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non linear models

3 messages · Christian Endter, Brian Ripley, John Fox

Original
Christian Endter
# 
Dear Members of the Help List,

Honestly, I feel a little bit stupid - I would like to do something rather
simple: fit a non linear model to existing data, to be more precise I wanted
to start with simple higher order polynomials.

Unfortunately, I do not quite understand the examples in the helpfiles for
the nlm, nls and nlsModel commands.

Could anyone please provide a simple example to get me started (i.e. y = p +
x^2 fitted to x= -1 0 1 y = 2 1 2; a simple parabola p should turn out to be
1). How do I do this and how do I do the same for something like y = a + bx
+ cx^2 + dx^3 ??


Thank you very much,

Christian Endter

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Brian Ripley
#
On Sun, 30 Sep 2001, Christian Endter wrote:

            

      
      
      
    

        
Which in that sense are not non-linear models...

            

      
      
      
    

        
Fitting to noise-free data is not a good test, so I've added some more
observations (or the cubic is not determined) and some noise.

set.seed(1)
x <- c(-1:1, runif(10, -1, 1))
y <- 1+ x^2+ rnorm(13, 0, 0.1)

quadratic:

lm(y ~ offset(x^2))
Coefficients:
(Intercept)
      0.977

via nls:

nls(y ~ p + I(x^2), start=list(p=0))
Nonlinear regression model
  model:  y ~ p + I(x^2)
   data:  parent.frame
        p
0.9769938
 residual sum-of-squares:  0.1429682

General cubic:

lm(y ~ x + I(x^2) + I(x^3))
Coefficients:
(Intercept)            x       I(x^2)       I(x^3)
     1.0224      -0.1833       0.9425       0.2188

or (better) use orthogonal polynomials

lm(y ~ poly(x, 3))
Coefficients:
(Intercept)  poly(x, 3)1  poly(x, 3)2  poly(x, 3)3
     1.4543      -0.8127       1.2372       0.1289

or via nls

nls(y ~ a + b*x +  c*x^2 + d*x^3, start=list(a=0, b=0, c=0, d=0))
Nonlinear regression model
  model:  y ~ a + b * x + c * x^2 + d * x^3
   data:  parent.frame
         a          b          c          d
 1.0224028 -0.1833263  0.9425109  0.2187559
 residual sum-of-squares:  0.1213026

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      John Fox
    

    

      
      #
    

  


  

      
Dear Christian,

Polynomial-regression models are linear in the parameters, so they can be 
fit by lm. There are several ways to do your example:

 > x <- c(-1, 0, 1)
 > y <- c(2, 1, 2)

 > lm(y ~ I(x^2))

Call:
lm(formula = y ~ I(x^2))

Coefficients:
(Intercept)       I(x^2)
           1            1

 > lm(y ~ x + I(x^2))  # this fits a term in x as well as x^2

Call:
lm(formula = y ~ x + I(x^2))

Coefficients:
(Intercept)            x       I(x^2)
    1.00e+00    -7.85e-17     1.00e+00

 > lm(y ~ poly(x, 2)) # this fits order-2 orthogonal polynomials

Call:
lm(formula = y ~ poly(x, 2))

Coefficients:
(Intercept)  poly(x, 2)1  poly(x, 2)2
   1.667e+00   -3.127e-16    8.165e-01

I hope that this helps,
  John

        
At 11:59 AM 30/09/2001 -0400, Christian Endter wrote:

            

      
      
      
    

        
-----------------------------------------------------
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario, Canada L8S 4M4
email: jfox at mcmaster.ca
phone: 905-525-9140x23604
web: www.socsci.mcmaster.ca/jfox
-----------------------------------------------------

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