EMA

Exponential smoothing or exponential moving average

import numpy as np
import matplotlib.pyplot as plt

# Generate noisy data
np.random.seed(42)  # For reproducibility
time = np.arange(1, 101)
values = 50 + np.cumsum(np.random.randn(100) * 10)  # Large noise

# Calculate EMA
alpha = 0.3  # Smoothing factor
ema_direct = []
alpha = 0.3
for i, value in enumerate(values):
    if i == 0:
        ema_direct.append(value)  # Initialize EMA with the first value
    else:
        ema_direct.append(alpha * value + (1 - alpha) * ema_direct[-1])

# Plot the noisy data and EMA
plt.figure(figsize=(10, 6))
plt.plot(time, values, label='Noisy Data', alpha=0.7)
plt.plot(time, ema_direct, label='Exponential Moving Average (EMA)', linewidth=2, color='orange')
plt.title('Noisy Data with Exponential Moving Average')
plt.xlabel('Time')
plt.ylabel('Value')
plt.legend()
plt.grid(True)
plt.show()

• $EMA_0 = x_0$

• EMA applies more weight to the recent data exponentially for interpolation

Exponential smoothing

Exponential smoothing or exponential moving average (EMA) is a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data.

https://en.wikipedia.org/wiki/Exponential_smoothing