monspline 0.1.3

monspline

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Python package for efficient spline interpolation using C++ (via CFFI),

with a strong focus on shape preservation.

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Overview

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This project implements and evaluates spline interpolation methods,

including:

• Cubic spline interpolation

• Monotone-preserving spline interpolation

• Taut exponential spline interpolation

• Tensor-product splines (multivariate, monotone)

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Motivation

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Spline interpolation is widely used in numerical analysis and machine

learning. However:

• Cubic splines may introduce oscillations

• Monotone splines improve stability

• Real-world datasets often contain noise and discontinuities

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[Methods and examples](https://gleb.cgps.ch/Assets/Documents/Shape_preserving_approximation_using_least_squares_splines.pdf, https://gleb.cgps.ch/Assets/Documents/Monotone_approximation_of_aggregation_operators_using_least_squares_splines.pdf)

Architecture

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Python - CFFI - C wrapper - C++ classes

• Python: User API and plotting

• CFFI: Interface layer

• C wrapper: Object management using handles

• C++: Core spline algorithms

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Features

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Cubic Spline

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• Smooth interpolation (C^2 continuity)

• Efficient C++ implementation

Monotone Spline

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• Preserves monotonicity

• Prevents overshooting

• Better for noisy / real-world data

• Taut spline remove unnecessary inflection points

Tensor (Multivariate) Spline

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• Supports higher-dimensional approximation

• Used for 3D surface modelling

• Support monotonicity preservation

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Build

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python build.py

 Run Tests

python tests/testcs.py

python tests/test_tensor_spline.py

Observation: Cubic splines introduce oscillations, while monotone

splines preserve structure.

Observation: Cubic splines overshoot near discontinuities. Monotone

splines provide stable approximation.

3D Tensor Spline

Observation: Tensor splines approximate smooth surfaces effectively in

2D.

Key Insights

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Cubic splines - smooth but unstable near discontinuities

Monotone splines - stable and shape-preserving

Taut splines - remove extraneous inflection points, no monotonicity

assumed

Tensor splines - extend interpolation to higher dimensions

References

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3. de Boor, A Practical Guide to Splines

Fritsch & Carlson, Monotone Piecewise Cubic Interpolation

G.Beliakov, (2000) Shape preserving approximation using least squares

splines. Analysis in theory and applications, 16(4), 80-98

G.Beliakov, (2002) Monotone approximation of aggregation operators using

least squares splines. International journal of uncertainty, fuzziness,

and knowledge-based systems, 10(6), 659-676.

Installation

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To install type:

$ pip install monspline

Usage of monspline

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from monspline import TensorSpline

from monspline import CubicSpline

 generate/read data tables

x = [0,1,2,3,4,5]

y = [0,0.1,0.2,0.3,0.4,0.5]

s = CubicSpline(x, y, type=1)

y=s.value(0.5)

s1= = TensorSpline(kind=0, dim=2, knots=knots, data=data, exactdata=exactdata)

y = s1.value([0.5, 0.5])

Usage of CubicSpline(x,y type, tau=0)

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Construction of a cubic/monotone/taut univariate spline

Parameters

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Input parameters:

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x[]: NumPy array of size n, float

w[]: NumPy array of size n, float

type: int, can be 1 (cubic),2 (taut exponential, required tau>0.01) , 3

monotone increasing

tau: float, tension parameter in taut splines to straighten inflection

points

Output parameters

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s: spline object

Usage of value(x)

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Calculation of a cubic/monotone/taut spline value

Parameters

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Input parameters:

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t: float, the query point

Output

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y: float, spline value at t

Usage of TensorSpline(kind, dim, knots, data, exactdata)

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Construction of a tensor product spline of dimension dim

Parameters

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Input parameters:

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kind: int, type of spline, see below

dim: int, dimension

knots: array of size(dim,knotN), float, contains spline knots for each

variable

data[]: NumPy array of size (n,dim+1), float, contains n data (x and y)

to approximate

exactdata[]: NumPy array of size (m,dim+1), float, contains m data (x

and y) to interpolate (exact fit)

Output parameters

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s: spline object

Interpretation of kind

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For each variable: 0: dim-dimensional tensor product spline of order 2

(linear), no constraints

1: monotone increasing

2: monotone decreasing

When dim >1, concatenate the values for each variable starting from the

right.

Example: dim=2

12: means increasing in the second variable (1) and decreasing in the

first variable (2)

dim=3

110: increasing in the third and second variable unrestricted in the

first variable

When dim==1:

kind=11,12,… means spline of order 3,4,… (order 4 means cubic spline),

i.e., kind-8 = spline order otherwise order is forced to be 2

Usage of value(x)

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Calculation of tensor spline value at x

Parameters

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Input parameters:

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x: float, when dim==1 and array of size dim otherwise, the query point

Output

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y: float, spline value at x

Usage of value_der(x,var)

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Calculation of tensor spline derivative value at x

Parameters

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Input parameters:

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x: float, when dim==1 and array of size dim otherwise, the query point

var: int, with respect to which variable, starting from 0 to dim-1

Output

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y: float, spline partial derivative value at x