QuanTAlib/lib/oscillators/smi/Smi.md

SMI: Stochastic Momentum Index

The stochastic tells you where price is in the range. The SMI tells you how enthusiastically it got there.

Property	Value
Category	Oscillator
Inputs	Bar series (High, Low, Close)
Parameters	kPeriod (default 10), kSmooth (default 3), dSmooth (default 3), blau (default true)
Outputs	Dual series (K line, D signal line)
Output range	-100 to +100
Warmup	kPeriod + kSmooth + dSmooth bars
PineScript	smi.pine

Key takeaways

• Measures where the close sits relative to the midpoint of the recent range, not the boundary. Zero means neutral; positive means above midpoint; negative means below.
• Two methods: Blau (default) smooths the ratio; Chande/Kroll smooths numerator and denominator separately before computing the ratio.
• Double EMA smoothing with warmup compensation produces clean crossover signals with controlled lag.
• Unlike classic Stochastic ([0, 100]), SMI ranges [-100, +100], centered on zero. Traditional Stochastic thresholds do not apply.
• Uses MonotonicDeque for O(1) amortized highest/lowest tracking and Math.FusedMultiplyAdd for EMA stages.

Historical Context

William Blau introduced the Stochastic Momentum Index in Momentum, Direction, and Divergence (1995) as an improvement over George Lane's classic Stochastic Oscillator. Blau's key insight: measuring distance from the range midpoint rather than from the low eliminates the asymmetric bias inherent in traditional stochastics. When price closes at the exact middle of its range, classic Stochastic reads 50—an arbitrary number that says nothing. SMI reads 0—neutral, centered, semantically honest.

Tushar Chande and Stanley Kroll proposed a variant in The New Technical Trader (1994) that smooths numerator and denominator separately before computing the ratio. This subtle difference in order of operations produces different behavior during volatile periods: Blau's method smooths the ratio directly, which compresses extreme values; Chande/Kroll preserves the ratio's sensitivity by smoothing its components independently.

QuanTAlib implements both methods via the blau parameter. The Blau method (default) suits trend-following; the Chande/Kroll method suits mean-reversion. Neither is universally better.

What It Measures and Why It Matters

SMI measures the closing price's distance from the midpoint of the highest-high to lowest-low range, normalized by half that range, then double-smoothed with cascaded EMAs. The result is a zero-centered oscillator bounded by [-100, +100].

The centering around zero gives SMI cleaner semantics than classic Stochastic. Positive values mean the close is above the range midpoint; negative values mean it is below. The magnitude indicates the strength of the displacement. Values beyond \pm 40 indicate extreme momentum; values near zero indicate no meaningful displacement.

The double EMA smoothing (-12 dB/octave rolloff) attenuates noise more aggressively than a single EMA of equivalent period, at the cost of additional group delay. This makes SMI better at filtering whipsaws than raw Stochastic while remaining responsive enough for momentum detection.

Mathematical Foundation

Core Formula

Rolling extremes:

HH_t = \max_{i=0}^{N-1} H_{t-i}, \quad LL_t = \min_{i=0}^{N-1} L_{t-i}

Midpoint and half-range:

\text{mid}_t = \frac{HH_t + LL_t}{2}, \quad \text{rh}_t = \frac{HH_t - LL_t}{2}

Blau method (smooth the ratio):

\text{raw}_t = \begin{cases} 100 \times \frac{C_t - \text{mid}_t}{\text{rh}_t} & \text{if } \text{rh}_t > 0 \\ 0 & \text{otherwise} \end{cases}

K_t = \text{EMA}_2(\text{EMA}_1(\text{raw}_t))

Chande/Kroll method (smooth the components):

K_t = 100 \times \frac{\text{EMA}_2(\text{EMA}_1(C_t - \text{mid}_t))}{\text{EMA}_2(\text{EMA}_1(\text{rh}_t))}

Signal line (both methods):

D_t = \text{EMA}(K_t, d)

All EMA stages use \alpha = 2/(N+1) with exponential warmup compensation.

Parameter Mapping

Parameter	Code	Default	Constraints
K Period	kPeriod	10	> 0
K Smoothing	kSmooth	3	> 0
D Smoothing	dSmooth	3	> 0
Method	blau	true	Blau or Chande/Kroll

Warmup Period

W = N + k_s + d_s

IsHot fires after kPeriod bars (sufficient for the deque window). Full EMA convergence requires the complete warmup period.

Architecture & Physics

1. MonotonicDeque Streaming

Two MonotonicDeque instances track the sliding highest-high and lowest-low over kPeriod bars. Circular buffers (_hBuf, _lBuf) store raw H/L values for deque rebuild on bar correction.

2. Dual-Path EMA Cascade

Blau path: Computes raw = 100 * (close - mid) / rh, then applies two cascaded EMAs with the same alpha (kSmooth), followed by an EMA with dSmooth alpha for the signal line.

Chande/Kroll path: Maintains four EMA accumulators (two per layer) for numerator and denominator independently, then computes the ratio.

3. Warmup Compensation

Each EMA stage uses exponential warmup compensators (e_t = d \cdot e_{t-1}, c_t = 1/(1 - e_t)) that correct initialization bias. The compensator converges to 1.0 as e_t \to 0, after which the hot path skips compensation for performance.

4. Edge Cases

Condition	Behavior
Any parameter <= 0	ArgumentException with nameof()
NaN / Infinity input	Substitutes last valid value per channel (H/L/C)
All NaN (no valid data yet)	Returns NaN for K and D
Zero range (HH = LL)	Returns 0 (Blau) or 0 (Chande/Kroll when denominator is 0)
isNew = false	Restores _ps, rebuilds both deques from circular buffer

Interpretation and Signals

Signal Zones

Zone	Condition	Interpretation
Overbought	K > +40	Close significantly above range midpoint
Neutral	-20 ≤ K ≤ +20	No strong momentum bias
Oversold	K < -40	Close significantly below range midpoint

Signal Patterns

• K/D crossover: K crossing above D signals bullish momentum shift; K crossing below D signals bearish shift.
• Zero-line crossover: K crossing above zero confirms upward momentum; crossing below confirms downward.
• Divergence: Price makes new highs while K makes lower highs (bearish) or price makes new lows while K makes higher lows (bullish).

Practical Notes

• Blau (default): Better for trend-following. Smoother output, fewer whipsaws. The ratio compression during high volatility acts as a natural dampener.
• Chande/Kroll: Better for mean-reversion. Preserves component oscillation sensitivity. More responsive during volatile reversals but noisier in trends.
• Do not use classic Stochastic thresholds (20/80) for SMI. The [-100, +100] range centered on zero requires \pm 40 thresholds.

Related Indicators

• Stoch: Measures distance from range boundary ([0, 100]); SMI measures distance from midpoint ([-100, +100]).
• KDJ: Extended stochastic with J-line divergence amplification.
• StochRSI: Applies stochastic formula to RSI instead of price.
• Willr: Measures distance from high, inverted scale [-100, 0].

Validation

Library	Status	Notes
Skender	✅	GetSmi() matches within 1e-6
Self-consistency	✅	Streaming/batch/span agree within 1e-6; both Blau and Chande/Kroll verified
TA-Lib	--	No SMI implementation
Tulip	--	No SMI implementation

Performance Profile

Key Optimizations

• O(1) amortized streaming: MonotonicDeque for highest/lowest tracking.
• FMA-optimized EMAs: Math.FusedMultiplyAdd for all EMA accumulations reduces rounding error and improves throughput on FMA-capable hardware.
• Warmup phase elimination: Once all compensators converge (e < 10^{-10}), the hot path skips compensation arithmetic.
• Zero allocation: Update uses pre-allocated buffers and record struct State.

Operation Count (Streaming Mode)

Operation	Count per bar
Deque push (amortized)	2-3 comparisons
Midpoint/rangeHalf	2 ops
EMA stage 1 (FMA)	1 FMA
EMA stage 2 (FMA)	1 FMA
Signal EMA (FMA)	1 FMA
Warmup compensators	3-6 (only during warmup)
Total	~10 ops (hot), ~16 ops (warmup)

SIMD Analysis (Batch Mode)

Property	Value
Vectorizable	Partially (via Highest.Batch / Lowest.Batch)
EMA cascade	Scalar (recursive dependency)
Fallback	Scalar loop with FMA for EMA stages

Common Pitfalls

1. Confusing SMI with classic Stochastic: SMI ranges [-100, +100] centered on zero. Classic Stochastic ranges [0, 100]. Using Stochastic thresholds (20/80) for SMI produces incorrect signals.
2. Ignoring the method parameter: Blau and Chande/Kroll produce meaningfully different results. Switching methods mid-analysis invalidates comparisons.
3. Zero range handling: When HH = LL, rangeHalf is zero. The division guard returns 0, but sustained zero readings may mask meaningful price action outside the deque window.
4. Cascaded EMA warmup: Three EMA stages each need convergence time. The first values after IsHot are less reliable than values after the full warmup period.
5. Parameter interaction: kPeriod controls reference range width; kSmooth controls noise filtering; dSmooth controls signal line lag. Increasing kPeriod without adjusting smoothing produces a wider reference range with insufficient filtering.
6. Bar correction cost: isNew=false triggers O(kPeriod) deque rebuild. Infrequent in normal streaming but visible with rapid corrections.

FAQ

Q: Should I use Blau or Chande/Kroll? A: Blau (default) for trend-following. Chande/Kroll for mean-reversion in volatile markets. Blau compresses extreme ratio values through smoothing; Chande/Kroll preserves component oscillation by smoothing numerator and denominator independently.

Q: Why does SMI use ±40 thresholds instead of 20/80? A: Because SMI is centered on zero with range [-100, +100]. The ±40 thresholds correspond to approximately the same distance from neutral as 20/80 in classic Stochastic's [0, 100] range.

Q: How does SMI compare to Stochastic for crossover signals? A: SMI's zero-centered output produces cleaner crossovers because the neutral point is semantically meaningful (close equals range midpoint). Classic Stochastic's 50 level has no equivalent semantic clarity.

References

• Blau, W. Momentum, Direction, and Divergence. Wiley, 1995.
• Chande, T. S.; Kroll, S. The New Technical Trader. Wiley, 1994.
• Lane, G. C. "Lane's Stochastics." Technical Analysis of Stocks & Commodities, 1984.