Impara Weighted and Exponential Moving Averages

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Weighted and exponential moving averages are powerful tools for analyzing time series data, offering more flexibility and responsiveness than the simple moving average (SMA). The weighted moving average (WMA) assigns different weights to each data point in the window, allowing you to emphasize more recent observations or any pattern of importance you choose. The mathematical formula for the WMA of a series $x$ with weights $w$ over $n$ periods is:

WMA = \frac{(w₁x₁ + w₂x₂ + ... + wₙxₙ)}{(w₁ + w₂ + ... + wₙ)}

The exponential moving average (EMA) goes a step further by applying exponentially decreasing weights to past observations. This means recent data points have much more influence on the average, making the EMA especially useful for tracking trends in volatile data. The EMA is calculated recursively as:

EMAₜ = αxₜ + (1 - α)EMAₜ₋₁

where:

$EMAₜ$ is the current EMA value;
$xₜ$ is the current observation;
$α$ is the smoothing factor, typically $2 / (n + 1)$ or a window of size $n$ .

These approaches help you capture recent trends and changes in your data more effectively than the SMA, which simply averages all points in the window equally.


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import pandas as pd
import matplotlib.pyplot as plt

# Create a time series
data = {
    "value": [10, 12, 13, 15, 14, 16, 18, 20, 19, 21, 22, 24, 23, 25, 27]
}
df = pd.DataFrame(data)

# Calculate the Exponential Moving Average (EMA) with a span of 5
df["EMA_5"] = df["value"].ewm(span=5, adjust=False).mean()

# Plot the original series and EMA
plt.figure(figsize=(8, 4))
plt.plot(df["value"], label="Original")
plt.plot(df["EMA_5"], label="EMA (span=5)", linestyle="--")
plt.title("Exponential Moving Average (EMA) Example")
plt.xlabel("Time")
plt.ylabel("Value")
plt.legend()
plt.tight_layout()
plt.show()

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