QuanTAlib/lib/numerics/fdist/Fdist.md

FDIST: F-Distribution CDF

The F-distribution CDF tests variance ratios — a cornerstone of hypothesis testing built from two chi-squared variables.

Property	Value
Category	Numeric
Inputs	Source (close)
Parameters	d1 (default 1), d2 (default 1), period (default 14)
Outputs	Single series (Fdist)
Output range	Varies (see docs)
Warmup	period bars
PineScript	fdist.pine

• The F-Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the F-distribution (Fisher-Snedecor distr...
• Trading note: F-distribution; used in ANOVA and regression significance tests. Compares model variances.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The F-Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the F-distribution (Fisher-Snedecor distribution), producing an output in [0, 1]. The F-distribution arises as the ratio of two chi-squared random variables divided by their respective degrees of freedom, making it the natural distribution for variance ratio tests. By mapping normalized price through the regularized incomplete beta function with parameters tied to degrees of freedom d_1 and d_2, FDIST provides a probabilistic ranking that is asymmetric: the CDF shape changes qualitatively depending on whether d_1 < d_2, d_1 = d_2, or d_1 > d_2, giving traders control over the nonlinear response curve.

Historical Context

The F-distribution was developed independently by George Snedecor (1934) and Ronald Fisher (1924), though Fisher's earlier work on variance ratios laid the theoretical foundation. The distribution is named in Fisher's honor by Snedecor. Its primary statistical application is the F-test for comparing variances of two populations, and it forms the backbone of ANOVA (Analysis of Variance), one of the most widely used statistical procedures.

In financial applications, the F-distribution appears in variance ratio tests (Lo and MacKinlay, 1988) used to test the random walk hypothesis. The CDF form used here repurposes the distribution's shape as a nonlinear mapping: with equal degrees of freedom (d_1 = d_2), the CDF is approximately symmetric around 0.5; with d_1 \gg d_2, the curve shifts left (more probability mass near zero); with d_1 \ll d_2, it shifts right. This parameter-controlled asymmetry distinguishes FDIST from simpler sigmoid-like transformations.

The implementation uses the same Lanczos log-gamma and Lentz continued fraction machinery as BETADIST, since the F-distribution CDF reduces to a regularized incomplete beta function through a variable substitution.

Architecture and Physics

The computation follows a three-phase pipeline:

Phase 1: Min-max normalization scans period bars to find extrema, then maps the current source to x \in [0, 1]. Zero-range defaults to 0.5.

Phase 2: Variable transformation converts the normalized value x to the beta function argument:

t = \frac{d_1 \cdot x}{d_1 \cdot x + d_2}

This maps x \in [0, \infty) to t \in [0, 1), which is the domain of the regularized incomplete beta function. Since input x is already in [0, 1], the effective range of t is [0, d_1/(d_1 + d_2)].

Phase 3: Regularized incomplete beta evaluates I_t(d_1/2, d_2/2) using the Lentz continued fraction algorithm. The implementation includes a reflection step when x > (a+1)/(a+b+2) to ensure the continued fraction converges from the faster side. Convergence typically requires 10-20 iterations to reach \epsilon = 10^{-10}.

Shared infrastructure: The lnGamma() function uses the Lanczos approximation with g = 7 and 9 coefficients, identical to the implementation in BETADIST and other distribution indicators. The betaReg() continued fraction is likewise shared.

Mathematical Foundation

The F-distribution with d_1 numerator and d_2 denominator degrees of freedom has PDF:

f(x; d_1, d_2) = \frac{1}{B(d_1/2, d_2/2)} \cdot \left(\frac{d_1}{d_2}\right)^{d_1/2} \cdot \frac{x^{d_1/2 - 1}}{(1 + d_1 x / d_2)^{(d_1+d_2)/2}}

The CDF is expressed via the regularized incomplete beta function:

F(x; d_1, d_2) = I_t\!\left(\frac{d_1}{2}, \frac{d_2}{2}\right), \quad t = \frac{d_1 x}{d_1 x + d_2}

where the regularized incomplete beta function is:

I_x(a, b) = \frac{B(x; a, b)}{B(a, b)} = \frac{1}{B(a, b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt

Lentz continued fraction for I_x(a, b):

I_x(a,b) = \frac{x^a (1-x)^b}{a \cdot B(a,b)} \cdot \cfrac{1}{1 + \cfrac{d_1}{1 + \cfrac{d_2}{1 + \cdots}}}

with convergents d_m defined by the even/odd recurrence involving a, b, and x.

Parameter constraints: period > 0, d_1 > 0, d_2 > 0. Output is bounded [0, 1].

FDIST(source, period, d1, d2):
    // Phase 1: min-max normalization
    min_val = min(source[0..period-1])
    max_val = max(source[0..period-1])
    range = max_val - min_val
    x = range > 0 ? (source - min_val) / range : 0.5

    // Phase 2: variable transformation
    safe_x = max(0, x)
    t = d1 * safe_x / (d1 * safe_x + d2)

    // Phase 3: regularized incomplete beta via Lentz CF
    return betaReg(t, d1/2, d2/2)

Performance Profile

Operation Count (Streaming Mode)

F-distribution CDF uses regularized incomplete beta function — same cost structure as BetaDist.

Operation	Count	Cost (cycles)	Subtotal
Input validation (d1, d2 > 0; x >= 0)	3	2 cy	~6 cy
Transform x to beta variable	1	3 cy	~3 cy
Regularized incomplete beta (Lentz CF, ~20 iter)	~20	15 cy	~300 cy
NaN guard + state update	1	2 cy	~2 cy
Total	O(1)	—	~311 cy

O(1) per evaluation. Dominated by the continued fraction solver, same as Beta/T distributions. Degrees-of-freedom parameters affect convergence speed slightly.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
x transformation	Yes	Vector arithmetic
Continued fraction	No	Sequential convergence
Output assignment	Yes	Trivial

No SIMD benefit for the core evaluation. Outer loop across observations parallelizable with PLINQ for bulk p-value computation.

Resources

• Fisher, R.A. "On a Distribution Yielding the Error Functions of Several Well Known Statistics." Proc. International Mathematical Congress, Toronto, 1924.
• Snedecor, G.W. "Calculation and Interpretation of Analysis of Variance and Covariance." Collegiate Press, 1934.
• Press, W.H. et al. "Numerical Recipes: The Art of Scientific Computing." 3rd edition, Cambridge University Press, 2007. Chapter 6.4 (Incomplete Beta Function).
• Lo, A. & MacKinlay, A.C. "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test." Review of Financial Studies, 1988.
• Lentz, W.J. "Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions." Applied Optics, 1976.