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Miha Kralj
121 lines
7.7 KiB
Markdown
121 lines
7.7 KiB
Markdown
# GAMMADIST: Gamma Distribution CDF
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> *The Gamma distribution generalizes the exponential, modeling the sum of waiting times with a shape that bends to fit.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Numeric |
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| **Inputs** | Source (close) |
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| **Parameters** | `alpha` (default 2.0), `beta` (default 1.0), `period` (default 14) |
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| **Outputs** | Single series (Gammadist) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `period` bars |
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| **PineScript** | [gammadist.pine](gammadist.pine) |
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- The Gamma Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the gamma distribution, producing an ...
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- **Trading note:** Gamma distribution; models waiting times and aggregate claims. Used in risk modeling.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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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.
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## Historical Context
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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.
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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.
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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.
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## Architecture and Physics
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The computation follows a three-phase pipeline:
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**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: $\text{scaled} = \max(0, x \cdot \beta)$.
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**Phase 2: Algorithm selection** chooses between series and continued fraction based on the relationship between the scaled input and the shape parameter:
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- If $\text{scaled} < \alpha + 1$: use the series expansion (converges from below)
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- If $\text{scaled} \ge \alpha + 1$: use $1 - Q(\alpha, \text{scaled})$ via continued fraction (converges from above)
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**Phase 3: CDF evaluation** computes the regularized lower incomplete gamma function $P(\alpha, \text{scaled})$.
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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 $x^\alpha e^{-x} / \Gamma(\alpha)$.
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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 $b_i = x + 2i + 1 - \alpha$.
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**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.
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## Mathematical Foundation
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The gamma distribution with shape $\alpha > 0$ and rate $\beta > 0$ has PDF:
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$$f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0$$
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The CDF is the **regularized lower incomplete gamma function**:
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$$F(x; \alpha, \beta) = P(\alpha, \beta x) = \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)} = \frac{1}{\Gamma(\alpha)} \int_0^{\beta x} t^{\alpha-1} e^{-t}\,dt$$
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**Series expansion** for $P(a, x)$ when $x < a + 1$:
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$$P(a, x) = e^{-x} x^a \sum_{n=0}^{\infty} \frac{x^n}{a(a+1)\cdots(a+n)}$$
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**Continued fraction** for $Q(a, x) = 1 - P(a, x)$ when $x \ge a + 1$:
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$$Q(a, x) = e^{-x} x^a \cdot \cfrac{1}{x + 1 - a + \cfrac{1 \cdot (1-a)}{x + 3 - a + \cfrac{2 \cdot (2-a)}{x + 5 - a + \cdots}}}$$
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**Log-gamma** via Lanczos approximation ($g = 7$, 9 coefficients):
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$$\ln\Gamma(z) = \frac{1}{2}\ln(2\pi) + \left(z - \frac{1}{2}\right)\ln(z + g - \frac{1}{2}) - (z + g - \frac{1}{2}) + \ln\!\left(\sum_{k=0}^{8} \frac{c_k}{z + k}\right)$$
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**Parameter constraints**: `period` $> 0$, $\alpha > 0$, $\beta > 0$. Output is bounded $[0, 1]$.
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```
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GAMMADIST(source, period, shape, rate):
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// Phase 1: min-max normalization + scaling
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min_val = min(source[0..period-1])
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max_val = max(source[0..period-1])
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range = max_val - min_val
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x = range > 0 ? (source - min_val) / range : 0.5
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scaled = max(0, x * rate)
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// Phase 2-3: regularized lower incomplete gamma
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if scaled <= 0: return 0.0
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if scaled < shape + 1:
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return gammaSeries(shape, scaled) // series expansion
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else:
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return 1.0 - gammaCF(shape, scaled) // continued fraction
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```
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## Performance Profile
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### Operation Count (Streaming Mode)
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Gamma distribution CDF uses regularized incomplete gamma function via series or continued fraction.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Input validation (alpha, beta > 0; x >= 0) | 3 | 2 cy | ~6 cy |
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| Log-Gamma normalization (lgamma) | 1 | 25 cy | ~25 cy |
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| Regularized incomplete gamma (series, ~20 iter) | ~20 | 12 cy | ~240 cy |
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| NaN guard + state update | 1 | 2 cy | ~2 cy |
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| **Total** | **O(1)** | — | **~273 cy** |
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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.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| lgamma() | No | Transcendental; scalar |
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| Series/CF iteration | No | Sequential convergence |
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| Output assignment | Yes | Trivial |
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No practical SIMD benefit. Parallelism via PLINQ on the outer observation loop.
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## Resources
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- Pearson, K. "Contributions to the Mathematical Theory of Evolution." Phil. Trans. Royal Society, 1893.
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- Lanczos, C. "A Precision Approximation of the Gamma Function." SIAM J. Numerical Analysis B, 1964.
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- Press, W.H. et al. "Numerical Recipes: The Art of Scientific Computing." 3rd edition, Cambridge University Press, 2007. Chapter 6.2 (Incomplete Gamma Function).
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- Lentz, W.J. "Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions." Applied Optics, 1976.
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- Johnson, N.L., Kotz, S. & Balakrishnan, N. "Continuous Univariate Distributions, Vol. 1." Wiley, 1994. |