# adjacency.polyReg: Adjacency matrix based on polynomial regression In WGCNA: Weighted Correlation Network Analysis

## Adjacency matrix based on polynomial regression

### Description

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`.

### Usage

```adjacency.polyReg(datExpr, degree=3, symmetrizationMethod = "mean")
```

### Arguments

 `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).

### 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)).

### Value

An adjacency matrix of dimensions ncol(datExpr) times ncol(datExpr).

### Author(s)

Lin Song, Steve Horvath

### References

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

For more information about polynomial regression, please refer to functions `poly` and `glm`

### Examples

```#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