# Natural cubic splines¶

Natural cubic splines (NCS) are an interpolation tool that is of interest to many applications. They are a special case of spline models, which are favoured for their smoothness and are relevant in particular applications where the degree of smoothness of the interpolant is important. This page presents the NCS linear system of equations and an implementation plus demo on toy data.

## Spline models¶

In interpolation problems, we are given some datapoints and want to produce a function that passes through them. One choice for achieving this would be to use a piecewise linear function, and constrain the slopes and intercepts of the piecewise linear segments, such that it passes through the datapoints. This would give a rough approximation to the underlying function. However, in many settings it may be appropriate to impose further, or different conditions on the interpolating curve. For example, we may know that the underlying function is differentiable (the piecewise linear spline is not differentiable), or that it is $$n$$-times differentiable and so on.

To achieve further regularity, we can choose a piecewise polynomial as the interpolating function. We can then constrain the piecewise segments to pass through the datapoints and (if necessary) to have equal derivatives at the datapoints. Such piecewise polynomials are called splines. Cubic splines are splines in which the piecewise polynomials are of third-order. NCS have the further constraint that the curvature, that is the second derivative, of the interpolating function at the first and last point is $$0$$.

In particular,[Rid00] given $$N$$ datapoints a cubic piecewise polynomial has $$N-1$$ piecewise segments, each with four coeffcients, giving $$4(N - 1)$$ degrees of freedom. Constraining the function values, first derivatives and second derivatives to be equal for the piecewise polynomials at all points except the first and last points gives $$3(N-2)$$ equations. Constraining the function values to be equal to the datapoint values gives $$N$$ more constraints. We are left with $$2$$ degrees of freedom, which can be dealt in different ways, giving different spline interpolants. If we constrain the second derivatives to be zero at the first and last points, we obtain NCS. Since all the constraints are linear, solving for the coefficients of a spline interpolant is a linear equation. Because the constraints relate pairs of segments only, the matrix corresponding to these linear constraints is tri-diagonal and can therefore be solved in $$\mathcal{O}(N)$$ time. Below are the expressions[Rid00] for the NCS system of equations.

Result (Natural cubic spline) Given data $$\{x_n, y_n\}_{n = 0}^N$$ with $$x_n < x_{n+1}$$, the natural cubic spline interpolating the data in the $$n^{th}$$ interval $$[x_n, x_{n+1}]$$ is

\begin{align} S_n(x) = &~\alpha_n (x - x_n)^3 + \beta_n (x_{n + 1} -x)^3 + \gamma_n (x - x_n) + \delta_n (x_{n + 1} - x). \end{align}

where the constants are given by

\begin{align} \alpha_n &= \frac{z_{n+1}}{6h_n},~~\beta_n = \frac{z_n}{6h_n},~~\gamma_n = \left(\frac{y_{n + 1}}{h_n} - \frac{y_{n + 1}h_n}{6} \right), ~~\delta_n = \left(\frac{y_n}{h_n} - \frac{y_nh_n}{6}\right), \end{align}

and the curvatures $$z_n$$ are the solution to the linear problem

\begin{split}\begin{align} \begin{pmatrix} v_1 & h_1 & 0 & \dots & 0 & 0 & 0 \\ h_1 & v_2 & h_2 & \dots & 0 & 0 & 0 \\ 0 & h_2 & v_3 & \dots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & v_{N-3} & h_{N-3} & 0 \\ 0 & 0 & 0 & \dots & h_{N-3} & v_{N-2} & h_{N-2} \\ 0 & 0 & 0 & \dots & 0 & h_{N-2} & v_{N-1} \\ \end{pmatrix}\begin{pmatrix} z_{1} \\ z_{2} \\ \vdots \\ \vdots \\ z_{N-2} \\ z_{N-1} \\ \end{pmatrix}=\begin{pmatrix} u_{1} \\ u_{2} \\ \vdots \\ \vdots \\ u_{N-2} \\ u_{N-1} \\ \end{pmatrix} \end{align}\end{split}

where $$z_0 = z_n = 0$$ and the linear problem is defined by

$h_n = x_{n + 1} - x_n,~b_n = \frac{1}{h_n} (y_{n + 1} - y_n),~v_n = 2 (h_n + h_{n - 1}),~u_n = 6 (b_n - b_{n - 1})$

As a further motivating point, it can be shown[Rid00] that out of all curves NCS is the smoothest interpolator, where smoothness is quantified by an appropriate metric based on the second derivative of the interpolant.

Result (NCS are the smoothest interpolator) Given data $$x_n, y_n \in \mathbb{R}$$ for $$n = 0, 1, 2, ..., N$$, among all twice continuously differentiable functions functions $$f : [a, b] \to \mathbb{R}$$ which interpolate $$\left\{x_n, y_n\right\}_{n = 0}^N$$, the natural cubic spline minimises the curvature metric

$C(f) = \int^b_a \left(f''(x)\right)^2 dx.$

## Implementation¶

Below is two functions implementing NCS. The first takes in data x_data, y_data computes $$h$$ and solves the NCS equations for $$z$$. This is the costlier part of the algorithm ($$\mathcal{O}(N)$$ in the number of datapoints) but only needs to be run once.

import numpy as np
from scipy.linalg import solve_banded

def solve_for_zh(x_data, y_data):

N = x_data.shape[0]

h = x_data[1:] - x_data[:-1]
b = (y_data[1:] - y_data[:-1]) / h
v = 2 * (h[1:] + h[:-1])
u = 6 * (b[1:] - b[:-1])

A = np.zeros((N - 2, N - 2))
A[np.arange(N-2), np.arange(N-2)] = v
A[np.arange(N-3)+1, np.arange(N-3)] = h[1:-1]
A[np.arange(N-3), np.arange(N-3)+1] = h[1:-1]

z = np.linalg.solve(A, u)
z = np.concatenate([[0.], z, [0.]])

return z, h


Once the $$z$$ and $$h$$ values have been computed, we can use them to compute $$S(x)$$. The function below takes in the data, the $$z$$ and $$h$$ values computed and as well as an input location and returns the value of the NCS at the input location.

def S(x, x_data, y_data, z, h):

i = np.searchsorted(x_data, x) - 1

if i >= 0 and i <= x_data.shape[0] - 2:

S = z[i+1] / (6 * h[i]) * (x - x_data[i]) ** 3
S = S + z[i] / (6 * h[i]) * (x_data[i+1] - x) ** 3
S = S + (y_data[i+1] / h[i] - z[i+1] * h[i] / 6) * (x - x_data[i])
S = S + (y_data[i] / h[i] - z[i] * h[i] / 6) * (x_data[i+1] - x)

return S

elif i < 0:

slope = z[0] / (2 * h[0]) * (x_data[1] - x) ** 2
slope = slope + (y_data[1] / h[0] - z[1] * h[0] / 6)
slope = slope - (y_data[0] / h[0] - z[0] * h[0] / 6)

S = y_data[0] + slope * (x - x_data[0])

return S

elif i >= x_data.shape[0] - 2:

slope = z[-1] / (2 * h[-2]) * (x - x_data[-2]) ** 2
slope = slope + (y_data[-1] / h[-2] - z[-1] * h[-1] / 6)
slope = slope - (y_data[-2] / h[-2] - z[-2] * h[-1] / 6)

S = y_data[-1] + slope * (x - x_data[-1])

return S


We can now fit NCS to a set of datapoints.

# Set random seed
np.random.seed(0)

# Generate data from noiseless sinusoid, with random inputs
N = 10
x_data = np.cumsum(np.random.uniform(size=(N+1,)))
y_data = np.sin(2 * np.pi * x_data)

import matplotlib.pyplot as plt

# Plot interpolant and datapoints
plt.figure(figsize=(8, 3))
plt.scatter(x_data, y_data, marker='x', c='red', s=50, zorder=2)

# Format plots
plt.title('Example data', fontsize=24)
plt.xlim([-0.5, 8])
plt.ylim([-3, 2])
plt.xticks([])
plt.yticks([])
plt.show()

# Solve for z and h
z, h = solve_for_zh(x_data, y_data)

x_plot = np.linspace(-0.5, 8., 100)
spline = [S(x_, x_data, y_data, z, h) for x_ in x_plot]

# Plot interpolant and datapoints
plt.figure(figsize=(8, 3))
plt.plot(x_plot, spline, color='k', zorder=1)
plt.scatter(x_data, y_data, marker='x', c='red', s=50, zorder=2)

# Format plots
plt.title('NCS fit and data', fontsize=24)
plt.xlim([-0.5, 8])
plt.ylim([-3, 2])
plt.xticks([])
plt.yticks([])
plt.show()


## Conclusion¶

This page has presented the NCS equations and a demo fit of NCS. Splines, and in particular cubic splines are of interest to many applications because of they are cheap to evaluate and have attractive smoothness conditions. For example, I am interested in them because they are used in the context of Neural Controlled Differential Equations.[KMFL20] There, they are favoured because they have the minimal smoothness conditions required for the modelling problem of interest. NCS and more generally any spline model can also be extended to Bayesian models of data using Gaussian Processes.[RW05]

## References¶

KMFL20

Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equations for irregular time series. arXiv preprint arXiv:2005.08926, 2020.

RW05

Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, 2005. ISBN 026218253X.

Rid00(1,2,3)

Mustrum Ridcully. Kvarving, arne morten. Department of Mathematical Sciences, 2000.