pydelt: Advanced Numerical Differentiation & Stochastic Computing

pydelt is a comprehensive Python library for numerical differentiation, interpolation, and stochastic computing. From basic polynomial derivatives to advanced stochastic calculus, pydelt provides a unified framework that scales from simple data analysis to cutting-edge research in dynamical systems, financial modeling, and scientific computing.

🎯 Key Strengths & Applications

Scientific Computing * Dynamical Systems: Reconstruct differential equations from noisy time series data * Phase Space Analysis: Compute derivatives for embedding dimension analysis and chaos detection * Fluid Dynamics: Analyze velocity fields, compute vorticity and strain tensors * Signal Processing: Extract instantaneous frequency and phase derivatives from complex signals

Financial & Risk Modeling * Stochastic Derivatives: Apply Itô’s lemma and Stratonovich corrections to derivative computations * Option Pricing: Compute Greeks (delta, gamma, theta) with stochastic link functions * Risk Management: Model volatility surfaces and correlation structures with multivariate derivatives * Algorithmic Trading: Real-time derivative computation for momentum and mean-reversion strategies

Engineering & Control * System Identification: Extract governing equations from experimental data * Control Theory: Design controllers using derivative information from system responses * Optimization: Gradient-based methods with automatic differentiation support * Machine Learning: Custom loss functions with higher-order derivative constraints

πŸš€ Progressive Feature Set

Level 1: Basic Interpolation & Derivatives * Classical Methods: Splines, polynomial fitting, local linear approximation * Universal API: Consistent .fit().differentiate() interface across all methods * Higher-Order Support: Analytical and numerical derivatives up to arbitrary order

Level 2: Advanced Interpolation * Robust Methods: LOWESS, LOESS for noisy data and outlier resistance * Neural Networks: Deep learning-based interpolation with automatic differentiation * Adaptive Methods: LLA, GLLA with local bandwidth selection

Level 3: Multivariate Calculus * Vector Calculus: Gradient (βˆ‡f), Jacobian (βˆ‚f/βˆ‚x), Hessian (βˆ‚Β²f/βˆ‚xΒ²), Laplacian (βˆ‡Β²f) * Tensor Operations: Full support for vector-valued functions and tensor calculus * Mixed Derivatives: Cross-partial derivatives for multivariate analysis

Level 4: Stochastic Computing ⭐ New Feature * Stochastic Link Functions: 6 probability distributions (Normal, Log-Normal, Gamma, Beta, Exponential, Poisson) * Stochastic Calculus: Itô’s lemma and Stratonovich integral corrections * Financial Applications: Geometric Brownian motion, volatility modeling, option pricing * Risk Analysis: Uncertainty propagation through derivative computations

πŸ“¦ Installation

Install pydelt from PyPI:

pip install pydelt

πŸ”§ Quick Start Examples

1. Universal Differentiation Interface

import numpy as np
from pydelt.interpolation import SplineInterpolator

# Generate sample data: f(t) = sin(t)
time = np.linspace(0, 2*np.pi, 100)
signal = np.sin(time)

# Universal API: fit interpolator and compute derivatives
interpolator = SplineInterpolator(smoothing=0.1)
interpolator.fit(time, signal)
derivative_func = interpolator.differentiate(order=1)

# Evaluate derivative at any points
derivatives = derivative_func(time)
print(f"Max error vs cos(t): {np.max(np.abs(derivatives - np.cos(time))):.4f}")

2. Multivariate Calculus

from pydelt.multivariate import MultivariateDerivatives

# Generate 2D data: f(x,y) = xΒ² + yΒ²
x = np.linspace(-2, 2, 50)
y = np.linspace(-2, 2, 50)
X, Y = np.meshgrid(x, y)
Z = X**2 + Y**2

# Fit multivariate derivatives
input_data = np.column_stack([X.flatten(), Y.flatten()])
output_data = Z.flatten()

mv = MultivariateDerivatives(SplineInterpolator, smoothing=0.1)
mv.fit(input_data, output_data)

# Compute gradient: βˆ‡f = [2x, 2y]
gradient_func = mv.gradient()
test_point = np.array([[1.0, 1.0]])
gradient = gradient_func(test_point)
print(f"Gradient at (1,1): {gradient[0]} (expected: [2, 2])")

3. Stochastic Derivatives for Financial Modeling

from pydelt.interpolation import SplineInterpolator

# Stock price data following geometric Brownian motion
time = np.linspace(0, 1, 252)  # 1 year of daily data
stock_prices = 100 * np.exp(0.1*time + 0.2*np.random.randn(252).cumsum()/np.sqrt(252))

# Fit interpolator with log-normal stochastic link
interpolator = SplineInterpolator(smoothing=0.1)
interpolator.fit(time, stock_prices)
interpolator.set_stochastic_link('lognormal', sigma=0.2, method='ito')

# Compute stochastic derivatives (automatically applies ItΓ΄ correction)
stochastic_deriv = interpolator.differentiate(order=1)
derivatives = stochastic_deriv(time)

print(f"Regular vs Stochastic derivative difference: {np.mean(np.abs(derivatives - regular_derivatives)):.2f}")

🌌 Applications in Dynamical Systems

pydelt excels in analyzing dynamical systems and differential equations from data:

  • System Identification: Reconstruct differential equations from time series observations

  • Phase Space Reconstruction: Compute derivatives for embedding dimension analysis

  • Stability Analysis: Calculate Jacobians and eigenvalues for equilibrium point classification

  • Bifurcation Analysis: Track parameter-dependent changes in system behavior

  • Control Theory: Design controllers using derivative information from system responses

  • Fluid Dynamics: Analyze velocity fields and compute vorticity, divergence, and strain tensors

  • Continuum Mechanics: Calculate stress and strain derivatives for material property estimation

  • Signal Processing: Extract instantaneous frequency and phase derivatives from complex signals

Time Series as a Special Case: Traditional time series derivative analysis is just one application of our broader dynamical systems framework. The universal differentiation interface seamlessly handles both temporal data and general multivariate functions.

πŸ“š Documentation Contents

Getting Started:

Progressive Learning Path:

Reference:

πŸ“‹ Indices and Tables