Using patsy, I understand how to turn intercepts on or off. But I haven't managed to get horizontal offsets. For instance, I would like to be able to fit, in essence
y = alpha + beta * abs(x_opt - x_obs)
with x_opt free in the fit. I tried write this like so:
y ~ 1 + np.abs(y - x)
using a constant column for y. But within the np.abs() parentheses, patsy "turns off," and y - x is just interpreted as a number. If I shift y to 1 or 20, I get different answers.
A similar question applies for e.g., np.pow(1-x, 2) or a sine wave. Being able to fit for the x offset would be extremely helpful. Is this possible? Or is this precisely what is meant that patsy doesn't do non-linear?
patsy and most of statsmodels only handle models that are linear in parameters. Or more precisely, models where the design matrix and estimated parameters are combined in a linear way, x * beta.
Polynomials and splines are nonlinear in the underlying variables but have a linear representation in terms of basis function and are therefore linear in parameters.
The only non-linearities in the models that are currently implemented in statsmodels are predefined nonlinearities like link functions in GLM or discrete models, shape parameters in models like NegativeBinomial, or covariances in mixed models and GEE.
The best Python package for nonlinear least squares is currently lmfit https://pypi.python.org/pypi/lmfit/