QuanTAlib/lib/oscillators/stc/stc.md

STC: Schaff Trend Cycle

Schaff Trend Cycle applies double stochastic smoothing to MACD, compressing a trend indicator into an oscillator's bounded range.

Property	Value
Category	Oscillator
Inputs	Source (close)
Parameters	kPeriod (default 10), dPeriod (default 3), fastLength (default 23), slowLength (default 50), smoothing (default StcSmoothing.Ema)
Outputs	Single series (Stc)
Output range	0 to 100
Warmup	1 bar
PineScript	stc.pine

• The Schaff Trend Cycle is a cyclometric oscillator that applies double-Stochastic normalization to MACD, extracting the cyclical phase hidden withi...
• Similar: MACD, Stoch | Complementary: ADX | Trading note: Schaff Trend Cycle; double Stochastic of MACD. Faster trend detection. 25/75 signal levels.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Schaff Trend Cycle is a cyclometric oscillator that applies double-Stochastic normalization to MACD, extracting the cyclical phase hidden within the trend itself. The recursive normalization produces a bounded 0–100 output that reaches extremes earlier than raw MACD while suppressing Stochastic jitter. Developed for currency markets, STC's tendency to flatline at 0 or 100 during strong trends signals continuation rather than reversal — a feature that distinguishes it from conventional momentum oscillators. Output converges toward a square wave in steady-state trending conditions.

Historical Context

Doug Schaff developed STC in the 1990s while trading currency markets. His diagnosis: MACD identified trends correctly but with unacceptable lag — by signal time, much of the move had elapsed. The Stochastic oscillator was fast but noisy, generating false signals in trending markets. Schaff's synthesis recognized that trends themselves move in cycles. Rather than choosing between lagging trend detection and noisy cycle extraction, he piped MACD through the Stochastic twice. The first pass normalizes MACD within its recent range, collapsing the unbounded trend signal into a 0–100 band. The second pass normalizes the smoothed first pass, further compressing the cycle information and creating a self-normalizing oscillator. The double normalization acts as a nonlinear filter that amplifies transitions and suppresses noise during sustained moves. STC found particular traction in forex trading where the 24-hour market rewarded speed advantages over MACD. The flatline behavior at extremes — initially dismissed as a limitation — became recognized as a defining feature: sustained 0 or 100 readings indicate trend continuation with high confidence, equivalent to a digital "trend on" signal.

Architecture & Physics

1. MACD Construction

Fast and slow EMAs generate the raw trend signal:

\alpha_f = \frac{2}{\text{fastLength} + 1}, \quad \alpha_s = \frac{2}{\text{slowLength} + 1} \text{EMA}_{f,t} = \alpha_f \cdot P_t + (1 - \alpha_f) \cdot \text{EMA}_{f,t-1} \text{EMA}_{s,t} = \alpha_s \cdot P_t + (1 - \alpha_s) \cdot \text{EMA}_{s,t-1} \text{MACD}_t = \text{EMA}_{f,t} - \text{EMA}_{s,t}

2. First Stochastic (%K₁)

Normalize MACD within its recent $k$-bar range:

\%K_1 = 100 \times \frac{\text{MACD}_t - \min(\text{MACD}_{t-k+1:t})}{\max(\text{MACD}_{t-k+1:t}) - \min(\text{MACD}_{t-k+1:t})}

When \max = \min (flat MACD), \%K_1 holds its previous value. This collapses the unbounded MACD into [0, 100].

3. First Smoothing (%D₁)

EMA smooth the first Stochastic to reduce whipsaw:

\alpha_d = \frac{2}{d\text{Period} + 1} \%D_{1,t} = \alpha_d \cdot \%K_{1,t} + (1 - \alpha_d) \cdot \%D_{1,t-1}

4. Second Stochastic (%K₂)

Apply Stochastic normalization again to %D₁, using the same $k$-bar window:

\%K_2 = 100 \times \frac{\%D_{1,t} - \min(\%D_{1,t-k+1:t})}{\max(\%D_{1,t-k+1:t}) - \min(\%D_{1,t-k+1:t})}

This second pass further compresses the signal, amplifying transitions between trend phases.

5. Final Smoothing

Apply selected smoothing method to %K₂:

\text{STC}_t = \text{Smooth}(\%K_{2,t})

Smoothing options:

• None: Raw %K₂ output
• EMA: Standard EMA smoothing with \alpha_d
• Sigmoid: S(x) = \frac{100}{1 + e^{-0.1(x - 50)}} — S-curve compression
• Digital: Threshold at 50 → output snaps to 0 or 100 (square wave)

6. Complexity

• Time: O(k) per bar for min/max scanning over both Stochastic windows
• Space: O(k) — two ring buffers of size kPeriod (MACD values and %D₁ values)
• Warmup: slowLength + kPeriod bars before output stabilizes

Mathematical Foundation

Parameters

Symbol	Parameter	Default	Constraint
k	kPeriod	10	k \geq 2
d	dPeriod	3	d \geq 1
f	fastLength	23	f \geq 1
s	slowLength	50	s > f
—	smoothing	EMA	None / EMA / Sigmoid / Digital

Signal Characteristics

Condition	Output Behavior
Strong uptrend	Flatlines at 100 (square wave high)
Strong downtrend	Flatlines at 0 (square wave low)
Trend transition	Rapid swing between extremes
Ranging market	Oscillates mid-range (25–75)
Above 75	Overbought zone
Below 25	Oversold zone

Cycle Length Heuristic

Setting k \approx f/2 targets the half-cycle of the MACD's dominant frequency, aligning the Stochastic window with the trend's internal oscillation period.

SIMD Applicability

The recursive EMA dependencies and sequential min/max ring buffer updates prevent SIMD vectorization of the streaming path. The Calculate(Span) path can parallelize independent MACD computations but must serialize the double-Stochastic pipeline.

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
Fast EMA	~3	1 FMA + 1 MUL
Slow EMA	~3	1 FMA + 1 MUL
MACD subtraction	~1	1 SUB
Ring buffer add (MACD)	~1	1 write + index update
Min/Max scan (MACD buf)	~2k	Linear scan of k elements × 2 (min + max)
First Stochastic (%K₁)	~4	1 SUB + 1 DIV + 1 MUL + 1 branch
First EMA smoothing (%D₁)	~3	1 FMA + 1 MUL
Ring buffer add (%D₁)	~1	1 write + index update
Min/Max scan (%D₁ buf)	~2k	Linear scan of k elements × 2
Second Stochastic (%K₂)	~4	1 SUB + 1 DIV + 1 MUL + 1 branch
Final smoothing (EMA)	~3	1 FMA + 1 MUL
Clamp	~2	2 comparisons
Total (k=10 default)	~65	O(k) dominated by dual min/max scans
Total (k=50 worst)	~225	Linear growth with kPeriod

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: recursive EMAs + sequential ring buffer min/max prevent vectorization
Bottleneck	Dual min/max scans over ring buffers (2×k comparisons per bar)
Parallelism	MACD EMA computation is independent of Stochastic pipeline but still sequential IIR
Memory	O(k): two ring buffers of kPeriod doubles + 6 scalar EMA states (~200 bytes at k=10)
Throughput	Moderate; faster than HT family (no transcendentals) but slower than pure IIR (min/max scans)

Resources

• Schaff, D. — "Schaff Trend Cycle" (currency trading methodology, 1990s)
• PineScript reference: stc.pine in indicator directory
• Ehlers, J.F. — Cybernetic Analysis for Stocks and Futures (cycle extraction theory)