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FISHER04: Ehlers Fisher Transform (2004 Cybernetic Analysis)

The Fisher Transform provides clear, unambiguous turning points that make it possible to identify trend reversals.

Introduction

The Fisher04 indicator implements the revised Fisher Transform from Chapter 1 of Ehlers' 2004 book Cybernetic Analysis for Stocks and Futures. It converts price data into a Gaussian normal distribution using the inverse hyperbolic tangent (arctanh), producing sharp turning-point signals. This 2004 revision uses wider normalization bandwidth, gentler IIR smoothing, and a reduced arctanh multiplier compared to the original 2002 TASC article, resulting in a smoother oscillator with less noise.

Historical Context

Ehlers first published the Fisher Transform in a November 2002 Stocks & Commodities article titled "Using The Fisher Transform." That version used a 0.66 normalization coefficient and 0.67 IIR feedback. Two years later, in Cybernetic Analysis for Stocks and Futures (Wiley, 2004), Ehlers revised the coefficients. The 2004 version normalizes with a full 1.0 coefficient and 0.5 IIR feedback, tightens the clamp to 0.9999, and halves the arctanh multiplier from 0.5 to 0.25. No major external library (Skender, TA-Lib, Tulip, Ooples) implements this specific 2004 variant; they all use the 2002 formulation.

Architecture

1. Min/Max Normalization

The lookback window tracks the highest high and lowest low over period bars using a RingBuffer. The raw price is mapped to [-0.5, 0.5]:

$$\text{norm} = \frac{\text{price} - \text{lowest}}{\text{highest} - \text{lowest}} - 0.5$$

When range is zero (flat price), Value1 resets to 0.

2. IIR Smoothing (Value1)

The normalized value is smoothed with a single-pole IIR filter:

$$\text{Value1}_t = 1.0 \times \text{norm}_t + 0.5 \times \text{Value1}_{t-1}$$

Compare with Fisher (2002): $\text{Value1}_t = 0.66 \times \text{norm}t + 0.67 \times \text{Value1}{t-1}$

3. Clamping

Value1 is clamped to $(-0.9999, 0.9999)$ to prevent arctanh singularity:

$$\text{Value1} = \text{clamp}(\text{Value1}, -0.9999, 0.9999)$$

The clamped value is stored back for next iteration's IIR feedback.

4. Fisher Transform

The Fisher Transform applies arctanh with IIR feedback:

$$\text{Fish}_t = 0.25 \times \ln!\left(\frac{1 + \text{Value1}}{1 - \text{Value1}}\right) + 0.5 \times \text{Fish}_{t-1}$$

The 0.25 multiplier (vs 0.5 in 2002) produces approximately half the amplitude, reducing false signals.

5. Signal Line

The signal line is the previous bar's Fisher value: $\text{Signal}t = \text{Fish}{t-1}$

Coefficient Comparison

Parameter Fisher (2002) Fisher04 (2004)
Normalization 0.66 1.0
IIR feedback (Value1) 0.67 0.5
Clamp threshold 0.99 → 0.999 0.9999
Arctanh multiplier 0.5 0.25
Fisher IIR 0.5 0.5

Performance Profile

Key Optimizations

  • FMA in IIR updates: Both Value1 IIR and Fisher IIR use Math.FusedMultiplyAdd for the feedback * prev + coeff * input pattern.
  • Precomputed constants: Normalization coefficient (1.0), IIR feedback (0.5), clamp threshold (0.9999), arctanh multiplier (0.25) are all const fields, avoiding repeated literal encoding.
  • RingBuffer for O(1) update: Add and UpdateNewest are constant-time; only the min/max scan is O(period).
  • State copy pattern: _state/_p_state record struct enables bar correction without allocation.
  • Zero allocation: No heap allocation in the Update hot path; all state is stack-promoted via local copy.

Operation Count (Streaming Mode)

Operation Count per bar
Comparisons 2 x period (min/max scan)
Multiplications 2 (normalize + arctanh multiplier)
Additions 3 (normalize offset + 2x IIR)
FMA calls 2 (Value1 IIR, Fisher IIR)
Log 1 (arctanh via Math.Log)
Clamp 1
Division 1 (normalization)

SIMD Analysis (Batch Mode)

Aspect Status
Min/max scan Scalar (RingBuffer-based, O(period) per bar)
Normalization Scalar (data-dependent division)
Value1 IIR smoothing Scalar (sequential IIR dependency)
arctanh Scalar (Math.Log, not vectorizable)
Fisher IIR Scalar (sequential dependency on previous Fisher)
Vectorization potential Low: dual IIR chain + logarithm prevents SIMD

Validation

No external library implements the 2004 Ehlers variant. Validation is performed against:

  • Manual step-by-step computation matching the published algorithm
  • Batch vs streaming consistency (tolerance: 1e-12)
  • Span vs streaming consistency (tolerance: 1e-12)
  • Coefficient difference verification against Fisher (2002)
  • Amplitude reduction verification (Fisher04 < Fisher in avg absolute value)

Common Pitfalls

  1. Confusing 2002 and 2004 versions. The coefficient differences are subtle but produce measurably different outputs. Using 2002 coefficients with 2004 labels (or vice versa) produces incorrect results.
  2. Not storing clamped Value1 back. The IIR feedback must use the clamped value, not the pre-clamp value. Failing to store back causes drift.
  3. Expecting identical results to Fisher. Fisher04 uses 0.25x arctanh multiplier vs 0.5x; the amplitude is roughly halved.
  4. Using Fisher04 for high-frequency scalping. The gentler coefficients make it slower to react than Fisher (2002). Better suited for swing trading.
  5. Ignoring the signal line crossover. The primary trading signal is Fisher crossing above/below its one-bar-lagged signal line.

References

  1. Ehlers, J. F. (2004). Cybernetic Analysis for Stocks and Futures. Wiley. Chapter 1.
  2. Ehlers, J. F. (2002). "Using The Fisher Transform." Technical Analysis of Stocks & Commodities, November 2002.
  3. MESA Software. "The Inverse Fisher Transform." mesasoftware.com