import numpy as np
from scipy.stats import norm
# ----------------- with tau ------- Rookley + Haerdle (Applied Quant. Finance)
[docs]def gaussian_kernel(x, Xi, h):
"""Gaussian Kernel.
:math:`K(u) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{1}{2} u^2}` where :math:`u = \\frac {x - X_i}{h}`
Args:
x (float): point of interest
Xi (array): data points, surrounding of x
h (float): bandwidth
Returns:
ndarray: Kernel function for the point x
"""
u = (x - Xi) / h
return norm.pdf(u)
[docs]def epanechnikov_kernel(x, Xi, h):
"""Epanechnikov kernel
:math:`K(u) = \\frac{3}{4} \\left( 1 - u^2 \\right)` where :math:`u = \\frac {x - X_i}{h}`
Args:
x (float): point of interest
Xi (array): data points, surrounding of x
h (float): bandwidth
Returns:
ndarray: Kernel function for the point x
"""
u = (x - Xi) / h
indicator = np.where(abs(u) <= 1, 1, 0)
k = 0.75 * (1 - u ** 2)
return k * indicator
kernel_dict = {"gaussian": gaussian_kernel, "epanechnikov": epanechnikov_kernel}