QuanTAlib/lib/dynamics/plusdi/PlusDi.md

PLUS_DI: Plus Directional Indicator

+DI isolates upward directional thrust as a fraction of true range — the bullish arm of Wilder's directional system.

Property	Value
Category	Dynamic
Inputs	OHLCV bar (TBar)
Parameters	period (default 14)
Outputs	Single series
Output range	0 to 100
Warmup	period bars
PineScript	plusdi.pine

• The Plus Directional Indicator measures the strength of upward price movement relative to true range.
• Similar: MinusDi, ADX | Complementary: ADX for trend strength | Trading note: Plus Directional Indicator; measures upward movement strength. Part of Wilder's DM system.
• Validated against TA-Lib, Skender, and Dx equivalence.

The Plus Directional Indicator (+DI) is one component of J. Welles Wilder Jr.'s Directional Movement System. It quantifies the fraction of recent true range attributable to upward price extension. The computation smooths both +DM (plus directional movement) and TR (true range) with Wilder's RMA (\alpha = 1/N), then divides: +DI = 100 \times \text{Smooth}(+DM) / \text{Smooth}(TR). When +DI rises, upward price pressure is increasing. When +DI crosses above -DI, it signals a potential bullish trend. The +DI line is commonly plotted alongside -DI to visualize directional balance.

Historical Context

J. Welles Wilder Jr. introduced the Directional Movement System in New Concepts in Technical Trading Systems (1978). The system decomposes price range into directional components. +DI and -DI are the normalized indicators from which DX and ADX are derived. While most traders focus on ADX for trend strength, +DI and -DI remain essential for determining trend direction — a bullish signal occurs when +DI crosses above -DI, bearish when -DI crosses above +DI.

Architecture & Physics

1. Plus Directional Movement

\text{UpMove} = H_t - H_{t-1}, \quad \text{DownMove} = L_{t-1} - L_t +DM = \begin{cases} \text{UpMove} & \text{if UpMove} > \text{DownMove and UpMove} > 0 \\ 0 & \text{otherwise} \end{cases}

2. True Range

TR = \max(H_t - L_t,\; |H_t - C_{t-1}|,\; |L_t - C_{t-1}|)

3. Wilder Smoothing (RMA)

+DM_{\text{smooth}} = \text{RMA}(+DM, N), \quad TR_{\text{smooth}} = \text{RMA}(TR, N)

4. Plus Directional Indicator

+DI = 100 \times \frac{+DM_{\text{smooth}}}{TR_{\text{smooth}}}

When TR_{\text{smooth}} = 0 (no price movement), +DI = 0.

5. Complexity

• Time: O(1) per bar — all RMA updates are recursive
• Space: O(1) — scalar state only (delegates to Dx)
• Warmup: N bars

Mathematical Foundation

Parameters

Symbol	Parameter	Default	Constraint
N	period	14	N \geq 2

Interpretation

+DI Value	Signal
Rising +DI	Strengthening upward movement
+DI > -DI	Bulls dominate; potential uptrend
+DI crossover above -DI	Bullish signal
High +DI (>40)	Strong upward momentum

+DI measures directional strength, not absolute direction. Compare +DI vs -DI for directional bias: if +DI > -DI, the trend is up.

Performance Profile

Operation Count (Streaming Mode)

+DI is a thin wrapper around Dx. The per-bar cost is identical to Dx (one property extraction after Dx completes its update).

Post-warmup steady state (per bar):

Operation	Count	Cost (cycles)	Subtotal
Dx.Update (full pipeline)	1	75	75
Property extraction	1	1	1
Total	2	—	~76 cycles

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Exact Dx delegation; FMA-precise RMA smoothing
Timeliness	7/10	N-bar warmup; responds to bar-level changes
Smoothness	7/10	Single RMA layer; moderate noise suppression
Noise Rejection	7/10	Wilder smoothing filters transient spikes

Resources

• Wilder, J.W. — New Concepts in Technical Trading Systems (Trend Research, 1978)
• PineScript reference: plusdi.pine in indicator directory