Miha Kralj
7.7 KiB
GAMMADIST: Gamma Distribution CDF
The Gamma distribution generalizes the exponential, modeling the sum of waiting times with a shape that bends to fit.
| Property | Value |
|---|---|
| Category | Numeric |
| Inputs | Source (close) |
| Parameters | alpha (default 2.0), beta (default 1.0), period (default 14) |
| Outputs | Single series (Gammadist) |
| Output range | Varies (see docs) |
| Warmup | period bars |
| PineScript | gammadist.pine |
- The Gamma Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the gamma distribution, producing an ...
- Trading note: Gamma distribution; models waiting times and aggregate claims. Used in risk modeling.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Gamma Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the gamma distribution, producing an output in [0, 1]. The gamma distribution generalizes the exponential distribution by adding a shape parameter \alpha that controls whether the PDF is monotonically decreasing (\alpha < 1), exponential (\alpha = 1), or bell-shaped with a right skew (\alpha > 1). Combined with a rate parameter \beta that scales the normalized input, GAMMADIST provides a flexible nonlinear mapping with controllable asymmetry. The CDF is computed via the regularized lower incomplete gamma function using series expansion or Lentz continued fraction, selecting the faster-converging method based on the argument relative to the shape parameter.
Historical Context
The gamma distribution was first studied by Leonard Euler (1729) through his generalization of the factorial function to the gamma function \Gamma(z). The distribution itself was formalized by Karl Pearson (1893) as part of his system of frequency curves, where it appears as a Type III distribution. The incomplete gamma function, central to computing the CDF, was tabulated extensively by Pearson (1922) before computational methods made tables obsolete.
In finance, the gamma distribution models positively-skewed quantities: waiting times between events (generalizing the exponential), aggregate claim sizes in insurance (actuarial science), and the distribution of realized volatility (which is approximately gamma-distributed under certain stochastic volatility models). The chi-squared distribution is a special case with \alpha = k/2 and \beta = 2 (where k is degrees of freedom), connecting GAMMADIST to variance-based statistical tests.
The implementation uses two complementary algorithms for the regularized incomplete gamma function: a series expansion that converges rapidly for x < \alpha + 1, and a Lentz continued fraction for x \ge \alpha + 1. This split ensures convergence in approximately 10-30 iterations across the entire parameter space.
Architecture and Physics
The computation follows a three-phase pipeline:
Phase 1: Min-max normalization scans period bars for extrema, maps the current source to x \in [0, 1], then scales by the rate parameter: \text{scaled} = \max(0, x \cdot \beta).
Phase 2: Algorithm selection chooses between series and continued fraction based on the relationship between the scaled input and the shape parameter:
- If
\text{scaled} < \alpha + 1: use the series expansion (converges from below) - If
\text{scaled} \ge \alpha + 1: use1 - Q(\alpha, \text{scaled})via continued fraction (converges from above)
Phase 3: CDF evaluation computes the regularized lower incomplete gamma function P(\alpha, \text{scaled}).
The series expansion accumulates terms \delta_n = x^n / (\alpha(\alpha+1)\cdots(\alpha+n)) until the relative change drops below 10^{-10}, then multiplies by the normalization factor x^\alpha e^{-x} / \Gamma(\alpha).
The continued fraction (Lentz algorithm) evaluates the complementary function Q(\alpha, x) = 1 - P(\alpha, x) using the recurrence with convergents a_i = -i(i - \alpha) and b_i = x + 2i + 1 - \alpha.
Shape parameter effects: \alpha = 1 reduces to exponential distribution. \alpha = 2, \beta = 3 (default) gives a moderate right-skewed S-curve. Large \alpha approaches a normal CDF shape.
Mathematical Foundation
The gamma distribution with shape \alpha > 0 and rate \beta > 0 has PDF:
f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0
The CDF is the regularized lower incomplete gamma function:
F(x; \alpha, \beta) = P(\alpha, \beta x) = \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)} = \frac{1}{\Gamma(\alpha)} \int_0^{\beta x} t^{\alpha-1} e^{-t}\,dt
Series expansion for P(a, x) when x < a + 1:
P(a, x) = e^{-x} x^a \sum_{n=0}^{\infty} \frac{x^n}{a(a+1)\cdots(a+n)}
Continued fraction for Q(a, x) = 1 - P(a, x) when x \ge a + 1:
Q(a, x) = e^{-x} x^a \cdot \cfrac{1}{x + 1 - a + \cfrac{1 \cdot (1-a)}{x + 3 - a + \cfrac{2 \cdot (2-a)}{x + 5 - a + \cdots}}}
Log-gamma via Lanczos approximation (g = 7, 9 coefficients):
\ln\Gamma(z) = \frac{1}{2}\ln(2\pi) + \left(z - \frac{1}{2}\right)\ln(z + g - \frac{1}{2}) - (z + g - \frac{1}{2}) + \ln\!\left(\sum_{k=0}^{8} \frac{c_k}{z + k}\right)
Parameter constraints: period > 0, \alpha > 0, \beta > 0. Output is bounded [0, 1].
GAMMADIST(source, period, shape, rate):
// Phase 1: min-max normalization + scaling
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
scaled = max(0, x * rate)
// Phase 2-3: regularized lower incomplete gamma
if scaled <= 0: return 0.0
if scaled < shape + 1:
return gammaSeries(shape, scaled) // series expansion
else:
return 1.0 - gammaCF(shape, scaled) // continued fraction
Performance Profile
Operation Count (Streaming Mode)
Gamma distribution CDF uses regularized incomplete gamma function via series or continued fraction.
| Operation | Count | Cost (cycles) | Subtotal |
|---|---|---|---|
| Input validation (alpha, beta > 0; x >= 0) | 3 | 2 cy | ~6 cy |
| Log-Gamma normalization (lgamma) | 1 | 25 cy | ~25 cy |
| Regularized incomplete gamma (series, ~20 iter) | ~20 | 12 cy | ~240 cy |
| NaN guard + state update | 1 | 2 cy | ~2 cy |
| Total | O(1) | — | ~273 cy |
O(1) per evaluation. Switches between series expansion (x <= alpha+1) and continued fraction (x > alpha+1) for numerical stability. lgamma() is the setup cost.
Batch Mode (SIMD Analysis)
| Operation | Vectorizable? | Notes |
|---|---|---|
| lgamma() | No | Transcendental; scalar |
| Series/CF iteration | No | Sequential convergence |
| Output assignment | Yes | Trivial |
No practical SIMD benefit. Parallelism via PLINQ on the outer observation loop.
Resources
- Pearson, K. "Contributions to the Mathematical Theory of Evolution." Phil. Trans. Royal Society, 1893.
- Lanczos, C. "A Precision Approximation of the Gamma Function." SIAM J. Numerical Analysis B, 1964.
- Press, W.H. et al. "Numerical Recipes: The Art of Scientific Computing." 3rd edition, Cambridge University Press, 2007. Chapter 6.2 (Incomplete Gamma Function).
- Lentz, W.J. "Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions." Applied Optics, 1976.
- Johnson, N.L., Kotz, S. & Balakrishnan, N. "Continuous Univariate Distributions, Vol. 1." Wiley, 1994.