Basis Terms

A basis term expands one predictor into several columns and gives that expansion its own vector of coefficients. This differs from a transform, which transforms one column before it receives the equation’s usual scalar coefficient. Basis terms are useful when a relationship needs a flexible but structured shape.

Basis	Syntax	Coefficients	Current support
HSGP	hsgp(x, m=20, c=1.5)	Kernel-scaled basis weights	Cross-sectional, exogenous 1-D input
Fourier	fourier(week, n=3, period=52)	2n independent harmonic weights	Cross-sectional; observed or latent 1-D input

For example, a weekly seasonal pattern with three harmonics is a Fourier basis; the Fourier seasonality example shows a complete workflow:

seasonal = pathmc.model(
    "sales ~ fourier(week, n=3, period=52) + adstock(tv, decay=theta_tv)",
    data=df,
)

The term owns beta_fourier_sales_week, so it does not take a coefficient prefix such as b_season*fourier(...). Its sine and cosine columns are symbolic in the PyMC graph, which means the term is recomputed when do() changes week or when its input is a latent mediator:

model = pathmc.model(
    """
    awareness ~ spend
    sales ~ fourier(awareness, n=2, period=10)
    """,
    data=df,
)

hsgp() uses the same owned-coefficient mechanism but has a kernel-informed prior and a boundary frozen from the fitted data. Its do() interventions must remain inside that boundary; see the HSGP worked example. Fourier has no fitted support boundary, so it remains well-defined outside the observed range.

Basis terms currently cannot be used in panel/scan models, inside residual-covariance blocks, or nested in transforms. These limits are checked before compilation and name the equation or variable that needs changing.