QuanTAlib/lib/numerics/decay/Decay.md

DECAY: Linear Decay

A ratchet that only moves down slowly: price can push it up instantly, but gravity pulls it back at a steady, linear pace.

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

• DECAY (Linear Decay) tracks the maximum of the current input and the previous output minus a fixed absolute step of 1/period.
• Similar: EDecay | Trading note: Linear decay function; models signal fading over time. Used for recency-weighted calculations.
• Validated against Tulip Indicators ti_decay reference algorithm.

DECAY implements the Tulip Indicators ti_decay function. When price is above the decayed level, output snaps to price. When price falls below, the output decays linearly at a rate of 1/period per bar, creating a ceiling that gradually descends. This produces a one-sided envelope that hugs price from above.

Historical Context

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

• Trailing stops: The decaying level acts as a simple trailing stop that descends at a fixed rate.
• 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. Linear Decay Calculation

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

where:

• x_t = current input value
• y_{t-1} = previous output value
• p = period parameter
• \frac{1}{p} = fixed decay step 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} - s)

where s = \frac{1}{p} is the fixed linear decay rate.

Decay Behavior

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

y_{t+k} = v - k \cdot s

reaching zero after k = v \cdot p bars (assuming v > 0).

Properties

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

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Notes
SUB	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)	Sub + 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	2/10	No smoothing — linear staircase
Simplicity	10/10	Single subtraction + compare

Validation

Library	Status	Notes
Tulip	✅	Manual ti_decay algorithm matches exactly

Common Pitfalls

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

2. Absolute decay rate: The decay step is 1/period in absolute terms, regardless of price level. For a stock at $100 with period=5, the decay is $0.20/bar; for a stock at $10, it's the same $0.20/bar. Consider normalizing if comparing across instruments.

3. Period interpretation: Period=5 means the output decays by 1.0 over 5 bars (0.2 per bar), not that it looks back 5 bars.

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 1/period per bar.

References

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