=(R.squared(ij)+R.squared(ji))/2,
A.max(ij)=max(R.squared(ij),R.squared(ji)).
An adjacency matrix of dimensions ncol(datExpr) times ncol(datExpr).
Lin Song, Steve Horvath
Song L, Langfelder P, Horvath S Avoiding mutual information based co-expression measures (to appear).
Horvath S (2011) Weighted Network Analysis. Applications in Genomics and Systems Biology. Springer Book. ISBN: 978-1-4419-8818-8
#Simulate a data frame datE which contains 5 columns and 50 observations m=50 x1=rnorm(m) r=.5; x2=r*x1+sqrt(1-r^2)*rnorm(m) r=.3; x3=r*(x1-.5)^2+sqrt(1-r^2)*rnorm(m) x4=rnorm(m) r=.3; x5=r*x4+sqrt(1-r^2)*rnorm(m) datE=data.frame(x1,x2,x3,x4,x5) #calculate adjacency by symmetrizing using max A.max=adjacency.polyReg(datE, symmetrizationMethod="max") A.max #calculate adjacency by symmetrizing using max A.mean=adjacency.polyReg(datE, symmetrizationMethod="mean") A.mean # output the unsymmetrized pairwise model fitting indices R.squared R.squared=adjacency.polyReg(datE, symmetrizationMethod="none") R.squared
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