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Create multivariate functions

We can create multivariate functions by different methods. One of the methods is to gather multivariate functions. The other one is to create a function having multivariate input using the tensor product of univariate functions. In this example, we present both methods.

Description

We want to build some multivariate functions f : \mathbb{R}^d \mapsto \mathbb{R}^q.

We can do that:

  • Case 1: using a finite collection of multivariate functions f_i: \mathbb{R}^d \mapsto \mathbb{R}^q,

  • Case 2: using the tensor product of univariate functions,

  • Case 3: using the tensor product of orthogonal univariate polynomials,

  • Case 4: using the tensor product of orthogonal univariate functions.

Case 1: Finite collection of multivariate functions

In that case, we have a finite collection of multivariate functions f_i : \mathbb{R}^d \mapsto \mathbb{R}^q. We use the Basis class.

For example, we consider:

\begin{array}{lcl} f_1(x_1, x_2) & = & (x_1^2, x_2^2)\\ f_2(x_1, x_2) & = & (x_1+1, 2x_2)\\ f_3(x_1, x_2) & = & (\cos(x_1x_2), x_2^3) \end{array}

import openturns as ot

f1 = ot.SymbolicFunction(["x1", "x2"], ["x1^2", "x2^2"])
f2 = ot.SymbolicFunction(["x1", "x2"], ["x1+1.0", "2.0*x2"])
f3 = ot.SymbolicFunction(["x1", "x2"], ["cos(x1*x2)", "x2^3"])

myBasis = ot.Basis([f1, f2, f3])
f = myBasis.build(1)

Case 2: Tensor product of univariate functions

In that case, the univariate functions are not necessarily orthogonal with respect to a measure.

We have the family of functions: (x_i \mapsto \phi^i_k(x_i))_{k \geq 0} for 1 \leq i \leq d.. The EnumerateFunction class stores the way the multivariate basis is enumerated: enumerate(n) is a multi-index \boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_d) such that the function number n is:

f_n(\vect{x}) = \prod_{i=1}^d \phi^i_{\alpha_i}(x_i).

We use the TensorizedUniVariateFunctionFactory class.

For example, we consider: f: \mathbb{R}^2 \mapsto \mathbb{R}, and:

  • dimension x_1: the family of monomials: (x_1 \mapsto x_1^k)_{k \geq 0},

  • dimension x_2: the family of Haar wavelets: (x_2 \mapsto \phi^2_k(x_2))_{k \geq 0}.

univFuncFamily_Mon = ot.MonomialFunctionFactory()
univFuncFamily_UnivPol = ot.OrthogonalUniVariatePolynomialFunctionFactory(
    ot.JacobiFactory()
)
univFuncFamily_Haar = ot.HaarWaveletFactory()
univFuncFamily_Fourier = ot.FourierSeriesFactory()

familyColl = [
    univFuncFamily_Mon,
    univFuncFamily_UnivPol,
    univFuncFamily_Haar,
    univFuncFamily_Fourier,
]
enumerateFunction = ot.LinearEnumerateFunction(len(familyColl))

familyFunction = ot.TensorizedUniVariateFunctionFactory(familyColl, enumerateFunction)
k = 3
f = familyFunction.build(k)
print("input dimension = ", f.getInputDimension())
print("output dimension = ", f.getOutputDimension())

input dimension =  4
output dimension =  1

If we want to use an orthogonal univariate polynomials family, then we have to cast the family in the OrthogonalUniVariatePolynomialFunctionFactory class.

For example, we use the Jacobi orthogonal univariate polynomials family.

univFuncFamily_Jacobi = ot.OrthogonalUniVariatePolynomialFunctionFactory(
    ot.JacobiFactory()
)

Case 3: Tensor product of orthogonal univariate polynomials

In that case, the univariate polynomials are orthogonal with respect to a measure \mu.

We have the family of polynomials: (x_i \mapsto \phi^i_k(x_i))_{k \geq 0} for 1 \leq i \leq d. The EnumerateFunction class decides the way the multivariate basis is enumerated: enumerate(n) is a multi-index (\ell_1, \dots, \ell_d) such that the function number n is:

f_n(\vect{x}) = \prod_{i=1}^d \phi^i_{\ell_i}(x_i).

We use the OrthogonalProductPolynomialFactory class.

For example, we consider : f: \mathbb{R}^2 \mapsto \mathbb{R}, and:

  • dimension of x_1: the family of Jacobi polynomials: (x_1 \mapsto x_1^k){k \geq 0},

  • dimension of x_2: the family of Hermite polynomials: (x_2 \mapsto \phi^2_k(x_2))_{k \geq 0}.

univFuncFamily_Jacobi = ot.JacobiFactory()
univFuncFamily_Hermite = ot.HermiteFactory()
familyColl = [univFuncFamily_Jacobi, univFuncFamily_Hermite]
enumerateFunction = ot.LinearEnumerateFunction(len(familyColl))
familyFunction = ot.OrthogonalProductPolynomialFactory(familyColl, enumerateFunction)
f = familyFunction.build(3)
print("input dimension = ", f.getInputDimension())
print("output dimension = ", f.getOutputDimension())

input dimension =  2
output dimension =  1

We get the measure:

measure_Jacobi = ot.JacobiFactory().getMeasure()
measure_Hermite = ot.HermiteFactory().getMeasure()
print("Measure orthogonal to Jacobi polynomials = ", measure_Jacobi)
print("Measure orthogonal to Hermite polynomials = ", measure_Hermite)

Measure orthogonal to Jacobi polynomials =  Beta(alpha = 2, beta = 2, a = -1, b = 1)
Measure orthogonal to Hermite polynomials =  Normal(mu = 0, sigma = 1)

Case 4: Tensor product of orthogonal univariate functions

In that case, the univariate functions are orthogonal with respect to a measure \mu.

We have the family of functions: (x_i \mapsto \phi^i_k(x_i))_{k \geq 0} for 1 \leq i \leq d. The EnumerateFunction class stores the way the multivariate basis is enumerated: enumerate(n) is a multi-index \vect{\alpha} = (\alpha_1, \dots, \alpha_d) such that the function number n is:

f_n(\vect{x}) = \prod_{i=1}^d \phi^i_{\alpha_i}(x_i)

We use the OrthogonalProductFunctionFactory class.

For example, we consider : f: \mathbb{R}^2 \mapsto \mathbb{R}, and:

  • dimension x_1: the family of Haar wavelets: (x_1 \mapsto \phi^1_k(x_1))_{k \geq 0},

  • dimension x_2: the family of Fourier series functions: (x_2 \mapsto \phi^2_k(x_2)){k \geq 0}.

univFuncFamily_Haar = ot.HaarWaveletFactory()
univFuncFamily_Fourier = ot.FourierSeriesFactory()
familyColl = [univFuncFamily_Haar, univFuncFamily_Fourier]
enumerateFunction = ot.LinearEnumerateFunction(len(familyColl))

familyFunction = ot.OrthogonalProductFunctionFactory(familyColl, enumerateFunction)
k = 3
f = familyFunction.build(k)

We get the measure:

measure_Haar = ot.HaarWaveletFactory().getMeasure()
measure_Fourier = ot.FourierSeriesFactory().getMeasure()
print("Measure orthogonal to Haar wavelets = ", measure_Haar)
print("Measure orthogonal to Fourier series = ", measure_Fourier)

Measure orthogonal to Haar wavelets =  Uniform(a = 0, b = 1)
Measure orthogonal to Fourier series =  Uniform(a = -3.14159, b = 3.14159)