sverdoot/pointwise-derivative-estimation

Final project for the course Advanced Statistical Methods in Skoltech

Pointwise derivative estimation

• Pointwise derivative estimation
  • Getting started
  • Usage
  • Overview
  • Results

Getting started

conda create --name local_smooth  python==3.9

if poetry is not installed:

curl -sSL https://install.python-poetry.org | python3 -
poetry config virtualenvs.create false --local

install the project:

conda activate local_smooth
poetry install

chmod +x run.sh

Usage

python local_smooth/main.py --config configs/loc_lin_gauss_ker.yml

run all experiments:

./run.sh

Overview

$Y_i = f(X_i) + \varepsilon_i, ~\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$

Unbiased risk estimate is computed as:

$\hat{\mathcal{R}}(\hat{f}) = |\mathcal{K}_{h} \mathbf{Y} - \mathbf{Y}|^2 + 2\sigma^2\text{tr}(\mathcal{K}_h)$

True risk is computed as:

$\mathcal{R}(\hat{f}) = \mathbb{E}((f^)'(x_0) - \hat{f}'(x_0))^2 = ((f^)'(x_0) - \mathcal{S}_1^{\top}\mathbf{f})^2 + \sigma^2 |\mathcal{S}_1|^2$,

where $\mathcal{S} = {\Psi(x_0)W(x_0)\Psi(x_0)^{\top}}^{-1}\Psi(x_0)W(x_0)$

Results

• Generated data:

• Risk (Locally linear estimate, Gaussian kernel):