Logit Transform -EasyNeuro-

Logit Transform

This script implements a novel indicator inspired by the Fisher Transform, replacing its core arctanh-based mapping with the logit transform. It is designed to highlight extreme values in bounded inputs from a probabilistic and statistical perspective.

Background: Fisher Transform

The Fisher Transform, introduced by John Ehlers, is a statistical technique that maps a bounded variable x (between a and b) to a variable approximately following a Gaussian distribution. The standard form for a normalized input y (between -1 and 1) is F(y) = 0.5 * ln((1 + y)/(1 - y)) = arctanh(y).

This transformation has the following properties:

Linearization of extremes:
Small deviations around the mean are smooth, while movements near the boundaries are sharply amplified.

Gaussian approximation:
After transformation, the variable approximates a normal distribution, enabling analytical techniques that assume normality.

Probabilistic interpretation:
The Fisher Transform can be linked to likelihood ratio tests, where the transform emphasizes deviations from median or expected values in a statistically meaningful way.

In technical analysis, this allows traders to detect turning points or extreme market conditions more clearly than raw oscillators alone.

Logit Transform as a Generalization

The logit function is defined for p between 0 and 1 as logit(p) = ln(p / (1 - p)).
Key properties of the logit transform:

• Maps probabilities in (0, 1) to the entire real line, similar to the Fisher Transform.
  
• Emphasizes values near 0 and 1, providing sharp differentiation of extreme states.
  
• Directly interpretable in terms of odds and likelihood ratios: logit(p) = ln(odds).
  

From a statistical viewpoint, the logit transform corresponds to the canonical link function in binomial generalized linear models (GLMs). This provides a natural interpretation of the transformed variable as the logarithm of the likelihood ratio between success and failure states, giving a rigorous probabilistic framework for extreme value detection.

Theoretical Advantages

Distributional linearization:
For inputs that can be interpreted as probabilities, the logit transform creates a variable approximately linear in log-odds, similar to Fisher’s goal of Gaussianization but with a probabilistic foundation.

Extreme sensitivity:
By amplifying small differences near 0 or 1, it allows for sharper detection of market extremes or overbought/oversold conditions.

Statistical interpretability:
Provides a link to statistical hypothesis testing via likelihood ratios, enabling integration with probabilistic models or risk metrics.

Applications in Technical Analysis

Oscillator enhancement:
Apply to RSI, Stochastic Oscillators, or other bounded indicators to accentuate extreme values with a well-defined probabilistic interpretation.

Comparative study:
Use alongside the Fisher Transform to analyze the effect of different nonlinear mappings on market signals, helping to uncover subtle nonlinearity in price behavior.

Probabilistic risk assessment:
Transforming input series into log-odds allows incorporation into statistical risk models or volatility estimation frameworks.

Practical Considerations

The logit diverges near 0 and 1, requiring careful scaling or smoothing to avoid numerical instability. As with the Fisher Transform, this indicator is not a standalone trading signal and should be combined with complementary technical or statistical indicators.

In summary, the Logit Transform builds upon the Fisher Transform’s theoretical foundation while introducing a probabilistically rigorous mapping. By connecting extreme-value detection to odds ratios and likelihood principles, it provides traders and analysts with a mathematically grounded tool for examining market dynamics.