weighted adjacency matrix

Q: 0 means no link. but small value means a very close link.

In igraph, direction is from Column to row. The following example show arrow from 2nd and 3rd to 1st.

In Yuan, network exact control paper, the directions are from row to columns. So, is the transpose of the igraph adjacency matrix.

Fitting with weights, gnls, weights and form, in R

gnls with weights (form)
http://stackoverflow.com/questions/10508474/step-halving-issue-in-gnlsnlme

dat:

Site_Code   SF
5   3
5   0
5   2
5   0
5   0
5   0

library(nlme)
g0 <- gnls(SF ~ a * Site_Code^b, data = dat,
           weights = varPower(form = ~Site_Code),
           start=list(a=30,b=-0.5))

This example indicates that averaged data should NOT be used for regression if weights

are used. Instead, original data should be used, because the noises in the un-averaged

data can be used as weights. 

The following example shows varPower() can bring the fitting model closer to 'truth'. 

> require(nlme)
> x1 = rnorm(20)
> y1 = x1 + rnorm(20)/10
> 
> #weird ones
> x2 = rnorm(5)+5
> y2 = rnorm(5)
> 
> y=c(y1,y2)
> x=c(x1,x2)
> mydata = data.frame(cbind(y,x))
> 
> foo = function(a,b) { y = x*a + b }

> model1 = gnls( y ~foo(a,b), start=list(a=1,b=0))
> summary(model1)
Generalized nonlinear least squares fit
  Model: y ~ foo(a, b) 
  Data: NULL 
       AIC      BIC    logLik
  72.21925 75.87587 -33.10962

Coefficients:
       Value  Std.Error    t-value p-value
a -0.0159375 0.08311006 -0.1917637  0.8496
b -0.3550085 0.20216324 -1.7560486  0.0924

 Correlation: 
  a     
b -0.346

Standardized residuals:
       Min         Q1        Med         Q3        Max 
-1.8712584 -0.7939610  0.2684611  0.7910997  1.2959748 

Residual standard error: 0.9485101 
Degrees of freedom: 25 total; 23 residual
> 
> model2 = gnls( y ~foo(a,b), data=mydata, start=list(a=1,b=0), weights=varPower(form = ~x))
> summary(model2)
Generalized nonlinear least squares fit
  Model: y ~ foo(a, b) 
  Data: mydata 
       AIC      BIC    logLik
  20.85874 25.73424 -6.429369

Variance function:
 Structure: Power of variance covariate
 Formula: ~x 
 Parameter estimates:
   power 
1.468287 

Coefficients:
       Value  Std.Error   t-value p-value
a  0.9281320 0.06104714 15.203530  0.0000
b -0.0255823 0.00891448 -2.869751  0.0087

 Correlation: 
  a   
b 0.17

Standardized residuals:
       Min         Q1        Med         Q3        Max 
-2.0273323 -0.8480791 -0.1211550  0.3073090  2.2141947 

Residual standard error: 0.4204751 
Degrees of freedom: 25 total; 23 residual
> model1
Generalized nonlinear least squares fit
  Model: y ~ foo(a, b) 
  Data: NULL 
  Log-likelihood: -33.10962

Coefficients:
         a          b 
-0.0159375 -0.3550085 

Degrees of freedom: 25 total; 23 residual
Residual standard error: 0.9485101 
> model2
Generalized nonlinear least squares fit
  Model: y ~ foo(a, b) 
  Data: mydata 
  Log-likelihood: -6.429369

Coefficients:
          a           b 
 0.92813202 -0.02558234 

Variance function:
 Structure: Power of variance covariate
 Formula: ~x 
 Parameter estimates:
   power 
1.468287 
Degrees of freedom: 25 total; 23 residual
Residual standard error: 0.4204751