Basis Functions

In TransportMaps.jl, the basis must match the reference distribution used by the map. The reference should have the same support as the target distribution. If supports do not match, consider an isoprobabilistic transformation before constructing the map.

Supported reference-basis pairings are:

Reference distribution	Support	Basis family
		Hermite-based bases
		Legendre polynomials
		Shifted Legendre polynomials

This compatibility is enforced by the map constructor.

Standard Normal Reference

Often, a standard normal reference with the probability density function

is chosen when the reference measure should have an infinite support. In this case, a probabilistic Hermite polynomial basis is a natural choice since they are orthogonal with respect to this density and satisfy the three-term recurrence

with   and  .

Orthonormal polynomial bases are useful because projections are stable and truncated expansions give the best approximation under the chosen reference measure. In the Gaussian case, Hermite polynomials are the natural choice in the Wiener-Askey scheme [10].

We also support edge-controlled Hermite variants that improve tail behavior while keeping the Gaussian reference [3], [2]:

• Probabilistic Hermite basis

• Linearized Hermite basis

• Gaussian-weighted Hermite basis

• Cubic-spline-weighted Hermite basis

Probabilistic Hermite Basis

We first visualize the standard probabilistic Hermite basis.

Construct the basis with HermiteBasis.

basis = HermiteBasis()
z = -3:0.01:3

p1 = plot(xlabel="z", ylabel="Basis function", title="Standard Hermite Basis")
for degree in 0:4
    plot!(p1, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

If we zoom out, we can see that the tails grow quickly for large :

z = -7:0.1:7

p2 = plot(xlabel="z", ylabel="Basis function", title="Standard Hermite Basis")
for degree in 0:4
    plot!(p2, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

Linearized Hermite Basis

Linearized Hermite polynomials (edge-linearized basis) were introduced in [3] to control growth for large by replacing the polynomial with a tangent line outside data-dependent bounds :

The bounds are chosen here as the 0.01 and 0.99 quantiles of the reference. The normalization follows [3]:   for   and   .

basis = LinearizedHermiteBasis(Normal(), 4, 1)
println("Linearization bounds: ", basis.linearizationbounds)

p3 = plot(xlabel="z", ylabel="Basis function", title="Linearized Hermite Basis")
for degree in 0:4
    plot!(p3, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

Linearization bounds: [-2.326347874040846, 2.326347874040846]

Edge-Controlled (Weighted) Hermite Basis: Gaussian Weight

Edge control modifies each Hermite polynomial with a decaying weight to reduce growth in the tails [2]. Using a Gaussian weight gives:

basis = GaussianWeightedHermiteBasis()

p4 = plot(xlabel="z", ylabel="Basis function", title="Gaussian-Weighted Hermite Basis")
for degree in 0:4
    plot!(p4, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

Note

In order to preserve some extrapolation properties, the weights are only applied to polynomials of degree  , as noted in [2].

Edge-Controlled Hermite Basis: Cubic Spline Weight

A cubic spline weight smoothly damps the polynomials outside a radius [2]. Here, is based on 0.01 and 0.99 quantiles .

basis = CubicSplineHermiteBasis(Normal())

p5 = plot(xlabel="z", ylabel="Basis function", title="Cubic Spline Weighted Hermite Basis")
for degree in 0:4
    plot!(p5, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

Uniform[-1, 1] Reference

For a uniform reference on  , the matching family is Legendre. The Legendre polynomials are orthogonal with respect to the uniform measure on   and satisfy

with   and  .

Use this basis with Uniform(-1, 1).

basis = LegendreBasis()

z = -1:0.01:1
p = plot(xlabel="z", ylabel="Basis function", title="Legendre Basis on [-1, 1]")
for degree in 0:4
    plot!(p, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

Uniform[0, 1] Reference

For a uniform reference on , the matching family is shifted Legendre. These polynomials are obtained from Legendre polynomials via

Use this basis with Uniform(0, 1).

basis = ShiftedLegendreBasis()

z = 0:0.01:1
p = plot(xlabel="z", ylabel="Basis function", title="Shifted Legendre Basis on [0, 1]")
for degree in 0:4
    plot!(p, z, map(x -> basisfunction(basis, degree, x), z), label="degree $degree")
end

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