DataCollocations.jl

DataCollocations.jl provides non-parametric data collocation functionality for smoothing timeseries data and estimating derivatives.

Two Approaches for Data Collocation

DataCollocations.jl offers two distinct methodologies for data collocation, each optimized for different data characteristics:

1. Kernel Smoothing Methods (For Noisy Data)

Robust regression-based approach for handling noisy measurements:

• Multiple kernel functions: Epanechnikov, Triangular, Gaussian, Quartic, Triweight, Tricube, Cosine, Logistic, Sigmoid, Silverman
• Automatic bandwidth selection: Optimally balances bias-variance tradeoff
• Noise robustness: Designed to handle measurement noise and outliers
• Regression splines: Smoothed fit that doesn't necessarily pass through data points
• Best for: Experimental data with significant measurement noise

2. DataInterpolations.jl Integration (For Clean Data)

Exact interpolation approach for high-quality data:

• Standard interpolation methods: CubicSpline, QuadraticInterpolation, BSpline, Akima, etc.
• Exact fitting: Interpolation curves pass exactly through data points
• Minimal noise assumption: Assumes data points are accurate measurements
• High efficiency: Fast computation for clean, well-sampled data
• Best for: Simulation data or high-precision measurements with minimal noise

When to Use Each Approach

Data Characteristics	Recommended Method	Reason
Experimental measurements with noise	Kernel smoothing	Robust to noise, provides smoothed estimates
Simulation results	DataInterpolations	Exact, efficient, preserves accuracy
Sparse, clean data	DataInterpolations (CubicSpline)	Exact interpolation between points
Dense, noisy data	Kernel smoothing (Epanechnikov)	Optimal noise handling
Very noisy data	NoiseRobustDifferentiation.jl	Specialized for heavy noise

Features

• Multiple kernel functions for data smoothing
• Automatic bandwidth selection
• Support for DataInterpolations.jl integration
• Derivative estimation from noisy data
• Efficient implementation with pre-allocated arrays

Installation

Since this package is not yet registered, install it directly from GitHub:

using Pkg
Pkg.add(url="https://github.com/SciML/DataCollocations.jl")

Once registered in the General registry:

using Pkg
Pkg.add("DataCollocations")

Quick Start

using DataCollocations
using OrdinaryDiffEq

# Generate some sample data
f(u, p, t) = p .* u
prob = ODEProblem(f, [1.0], (0.0, 10.0), [-0.1])
t = collect(0.0:0.1:10.0)
data = Array(solve(prob, Tsit5(); saveat=t))

# Perform collocation to estimate derivatives and smooth data
u′, u = collocate_data(data, t, TriangularKernel(), 0.1)

# u′ contains the estimated derivatives
# u contains the smoothed data

Available Kernels

DataCollocations.jl supports multiple kernel functions for noisy data:

Bounded Support Kernels (support on [-1, 1]):

• EpanechnikovKernel()
• UniformKernel()
• TriangularKernel() (default)
• QuarticKernel()
• TriweightKernel()
• TricubeKernel()
• CosineKernel()

Unbounded Support Kernels:

• GaussianKernel()
• LogisticKernel()
• SigmoidKernel()
• SilvermanKernel()

DataInterpolations.jl Integration

With DataInterpolations.jl loaded, you can use interpolation methods:

using DataInterpolations

# Use interpolation to generate data at intermediate timepoints
tpoints_sample = 0.05:0.1:9.95
u′, u = collocate_data(data, t, tpoints_sample, LinearInterpolation)

API Reference

collocate_data

u′, u = collocate_data(data, tpoints, kernel=TriangularKernel(), bandwidth=nothing)
u′, u = collocate_data(data, tpoints, tpoints_sample, interp, args...)

Arguments:

• data: Matrix where each column is a snapshot of the timeseries
• tpoints: Time points corresponding to data columns
• kernel: Kernel function for smoothing (default: TriangularKernel())
• bandwidth: Smoothing bandwidth (auto-selected if nothing)
• tpoints_sample: Sample points for interpolation method
• interp: Interpolation method from DataInterpolations.jl

Returns:

• u′: Estimated derivatives
• u: Smoothed data

Contributing

Contributions are welcome! Please see the contributing guidelines for more information.

Related Packages

• DiffEqFlux.jl - Neural differential equations
• DataInterpolations.jl - Interpolation methods
• OrdinaryDiffEq.jl - ODE solvers
• NoiseRobustDifferentiation.jl - Specialized library for estimating derivatives from very noisy data

Citation

If you use DataCollocations.jl in your research, please cite the collocation methodology paper:

@article{roesch2021collocation,
  title={Collocation based training of neural ordinary differential equations},
  author={Roesch, Elisabeth and Rackauckas, Christopher and Stumpf, Michael P. H.},
  journal={Statistical Applications in Genetics and Molecular Biology},
  volume={20},
  number={2},
  pages={37--49},
  year={2021},
  publisher={De Gruyter},
  doi={10.1515/sagmb-2020-0025},
  url={https://doi.org/10.1515/sagmb-2020-0025}
}