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Supported Approximation Methods

In addition the following approximation schemes are available, each of which can be used in any number of dimensions (subject to having enough computational power)

  • OLS regression - Performs an OLS regression of the data and generates a Sum_Of_Functions containing the resultant approximation. This should work well in many dimensions.
  • Chebyshev polynomials - Creates a Sum_Of_Functions that uses Chebyshev polynomials to approximate a certain function. Unlike the other approximation schemes this does not take in an arbitrary collection of datapoints but rather takes in a function that it evaluates at certain points in a grid to make an approximation function. This might be useful if the original function is expensive (so you want a cheaper one). You might also want to numerically integrate a function by getting a Chebyshev approximation function that can be analytically integrated. See Judd (1998) for details on how this is done.
  • Mars regression spline - Creates a Sum_Of_Piecewise_Functions representing a MARS regression spline. See Friedman (1991) for an explanation of the spline.
  • Monotonic MARS spline - A variant of MARS that guarantees monotonicity in each dimension. Uses only forward hinges max(0, x - t) (for increasing) or backward hinges max(0, t - x) (for decreasing), with non-negative coefficients enforced via NNLS. Each dimension can be independently specified as increasing or decreasing. An optional min_gradient parameter adds a guaranteed minimum slope in every dimension, eliminating flat regions to give strict monotonicity.

Iterative Fitting

The MultivariateFitter and MultivariateAdjustedFitter structs support iterative fitting of multivariate functions. Each call to fit! fits new data and blends with the previous model using an exponentially decaying weight. This is useful for applications where data arrives sequentially (e.g. daily signal-to-return mappings).

Supported methods: :mars, :recursive_partitioning, :monotonic_mars, :ols, :saturated_ols.

Key features:

  • Blending: weight = min(1/(times_through+1), weight_on_new) controls how much the new fit contributes.
  • Simplification: Periodic trimming via trim_mars_spline on synthetic data prevents unbounded complexity growth. The simplify_to parameter (target basis count after trimming) can be larger than MaxM (basis count per daily fit), allowing accumulated models to be richer than any single fit.
  • MultivariateAdjustedFitter: Fits a shared shape function f(x) with per-group affine adjustments y_g ≈ a_g + b_g * f(x). Useful when multiple groups share the same signal-to-response shape but at different scales.
  • Callable syntax: Both fitter types support callable syntax — fitter(dd) is equivalent to evaluate(fitter, dd), and fitter(dd, group) is equivalent to evaluate(fitter, dd, group).

Observation Weights

All data-fitting methods support an optional weights keyword argument for weighted least squares:

  • create_ols_approximation
  • create_saturated_ols_approximation
  • create_mars_spline
  • create_recursive_partitioning
  • create_monotonic_mars_spline
  • trim_mars_spline
  • fit! (for both MultivariateFitter and MultivariateAdjustedFitter)

Weights must be a Vector{Float64} with one entry per observation, with all values non-negative. They are interpreted as frequency weights (equivalent to replicating observations). Passing nothing (the default) is equivalent to uniform weights.

For the NNLS-based methods (monotonic MARS, monotonic trimming), weights are applied via a square-root scaling transformation to preserve the non-negativity constraints.