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lib/numerics/accel/Accel.md
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# ACCEL: Second Derivative (Acceleration)
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (ACCEL) |
| **Output range** | Varies (see docs) |
| **Warmup** | `3` bars |
### TL;DR
- ACCEL measures the rate of change of velocity—the acceleration of a time series.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `3` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "Velocity tells you where you're going. Acceleration tells you if you're getting there faster or slower."
ACCEL measures the rate of change of velocity—the acceleration of a time series. As the second derivative, it reveals momentum shifts before they manifest in price direction. Positive acceleration means velocity is increasing (trend strengthening); negative means velocity is decreasing (trend weakening). This O(1) streaming implementation uses FMA optimization and SIMD batch processing.
lib/numerics/betadist/Betadist.md
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# BETADIST: Beta Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` (default 50), `alpha` (default 2.0), `beta` (default 2.0) |
| **Outputs** | Single series (Betadist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- BETADIST computes the cumulative distribution function of the Beta distribution applied to a min-max normalized price series.
- Parameterized by `period` (default 50), `alpha` (default 2.0), `beta` (default 2.0).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
BETADIST computes the cumulative distribution function of the Beta distribution applied to a min-max normalized price series. The source price is first normalized to $[0, 1]$ over a lookback window, then passed through the regularized incomplete beta function $I_x(\alpha, \beta)$ to produce a probability-mapped oscillator. The two shape parameters $\alpha$ and $\beta$ control the nonlinear mapping: symmetric parameters ($\alpha = \beta$) produce a sigmoid-like transformation centered at 0.5, while asymmetric parameters skew the mapping to emphasize extremes in either direction.
## Historical Context
lib/numerics/binomdist/Binomdist.md
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# BINOMDIST: Binomial Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` (default 50), `trials` (default 20), `threshold` (default 10) |
| **Outputs** | Single series (Binomdist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- BINOMDIST computes the cumulative distribution function of the Binomial distribution, mapping a min-max normalized price to a success probability $...
- Parameterized by `period` (default 50), `trials` (default 20), `threshold` (default 10).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
BINOMDIST computes the cumulative distribution function of the Binomial distribution, mapping a min-max normalized price to a success probability $p$ and evaluating $P(X \leq k)$ for $X \sim \text{Binomial}(n, p)$. The normalized price position within its lookback range determines the probability of success per trial, while the trial count $n$ and threshold $k$ control the shape of the CDF response. The output is a $[0, 1]$ bounded oscillator where values near 0 indicate the price-derived probability makes $k$ or fewer successes very unlikely (bullish pressure), and values near 1 indicate $k$ successes are very likely (established range).
## Historical Context
lib/numerics/change/Change.md
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# CHANGE: Relative Price Change
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` (default 1) |
| **Outputs** | Single series (Change) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period + 1` bars |
### TL;DR
- CHANGE calculates the percentage change between the current value and a value N periods ago.
- Parameterized by `period` (default 1).
- Output range: Varies (see docs).
- Requires `period + 1` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The simplest measure of movement is often the most powerful."
CHANGE calculates the percentage change between the current value and a value N periods ago. This fundamental indicator forms the basis for momentum analysis, rate of change calculations, and relative performance comparisons.
lib/numerics/cwt/Cwt.md
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# CWT: Continuous Wavelet Transform
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `scale` (default 10.0), `omega0` (default 6.0) |
| **Outputs** | Single series (Cwt) |
| **Output range** | Varies (see docs) |
| **Warmup** | 1 bar |
### TL;DR
- CWT computes the magnitude of the Continuous Wavelet Transform at a specified scale using the Morlet wavelet, providing a time-frequency decomposit...
- Parameterized by `scale` (default 10.0), `omega0` (default 6.0).
- Output range: Varies (see docs).
- Requires 1 bar of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
CWT computes the magnitude of the Continuous Wavelet Transform at a specified scale using the Morlet wavelet, providing a time-frequency decomposition that measures the energy content of a specific frequency band at each point in time. Unlike Fourier analysis which loses time localization, the wavelet transform maintains both time and frequency information simultaneously. The output is a non-negative magnitude series where peaks indicate strong presence of the target frequency (determined by the scale parameter) and troughs indicate absence of that frequency component.
## Historical Context
lib/numerics/dwt/Dwt.md
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# DWT: Discrete Wavelet Transform
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `levels` (default 4), `output` (default 0) |
| **Outputs** | Single series (Dwt) |
| **Output range** | Varies (see docs) |
| **Warmup** | `bufferSize` bars |
### TL;DR
- The Discrete Wavelet Transform decomposes a price series into multi-resolution frequency components using the a trous (with holes) stationary Haar ...
- Parameterized by `levels` (default 4), `output` (default 0).
- Output range: Varies (see docs).
- Requires `bufferSize` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Discrete Wavelet Transform decomposes a price series into multi-resolution frequency components using the a trous (with holes) stationary Haar wavelet. Unlike decimated DWT, the stationary variant preserves time alignment at every scale, producing an approximation (trend) and detail coefficients (noise/cycles) at each decomposition level. Each level doubles the effective receptive field: level $L$ captures structure at $2^L$ bars. With 1-8 levels and $O(L)$ per-bar cost, DWT provides a complete multi-scale decomposition that cleanly separates trend from noise without the phase distortion inherent in moving-average cascades.
## Historical Context
lib/numerics/expdist/Expdist.md
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# EXPDIST: Exponential Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` (default 50), `lambda` (default 3.0) |
| **Outputs** | Single series (Expdist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Exponential Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the exponential distribution, p...
- Parameterized by `period` (default 50), `lambda` (default 3.0).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Exponential Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the exponential distribution, producing an output in $[0, 1]$. The exponential distribution models memoryless waiting times: the probability that a normalized value falls below a threshold depends only on the rate parameter $\lambda$, not on any history. Higher $\lambda$ values compress the CDF curve toward zero, making the indicator more sensitive to small normalized deviations. With $O(N)$ normalization and $O(1)$ CDF evaluation, EXPDIST provides a nonlinear percentile ranking that emphasizes the lower end of the price range while compressing the upper end.
## Historical Context
lib/numerics/exptrans/Exptrans.md
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# EXPTRANS: Exponential Function
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (EXPTRANS) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The Exponential (EXP) transformer applies the natural exponential function $e^x$ to each value in a time series.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The exponential function is the only function that is its own derivative—a mathematical curiosity that makes it indispensable for modeling growth, decay, and everything compounding."
The Exponential (EXP) transformer applies the natural exponential function $e^x$ to each value in a time series. As the inverse of the natural logarithm, it converts additive relationships back to multiplicative ones, making it essential for reconstructing price levels from log-returns and implementing models that assume log-normal distributions.
lib/numerics/fdist/Fdist.md
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# FDIST: F-Distribution CDF
| 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 |
### TL;DR
- The F-Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the F-distribution (Fisher-Snedecor distr...
- Parameterized by `d1` (default 1), `d2` (default 1), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- 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
lib/numerics/fft/Fft.md
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# FFT: Fast Fourier Transform (Dominant Cycle Detector)
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `windowSize` (default 64), `minPeriod` (default 4), `maxPeriod` (default 32) |
| **Outputs** | Single series (Fft) |
| **Output range** | Varies (see docs) |
| **Warmup** | 1 bar |
### TL;DR
- The FFT indicator computes the dominant cycle period in a price series using a Discrete Fourier Transform with a Hanning window.
- Parameterized by `windowsize` (default 64), `minperiod` (default 4), `maxperiod` (default 32).
- Output range: Varies (see docs).
- Requires 1 bar of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The FFT indicator computes the dominant cycle period in a price series using a Discrete Fourier Transform with a Hanning window. Rather than outputting frequency-domain magnitudes, it returns the estimated dominant cycle period in bars, making it directly usable as an adaptive period input for other indicators. The implementation uses a brute-force DFT over a constrained frequency band (not a radix-2 FFT), with parabolic interpolation on the magnitude spectrum to achieve sub-bin frequency resolution. With window sizes of 32, 64, or 128 and $O(N \cdot N/2)$ complexity per bar, the indicator trades computational cost for precise cycle detection within user-specified period bounds.
## Historical Context
lib/numerics/gammadist/Gammadist.md
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# GAMMADIST: Gamma Distribution CDF
| 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 |
### TL;DR
- The Gamma Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the gamma distribution, producing an ...
- Parameterized by `alpha` (default 2.0), `beta` (default 1.0), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- 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
lib/numerics/highest/Highest.md
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# HIGHEST: Rolling Maximum
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` |
| **Outputs** | Single series (Highest) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- HIGHEST calculates the maximum value over a rolling lookback window.
- Parameterized by `period`.
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "What's the peak? The answer to that question defines support, resistance, and breakout levels."
HIGHEST calculates the maximum value over a rolling lookback window. This O(1) amortized streaming implementation uses a monotonic deque algorithm, enabling real-time updates without re-scanning the entire window. Validated against TA-Lib MAX and Tulip max functions.
lib/numerics/ifft/Ifft.md
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# IFFT: Inverse Fast Fourier Transform (Spectral Filter)
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `windowSize` (default 64), `numHarmonics` (default 5) |
| **Outputs** | Single series (Ifft) |
| **Output range** | Varies (see docs) |
| **Warmup** | 1 bar |
### TL;DR
- The Inverse FFT indicator reconstructs a smoothed version of the price series by performing a forward DFT, retaining only the lowest-frequency harm...
- Parameterized by `windowsize` (default 64), `numharmonics` (default 5).
- Output range: Varies (see docs).
- Requires 1 bar of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Inverse FFT indicator reconstructs a smoothed version of the price series by performing a forward DFT, retaining only the lowest-frequency harmonics, and synthesizing the output via inverse transform. The result is a spectral low-pass filter that preserves the dominant cyclical components while discarding high-frequency noise. By controlling the number of retained harmonics $H$, the user adjusts the smoothness/responsiveness trade-off: $H = 1$ yields a near-sinusoidal trend, while $H = N/2$ reproduces the original (windowed) signal. The indicator overlays on price and provides a frequency-domain alternative to conventional moving averages.
## Historical Context
lib/numerics/jerk/Jerk.md
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# JERK: Third Derivative
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (JERK) |
| **Output range** | Varies (see docs) |
| **Warmup** | `4` bars |
### TL;DR
- JERK measures the rate of change of acceleration—called "jerk" in physics.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `4` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "Acceleration tells you the trend is changing. Jerk tells you that change is itself changing—the earliest possible warning."
JERK measures the rate of change of acceleration—called "jerk" in physics. As the third derivative, it detects changes in momentum dynamics before they appear in acceleration, velocity, or price. A positive jerk means acceleration is increasing; negative means acceleration is decreasing. This O(1) streaming implementation uses dual FMA optimization and SIMD batch processing for four-point calculations.
lib/numerics/lineartrans/Lineartrans.md
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# LINEARTRANS: Linear Scaling Transformer
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `slope` (default 1.0), `intercept` (default 0.0) |
| **Outputs** | Single series (Lineartrans) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The Linear transformer applies an affine transformation $y = \text{slope} \cdot x + \text{intercept}$ to each value in a time series.
- Parameterized by `slope` (default 1.0), `intercept` (default 0.0).
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The simplest transformations are often the most powerful—linear scaling is the mathematical equivalent of adjusting the volume and tuning the dial."
The Linear transformer applies an affine transformation $y = \text{slope} \cdot x + \text{intercept}$ to each value in a time series. This fundamental operation enables scaling, offsetting, unit conversion, and normalization—the building blocks for preparing data for analysis or combining signals from different sources.
lib/numerics/lognormdist/Lognormdist.md
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# LOGNORMDIST: Log-Normal Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `mu` (default 0.0), `sigma` (default 1.0), `period` (default 14) |
| **Outputs** | Single series (Lognormdist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Log-Normal Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the log-normal distribution, pro...
- Parameterized by `mu` (default 0.0), `sigma` (default 1.0), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Log-Normal Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the log-normal distribution, producing an output in $[0, 1]$. A random variable $X$ is log-normally distributed when $\ln(X)$ follows a normal distribution. This makes the log-normal CDF natural for financial data, where multiplicative returns (log-returns) are approximately normally distributed. The indicator min-max normalizes the source to $(0, 1]$, takes the natural logarithm, standardizes by parameters $\mu$ and $\sigma$, then evaluates the standard normal CDF. The result emphasizes values near the bottom of the recent range (where the logarithm diverges) and compresses values near the top.
## Historical Context
lib/numerics/logtrans/Logtrans.md
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# LOGTRANS: Natural Logarithm Transformer
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (LOGTRANS) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The LOG transformer applies the natural logarithm function $\ln(x)$ to input values.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The logarithm is one of the most useful mathematical functions, turning multiplicative relationships into additive ones—a property that makes many financial calculations tractable."
The LOG transformer applies the natural logarithm function $\ln(x)$ to input values. This point-wise transformation compresses large values and expands small ones, making it essential for analyzing multiplicative processes like compounded returns.
lib/numerics/lowest/Lowest.md
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# LOWEST: Rolling Minimum
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` |
| **Outputs** | Single series (Lowest) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- LOWEST calculates the minimum value over a rolling lookback window.
- Parameterized by `period`.
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "Know your floor. Support levels are just historical minimums waiting to be tested."
LOWEST calculates the minimum value over a rolling lookback window. This O(1) amortized streaming implementation uses a monotonic deque algorithm, enabling real-time updates without re-scanning the entire window. Validated against TA-Lib MIN and Tulip min functions.
lib/numerics/normalize/Normalize.md
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# NORMALIZE: Min-Max Normalization
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `period` (default 14) |
| **Outputs** | Single series (Normalize) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Normalize transformer applies min-max scaling to map any value series into the bounded range [0, 1] based on the observed minimum and maximum w...
- Parameterized by `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "Normalization is the art of making apples and oranges comparable—by insisting that everything lives on the same scale from 0 to 1."
The Normalize transformer applies min-max scaling to map any value series into the bounded range [0, 1] based on the observed minimum and maximum within a rolling lookback window. This technique is fundamental for feature scaling, creating bounded oscillators, and comparing series with different magnitudes.
lib/numerics/normdist/Normdist.md
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# NORMDIST: Normal Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `mu` (default 0.0), `sigma` (default 1.0), `period` (default 14) |
| **Outputs** | Single series (Normdist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Normal Distribution CDF transforms a z-score normalized price into the cumulative distribution function of the Gaussian distribution, producing...
- Parameterized by `mu` (default 0.0), `sigma` (default 1.0), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Normal Distribution CDF transforms a z-score normalized price into the cumulative distribution function of the Gaussian distribution, producing an output in $[0, 1]$. Unlike other distribution indicators in this library that use min-max normalization, NORMDIST computes a rolling mean and standard deviation over the lookback window, converting the raw price to a z-score, then applies optional $\mu$ and $\sigma$ parameters for further shaping. The result represents the probability that a standard normal random variable would fall at or below the observed z-score. This makes NORMDIST a direct percentile ranking under the assumption of normally distributed returns, with the output naturally centered at 0.5 when the price is at its rolling mean.
## Historical Context
lib/numerics/poissondist/Poissondist.md
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# POISSONDIST: Poisson Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `lambda` (default 1.0), `period` (default 14), `threshold` (default 5) |
| **Outputs** | Single series (Poissondist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Poisson Distribution CDF computes the probability $P(X \le k)$ for a Poisson random variable whose rate parameter $\lambda$ is derived from the...
- Parameterized by `lambda` (default 1.0), `period` (default 14), `threshold` (default 5).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Poisson Distribution CDF computes the probability $P(X \le k)$ for a Poisson random variable whose rate parameter $\lambda$ is derived from the min-max normalized price. The Poisson distribution models the number of events in a fixed interval given a constant average rate, making it natural for count-based financial metrics (trade arrivals, tick counts, order flow). The implementation maps normalized price to $\lambda$ via a scale factor, then evaluates the CDF using the identity $P(X \le k) = 1 - P(k+1, \lambda)$ where $P(a, x)$ is the regularized lower incomplete gamma function. This reuses the same Lanczos log-gamma and series/continued-fraction infrastructure as GAMMADIST.
## Historical Context
lib/numerics/relu/Relu.md
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# RELU: Rectified Linear Unit
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (RELU) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The Rectified Linear Unit (ReLU) activation function applies `max(0, x)` to each value, passing positive inputs unchanged while zeroing negative ones.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The simplest non-linearity that works—ReLU's computational efficiency and gradient-friendly properties made deep learning practical."
The Rectified Linear Unit (ReLU) activation function applies `max(0, x)` to each value, passing positive inputs unchanged while zeroing negative ones. Its simplicity belies its importance: ReLU enabled the training of deep neural networks by mitigating vanishing gradients, and its computational efficiency makes it the default activation for most architectures.
lib/numerics/sigmoid/Sigmoid.md
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# SIGMOID: Logistic Function
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `k` (default 1.0), `x0` (default 0.0) |
| **Outputs** | Single series (Sigmoid) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The Sigmoid (Logistic) transformer maps any real-valued input to the bounded range (0, 1) using the standard logistic function.
- Parameterized by `k` (default 1.0), `x0` (default 0.0).
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The sigmoid function is the S-curve that turns messy reality into neat probabilities—a mathematical diplomat that insists every answer must be between 0 and 1."
The Sigmoid (Logistic) transformer maps any real-valued input to the bounded range (0, 1) using the standard logistic function. Its characteristic S-shaped curve makes it indispensable for probability estimation, neural network activations, and any scenario requiring bounded outputs from unbounded inputs.
lib/numerics/slope/Slope.md
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# SLOPE: First Derivative (Velocity)
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (SLOPE) |
| **Output range** | Varies (see docs) |
| **Warmup** | `2` bars |
### TL;DR
- SLOPE measures the instantaneous rate of change—the velocity of a time series.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `2` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The simplest measure of change reveals the most: is it going up, or going down?"
SLOPE measures the instantaneous rate of change—the velocity of a time series. As the first derivative, it answers the fundamental question: how fast is the value changing right now? A positive slope means ascending; negative means descending; zero means flat. This O(1) streaming implementation uses SIMD optimization for batch calculations and handles bar corrections via state rollback.
lib/numerics/sqrttrans/Sqrttrans.md
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# SQRTTRANS: Square Root Transform
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | None |
| **Outputs** | Single series (SQRTTRANS) |
| **Output range** | Varies (see docs) |
| **Warmup** | `0` bars |
### TL;DR
- The Square Root (SQRT) transformer applies $\sqrt{x}$ to each value in a time series.
- No configurable parameters; computation is stateless per bar.
- Output range: Varies (see docs).
- Requires `0` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "The square root is nature's variance-stabilizing trick—halving the exponent space while preserving monotonicity. When price volatility scales with level, sqrt compresses the noise."
The Square Root (SQRT) transformer applies $\sqrt{x}$ to each value in a time series. This variance-stabilizing transformation compresses ranges where volatility scales with magnitude, making it useful for heteroscedastic data where standard deviation increases with price level.
@@ -193,4 +210,4 @@ var recovered = Sqrttrans.Calculate(squared);
- Box, G.E.P., & Cox, D.R. (1964). "An Analysis of Transformations." *Journal of the Royal Statistical Society, Series B*, 26(2), 211-252.
- Tukey, J.W. (1977). *Exploratory Data Analysis*. Addison-Wesley. (Variance-stabilizing transformations)
- IEEE 754-2019. *Standard for Floating-Point Arithmetic*. (sqrt specification)
- IEEE 754-2019. *Standard for Floating-Point Arithmetic*. (sqrt specification)
lib/numerics/tdist/Tdist.md
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# TDIST: Student's t-Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `nu` (default 10), `period` (default 14) |
| **Outputs** | Single series (Tdist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Student's t-Distribution CDF transforms a min-max normalized price into the cumulative distribution function of Student's t-distribution, produ...
- Parameterized by `nu` (default 10), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Student's t-Distribution CDF transforms a min-max normalized price into the cumulative distribution function of Student's t-distribution, producing an output in $[0, 1]$. The t-distribution is the normal distribution's heavier-tailed cousin: as degrees of freedom $\nu$ increase, it converges to the Gaussian; at low $\nu$ it accommodates extreme values that the normal distribution would assign negligible probability. The implementation normalizes price to $[0, 1]$, maps to a t-statistic via linear scaling to $[-3, +3]$, then evaluates the CDF through the regularized incomplete beta function. This makes TDIST a robust percentile ranking that is less sensitive to outliers than NORMDIST.
## Historical Context
lib/numerics/weibulldist/Weibulldist.md
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# WEIBULLDIST: Weibull Distribution CDF
| Property | Value |
| ---------------- | -------------------------------- |
| **Category** | Numeric |
| **Inputs** | Source (close) |
| **Parameters** | `k` (default 1.5), `lambda` (default 1.0), `period` (default 14) |
| **Outputs** | Single series (Weibulldist) |
| **Output range** | Varies (see docs) |
| **Warmup** | `period` bars |
### TL;DR
- The Weibull Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the Weibull distribution, producing...
- Parameterized by `k` (default 1.5), `lambda` (default 1.0), `period` (default 14).
- Output range: Varies (see docs).
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
The Weibull Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the Weibull distribution, producing an output in $[0, 1]$. The Weibull distribution is a flexible two-parameter family that subsumes the exponential distribution ($k = 1$) and approximates the normal distribution ($k \approx 3.6$) as special cases. Its closed-form CDF requires only `pow` and `exp`, making it the computationally cheapest distribution indicator after EXPDIST. The shape parameter $k$ controls the CDF curvature: $k < 1$ produces a concave curve (rapid initial rise), $k = 1$ gives the exponential, $k = 2$ produces the Rayleigh distribution, and $k > 3$ creates an S-shaped curve approaching Gaussian behavior.
## Historical Context