FunctionalCurves

Research code for simulating dependent (mixing) time series and estimating Tukey's halfspace depth and the associated minimal direction at a point, both empirically (via simulation) and analytically (for Gaussian / VAR(1) processes). Used to study how fast empirical depth/direction estimates converge to their theoretical values as sample size grows.

Structure

FunctionalCurves/
├── pyproject.toml                 # Package metadata & dependencies (pip install -e .)
├── functionalcurves/               # Installable package
│   ├── __init__.py
│   ├── mixing_models.py            # Simulate mixing (weakly dependent) bivariate processes
│   └── depth.py                    # Compute Tukey depth / minimal direction, empirically and analytically
├── notebooks/                      # Interactive demos
│   ├── example_var1.ipynb          # Demo: VAR(1) process + empirical depth estimation & convergence
│   ├── example_polynomial.ipynb    # Demo: MixingLinearModel process + Gaussian analytic depth comparison
│   └── scratch_interactive.ipynb   # Scratch/interactive notebook, mostly duplicated commented-out code
└── scripts/
    └── donsker_cdf_scratch.py      # Scratch script: empirical CDF convergence (Donsker) plots

Modules

mixing_models.py

Generates synthetic bivariate sample paths with controllable dependence ("mixing rate").

• transition_markov(X, p, e) / transition_diff(X, p, e) — transition functions defining how the next state depends on the previous state and an innovation e.
• MixingMarkovModel — simple Markov-chain-style simulator: repeatedly applies a transition function to Gaussian innovations.
• MixingLinearModel — simulates a linearly-weighted mixing process where weights decay as k^-mixing_rate; includes an error-correction term (zeta/zeta_k, via scipy.special.zeta) to control approximation error from truncating an infinite sum, and a distribution() method giving the theoretical mean/covariance of the limiting process.
• CovHC_ — placeholder for a heteroscedasticity-consistent covariance estimator (unimplemented).
• if __name__ == '__main__' block — example script simulating a process and plotting estimated depth/direction vs. sample size (uses depth.py).

depth.py

Core depth/direction estimation logic.

• rad(v1, v2, ...) — signed angle in [0, 2π) between vectors (used throughout for directional comparisons).
• rad_wu, rad_dec — related/deprecated angle helpers.
• GaussianDepth(mean, cov, X0) — closed-form Tukey depth and minimal direction for a bivariate normal distribution.
• Estimator — empirical depth/direction estimator given a sample X and a point X0. Two methods:
  • 'deg' — scans candidate directions uniformly over [-π, π].
  • 'point_wise' — uses each sample point's own direction as a candidate. Returns the minimal depth, the corresponding direction, and the full arrays of angles/depths/directions considered.
• TDE — deprecated older counting-based depth estimator, superseded by Estimator.
• Analytic_Depth — computes the exact Tukey depth and minimal direction for a stationary VAR(1) process (X_t = A0 + A1 X_{t-1} + noise), by deriving the process's stationary mean/covariance and applying the Gaussian depth formula (TD_analytic).
• Depth class — thin wrapper stub intended to expose a halfspace depth method (currently unimplemented / placeholder).
• if __name__ == '__main__' block — simulates a VAR(1) process via statsmodels, computes the true analytic depth/direction, and compares against empirical estimates ('deg' vs 'point_wise') as sample size increases, plotting convergence.

scripts/donsker_cdf_scratch.py

Standalone scratch script unrelated to the depth machinery: plots the empirical CDF process √N (F_N(x) - F(x)) for uniform samples of increasing size, illustrating Donsker's theorem.

Notebooks

• notebooks/example_var1.ipynb — Simulates a VAR(1) process, computes the analytic Tukey depth/direction at a chosen point X0, then compares against empirical estimates (Estimator, both methods) across increasing sample sizes, with convergence and directional-vector plots. Also has a section using mixing_models.MixingModel (transition-based simulation).
• notebooks/example_polynomial.ipynb — Same style of analysis but driving the sample from MixingLinearModel instead of a VAR process, comparing empirical estimates against the Gaussian analytic depth (GaussianDepth) using the model's theoretical stationary distribution.
• notebooks/scratch_interactive.ipynb — Exploratory/scratch notebook; most cells are commented-out duplicates of the empirical-CDF script from scripts/donsker_cdf_scratch.py, plus an interactive version. Not core to the library.

Dependencies

numpy, scipy, matplotlib, statsmodels (for VARProcess), tqdm. A commented-out import suggests intended (but currently unused/unavailable) integration with a depth package (depth.multivariate) for halfspace depth via compiled libraries.

Installation

pip install -e .

Usage

Both functionalcurves/mixing_models.py and functionalcurves/depth.py are runnable as scripts (python -m functionalcurves.depth, python -m functionalcurves.mixing_models) and contain end-to-end examples: simulate a process, estimate depth/direction over growing sample sizes, and plot convergence against the true/analytic values. The notebooks in notebooks/ walk through the same workflow interactively.