Response geometry

gamfit.fit accepts manifold-valued responses via response_geometry= and response_columns=. Each training response row is mapped to the tangent space at the intrinsic Fréchet mean of the training responses. A shared-smoothing Gaussian REML fit is then run on the tangent coordinates, and predictions are mapped back to the response manifold with the exponential map.

Supported geometries:

response_geometry	Response space	Tangent coordinates
"spherical"	unit sphere	ambient tangent coordinates at the Karcher/Fréchet mean
"simplex" / "clr"	strictly positive simplex	centered log-ratio
"alr"	strictly positive simplex	additive log-ratio

For simplex responses, the base point is the Aitchison Fréchet mean (the closed componentwise geometric mean). For spherical responses, the base point is computed by Karcher iteration on the sphere.

import gamfit

model = gamfit.fit(
    train,
    "composition ~ s(x)",          # LHS is a label; the RHS is reused
    response_geometry="simplex",
    response_columns=["sand", "silt", "clay"],
)

pred = model.predict(test)          # columns: sand, silt, clay; rows sum to one
summary = model.summary()           # includes base_point and per-coordinate summaries

Pass response_coordinates="alr" (or response_geometry="alr") to fit a D − 1 dimensional ALR chart instead of the default D-column CLR representation. response_reference= selects the ALR denominator component and defaults to the last response column.