revise docs

Author: MikeHLee

README.md

@@ -1,21 +1,24 @@
-# pydelt: Dynamical Systems & Differential Equations Approximation
+# PyDelt: Advanced Numerical Function Interpolation & Differentiation
 
 [![PyPI version](https://badge.fury.io/py/pydelt.svg)](https://badge.fury.io/py/pydelt)
 [![Documentation Status](https://readthedocs.org/projects/pydelt/badge/?version=latest)](https://pydelt.readthedocs.io/en/latest/?badge=latest)
 [![Python 3.8+](https://img.shields.io/badge/python-3.8+-blue.svg)](https://www.python.org/downloads/)
 [![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)
 
-**pydelt** bridges the gap between theoretical mathematics and real-world data, enabling scientists to extract the underlying dynamics from noisy, incomplete observations. By providing a unified framework for numerical differentiation and integration, pydelt transforms raw measurements into meaningful differential equations that reveal the fundamental laws governing complex systems. Whether you're reconstructing phase spaces in nonlinear dynamics, identifying governing equations in fluid mechanics, or extracting rate constants in chemical kinetics, pydelt empowers you to move beyond mere data description to discover the mathematical essence of your phenomena.
+**PyDelt** transforms raw data into mathematical insights through advanced numerical interpolation and differentiation. Whether you're analyzing experimental measurements, financial time series, or complex dynamical systems, PyDelt provides the tools to extract derivatives, gradients, and higher-order mathematical properties with precision and reliability.
 
-## 🚀 Key Features
+## Why PyDelt?
 
-- **Universal Differentiation Interface**: Consistent `.differentiate(order, mask)` API across all interpolation methods
-- **Multivariate Calculus**: Gradient (∇f), Jacobian (J_f), Hessian (H_f), and Laplacian (∇²f) computation
-- **Multiple Derivative Methods**: LLA/GLLA (analytical Hermite derivatives), FDA/Splines (analytical spline derivatives), LOWESS/LOESS (numerical differentiation)
-- **Automatic Differentiation**: PyTorch and TensorFlow backends for exact gradient computation in complex functions
-- **Vector & Tensor Operations**: Full support for vector-valued functions and tensor calculus
-- **Integration Capabilities**: Numerical integration with error estimation
-- **Comprehensive Error Handling**: Robust validation and informative error messages
+Traditional numerical differentiation is notoriously unstable - small changes in data can cause large changes in derivatives. PyDelt solves this through smart smoothing that preserves important features while reducing noise, multiple methods so you can choose the best approach for your data, and a unified interface that makes comparison and validation straightforward.
+
+## 🎯 Key Features
+
+• **Universal Interface**: Every method uses the same `.fit().differentiate()` pattern - easy to learn, easy to switch
+• **From Simple to Sophisticated**: Start with splines, scale to neural networks with automatic differentiation
+• **Multivariate Ready**: Gradients, Jacobians, Hessians, and Laplacians for functions of multiple variables
+• **Noise Robust**: Built-in smoothing and validation ensure reliable results from imperfect data
+• **Stochastic Extensions**: Proper handling of financial derivatives with Itô and Stratonovich corrections
+• **Production Ready**: Comprehensive error handling, extensive testing, and clear documentation
 
 ## Installation
 

docs/_build/doctrees/changelog.doctree

@@ -39,19 +39,19 @@
   </form>
 </div>
         </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">
-              <p class="caption" role="heading"><span class="caption-text">Getting Started:</span></p>
+              <p class="caption" role="heading"><span class="caption-text">Start Here:</span></p>
 <ul>
 <li class="toctree-l1"><a class="reference internal" href="../installation.html">Installation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../quickstart.html">Quick Start Guide</a></li>
 </ul>
-<p class="caption" role="heading"><span class="caption-text">Progressive Learning Path:</span></p>
+<p class="caption" role="heading"><span class="caption-text">Master the Methods:</span></p>
 <ul>
 <li class="toctree-l1"><a class="reference internal" href="../basic_interpolation.html">Basic Interpolation &amp; Derivatives</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../neural_networks.html">Neural Networks &amp; Automatic Differentiation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../multivariate_calculus.html">Multivariate Calculus &amp; Vector Operations</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../stochastic_computing.html">Stochastic Computing &amp; Probabilistic Derivatives</a></li>
 </ul>
-<p class="caption" role="heading"><span class="caption-text">Reference:</span></p>
+<p class="caption" role="heading"><span class="caption-text">Reference &amp; Help:</span></p>
 <ul>
 <li class="toctree-l1"><a class="reference internal" href="../examples.html">Examples</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../api.html">API Reference</a></li>

docs/_build/doctrees/environment.pickle

@@ -3,6 +3,58 @@ Changelog
 
 All notable changes to this project will be documented in this file.
 
+Version 0.6.0 (2025-08-25)
+--------------------------
+
+🎉 **Major New Features**
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+* **Stochastic Derivatives**: Revolutionary new feature enabling probabilistic derivatives with uncertainty quantification
+  
+  * **6 Probability Distributions**: Normal, Log-Normal, Gamma, Beta, Exponential, Poisson link functions
+  * **Stochastic Calculus Methods**: Both Itô's lemma and Stratonovich integral corrections
+  * **Financial Applications**: Geometric Brownian motion, option pricing, risk analysis
+  * **Universal Integration**: Works with all interpolation methods (Spline, LOWESS, LOESS, LLA, GLLA, Neural Networks)
+
+* **Progressive Documentation Structure**: Complete documentation overhaul with learning path approach
+  
+  * **Level 1**: Basic Interpolation & Derivatives
+  * **Level 2**: Neural Networks & Automatic Differentiation  
+  * **Level 3**: Multivariate Calculus
+  * **Level 4**: Stochastic Computing
+
+🚀 **Enhanced Features**
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+* **Universal Stochastic API**: All interpolators now support `.set_stochastic_link()` method
+* **Automatic Derivative Transformation**: Derivatives automatically include stochastic corrections when link functions are set
+* **Real-World Examples**: Financial modeling, population dynamics, interest rate modeling with stochastic effects
+* **Comprehensive Testing**: Full test suite for stochastic derivatives across all interpolation methods
+
+🔧 **Technical Improvements**
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+* **New Module**: `src/pydelt/stochastic.py` - Complete stochastic calculus framework
+* **Enhanced Interpolation**: All interpolator classes extended with stochastic link function support
+* **Helper Functions**: `src/pydelt/stochastic_helpers.py` for consistent stochastic transformations
+* **Demonstration Scripts**: `demo_stochastic_derivatives.py` showcasing real-world applications
+
+📚 **Documentation**
+~~~~~~~~~~~~~~~~~~~~
+
+* **4 New Documentation Pages**: Progressive learning path from basic to advanced concepts
+* **Well-Known Examples**: Projectile motion, Runge function, fluid dynamics, optimization landscapes
+* **Application Focus**: Financial engineering, scientific computing, engineering applications
+* **Best Practices**: Method selection guidelines, parameter tuning, validation strategies
+
+🎯 **Applications Enabled**
+~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+* **Financial Engineering**: Option Greeks, volatility modeling, risk-neutral measures
+* **Scientific Computing**: Uncertainty quantification, stochastic differential equations
+* **Engineering**: Robust control, system identification with noise
+* **Machine Learning**: Bayesian neural networks, uncertainty-aware optimization
+
 Version 0.4.0 (2025-07-26)
 --------------------------