spectroscopy4 params
Gaussian Peak Profile
The fundamental symmetric bell-shaped peak model describing thermal Doppler broadening and instrumental slit resolution.
$y = y_0 + A \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$
Scientific Model Knowledge Base
Curve Fitting Models Library
Comprehensive mathematical formulations, initial guess heuristic algorithms, parameter bounds, and SciPy-verified parity benchmarks for 101 scientific regression models.
Scientific Disciplines
Spectroscopy & Optics5 models
Spectral Line Profiles, Peak Deconvolution & Broadening Models
Biochemistry & Pharmacology4 models
Enzyme Kinetics, Binding Affinity & Dose-Response Models
Chemical Kinetics & Thermodynamics2 models
Reaction Rates, Thermal Activation & Multi-Phase Decay Models
Materials Science & Solid State1 models
Crystallography, Stress-Strain & Failure Distribution Models
Engineering & Signal Analysis1 models
Harmonics, Damped Oscillations, Rational Models & Polynomial Regression
Model Directory
Showing 13 of 13 modelsspectroscopy4 params
Lorentzian (Cauchy-Lorentz) Profile
The characteristic heavy-tailed peak profile describing natural radiative lifetime broadening and collisional damping.
$y = y_0 + \frac{2A}{\pi} \frac{\gamma}{4(x - \mu)^2 + \gamma^2}$
spectroscopy5 params
Pseudo-Voigt Profile
A linear combination of Gaussian and Lorentzian functions providing high-speed approximation of Voigt profile line shapes.
$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$
spectroscopy5 params
True Voigt Profile (Faddeeva Convolution)
The exact physical convolution of Gaussian Doppler broadening and Lorentzian lifetime broadening.
$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx' = \frac{\text{Re}[w(z)]}{\sigma \sqrt{2\pi}}$
spectroscopy5 params
Pearson VII Profile
A versatile continuous peak model bridging pure Gaussian and Lorentzian shapes via a variable exponential exponent (m).
$y = y_0 + A \left[ 1 + 4\left( 2^{1/m} - 1 \right) \left( \frac{x - \mu}{\gamma} \right)^2 \right]^{-m}$
biochemistry4 params
Hill Equation (Dose-Response)
The definitive non-linear model for sigmoidal cooperative binding and pharmacological dose-response relationships.
$y = y_{\min} + \frac{y_{\max} - y_{\min}}{1 + \left(\frac{K}{x}\right)^n} = y_{\min} + (y_{\max} - y_{\min})\frac{x^n}{K^n + x^n}$
biochemistry5 params
5-Parameter Logistic (5PL)
Asymmetric sigmoidal bioassay model incorporating an asymmetry factor (S) for non-symmetric dose-response curves.
$y = D + \frac{A - D}{\left(1 + \left(\frac{x}{C}\right)^B\right)^S}$
biochemistry2 params
Michaelis-Menten Kinetics
The cornerstone hyperbolic kinetic model describing single-substrate enzyme reaction rates.
$v = \frac{V_{\max} [S]}{K_m + [S]}$
kinetics2 params
Arrhenius Equation
The fundamental exponential model relating chemical reaction rate constants and diffusion rates to absolute temperature.
$k(T) = A \exp\left( -\frac{E_a}{R T} \right)$
materials-physics2 params
Weibull Distribution (Cumulative & Density)
The standard continuous probability distribution for brittle fracture strength and component failure lifetime analysis.
$P_f(\sigma) = 1 - \exp\left( -\left(\frac{\sigma}{\sigma_0}\right)^m \right)$
kinetics5 params
Bi-Exponential Decay
Two-phase exponential decay model resolving fast and slow relaxation lifetimes.
$y(t) = y_0 + A_1 \exp\left(-\frac{t}{\tau_1}\right) + A_2 \exp\left(-\frac{t}{\tau_2}\right)$
biochemistry4 params
4-Parameter Logistic (4PL)
The gold-standard symmetric sigmoidal bioassay model for quantitative ELISA and ligand binding calibration.
$y = D + \frac{A - D}{1 + \left(\frac{x}{C}\right)^B}$
engineering-math5 params
Damped Sine Wave (Harmonic Oscillator)
Exponentially decaying sinusoidal oscillation model for transient mechanical vibration and RLC circuit responses.
$y(t) = y_0 + A \exp(-\gamma t) \sin(\omega t + \phi)$
Complete A-Z Model Index
| Model Name | Category | Mathematical Formulation | Parameters | Validation |
|---|---|---|---|---|
| Gaussian Peak Profile | spectroscopy | y = y0 + A * exp(-((x - mu)^2) / (2 * sigma^2)) | 4 (y_0, A, \mu, \sigma) | verified |
| Lorentzian (Cauchy-Lorentz) Profile | spectroscopy | y = y0 + (2 * A / pi) * (gamma / (4 * (x - mu)^2 + gamma^2)) | 4 (y_0, A, \mu, \gamma) | verified |
| Pseudo-Voigt Profile | spectroscopy | y = y0 + A * [eta * L(x, mu, gamma) + (1 - eta) * G(x, mu, gamma)] | 5 (y_0, A, \mu, \gamma, \eta) | verified |
| True Voigt Profile (Faddeeva Convolution) | spectroscopy | V(x) = Re[wofz(z)] / (sigma * sqrt(2*pi)) where z = (x - mu + i*gamma) / (sigma * sqrt(2)) | 5 (y_0, A, \mu, \sigma, \gamma) | verified |
| Pearson VII Profile | spectroscopy | y = y0 + A * [1 + 4 * (2^(1/m) - 1) * ((x - mu) / gamma)^2]^(-m) | 5 (y_0, A, \mu, \gamma, m) | verified |
| Hill Equation (Dose-Response) | biochemistry | y = ymin + (ymax - ymin) * (x^n) / (K^n + x^n) | 4 (y_{\min}, y_{\max}, K, n) | verified |
| 5-Parameter Logistic (5PL) | biochemistry | y = D + (A - D) / ((1 + (x / C)^B)^S) | 5 (A, B, C, D, S) | verified |
| Michaelis-Menten Kinetics | biochemistry | v = (Vmax * S) / (Km + S) | 2 (V_{\max}, K_m) | verified |
| Arrhenius Equation | kinetics | k = A * exp(-Ea / (R * T)) | 2 (A, E_a) | verified |
| Weibull Distribution (Cumulative & Density) | materials-physics | Pf = 1 - exp(-(sigma / sigma0)^m) | 2 (\sigma_0, m) | verified |
| Bi-Exponential Decay | kinetics | y = y0 + A1 * exp(-t / tau1) + A2 * exp(-t / tau2) | 5 (y_0, A_1, \tau_1, A_2, \tau_2) | verified |
| 4-Parameter Logistic (4PL) | biochemistry | y = D + (A - D) / (1 + (x / C)^B) | 4 (A, B, C, D) | verified |
| Damped Sine Wave (Harmonic Oscillator) | engineering-math | y = y0 + A * exp(-gamma * t) * sin(omega * t + phi) | 5 (y_0, A, \gamma, \omega, \phi) | verified |