QuanTAlib/lib/numerics/edecay/Edecay.md

EDECAY: Exponential Decay

A ratchet that only moves down gradually: price can push it up instantly, but gravity pulls it back at an exponential pace — faster when far from zero, slower as it approaches.

Property	Value
Category	Numerics
Inputs	Source (close)
Parameters	period (default 5)
Outputs	Single series (Edecay)
Output range	Same as input (overlay)
Warmup	1 bar
PineScript	edecay.pine

• EDECAY (Exponential Decay) tracks the maximum of the current input and the previous output multiplied by a decay factor of (period-1)/period.
• Similar: Decay | Trading note: Exponential decay; faster initial fading than linear. Natural half-life model for signal importance.
• Validated against Tulip Indicators ti_edecay reference algorithm.

EDECAY implements the exponential decaying function. When price is above the decayed level, output snaps to price. When price falls below, the output decays exponentially by multiplying by (period-1)/period per bar, creating a ceiling that gradually descends. Unlike linear DECAY which subtracts a fixed amount, EDECAY's multiplicative factor produces a proportional decay rate.

Historical Context

The exponential decay indicator originates from the Tulip Indicators library, a high-performance C library of technical indicators. It provides a peak-tracking mechanism where the tracked level decays at a proportional rate. The indicator is useful for:

• Trailing stops: The decaying level acts as a trailing stop that descends proportionally.
• Peak detection: Identifies when price last reached a new high relative to the decay rate.
• Signal filtering: Removes noise by requiring price to exceed the decayed level to register as significant.

Architecture & Physics

1. Pure IIR (No Buffer)

The indicator requires no history buffer — only the previous output value is needed:

\text{state} = \{y_{t-1}\}

This makes it O(1) in both time and space.

2. Exponential Decay Calculation

y_t = \max(x_t, \; y_{t-1} \cdot \frac{p-1}{p})

where:

• x_t = current input value
• y_{t-1} = previous output value
• p = period parameter
• \frac{p-1}{p} = multiplicative decay factor per bar

3. First Bar Initialization

y_0 = x_0

The first bar simply passes through the input value.

4. State Management

The indicator uses state rollback for bar correction:

if isNew:
    save current state as previous
else:
    restore previous state

Mathematical Foundation

Core Formula

y_t = \max(x_t, \; y_{t-1} \cdot s)

where s = \frac{p-1}{p} is the multiplicative decay factor.

Decay Behavior

After a peak at value v, with no new inputs exceeding the decayed level, the output follows:

y_{t+k} = v \cdot s^k = v \cdot \left(\frac{p-1}{p}\right)^k

The output asymptotically approaches zero but never reaches it (v > 0).

Comparison with Linear Decay

Property	DECAY (Linear)	EDECAY (Exponential)
Formula	y - \frac{1}{p}	y \cdot \frac{p-1}{p}
Decay rate	Constant absolute	Proportional to current value
Reaches zero	Yes, in finite time	No, asymptotic approach
Scale-invariant	No	Yes

Properties

Property	Value
Lookback	0
Output ≥ Input	Always (by construction)
Decay rate	Proportional \frac{p-1}{p}
Monotonic when decaying	Yes (strictly decreasing)
Scale-invariant	Yes

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Notes
MUL	1	prev_output × scale
MAX/CMP	1	max(input, decayed)
State copy	1	rollback support
Total	~3 ops	Extremely lightweight

Batch Mode (Span-based)

Operation	Complexity	Notes
Per-element	O(1)	Mul + compare
Total	O(n)	Linear scan
Memory	O(1)	No additional allocation

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Exact arithmetic, no approximation
Timeliness	10/10	Zero lag on upward moves
Smoothness	3/10	Exponential curve smoother than linear staircase
Simplicity	10/10	Single multiplication + compare

Validation

Library	Status	Notes
Tulip	✅	Manual ti_edecay algorithm matches exactly

Common Pitfalls

1. Not a moving average: Edecay is a peak-tracking/envelope indicator, not a smoothing filter. It only descends when price is below the decayed level.

2. Proportional decay rate: Unlike linear DECAY, EDECAY decays proportionally. For a stock at $100 with period=5, the first bar decays by $20; for a stock at $10, it decays by $2. This makes EDECAY scale-invariant.

3. Period interpretation: Period=5 means scale = 4/5 = 0.8, so each bar retains 80% of the previous value. After 5 bars, approximately 32.8% of the peak value remains.

4. First bar: The first bar always equals the input — there is no warmup period in the traditional sense.

5. Asymmetric behavior: Upward moves are instant (output = input), but downward moves are rate-limited to multiplication by (period-1)/period per bar.

6. Never reaches zero: Unlike linear DECAY, exponential decay asymptotically approaches zero but never reaches it (assuming positive values).

References

• Tulip Indicators Library: https://tulipindicators.org/edecay
• Kegel, L. "Tulip Indicators" — Open-source C library of technical indicators.