QuanTAlib/lib/dynamics/dx/Dx.md

DX: Directional Movement Index

Property	Value
Category	Dynamic
Inputs	OHLCV bar (TBar)
Parameters	period (default 14)
Outputs	Multiple series (DiPlus, DiMinus)
Output range	Varies (see docs)
Warmup	period bars

TL;DR

• The Directional Movement Index is the raw, unsmoothed measure of trend strength from Wilder's directional movement system.
• Parameterized by period (default 14).
• Output range: Varies (see docs).
• Requires period bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Directional Movement Index is the raw, unsmoothed measure of trend strength from Wilder's directional movement system. It decomposes price expansion into +DM and -DM, normalizes against True Range using RMA smoothing to produce +DI and -DI, then computes the ratio DX = 100 \times |{+DI - {-DI}}| / ({+DI + {-DI}}). Unlike ADX, which applies a final RMA pass to DX, the raw DX responds immediately to changes in directional dominance — making it noisier but approximately one full period faster. Output ranges from 0 to 100, where high values indicate strong directional movement regardless of up/down direction. DX is the building block from which ADX is derived.

Historical Context

J. Welles Wilder Jr. introduced the complete Directional Movement System in New Concepts in Technical Trading Systems (1978). The system's pipeline produces several intermediate values — +DM, -DM, TR, +DI, -DI, DX — before reaching the final ADX. Most traders skip directly to ADX, but DX occupies a useful middle ground: it contains all the directional normalization logic (the hard part) without the final smoothing layer (which adds lag). For traders who can tolerate more noise in exchange for faster response, DX provides trend strength signals roughly N bars ahead of ADX. The tradeoff is straightforward: DX spikes on volatile bars and can produce false readings during whipsaw, while ADX absorbs these transients through its additional RMA pass.

Architecture & Physics

1. 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} -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)

All three series use Wilder's smoothing with \alpha = 1/N:

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

4. Directional Indicators

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

5. DX (No Final Smoothing)

DX = 100 \times \frac{|+DI - (-DI)|}{+DI + (-DI)}

When +DI + (-DI) = 0 (no directional movement), DX = 0.

6. Complexity

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

Mathematical Foundation

Parameters

Symbol	Parameter	Default	Constraint
N	period	14	N \geq 2

Pseudo-code

Initialize:
  α = 1 / period
  smoothPlusDM = smoothMinusDM = smoothTR = 0
  prevHigh = prevLow = prevClose = NaN

On each bar (high, low, close, isNew):
  if !isNew: restore previous state

  // True Range
  TR = max(high - low, |high - prevClose|, |low - prevClose|)

  upMove = high - prevHigh
  downMove = prevLow - low

  +DM = (upMove > downMove AND upMove > 0) ? upMove : 0
  -DM = (downMove > upMove AND downMove > 0) ? downMove : 0

  // Wilder smoothing
  smoothPlusDM = FMA(smoothPlusDM, 1 - α, α × +DM)
  smoothMinusDM = FMA(smoothMinusDM, 1 - α, α × -DM)
  smoothTR = FMA(smoothTR, 1 - α, α × TR)

  // Directional Indicators
  +DI = smoothTR > 0 ? 100 × smoothPlusDM / smoothTR : 0
  -DI = smoothTR > 0 ? 100 × smoothMinusDM / smoothTR : 0

  // DX (raw, no final RMA)
  diSum = +DI + -DI
  DX = diSum > 0 ? 100 × |+DI - -DI| / diSum : 0

  prevHigh = high
  prevLow = low
  prevClose = close

  output:
    DX = DX          // trend strength (0-100)
    DiPlus = +DI     // bullish directional indicator
    DiMinus = -DI    // bearish directional indicator

DX vs ADX

Property	DX	ADX
Smoothing	RMA on components only	RMA on components + RMA on DX
Response	Immediate to bar-level changes	Lagged by \approx N bars
Noise	High; can spike on volatile bars	Low; smooth, stable signal
Use case	Fast trend detection, signal generation	Regime classification, filter

Interpretation

DX Value	Trend Strength
0-15	No meaningful trend
15-25	Developing trend
25-50	Strong trend
50-75	Very strong trend
75-100	Extreme (rare, usually transient)

DX measures trend strength, not direction. Direction is determined by comparing +DI vs -DI: if +DI > -DI, the trend is up; if -DI > +DI, the trend is down. DI crossovers signal potential trend reversals.

Performance Profile

Operation Count (Streaming Mode)

DX is an intermediate step in the ADX calculation: it computes the directional movement index without the final ADX smoothing pass.

Post-warmup steady state (per bar):

Operation	Count	Cost (cycles)	Subtotal
SUB × 5 (TR, DM moves)	5	1	5
ABS × 2 (TR components)	2	1	2
MAX × 2 (TR)	2	1	2
CMP × 2 (DM guards)	2	1	2
FMA × 3 (RMA TR, +DM, −DM)	3	4	12
DIV × 2 (+DI, −DI)	2	15	30
MUL × 2 (×100)	2	3	6
ABS + ADD + DIV (DX formula)	3	16	16
Total	23	—	~75 cycles

DX requires N bars warmup (vs 2N for ADX). ~75 cycles per bar.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
TR/DM initial computation	Yes	VSUBPD + VABSPD + VMAXPD
RMA smoothing × 3	No	Recursive IIR
DX formula	Yes	VABSPD + VADDPD + VDIVPD post-RMA

Same RMA bottleneck as ADX. The DX formula itself is fully vectorizable.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	FMA smoothing; exact TR computation
Timeliness	7/10	N-bar warmup; more responsive than ADX
Smoothness	6/10	Raw DX is noisier than ADX; typically used as input to ADX
Noise Rejection	6/10	Single RMA layer; moderate noise suppression

Resources

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