Exponential Moving Average (EMA)

The Problem with SMA First

Remember SMA — it gives equal weight to all values in the window.

For a 3-day SMA on sales:

Day 1: 200, Day 2: 450, Day 3: 180

SMA = (200 + 450 + 180) / 3 = 276.6

Here, Day 1 (old data) and Day 3 (today) are treated equally. But think about it — should 2-day-old data matter as much as today's data?

In most real cases — No. Recent data is more important.

That's exactly what EMA fixes.

What is EMA?

EMA gives MORE weight to recent values and LESS weight to older values.

The further back a value is, the less it influences the average. Recent values dominate.

Weight Concept — Simple Visual

For a 3-period EMA, weights look like this:

Data Point	Weight
Today (most recent)	Highest ⬆️
Yesterday	Medium
Day before	Low
Even older	Very Low (almost ignored)

Compare this to SMA where every day gets exactly equal weight.

The Formula

EMA today = (Today's Value × α) + (Yesterday's EMA × (1 - α))

Where α (alpha) is the smoothing factor:

α = 2 / (N + 1)

For N = 3:

α = 2 / (3 + 1) = 0.5

That means — 50% weight to today, 50% to the past EMA.

For N = 10:

α = 2 / (10 + 1) = 0.18

Smaller alpha = smoother = older data still matters more.

Manual Walkthrough — Step by Step

Daily Sales data:

Day	Sales
1	200
2	450
3	180
4	500
5	220

Using N = 3, so α = 0.5

Step 1 — Day 1: No previous EMA exists, so EMA = first value itself

EMA(1) = 200

Step 2 — Day 2:

EMA(2) = (450 × 0.5) + (200 × 0.5)
       = 225 + 100
       = 325

Step 3 — Day 3:

EMA(3) = (180 × 0.5) + (325 × 0.5)
       = 90 + 162.5
       = 252.5

Step 4 — Day 4:

EMA(4) = (500 × 0.5) + (252.5 × 0.5)
       = 250 + 126.25
       = 376.25

Step 5 — Day 5:

EMA(5) = (220 × 0.5) + (376.25 × 0.5)
       = 110 + 188.12
       = 298.12

Final result:

Day	Sales	EMA (N=3)
1	200	200
2	450	325
3	180	252.5
4	500	376.25
5	220	298.12

Notice — EMA reacts faster to the spike on Day 4 (500) compared to SMA. That's the power.

SMA vs EMA — Side by Side

Feature	SMA	EMA
Weight to all values	Equal	More to recent
Reacts to sudden change	Slow	Fast
Smoother line	Yes	Slightly less smooth
NaN at start	Yes (first N rows)	No
Best for	Long-term trend	Short-term, fast signals

Python Program


    import pandas as pd
    import matplotlib.pyplot as plt

    # --- Data ---
    data = {
        'day': list(range(1, 16)),
        'sales': [200, 450, 180, 500, 220, 480, 210, 460, 190, 510, 230, 490, 200, 470, 215]
    }

    df = pd.DataFrame(data)

    # --- Calculate SMA and EMA ---
    df['SMA_3'] = df['sales'].rolling(window=3).mean()
    df['EMA_3'] = df['sales'].ewm(span=3, adjust=False).mean()  # EMA with N=3
    df['EMA_7'] = df['sales'].ewm(span=7, adjust=False).mean()  # EMA with N=7

    print(df.to_string(index=False))

    # --- Plot ---
    plt.figure(figsize=(12, 5))
    plt.plot(df['day'], df['sales'], label='Raw Sales', marker='o', linewidth=1.5, alpha=0.6)
    plt.plot(df['day'], df['SMA_3'], label='SMA (3-day)', linewidth=2, linestyle='--')
    plt.plot(df['day'], df['EMA_3'], label='EMA (3-day)', linewidth=2)
    plt.plot(df['day'], df['EMA_7'], label='EMA (7-day)', linewidth=2)

    plt.title('SMA vs EMA Comparison')
    plt.xlabel('Day')
    plt.ylabel('Sales')
    plt.legend()
    plt.grid(True)
    plt.tight_layout()
    plt.savefig('ema_vs_sma.png')
    plt.show()
    print("Plot saved!")


Output:

 day  sales      SMA_3      EMA_3      EMA_7
   1    200        NaN 200.000000 200.000000
   2    450        NaN 325.000000 262.500000
   3    180 276.666667 252.500000 241.875000
   4    500 376.666667 376.250000 306.406250
   5    220 300.000000 298.125000 284.804688
   6    480 400.000000 389.062500 333.603516
   7    210 303.333333 299.531250 302.702637
   8    460 383.333333 379.765625 342.026978
   9    190 286.666667 284.882812 304.020233
  10    510 386.666667 397.441406 355.515175
  11    230 310.000000 313.720703 324.136381
  12    490 410.000000 401.860352 365.602286
  13    200 306.666667 300.930176 324.201714
  14    470 386.666667 385.465088 360.651286
  15    215 295.000000 300.232544 324.238464
Plot saved!

Key Things to Remember

ewm(span=3) — span is your N value, same as window in rolling

adjust=False — uses the recursive formula shown above (standard EMA). Always use this.

No NaN — EMA starts from Day 1 itself, unlike SMA which waits for N values

Where EMA is Used in ML


    # Feature Engineering with EMA
    df['ema_3']  = df['sales'].ewm(span=3,  adjust=False).mean()  # short trend
    df['ema_7']  = df['sales'].ewm(span=7,  adjust=False).mean()  # medium trend
    df['ema_21'] = df['sales'].ewm(span=21, adjust=False).mean()  # long trend

    # EMA reacts faster — great for detecting sudden changes (fraud, anomaly)
    df['deviation_from_ema'] = df['sales'] - df['ema_7']  # how far today is from trend

deviation_from_ema is a very powerful feature — if this value is very high or very low, it signals something unusual happening. Used heavily in anomaly detection and fraud detection models.

One Line Summary

EMA is a smarter Moving Average — it remembers the past but pays more attention to what just happened, making it faster to react to real changes in data.

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