QuanTAlib/lib/dynamics/minusdi/MinusDi.md

MINUS_DI: Minus Directional Indicator

-DI isolates downward directional thrust as a fraction of true range — the bearish 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	minusdi.pine

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

The Minus 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 downward price extension. The computation smooths both -DM (minus 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, downward price pressure is increasing. When -DI crosses above +DI, it signals a potential bearish 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 bearish signal occurs when -DI crosses above +DI, bullish when +DI crosses above -DI.

Architecture & Physics

1. Minus Directional Movement

\text{UpMove} = H_t - H_{t-1}, \quad \text{DownMove} = L_{t-1} - L_t -DM = \begin{cases} \text{DownMove} & \text{if DownMove} > \text{UpMove and DownMove} > 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. Minus 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 downward movement
-DI > +DI	Bears dominate; potential downtrend
-DI crossover above +DI	Bearish signal
High -DI (>40)	Strong downward momentum

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

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: minusdi.pine in indicator directory