B-spline Basis Manipulation

Introduction

This vignette covers the B-spline basis manipulation tools provided by the BsplineQuantReg package. B-splines are a powerful tool for nonparametric regression and function approximation. The package provides a comprehensive set of functions for building, manipulating, and evaluating B-spline bases. The calculations are standard and follow authors such as Deboor or Schumaker.

Building a B-spline Basis

Basic Construction

The primary function for building a B-spline basis is Bspline_base(), which takes an extended knot sequence and returns the basis functions in piecewise polynomial form.

# Extended knot sequence for cubic B-splines on [0,5]
# with internal knots at 1, 2, 3, 4
sn <- c(0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5)

# Build the basis
basis <- Bspline_base(sn, degree = 3)

# Inspect the basis structure
str(basis[1:4])

## List of 4
##  $ base    : num [1:8, 1:11, 1:4] 0 0 0 0 0 0 0 0 0 0 ...
##  $ base0   : num [1:8, 1:11, 1:4] 0 0 0 0 0 0 0 0 0 0 ...
##  $ ext_knot: num [1:12] 0 0 0 0 1 2 3 4 5 5 ...
##  $ knot    : num [1:6] 0 1 2 3 4 5

Visualizing the Basis

The view_basis() function provides a quick visualization of all basis functions.

# Visualize the basis
view_basis(basis)

Understanding the Extended Knot Sequence

For a B-spline of degree d with kn intervals, the extended knot sequence has length kn + 1 + 2*d:

# d = 3, kn = 5 (intervals: [0,1], [1,2], [2,3], [3,4], [4,5])
# Length: 5 + 1 + 6 = 12
sn_example <- c(0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5)

# The first and last d knots are repeated to enforce boundary conditions
basis_example <- Bspline_base(sn_example, degree = 3)
cat("Number of basis functions:", basis_example$n_splines, "\n")

## Number of basis functions: 8

cat("Effective knots:", basis_example$knot, "\n")

## Effective knots: 0 1 2 3 4 5

Visualizing B-spline Bases

Visualizing a B-spline Basis

The view_basis() function provides a quick visualization of all basis functions.

# Visualize the basis
view_basis(basis_example)

Visualizing Different Degrees

# piecewise constant (degree 0)
sn_cnst <- c(0, 1, 2, 3, 4, 5)
basis_cnst <- Bspline_base(sn_cnst, degree = 0)
view_basis(basis_cnst)

# Linear (degree 1)
sn_lin <- c(0, 0, 1, 2, 3, 4, 5, 5)
basis_lin <- Bspline_base(sn_lin, degree = 1)
view_basis(basis_lin)

# Quadratic (degree 2)
sn_quad <- c(0, 0, 0, 1, 2, 3, 4, 5, 5, 5)
basis_quad <- Bspline_base(sn_quad, degree = 2)
view_basis(basis_quad)

Customizing Visualization

# View with custom evaluation points
x_fine <- seq(-0.5, 5.5, length.out = 300)
view_basis(basis, x_values = x_fine)

Differentiating a B-spline Basis

The Bspline_base_deriv() function computes the basis of derivatives of a B-spline basis.

# Compute first derivative basis
basis_der1 <- Bspline_base_deriv(basis, der = 1)

# Compute second derivative basis
basis_der2 <- Bspline_base_deriv(basis, der = 2)

# Compute third derivative basis
basis_der3 <- Bspline_base_deriv(basis, der = 3)

# Check degrees
cat("Original degree:", basis$degree, "\n")

## Original degree: 3

cat("1st derivative degree:", basis_der1$degree, "\n")

## 1st derivative degree: 2

cat("2nd derivative degree:", basis_der2$degree, "\n")

## 2nd derivative degree: 1

cat("3rd derivative degree:", basis_der3$degree, "\n")

## 3rd derivative degree: 0

Visualizing Derivative Bases

par(mfrow = c(2, 2))
view_basis(basis, main = "Original Basis (deg 3)")
view_basis(basis_der1, main = "1st Derivative Basis (deg 2)")
view_basis(basis_der2, main = "2nd Derivative Basis (deg 1)")
view_basis(basis_der3, main = "3rd Derivative Basis (deg 0)")

par(mfrow = c(1, 1))

Differentiating a B-spline Function

There are two methods to compute derivatives of a B-spline function.

Method 1: Using `spline_eval()` with `der` Parameter

The simplest method is to use spline_eval() with the der argument. This function calculates the derivative of the Bspline basis, then the values of the differentiated basis, and finaly evaluates the function.

# Create a random B-spline function
bs_function<-basis
bs_function$coeff <- rnorm(basis$n_splines)

# Evaluate the function and its derivatives
x_plot <- seq(0, 5, length.out = 200)
y <- spline_eval(bs_function, x_plot, der = 0)
y1 <- spline_eval(bs_function, x_plot, der = 1)
y2 <- spline_eval(bs_function, x_plot, der = 2)
y3 <- spline_eval(bs_function, x_plot, der = 3)

# Plot
par(mfrow = c(2, 2))
plot(x_plot, y, type = "l", main = "Function", xlab = "x", ylab = "f(x)")
plot(x_plot, y1, type = "l", main = "1st Derivative", xlab = "x", ylab = "f'(x)")
plot(x_plot, y2, type = "l", main = "2nd Derivative", xlab = "x", ylab = "f''(x)")
plot(x_plot, y3, type = "l", main = "3rd Derivative", xlab = "x", ylab = "f'''(x)")

par(mfrow = c(1, 1))

Method 2: Using `Bspline_deriv()` to Get Derivative Coefficients

The second method computes the coefficients of the der-th derivative B-spline directly on a Bspline basis that is supposed with new degree=degree-der .

# Compute derivative coefficients
der1_func <- Bspline_deriv(bs_function, der = 1)
der2_func <- Bspline_deriv(bs_function, der = 2)

# Evaluate using the derivative B-spline
y1_coeff <- spline_eval(der1_func, x_plot)
y2_coeff <- spline_eval(der2_func, x_plot)

# Both methods should give the same result
max(abs(y1 - y1_coeff))

## [1] 2.967695e-10

max(abs(y2 - y2_coeff))

## [1] 6.057128e-10

Method 3: Step-by-step evaluation using basis derivatives

This method follows the same path as spline_eval:

1. Compute the derivative basis

2. Evaluate the basis at points using bs_direct

3. Use matrix multiplication with coefficients

# Create a B-spline function
sn <- c(0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5)
basis <- Bspline_base(sn, degree = 3)
basis$coeff <- c(1, -2, 3, -1, 2, 1, 0, 0.5)

# Step 1: Compute the derivative basis (derivative order = 1)
basis_der1 <- Bspline_base_deriv(basis, der = 1)

# Step 2: Evaluate the derivative basis at points using bs_direct
x_plot <- seq(0, 5, length.out = 100)
Bvalues_der1 <- bs_direct(basis_der1, x_plot)

# Step 3: Compute the derivative values using matrix multiplication
# Bvalues_der1 is (n_splines x n_points), coefficients is (n_splines x 1)
# Result is (1 x n_points) or vector of length n_points
y_der1 <- t(Bvalues_der1) %*% basis$coeff

# Step 4: Compare with spline_eval (direct method)
y_der1_direct <- spline_eval(basis, x_plot, der = 1)

# The results are identical
cat("Maximum difference:", max(abs(y_der1 - y_der1_direct)), "\n")

## Maximum difference: 0

# Plot to verify
plot(x_plot, y_der1, type = "l", col = "blue", lwd = 2,
     main = "First Derivative: Step-by-Step Method",
     xlab = "x", ylab = "f'(x)")
lines(x_plot, y_der1_direct, col = "red", lty = 2, lwd = 2)
legend("topright", legend = c("Matrix multiplication", "spline_eval"),
       col = c("blue", "red"), lty = c(1, 2), lwd = 2)

# For higher derivatives, repeat the process
basis_der2 <- Bspline_base_deriv(basis, der = 2)
Bvalues_der2 <- bs_direct(basis_der2, x_plot)
y_der2 <- t(Bvalues_der2) %*% basis$coeff

basis_der3 <- Bspline_base_deriv(basis, der = 3)
Bvalues_der3 <- bs_direct(basis_der3, x_plot)
y_der3 <- t(Bvalues_der3) %*% basis$coeff

# Plot all derivatives
par(mfrow = c(2, 2))
plot(x_plot, spline_eval(basis, x_plot), type = "l", col = "blue", lwd = 2,
     main = "Function", xlab = "x", ylab = "f(x)")
grid()

plot(x_plot, y_der1, type = "l", col = "darkgreen", lwd = 2,
     main = "1st Derivative", xlab = "x", ylab = "f'(x)")
grid()
abline(h = 0, col = "gray", lty = 3)

plot(x_plot, y_der2, type = "l", col = "purple", lwd = 2,
     main = "2nd Derivative", xlab = "x", ylab = "f''(x)")
grid()
abline(h = 0, col = "gray", lty = 3)

plot(x_plot, y_der3, type = "l", col = "orange", lwd = 2,
     main = "3rd Derivative", xlab = "x", ylab = "f'''(x)")
grid()
abline(h = 0, col = "gray", lty = 3)

par(mfrow = c(1, 1))

PP-form Conversion

B-splines can be converted to piecewise polynomial (PP) form using Bsplinetopp().

# Convert to PP form
pp <- Bsplinetopp(bs_function,Bsbasis=basis, callable = FALSE)
# PP form contains polynomial coefficients for each interval
print(pp)

## Piecewise Polynomial (PP) (non-callable)
## ================================
##   $degree: 3 
##   $knot: 0 1 2 3 4 5 
##  coefficients dimension: 5 x 4 
## 
##   $coeff:
##     Intervals 1 : -2.2792, 3.8426, -0.9429, -0.0381 
##     Intervals 2 : 1.8317, -2.9952, -0.0955, 0.5823 
##     Intervals 3 : -1.3953, 2.4999, -0.5907, -0.6767 
##     Intervals 4 : 0.8821, -1.6859, 0.2233, -0.1628 
##     Intervals 5 : 0.8088, 0.9604, -0.5021, -0.7432 
##  Usage: pp_eval(pp, x_values)

#or omit the basis (slower, re-calculate the basis)
pp <- Bsplinetopp(bs_function, callable = FALSE)
# PP form contains polynomial coefficients for each interval
print(pp)

## Piecewise Polynomial (PP) (non-callable)
## ================================
##   $degree: 3 
##   $knot: 0 1 2 3 4 5 
##  coefficients dimension: 5 x 4 
## 
##   $coeff:
##     Intervals 1 : -2.2792, 3.8426, -0.9429, -0.0381 
##     Intervals 2 : 1.8317, -2.9952, -0.0955, 0.5823 
##     Intervals 3 : -1.3953, 2.4999, -0.5907, -0.6767 
##     Intervals 4 : 0.8821, -1.6859, 0.2233, -0.1628 
##     Intervals 5 : 0.8088, 0.9604, -0.5021, -0.7432 
##  Usage: pp_eval(pp, x_values)

# Evaluate the PP form
y_pp <- evalpp(pp, x_plot)
# Recalculatethe
y <- spline_eval(bs_function,x_plot)

# Should match the original B-spline
max(abs(y - y_pp))

## [1] 6.661338e-16

Callable PP Objects

# Create a callable PP object
pp_call <- Bsplinetopp(bs_function, callable = TRUE)
class(pp_call)  # "callable_pp" "function"

## [1] "callable_pp" "function"

# Evaluate directly
y_call <- pp_call(x_plot)

# Access parameters
params <- get_parameters(pp_call)
print(params$degree)

## [1] 3

print(params$knot)

## [1] 0 1 2 3 4 5

# Print method
print(pp_call)

## Callable Piecewise Polynomial (PP) Object
## ==========================================
##   Degree: 3 
##   Intervals: 5 
##   Knots ( 6 ) : 0, 1, 2, 3, 4, 5 
##  coefficients dimension: 5 x 4 
## 
##   $coeff:
##     Intervals 1 : -2.2792, 3.8426, -0.9429, -0.0381 
##     Intervals 2 : 1.8317, -2.9952, -0.0955, 0.5823 
##     Intervals 3 : -1.3953, 2.4999, -0.5907, -0.6767 
##     Intervals 4 : 0.8821, -1.6859, 0.2233, -0.1628 
##     Intervals 5 : 0.8088, 0.9604, -0.5021, -0.7432 
##  Usage: pp(x_values) or evalpp(pp, x_values)

Callable Splines

Creating Callable Splines

The make_spline() function transforms a B-spline object into a callable function.

# Create a callable spline
spline_func <- make_spline(bs_function, callable = TRUE)
class(spline_func)  # "callable_spline" "function"

## [1] "callable_spline" "function"

# Evaluate directly
y_callable <- spline_func(x_plot)

# Access parameters
get_parameters(spline_func)$degree

## [1] 3

get_parameters(spline_func)$knot

## [1] 0 1 2 3 4 5

get_parameters(spline_func)$coeff

## [1] -0.03811505 -0.35241778  1.58067971 -1.51000705  0.39920081 -1.06333943
## [7] -0.75777936  0.52391407

# Print method
print(spline_func)

## callable_spline Object
## ======================
##   Degree: 3 
##   Knots ( 6 ):  0 1 2 3 4 5 
##   Coefficients ( 8 ):  -0.03811505 -0.3524178 1.58068 -1.510007 0.3992008 -1.063339 -0.7577794 0.5239141 
## for result, type get_parameter(spline)

Non-Callable Splines

# Create a non-callable spline
spline_list <- make_spline(bs_function, callable = FALSE)
class(spline_list)  # "non_callable_spline" "list"

## [1] "non_callable_spline" "list"

# Access components
spline_list$degree

## [1] 3

spline_list$knot

## [1] 0 1 2 3 4 5

spline_list$coeff

## [1] -0.03811505 -0.35241778  1.58067971 -1.51000705  0.39920081 -1.06333943
## [7] -0.75777936  0.52391407

# Print method
print(spline_list)

## Non callable Spline List 
## ======================== 
##  $degree:  3 
##  $knot : [ 0, 1, 2, 3, 4, 5 ] ( 6 knots ) 
##  $coeff (rounded 10^(-7)) :[ -0.0381151, -0.3524178, 1.5806797, -1.510007, 0.3992008, -1.0633394, -0.7577794, 0.5239141 ]  ( dim Basis is  8 )

Getting Parameters

The get_parameters() function extracts parameters from callable objects.

params <- get_parameters(spline_func)
print(params$degree)

## [1] 3

print(params$knot)

## [1] 0 1 2 3 4 5

print(head(params$coeff, 5))

## [1] -0.03811505 -0.35241778  1.58067971 -1.51000705  0.39920081

Advanced: Knot Multiplicity and Regularity

Knot multiplicity controls the smoothness of B-splines at knot points. The regularity at a knot is determined by the multiplicity m and the degree d: the spline is C^(d - m - 1) at that knot.

Creating B-splines with Multiple Knots

# Create a basis with a double knot at 3: accept discontinuity of the 2cnd derivative at 3
sn_mult <- c(0, 0, 0, 0, 1, 2, 3, 3, 4, 5, 5, 5, 5)
basis_mult <- Bspline_base(sn_mult, degree = 3)

# Visualize
view_basis(basis_mult)
abline(v = 3, col = "orange", lty = 2, lwd = 2)

Effect of Multiplicity on Regularity

# Single knot (m=1): C^2 continuity
# Double knot (m=2): C^1 continuity
# Triple knot (m=3): C^0 continuity
# Quadruple knot (m=4): Discontinuity

sn1 <- c(0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5)  # m=1 at 3
sn2 <- c(0, 0, 0, 0, 1, 2, 3, 3, 4, 5, 5, 5, 5)  # m=2 at 3
sn3 <- c(0, 0, 0, 0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 5)  # m=3 at 3
sn4 <- c(0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5)  # m=3 at 3
# Compare the bases
basis1 <- Bspline_base(sn1, degree = 3)
basis2 <- Bspline_base(sn2, degree = 3)
basis3 <- Bspline_base(sn3, degree = 3)
basis4 <- Bspline_base(sn4, degree = 3)

# Visualize the differences
par(mfrow = c(2, 1))
view_basis(basis1, main = "m=1 (only C² at x=3)")
view_basis(basis2, main = "m=2 (only C¹ at x=3)")

par(mfrow = c(2, 1))
view_basis(basis3, main = "m=3 (only C⁰ at x=3)")
view_basis(basis4, main = "m=4 (discontinuous at x=3)")

par(mfrow = c(1, 1))

Derivatives at Knots

The Spline_der_knot() function computes derivative values at knot points efficiently.

# Compute derivatives at knots
der_knots <- Spline_der_knot(basis, der = 1)
print(head(der_knots))

##      [,1]  [,2]  [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    0  0.00  0.00  0.0  0.0    0    0    0
## [2,]    0  0.00  0.00  0.0  0.0    0    0    0
## [3,]    0  0.00  0.00  0.0  0.0    0    0    0
## [4,]   -3  3.00  0.00  0.0  0.0    0    0    0
## [5,]    0 -0.75  0.25  0.5  0.0    0    0    0
## [6,]    0  0.00 -0.50  0.0  0.5    0    0    0

Simple Example: Coherence of Derivative Evaluation Methods (Degree 5)

# Create a degree 5 B-spline
sn5 <- c(0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5)
basis5 <- Bspline_base(sn5, degree = 5)
basis5$coeff <- c(1, -2, 3, -1, 2, 1, 0, -1, 2,5)

# Evaluation points
x <- seq(0, 5, length.out = 100)

# --- Method 1: spline_eval with der parameter ---
y1 <- spline_eval(basis5, x, der = 1)

# --- Method 2: Bspline_deriv + spline_eval ---
der_basis <- Bspline_deriv(basis5, der = 1)
y2 <- spline_eval(der_basis, x)

# --- Method 3: Step-by-step (basis derivative + bs_direct + matrix mult) ---
der_basis2 <- Bspline_base_deriv(basis5, der = 1)
Bvals <- bs_direct(der_basis2, x)
y3 <- t(Bvals) %*% basis5$coeff

# All three methods give identical results
cat("Max differences:\n")

## Max differences:

cat("  Method 1 vs Method 2:", max(abs(y1 - y2)), "\n")

##   Method 1 vs Method 2: 8.999983e-10

cat("  Method 1 vs Method 3:", max(abs(y1 - y3)), "\n")

##   Method 1 vs Method 3: 0

cat("  Method 2 vs Method 3:", max(abs(y2 - y3)), "\n")

##   Method 2 vs Method 3: 8.999983e-10

# Plot to verify visually
plot(x, y1, type = "l", col = "blue", lwd = 2,
     main = "First Derivative - All Methods Coincide (deg 5)",
     xlab = "x", ylab = "f'(x)")
lines(x, y2, col = "red", lty = 2, lwd = 2)
lines(x, y3, col = "green", lty = 3, lwd = 2)
legend("topright", 
       legend = c("spline_eval(der=1)", "Bspline_deriv", "Step-by-step"),
       col = c("blue", "red", "green"), lty = c(1, 2, 3), lwd = 2)
grid()

Summary

Feature	Function	Description
Build basis	`Bspline_base()`	Create B-spline basis
View basis	`view_basis()`	Visualize basis functions
Differentiate basis	`Bspline_base_deriv()`	Compute derivative basis
Differentiate function	`spline_eval(der=...)`	Evaluate derivatives
Derivative coefficients	`Bspline_deriv()`	Get derivative B-spline
PP conversion	`Bsplinetopp()`	Convert to PP form
Callable spline	`make_spline()`	Create callable function
Extract params	`get_parameters()`	Get parameters from callable
Derivatives at knots	`Spline_der_knot()`	Efficient knot derivatives

All Bsplines functions are implemented in pure R and support B-splines of any degree (greater than 0), with full derivative and PP-form capabilities. Limitation to degree 4 is only for regression with contsraints over intervals.