Nonlinear total least squares / Deming regression in R

Question

I've been using nls() to fit a custom model to my data, but I don't like how the model is fitting and I would like to use an approach that minimizes residuals in both x and y axes.

I've done a lot of searching, and have found solutions for fitting linear models:

• via the deming package,
• various stackoverflow posts:
  • total-least-square-method-using-r,
  • how-to-calculate-total-least-squares-in-r-orthogonal-regression,
• and this very nice CrossValidated post: How to perform orthogonal regression (total least squares) via PCA?.
• I've also found a MATLAB solution: Total least squares curve fit problem,

but these fit a second order polynomial and not a custom, user-defined model.

What I would like is something similar to nls() that does the x and y residual minimization. This would allow me to enter my custom model. Is anyone aware of any solution in R?

Here's an example, but please note that I'm seeking suggestions on a general solution for nonlinear total least squares regression, and not something specific to this dataset (this is just an example data from here):

df <- structure(list(x = c(3, 4, 5, 6, 7, 8, 9, 10, 11), y = c(1.0385, 
1.0195, 1.0176, 1.01, 1.009, 1.0079, 1.0068, 1.0099, 1.0038)), .Names = c("x", 
"y"), row.names = c(NA, -9L), class = "data.frame")

(nlsfit <- nls(y ~ a^b^x, data = df, start = c(a=0.9, b=0.6)))

library(ggplot2)
ggplot(df, aes(x=x, y=y)) + 
    geom_point() + 
    geom_smooth(method="nls", formula = y ~ a^b^x, se=F, start = list(a=0.9, b=0.6))

Does anyone have any suggestion for how I might proceed?

Answer 1

There is a technique called "Orthogonal Distance Regression" that does this. An implementation in R was recently released:

http://www.r-bloggers.com/introducing-orthogonal-nonlinear-least-squares-regression-in-r/