- **Similar:** [EDecay](../edecay/Edecay.md) | **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) |
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.
