Package {BsplineQuantReg}

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Description

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Usage

.onAttach(libname, pkgname)

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Build B-spline basis in piecewise polynomial form Computes local polynomial coefficients for each B-spline basis function on each interval. Polynomials are expressed in the canonical basis Uses De Boor's recursion formula.

Description

Build B-spline basis in piecewise polynomial form Computes local polynomial coefficients for each B-spline basis function on each interval. Polynomials are expressed in the canonical basis Uses De Boor's recursion formula.

Usage

Bspline_base(sn, degree = 3, der = NULL, verbose = FALSE)

Arguments

Value

A list containing:

Examples

sn <- c(0,0,0,0,1,2,3,4,5,5,5,5)
basis <- Bspline_base(sn, degree=3)
x=(0:(5*100))/100
y=bs_direct(basis,x)
matplot(x,t(y))
#or simple:
#view_basis(basis)

Differentiate a B-spline basis

Description

Computes the basis of order der derivatives of a B-spline basis.

Usage

Bspline_deriv(bspline, der = 2, verbose = FALSE)

Arguments

Value

A list similar to Bspline_base for the derivative basis

Omega function for De Boor recursion

Description

Computes the affine element used in the recursive De Boor algorithm for B-spline construction. for a given extended knot partition s(1),.. Omega(j,l)(x)=(s(j-x))/(s(j+l-1)-s(j)\), with l: order of the spline

Usage

Omega(s, j, o)

Arguments

Value

(Side function) A Vector of coefficients [alpha, beta] representing (alpha*t + beta)

Wrapper function for linear spline quantile regression

Description

This is a convenience wrapper for SplineLinearQuant.

Usage

SplineConstQuantRegBs1(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  solver = "OSQP",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

Same as SplineLinearQuant

Wrapper function for quadratic spline quantile regression Previous version notation style This is a convenience wrapper for SplineQuadraticQuant.

Description

Wrapper function for quadratic spline quantile regression Previous version notation style This is a convenience wrapper for SplineQuadraticQuant.

Usage

SplineConstQuantRegBs2(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  solver = "OSQP",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

Same as SplineQuadraticQuant

Wrapper function for Cubic spline quantile regression

Description

This is a convenience wrapper for SplineCubicQuant with convention of previous versions SplineConstQuantRegBs3.

Usage

SplineConstQuantRegBs3(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  der3cons = 0,
  solver = "CLARABEL",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

Same as SplineCubicQuant

Wrapper function for quartic spline quantile regression

Description

This is a convenience wrapper for SplineQuarticQuant that follows the same naming convention as SplineConstQuantRegBs3.

Usage

SplineConstQuantRegBs4(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  der3cons = 0,
  solver = "CLARABEL",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

Same as SplineQuarticQuant

Constrained quantile regression with cubic splines

Description

Performs quantile regression using cubic B-splines, with optional monotonicity constraints (via Karlin-Studden) and convexity constraints.

Usage

SplineCubicQuant(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  der3cons = 0,
  solver = "CLARABEL",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing:

References

• Abbes, A. (2025). Quantile regression with cubic polynomial splines under shape constraints with applications . Zenodo. doi:10.5281/zenodo.16999784

• de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag. doi:10.1007/978-1-4612-6333-3

• Karlin, S., & Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience.

• Koenker, R., & Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1), 33-50. doi:10.2307/1913643

• Koenker, R. (2025). quantreg: Quantile Regression. R package version 5.99. https://CRAN.R-project.org/package=quantreg

• Ng, P., & Maechler, M. (2024). cobs: Constrained B-Splines. R package version 1.3-8. https://CRAN.R-project.org/package=cobs

See Also

Related R packages:

• quantreg - Quantile regression with linear programming

• cobs - Constrained B-sines (linear and quadratic only)

Other implementations:

• MATLAB/Python versions: https://github.com/alexandreabbes/Constrained-Quantile-Regression-with-cubic-splines

Examples

#optional set.seed(42)
x <- seq(0, 1, length=100)
y <- 2*x + sin(6*pi*x)/2 + rnorm(100, 0, 0.05)
knot <- quantile(x, probs=seq(0,1,length.out=10))

# Median quantile regression without constraints
fit <- SplineConstQuantRegBs3(x, y, knot, tau=0.5)

# With increasing monotonicity constraint
fit_monot <- SplineConstQuantRegBs3(x, y, knot, tau=0.5, monot=1)

# With convexity constraint
fit_convex <- SplineConstQuantRegBs3(x, y, knot, tau=0.5, convcons=1)

Quantile regression with linear splines and monotonicity constraints

Description

Performs quantile regression using linear B-splines with monotonicity constraints. For linear splines, the derivative is constant on each interval, so monotonicity is a simple linear constraint on the derivative.

Usage

SplineLinearQuant(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  solver = "OSQP",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing coefficients, degree, and knots

Quantile regression with quadratic splines and shape constraints

Description

Performs quantile regression using quadratic B-splines with: - Monotonicity: Karlin-Studden constraints on the derivative (linear) - Convexity: Karlin-Studden constraints on the second derivative (constant)

Usage

SplineQuadraticQuant(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  solver = "OSQP",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing coefficients, degree, and knots

Quantile regression with quartic splines and shape constraints

Description

Performs quantile regression using quartic (degree 4) B-splines with monotonicity (Karlin-Studden constraints on the cubic derivative), convexity (Karlin-Studden constraints on the quadratic second derivative), and third derivative constraints (linear at knot).

Usage

SplineQuarticQuant(
  xtab,
  ytab,
  knot,
  tau,
  monot = 0,
  convcons = 0,
  der3cons = 0,
  solver = "OSQP",
  weight = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing coefficients, degree, and knot

Derivatives at knot of a B-spline

Description

Computes derivative values of a B-spline at knot (efficient because it directly uses polynomial coefficients).

Usage

Spline_der_knot(Bsbase, der = 1)

Arguments

Value

Matrix of derivative values (n_splines x n_knot)

Apply Karlin-Studden constraints for a cubic polynomial

Description

For a cubic polynomial p(u) = a*u^3 + b*u^2 + c*u + d on [0,1], this function applies the Karlin-Studden SOCP constraints to ensure p(u) >= 0 or p(u) <= 0 on [0,1].

Usage

apply_karlin_cubic(p3, p2, p1, p0, z0, z1, sign = 1)

Arguments

Value

List of CVXR constraints

Apply Karlin-Studden constraints for a quadratic polynomial

Description

For a quadratic polynomial p(u) = a*u^2 + b*u + c on [0,1], this function applies the Karlin-Studden SOCP constraints to ensure p(u) >= 0 or p(u) <= 0 on [0,1].

Usage

apply_karlin_quadratic(p2, p1, p0, z0, sign = 1)

Arguments

Value

List of CVXR constraints

Apply linear constraints at knot for the third derivative

Description

For a quartic spline, the third derivative is affine (linear) on each interval. This function applies sign constraints at the knot.

Usage

apply_linear_constraint(const_value, sign = 1)

Arguments

Value

List of CVXR constraints

Direct evaluation of a B-spline basis

Description

Computes the values of all B-spline basis functions at given points.

Usage

bs_direct(Basis, x_values)

Arguments

Value

Matrix of basis function values (n_splines x length(x_values))

normalized first and second derivative coefficients on each interval. for a cubic B-spline basis

Description

normalized first and second derivative coefficients on each interval. for a cubic B-spline basis

Usage

bspline_to_deriv_coeffs_cubic(tn, degree = 3, x_values = 0, verbose = FALSE)

Arguments

Value

A list containing:

Derivative coefficients for linear B-spline to

Description

Computes normalized first derivative coefficients for linear B-splines. For linear splines, the derivative is constant on each interval.

Usage

bspline_to_deriv_coeffs_lin(tn, degree = 1, x_values = 0, verbose = FALSE)

Arguments

Value

A list containing:

quadratic B-spline derivative coefficients

Description

Computes normalized first and second derivative coefficients for quadratic B-splines on each interval.

Usage

bspline_to_deriv_coeffs_quad(tn, degree = 2, x_values = 0, verbose = FALSE)

Arguments

Value

A list containing:

Convert quartic B-spline to derivative coefficients

Description

Computes normalized first, second, and third derivative coefficients for quartic B-splines on each interval.

Usage

bspline_to_deriv_coeffs_quart(knot, degree = 4, x_values = 0)

Arguments

Value

A list containing:

Change polynomial basis (Taylor expansion)

Description

Converts a polynomial expressed in the basis (t-a)^k to its representation in the basis (t-b)^k using Taylor's formula. P(t) = sum c_k (t-a)^k P(t) = sum c"_k (t-b)^k

Usage

change_polynomial_base_taylor(coeffs_a, a, b)

Arguments

Value

Coefficients in basis centered at b

Evaluate a piecewise polynomial (PP) form

Description

Evaluates a piecewise polynomial function at given points.

Usage

evalpp(p, x_values)

Arguments

Value

Function values at the requested points

Create a callable spline object

Description

Transforms a spline regression result into a callable function that can be evaluated at any point, while preserving access to parameters.

Usage

make_spline(result)

Arguments

Value

A function that can be called as 'result(x)' and has attributes for 'degree', 'knot', 'coefficients', and 'status'

Build a piecewise polynomial (PP) form

Description

Creates a PP structure from polynomial coefficients and knot.

Usage

makpp(coeff, tn)

Arguments

Value

List with components coefficients and knot

Evaluate polynomial

Description

Evaluates a polynomial at one or more points.

Usage

poly_eval(p, xvalues)

Arguments

Value

Vector of same length as xvalues : polynomial values at the points xvalues

Examples

# P(x) = 1 + x + x^2
poly_eval(c(1, 1, 1), c(0, 1, 2)) # returns c(1, 3, 7)

Polynomial addition

Description

Adds two polynomials represented by coefficients in decreasing power order.

Usage

polyadd(p1, p2, verbose = FALSE)

Arguments

Value

Coefficient vector of the sum

Examples

polyadd(c(1, 1), c(1, -1)) # returns c(2, 0)

Polynomial derivative Computes the derivative of order der of a polynomial.

Description

Polynomial derivative Computes the derivative of order der of a polynomial.

Usage

polyderiv(p, der = 1)

Arguments

Value

Coefficients of the derivative polynomial

Examples

# P(x) = x^2 -> P'(x) = 2x
polyderiv(c(1, 0, 0), 1) # returns c(2, 0)

Polynomial multiplication

Description

Multiplies two polynomials represented by coefficients in decreasing power order.

Usage

polymul(p1, p2, ord = 0, verbose = FALSE)

Arguments

Value

Coefficient vector of the product polynomial (decreasing powers)

Examples

# (1 + x) * (1 + x) = 1 + 2x + x^2
polymul(c(1, 1), c(1, 1)) # returns c(1, 2, 1)

Print method for callable spline

Description

Print method for callable spline

Usage

## S3 method for class 'callable_spline'
print(x, ...)

Arguments

Print method for quantile_spline results

Description

Print method for quantile_spline results

Usage

## S3 method for class 'quantile_spline'
print(x, ...)

Arguments

Unified Quantile Regression with B-Splines of Any Degree (1 to 4)

Description

This function provides a unified interface for quantile regression with B-splines of degrees 1 to 4, with shape constraints (monotonicity, convexity, third derivative).

Usage

quantile_spline(
  xtab,
  ytab,
  knot,
  tau,
  degree = 3,
  monot = 0,
  convcons = 0,
  der3cons = 0,
  solver = "CLARABEL",
  weight = NULL,
  verbose = FALSE,
  callable = FALSE
)

Arguments

Value

A list containing coefficients, degree, and knots

References

• Abbes, A. (2025). Quantile regression with cubic polynomial splines under shape constraints with applications. Zenodo. doi:10.5281/zenodo.16999784

• Karlin, S., & Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience.

See Also

SplineLinearQuant, SplineQuadraticQuant, SplineCubicQuant, SplineQuarticQuant

Examples

# Generate data
set.seed(42)
x <- seq(0, 1, length.out = 100)
y <- 2*x + sin(6*pi*x)/2 + rnorm(100, 0, 0.05)
knot <- quantile(x, probs = seq(0, 1, length.out = 10))

# Linear spline (degree 1) with increasing constraint
fit_lin <- quantile_spline(x, y, knot, tau = 0.5, degree = 1, monot = 1)

# Quadratic spline (degree 2) with convexity constraint
fit_quad <- quantile_spline(x, y, knot, tau = 0.5, degree = 2, convcons = 1)

# Cubic spline (degree 3) with both constraints
fit_cubic <- quantile_spline(x, y, knot, tau = 0.5, degree = 3,
                             monot = 1, convcons = 1)

# Quartic spline (degree 4) with all constraints
fit_quart <- quantile_spline(x, y, knot, tau = 0.5, degree = 4,
                             monot = 1, convcons = 1, der3cons = 1)

Reduce polynomial

Description

Removes leading zeros from a polynomial coefficient vector.

Usage

reduce_pol(p, verbose = FALSE)

Arguments

Value

Reduced vector (without leading zeros)

Examples

reduce_pol(c(0,0, 1, 1))

Evaluate a B-spline

Description

Evaluates a spline (linear combination of B-splines) at given points.

Usage

spline_eval(Bspline, x_values = NULL, Bvalues = NULL)

Arguments

Value

vector of same length as x_values, with Spline values at the requested points

Examples

{ # Create and evaluate a spline
# This function is also used when a Bspline object is rendered callable as a function
# with make_spline()
tn<-c(0,1,2,3,4,5)
x_values<-seq(0,5,length=100)
Bspline=list(degree=3, knot=tn,coefficients=runif(length(tn)+3-1))
y <- spline_eval(Bspline, x_values)
#alternatively compute the base before to optimize if needed
sn <- c(0,0,0,0,1,2,3,4,5,5,5,5)
Bsbase <- Bspline_base(sn, degree=3)
Bvalues <-bs_direct(Bsbase,x_values)
y <- spline_eval(Bspline=Bspline,x_values=x_values, Bvalues=Bvalues )}

Comprehensive test function

Description

Runs quantile regression tests with and without constraints, and displays results. Demo function.

Usage

test_karlin_simple(verbose = FALSE, degree = 3, seed = NULL)

Arguments

Value

No return value, produces plots.

Examples

test_karlin_simple()

Visualize a B-spline functions basis

Description

Plots all functions of a B-spline basis.

Usage

view_basis(BB, x_values = 0)

Arguments

Value

No return value, called for side effects (generates a plot)