ema_demo.ipynb

Kernel: Unknown Kernel

# Illustrate expoentnially-weighted moving average
# Based on
# http://people.duke.edu/~ccc14/sta-663-2019/notebook/S09G_Gradient_Descent_Optimization.html#Smoothing-with-exponentially-weighted-averages
# http://people.duke.edu/~ccc14/sta-663-2018/notebooks/S09G_Gradient_Descent_Optimization.html


import numpy as np
import matplotlib.pyplot as plt

try:
    from probml_utils import savefig
except ModuleNotFoundError:
    %pip install -qq git+https://github.com/probml/probml-utils.git
    from probml_utils import savefig


def ema(y, beta):
    """Exponentially weighted average."""
    n = len(y)
    zs = np.zeros(n)
    z = 0
    for i in range(n):
        z = beta * z + (1 - beta) * y[i]
        zs[i] = z
    return zs


def ema_debiased(y, beta):
    """Exponentially weighted average with hias correction."""
    n = len(y)
    zs = np.zeros(n)
    z = 0
    for i in range(n):
        z = beta * z + (1 - beta) * y[i]
        zc = z / (1 - beta ** (i + 1))
        zs[i] = zc
    return zs


np.random.seed(0)
n = 50
x = np.arange(n) * np.pi
y = np.cos(x) * np.exp(x / 100) - 10 * np.exp(-0.01 * x)

betas = [0.9, 0.99]
for i, beta in enumerate(betas):
    plt.figure()
    plt.plot(x, y, "o-")
    plt.plot(x, ema(y, beta), c="red", label="EMA")
    plt.plot(x, ema_debiased(y, beta), c="orange", label="EMA with bias correction")
    plt.title("beta = {:0.2f}".format(beta))
    plt.legend()
    name = "EMA{}.pdf".format(i)
    savefig(name)
    plt.show()