CurveFit Pro

Curve fitting online with CurveFit Pro. The world's most advanced curve fitting application right in your browser.

Fit your data in seconds: choose a model, input data, and analyze. Updated: 6/10/2026

1. Select Model

Model function:

Sample data:

2. Enter Data

Paste or input (X, Y) data:

Min points: (parameters + 1). Paste direct from Excel, e.g.: 1.2, 3.4

3. Fit Options

Loss function:

(“c”):

Max fit iterations:
(per-fit refinement steps)

Max diag. iterations:
(bootstrap / cross-validation)

Multistart (recommended) (only runs when needed)

Use built-in initial-guess

Max starts: Sensitivity:

Show 95% CI

Show residuals

Auto-grow bounds

AutoFit tests all 166 models and ranks them by goodness-of-fit.

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Frequently Asked Questions

What is CurveFit Pro?

CurveFit Pro is an advanced, 100% free online curve fitting tool that allows you to fit data to over 100 linear and nonlinear models directly in your browser.

Is my data private?

Yes. All calculations are performed locally in your browser. Your data never leaves your device, ensuring complete privacy.

How CurveFit Pro Works

We use nonlinear least-squares regression to find the parameters that best align your model to your data. Our engine follows a robust three-step approach:

Direct Initial Fit: We derive a sensible starting guess from your data and run a high-speed Levenberg–Marquardt optimization.
Intelligent Multistart: If the initial fit doesn't meet the target noise floor, the solver automatically restarts from various parameter sets to ensure we don't get trapped in a local minimum.
Model Comparison: Click AutoFit to automatically compare all 100+ models, ranking them by goodness-of-fit error.

Model Reference — 100+ Scientific Equations

Our library contains 100+ professional-grade models, each optimized with sensible default bounds, closed-form initial guesses, and hand-derived Jacobians for maximum solver stability.

How to choose: Scan by curve family (e.g., Growth, Spectroscopy, Reliability), then review the model’s physical context to ensure it aligns with your underlying system.

Pro Tip: Use AutoFit to rank all 100+ models against your data. Our smart ranking system doesn't just look at error—it intelligently blends AICc, parameter count, and model complexity to prioritize the most physically meaningful fits, ensuring you get the highest accuracy with the simplest model possible.

1. Linear and polynomial baselines

Linear (`y = a·x + b`)

Straight-line baseline — sanity check before reaching for anything more complex. Use for calibration curves, Hooke-region stress–strain, log-log power-law verification (after taking logs).

Quadratic (`y = a·x^2 + b·x + c`)

Single-bend parabola with one extremum — projectile trajectories, drag vs. speed, dose-response with an internal optimum. For data with more than one turning point, jump to cubic or higher.

Cubic (`y = a·x^3 + b·x^2 + c·x + d`)

Two bends and an inflection — gentle S-shapes without a plateau, monotone-with-a-twist trends, short interpolation patches. Not a growth model: cubics fly off to ±∞, so do not extrapolate past your data range.

Polynomial degree 4 (`y = a·x^4 + b·x^3 + c·x^2 + d·x + e`)

Up to three bends — peak-then-valley or W-shape that cubic cannot capture. Useful for empirical interpolation across dense data; for a true peak shape use Gaussian or Lorentzian instead.

2. Rational functions

Reciprocal (`y = a/(x + b) + c`)

Simplest 1/x decay with shift and offset — dilution curves, attenuation with distance, rough Coulomb-style 1/r intensity. For a saturating rise to a horizontal plateau, use Hyperbolic Decay or Michaelis–Menten.

Rational (1/1) (`y = (a·x + b) / (c·x + d)`)

Linear-over-linear Padé approximant — sigmoidal-ish transition with one vertical and one horizontal asymptote. Often captures saturation with an offset baseline more naturally than Michaelis–Menten; watch for poles if c·x + d crosses zero in your x range.

3. Exponential growth and decay

Exponential (`y = a·exp(b·x) + c`)

Textbook continuous-growth / radioactive-decay / Newton-cooling fit with a baseline offset. Sign of b sets growth vs. decay; c sets the asymptote. For two timescales, use one of the sum-of-exponentials variants below.

Exponential Decay + Linear (`y = a·exp(b·x) + c·x + d`)

Fast exponential transient riding on a linear baseline — fluorescence decay against photobleaching drift, sensor settling under thermal drift, fast cooldown then slow equilibration.

Exponential Rise (to baseline) (`y = A·(1 − exp(−k·(x−x0))) + C`)

Saturating exponential approach to a plateau — RC charging, first-order approach to thermal equilibrium, drug uptake to steady state. Concave-down everywhere; for a delayed-start S-shape use Logistic Growth or Baranyi.

Leaky Integrator (`y = A·(1 − exp(−(x−x0)/τ)) + B`)

Single-pole step response with explicit time constant τ and step onset x0 — neuron membrane charging, capacitor charging through a series resistor, control-system step response.

Sum of Two Exponentials (`y = a1·exp(b1·x) + a2·exp(b2·x) + c`)

Two superimposed timescales — pharmacokinetics (distribution + elimination), reaction kinetics with parallel pathways, dielectric relaxation. Identifiability is poor when the two rates differ by less than ~5×; for a continuum of rates use Stretched Exponential.

Sum of Two Exponentials (Decay) (`y = a1·exp(−k1·x) + a2·exp(−k2·x) + c`)

Two-component decay with explicit positive rate constants k1, k2 — fluorescence lifetimes, double-compartment drug elimination, phosphorescence afterglow. Same form as Sum of Two Exponentials but with sign-locked decay parameters for cleaner reporting.

Sum of Two Exponentials (Rise) (`y = a1·(1 − exp(−k1·x)) + a2·(1 − exp(−k2·x)) + c`)

Two saturating timescales rising to a combined asymptote — two-compartment drug uptake, sequential thermal equilibration, two-RC charging. Keep the two rate constants well separated in the initial guess.

Sum of Three Exponentials (`y = a1·exp(b1·x) + a2·exp(b2·x) + a3·exp(b3·x) + c`)

Three timescales — multi-compartment pharmacology, photophysics with intersystem crossing, multi-decay nuclear spectra. Justify only with a clear visual sign of three regimes; multistart is essential because the SSR surface has many shallow local minima.

Polynomial × Exponential (`y = (a·x² + b·x + c)·exp(d·x) + e`)

Quadratic prefactor times an exponential — chemical reaction profiles with induction period, gamma-function-like distributions, diffusion-relaxation hybrids. For peaks with a long exponential tail, EMG is usually a better match.

Stretched Exponential (`y = a·exp(−|(x−x0)/b|ᶜ) + d`)

Kohlrausch–Williams–Watts relaxation — generalizes Gaussian (c=2), Laplace (c=1), and box-like (c→∞) decay in one knob. Standard for glassy / disordered-material relaxation, polymer creep, anomalous diffusion; pinning c needs decades of dynamic range.

References
[1] Wikipedia, “Stretched exponential function” (Kohlrausch–Williams–Watts). en.wikipedia.org/wiki/Stretched_exponential_function
[2] G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80–85 (1970). doi:10.1039/tf9706600080

Double Exponential Difference (`y = a·exp(−k1·x) − b·exp(−k2·x) + c`)

Subtraction produces a rise-then-decay impulse — absorption-then-elimination drug profiles, two-compartment impulse response, RC-coupled inrush current. Non-identifiable when the two rates are equal; seed them clearly apart.

4. Power-law and allometric

Power Law

The power law (y = a·x^b + c) is a scale-free mathematical relationship where multiplying one quantity (x) by a factor rescales the other (y) by a fixed power. Because of this scale invariance, the same relationship holds across many decades of data.

The Variables Explained

b (Exponent): Dictates the primary behavior—positive for growth, negative for decay.
a (Amplitude): Fixes the overall normalization or scale.
c (Offset): Tolerates a nonzero baseline.

Common Applications

Biology: Allometric scaling (e.g., metabolic rate vs. body mass).
Physics: Energy spectra of turbulent cascades and fluid drag laws.
Social Systems: Heavy-tailed frequency distributions.

Diagnostic Caution

The signature diagnostic of a power law is that plotting log y against log x yields roughly a straight line with a slope of b. However, limited dynamic ranges or competing distributions can easily masquerade as a straight line on log–log axes. Always treat a fitted exponent as suggestive rather than definitive without a wide range of data.

References:
[1] Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf’s law. Open preprint. arXiv:cond-mat/0412004
[2] Newman, M. E. J. (2005). Contemp. Phys. 46, 323–351. doi:10.1080/00107510500052444

Shifted Power Law (`y = a·(x − x0)ᵇ + c`)

Power law with a threshold onset — yield stress in materials, percolation above critical density, post-pulse afterglow with quiescent floor. Returns c for x < x0.

Inverse Quadratic (`y = a/(x − x0)² + c`)

Inverse-square law with a shift — gravitational potential, dipole field magnitude, point-source intensity vs. distance. Has a vertical asymptote at x = x0; keep your data on one side of it.

Hyperbolic Decay (`y = a/(b + x) + c`)

Finite-amplitude 1/x form — drops fast near x = −b and asymptotes to c. Standard for titration curves and saturating dose-response with a non-zero floor.

Double Power + Constant (`y = a0·xᵉ⁰ + a1·xᵉ¹ + c`)

Two power-law regimes crossing over — near-field plus far-field decay, short-range vs. long-range scaling. Identifiability is poor when the two exponents are close; seed them well apart.

Sum of Three Powers (`y = a0·xᵉ⁰ + a1·xᵉ¹ + a2·xᵉ² + c`)

Three power-law regimes — justify only when the log-log plot shows three distinct slopes. Heavy on parameters; bounds and multistart are essential.

5. Saturation, binding, and approach-to-limit

Logarithmic (`y = a·ln(b·x) + c`)

Slow-growing, never-saturating curve — Weber–Fechner perception scaling, Shannon entropy vs. count, sound-pressure level (dB) vs. intensity. Cannot represent a plateau; for saturating log-like behavior reach for Hill or Logistic. Undefined at x ≤ 0.

Saturating Hyperbola (`y = a·x / (b + x) + c`)

Michaelis–Menten / Langmuir form with an offset — dose-response, sensor saturation, enzyme rate vs. substrate, gas-surface adsorption. For a sharper rise than 1/(1+x), use Hill.

Michaelis-Menten / Langmuir Model

This model (y = a·b·x / (1 + b·x) + c) describes a rectangular-hyperbola saturation curve. At small x values, the response climbs almost linearly, then gradually bends over toward a maximum plateau.

The Variables Explained

a (Plateau / V_max): The maximum limit the curve approaches.
b (Affinity / Binding Constant): The reciprocal of the half-saturation point (like the Michaelis constant, K_m). A larger b means tighter binding and a faster approach to the plateau.
c (Offset): Tolerates a nonzero baseline.

Two Contexts, Identical Algebra

Enzyme Kinetics (Michaelis-Menten): Describes molecules reacting at an enzyme's active site.
Surface Chemistry (Langmuir Isotherm): Describes molecules sticking to a surface, where b relates surface coverage to concentration.

When to use this model

Pick this specific parameterization when the binding or affinity constant itself is the exact quantity you want to read directly from the fitted parameters. If you only care about the plateau and half-saturation point, the simpler standard saturation form is usually sufficient.

References:
[1] Chemistry LibreTexts, “Michaelis–Menten Kinetics” — derivation of K_m and V_max. chem.libretexts.org
[2] I. Langmuir, “The adsorption of gases on plane surfaces of glass, mica and platinum,” J. Am. Chem. Soc. 40, 1361–1403 (1918). zenodo.org/records/1429050

Michaelis-Menten with Inhibition (`y = V_max·x / (K_m + x + x²/K_i)`)

Saturating rise that bends back down at high substrate concentration — substrate inhibition in enzymology, dose-response curves that overshoot and fall. For pure saturation without rollover use Michaelis-Menten / Langmuir.

References
[1] MIT OpenCourseWare 10.492, “Enzyme Inhibition and Toxicity” — derives the substrate-inhibition (Haldane) rate law. ocw.mit.edu/…/lecture4.pdf
[2] UC Davis PLB 105, “Enzyme inhibition kinetics” lecture notes. labs.plb.ucdavis.edu/…/EnzKinetics2.pdf

Exponential-Sigmoid Saturation (`y = L·(1−exp(−k·(x−x0))) / (1 + exp(−b(x−w))) + C`)

Composite of an exponential rise and a logistic gate — biological systems with a refractory period, gated photoresponse, two-stage instrument warm-up. Six tightly-coupled parameters; supply bounds.

Double Michaelis–Menten (`y = V1·x/(K1 + x) + V2·x/(K2 + x) + C`)

Two saturating sites with different affinities — enzymes with two binding sites, transporters with high- and low-affinity modes, two-state ion-channel kinetics. Identifiable only when K1 and K2 differ by ~5× or more.

Exponential-Linear Crossover (`y = a·(1−exp(−b(x−x0)))` then linear)

Saturating exponential rise that hands off to a linear growth phase at a knee — bacterial-growth lag-phase to balanced linear growth, two-stage instrument settling. More restrictive than the full Baranyi model but fewer parameters.

BET Isotherm (`y = (a·K·x) / [(1 − x)(1 + (K − 1)·x)] + c`)

Brunauer-Emmett-Teller multilayer-adsorption isotherm — gas adsorption on porous solids (N₂ at 77 K), specific-surface-area measurements in catalysis and battery materials, water-vapor sorption in foods. Extends Langmuir past monolayer coverage.

References
[1] Wikipedia, “BET theory” — multilayer adsorption model. en.wikipedia.org/wiki/BET_theory
[2] S. Brunauer, P. H. Emmett, and E. Teller, “Adsorption of Gases in Multimolecular Layers,” J. Am. Chem. Soc. 60, 309–319 (1938). doi:10.1021/ja01269a023
[3] ISO 9277:2022, “Determination of the specific surface area of solids by gas adsorption — BET method.”

Hill Equation (Baseline)

This model (y = y0 + (y_max − y0) / (1 + (c/x)ⁿ)) describes a sigmoidal dose–response curve. It models a response that transitions from a baseline state to a maximum saturated state, tracing its origins back to A. V. Hill's 1910 work on hemoglobin oxygen binding.

The Variables Explained

y0 (Baseline): The starting, minimum response.
y_max (Maximum): The upper plateau or saturated response.
c (Midpoint / EC₅₀): The x value where the response is exactly half-maximal. This is your potency measure.
n (Hill Coefficient): The exponent that defines the steepness of the curve.

Interpreting the Hill Coefficient (`n`)

n ≈ 1: Independent, non-cooperative binding (curve reduces to a simple hyperbola).
n > 1: Positive cooperativity. Binding at one site increases affinity at others, creating a steep, switch-like response.
n < 1: Negative cooperativity. Binding decreases subsequent affinity.

Relationship to 4PL

Mathematically, this is the exact same equation as the standard Four-Parameter Logistic (4PL) curve. However, reparameterizing the half-maximal term as (c/x)ⁿ (instead of x/c) creates a highly convenient framing when your primary goal is to report activator or agonist potency (EC₅₀) that acts at low concentrations.

References:
[1] Wikipedia, “Hill equation (biochemistry)” — cooperativity, EC₅₀ and the Hill coefficient. en.wikipedia.org/wiki/Hill_equation_(biochemistry)
[2] A. V. Hill, “The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves,” J. Physiol. 40, iv–vii (1910). doi:10.1113/jphysiol.1910.sp001386
[3] D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications, Wiley (1988).

Hill Inhibition (`y = y0 + (y_max − y0) / (1 + (x/c)^n)`)

Hill curve oriented downward — y starts at y_max and drops as x increases, asymptoting at y0. The standard fit for IC50 / inhibition assays.

References
[1] Wikipedia, “Hill equation (biochemistry)” — cooperativity, IC50 and the Hill coefficient. en.wikipedia.org/wiki/Hill_equation_(biochemistry)
[2] A. V. Hill, “The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves,” J. Physiol. 40, iv–vii (1910). doi:10.1113/jphysiol.1910.sp001386
[3] D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications, Wiley (1988).

6. Symmetric sigmoid / S-curve

Logistic (3PL) Model

This model (y = a / (1 + exp(−k·(x−x0)))) produces the textbook symmetric S-curve. Introduced by Pierre-François Verhulst in the 1830s, it remains the canonical description for phenomena like self-limiting population growth, learning curves, and market adoption.

The Variables Explained

a (Upper Asymptote): The carrying capacity or ultimate ceiling of the curve.
x0 (Midpoint): The exact center of the curve where the response passes through a/2.
k (Rate Constant): Dictates the steepness of the climb.

Key Assumptions

Zero Baseline: The curve implicitly rises from a lower asymptote of exactly 0.
Perfect Symmetry: The inflection point sits exactly at the halfway mark (a/2). The approach to the ceiling perfectly mirrors the climb away from the floor.

Which S-Curve should you choose?

Use this three-parameter (3PL) form only when your data naturally starts near zero and exhibits symmetric growth.

Upgrading: If your curve plateaus at a nonzero baseline, switch to the Hill/4PL variant. If the "foot" and "shoulder" curvatures of your data are visibly unequal, you need the asymmetric Richards (5PL) generalization.

References:
[1] Wikipedia, “Logistic function.” en.wikipedia.org/wiki/Logistic_function
[2] P.-F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” Corresp. Math. Phys. 10, 113–121 (1838). ciencias.ulisboa.pt/…/Verhulst.pdf
[3] J. S. Cramer, “The origins of logistic regression,” Tinbergen Institute Discussion Paper (2002). papers.tinbergen.nl/02119.pdf

Smoothed Heaviside Step (k=100) (`y = a/(1+exp(−100·(x−x0))) + b`)

Logistic with a fixed steep slope (k=100) — represents an idealized jump with smooth edges so the solver remains well-behaved. Use for phase transitions, on/off events, detector thresholds; for a softer transition let the standard Logistic fit the slope.

Sigmoid + Linear (`y = L/(1+exp(−k·(x−x0))) + a·x + b`)

Logistic transition on top of a linear background — drift through a threshold, sensor calibration with ramp, dose-response with fluence-dependent baseline. Cleanly separates the step from the drift.

Hyperbolic Tangent (`y = a·tanh(b·(x−x0)) + c`)

Mathematically the same S-shape as a logistic but odd-symmetric, ranging from −a to +a — magnetization curves, neural-network activations, smoothed sign functions. Switch to Logistic if your data is one-sided.

Arctangent Step (`y = a·arctan(b·(x−x0)) + c`)

A heavier-tailed symmetric S-shape — falls to its asymptote more slowly than tanh. Used for psychometric stimulus-response, gradual saturation in the wings of physical systems.

Generalized Lorentzian (Power Bell) (`y = a / (1 + |b·(x − x0)|ⁿ) + c`)

Symmetric bell with adjustable tail sharpness via exponent n — sharp peak with heavy tails for small n, narrower Gaussian-like for large n. A peak shape (max at x0), not an S-curve; reach for it when neither Gaussian nor Lorentzian matches the tail.

Softplus (Smooth Rectifier) (`y = a·ln(1 + exp(b·(x−x0))) + c`)

Smooth one-sided rectifier — flat for small x, linear with slope a·b for large x. Smoother and more solver-friendly than ReLU when the elbow needs curvature; standard activation in ML.

References
[1] C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating Second-Order Functional Knowledge for Better Option Pricing,” Advances in Neural Information Processing Systems 13 (2000) — free PDF. papers.nips.cc/paper/1920

7. Asymmetric and generalized growth (Richards, Gompertz, Hill)

Generalized Logistic (Richards 5PL)

This model (y = A + (K − A) / (1 + Q·exp(−B·(x − v_x)))^(1/ν)) is the most flexible asymmetric S-curve in the family. Introduced by F. J. Richards in 1959 for plant-growth analysis, it adds a critical shape parameter to overcome the rigid symmetry of standard logistic curves.

The 5 Parameters Explained

A & K (Asymptotes): The independent lower (A) and upper (K) bounds.
B (Slope): The primary growth rate.
ν (Nu - Shape/Skew): The crucial 5th parameter governing asymmetry. It dictates near which asymptote the maximum growth rate occurs.
v_x & Q: Modifiers that handle the horizontal placement of the curve.

The Power of Asymmetry

ν = 1: The ordinary symmetric logistic curve is recovered.
ν ≠ 1: The inflection point shifts off-center. The transition from fast to slow growth leans toward either the upper "shoulder" or the lower "foot".
Ideal Use Cases: Biological growth, ecological abundance, and tech-adoption curves where the start and end phases behave differently.

The Cost of Flexibility (Overfitting Risk)

Because this model has five parameters, they are highly correlated and mathematically harder to fit reliably. If your data lacks distinct asymmetry, or if you don't have enough data points in both the "foot" and "shoulder" regions, the simpler 3PL or 4PL models will yield much more robust and reproducible results.

References:
[1] F. J. Richards, “A Flexible Growth Function for Empirical Use” (1959) — full-text PDF (MIT mirror). web.mit.edu/~kardar/.../Richards.pdf
[2] F. J. Richards, J. Exp. Bot. 10, 290–300 (1959). doi:10.1093/jxb/10.2.290

Generalized Logistic (Richards 4PL)

This model (y = K / (1 + Q·exp(−B·(x − x₀)))^(1/ν)) is the zero-baseline variant of the Richards growth function. By pinning the lower asymptote to exactly 0, it reduces the number of free parameters while retaining the signature flexibility of the Richards skew.

The 4 Parameters Explained

K (Asymptote): The final upper capacity (carrying capacity).
B (Growth Rate): Governs the steepness of the transition.
ν (Nu - Shape): The skew parameter that controls the asymmetry.
x₀: The horizontal offset determining the position of the inflection point.

Why use a 4PL Constraint?

Scientific Reality: In many systems, such as population growth from an initial seed or yield accumulation in forestry, the process logically starts at zero.
Computational Stability: By removing the lower asymptote (A) from the fit, the algorithm converges significantly faster and is less likely to produce unphysical "negative floor" fits.

Best Practice: 4PL vs 5PL

Always prefer this 4PL variant if your dataset is known to start at zero. Adding an unneeded 5th parameter (an unknown lower floor) introduces unnecessary correlation between your variables, which can increase the uncertainty (standard error) of your other parameters. Use the 5PL only when the baseline level is truly unknown or contains significant measurement offset.

Physical Context:
This model is a mainstay in fisheries biology and forestry yield-tables, where growth is measured from a clear starting point of zero. Because it forces the baseline to zero, it is inherently more stable and provides cleaner, more interpretable results for yield-projection analysis than the more permissive 5PL variant.

References:
[1] F. J. Richards, “A Flexible Growth Function for Empirical Use” (1959). web.mit.edu/.../Richards.pdf
[2] F. J. Richards, J. Exp. Bot. 10, 290–300 (1959). doi.org/10.1093/jxb/10.2.290

Gompertz Growth Model

This model (y = a·exp(−b·exp(−c·x))) describes a right-skewed S-curve where growth is fastest early on and then makes a long, slow approach to the upper plateau a. Originally proposed by Benjamin Gompertz in 1825 to model human mortality, it is now a standard for growth dynamics.

The Core Variables

a (Asymptote): The final plateau or carrying capacity.
b (Displacement): Sets the horizontal position (timing) of the curve.
c (Growth Rate): Governs the steepness of the initial climb.

Fixed Asymmetry

The Inflection Point: Unlike the symmetric logistic curve (50%), the fastest growth for Gompertz always occurs at 1/e ≈ 36.8% of the final asymptote.
Why it matters: This fixed "front-loading" makes it ideal for processes that start aggressively and taper off slowly.

Gompertz vs. Richards (5PL)

The Gompertz model is more stable and easier to fit than the 5PL Richards model because it has one fewer parameter. However, its asymmetry is fixed. Use Gompertz for tumor growth, microbial colonies, or tech adoption—but if your data's "foot" and "shoulder" curvatures don't match this fixed 36.8% inflection, upgrade to the Richards (5PL) form for extra flexibility.

Physical Context:
Benjamin Gompertz originally designed this to show that adult age-specific mortality rises exponentially with age. Modern applications have adapted this to almost any scenario where growth begins rapidly (like an initial cell population explosion) and is constrained by gradual resource exhaustion or market saturation.

References:
[1] J. M. Mahaffy, “Gompertz Model for Tumor Growth,” San Diego State University course notes. jmahaffy.sdsu.edu/.../product.html
[2] A. Tjørve and K. M. C. Tjørve, “The use of Gompertz models in growth analyses,” PLoS ONE 12, e0178691 (2017). doi.org/10.1371/journal.pone.0178691

Logistic-Hill (4PL) (`y = d + (a − d) / (1 + (x/c)ᵇ)`)

Biostatistics standard for dose-response — explicit bottom (d), top (a), EC50 (c), Hill slope (b). Use for any saturable dose-response curve where IC50/EC50 must appear in the report.

References
[1] H. Gottschalk and J. R. Dunn et al., “Nonlinear Calibration Model Choice between the Four- and Five-Parameter Logistic Models,” J. Biopharm. Stat. (2014) — open access. ncbi.nlm.nih.gov/pmc/articles/PMC4263697
[2] P. G. Gottschalk and J. R. Dunn, “The five-parameter logistic…,” Anal. Biochem. 343, 54–65 (2005). doi:10.1016/j.ab.2005.04.035

Logistic (5PL) (`y = d + (a − d) / (1 + (x/c)ᵇ)ˢ`)

Hill 4PL with an extra asymmetry exponent s — required by FDA-validated immunoassay calibration when the curve is provably skewed. Spend the extra parameter only when 4PL leaves systematic residuals on one shoulder.

References
[1] “Nonlinear Calibration Model Choice between the Four- and Five-Parameter Logistic Models,” open-access review (PMC). ncbi.nlm.nih.gov/pmc/articles/PMC4263697
[2] P. G. Gottschalk and J. R. Dunn, “The five-parameter logistic…,” Anal. Biochem. 343, 54–65 (2005). doi:10.1016/j.ab.2005.04.035

Logistic (5PL, skewed) (`y = d + (a − d) / (1 + (x/c)ᵇ)ᵍ`)

Alternate 5PL parameterization with the asymmetry knob named g — pick the entry whose parameter naming matches your assay software.

Baranyi (lag-growth) (`y = y0 + (L − y0) / (1 + exp(−k·(x − D)))`)

Lag-then-exponential growth widely used in predictive food microbiology — flat lag phase before the population starts growing, then S-curve toward L. Use over plain logistic when there is a clear delay before onset.

References
[1] “Predictive Modeling of Microbial Behavior in Food,” Foods 8, 654 (2019) — open-access review covering the Baranyi growth model. pmc.ncbi.nlm.nih.gov/articles/PMC6963536
[2] J. Baranyi and T. A. Roberts, “A dynamic approach to predicting bacterial growth in food,” Int. J. Food Microbiol. 23, 277–294 (1994). doi:10.1016/0168-1605(94)90157-0

Logistic + Exponential Tail (`y = L/(1+exp(−k·(x−x0))) + A·exp(b·x) + C`)

Logistic transition superimposed on a continuing exponential — viral-load curves with prolonged shedding, market adoption with long-tail growth. Six tightly-coupled parameters; supply bounds.

Logistic + Exponential Mixture (`y = L/(1 + exp(−k·(x − x0))) + A·exp(−b·x) + c`)

Sigmoid plus a decaying exponential — drug rebound on top of saturating dose-response, transient instrument settle time during a step calibration.

Generalized Logistic + Exponential Tail (`y = A/(1+exp(−k(x−x0)))ᵛ + B·exp(−λ(x−t0)) + c`)

Asymmetric Richards-style sigmoid with an exponential addition — biological systems with both development and decay phases. Many free parameters; use only when simpler models clearly fail.

Logistic + Linear Drift (`y = a/(1 + exp(−b·(x−x0))) + m·x + c`)

Logistic step bounded between 0 and a on top of a linear baseline — fraction-positive measurements drifting through a threshold, temperature-corrected calibration curves.

Logistic Hazard (`y = 1 − [1/(1 + exp(k·(x − x0)))]ᵐ`)

Logistic-style survival/hazard function with shape exponent m — reliability time-to-failure, epidemiological survival analysis, accelerated-life-test data.

8. Peaks, bumps, and bells

Gaussian (Normal) Distribution

This model (y = a·exp(−(x−μ)²/(2σ²)) + c) describes a perfectly symmetric bell curve. It is the natural choice for instrument response, beam profiles, particle-size histograms, and serves as the generic default for peak-modeling.

The Variables Explained

μ (Mean / Center): The x position where the bell reaches its exact center and peak.
σ (Scale / Broadness): The standard deviation, setting how wide or narrow the bell is.
a (Amplitude): The peak height of the bell above the baseline.
c (Baseline): Tolerates a nonzero floor or background level.

The FWHM Rule (Estimating `σ`)

The Math: The Full Width at Half Maximum (FWHM) is exactly 2√(2 ln 2)·σ ≈ 2.355·σ.
Visual Estimation: You can read σ straight off a plot simply by measuring the total width of the peak at exactly half of its height, independent of the center or overall scale.
Area Rule: The band within that FWHM holds roughly 76% of the total area under the peak.

Beware of Tailing (Asymmetry)

The defining feature of a pure Gaussian is perfect symmetry. If your data exhibits a "tail" (where one side trails off slower than the other, common in chromatography or trailing instrument responses), this model will fail to capture the area accurately. For asymmetric peaks, switch to the Skew-Normal or EMG (Exponentially Modified Gaussian) models.

References:
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution. itl.nist.gov/.../eda3661.htm
[2] Wolfram MathWorld, “Gaussian Function.” mathworld.wolfram.com/GaussianFunction.html

Lorentzian (Cauchy)

This model (y = a / (1 + ((x−x0)/γ)²) + c) describes a bell shape that looks superficially Gaussian near its summit, but falls off far more slowly. Its "wings" decay only as 1/x². As the Cauchy distribution, it is the textbook example of a curve whose mean and variance are mathematically undefined because the variance diverges.

The Variables Explained

x₀ (Center): The position of the peak.
γ (Gamma / Half-Width): The half-width at half-maximum (HWHM). This makes estimating the full width incredibly simple: FWHM = 2γ.
a (Amplitude): The peak height.
c (Baseline): The nonzero floor.

Physical Signatures

Homogeneous Broadening: The hallmark physical process for this curve.
Common Origins: Finite excited-state lifetimes, pressure or collisional broadening of atomic transitions.
Resonance: Damped-resonance power spectra and spin–spin relaxation in NMR.

When to upgrade to a Voigt Profile

Reach for the pure Lorentzian whenever the underlying physics points to a single relaxation or damping process. However, if your system also experiences inhomogeneous broadening (like Doppler broadening, which produces a Gaussian shape), a pure Lorentzian will fail to fit the data perfectly. In those cases, you must upgrade to the Voigt or Pseudo-Voigt model, which elegantly blends both shapes together.

References:
[1] Wolfram MathWorld, “Lorentzian Function.” mathworld.wolfram.com/LorentzianFunction.html

Pseudo-Voigt

This model (y = a·[η/(1+((x−x0)/γ)²) + (1−η)·exp(−(x−x0)²/(2σ²))] + c) is a practical stand-in for the true Voigt profile. Instead of performing a mathematically expensive convolution, it uses a simple weighted sum of a Lorentzian and a Gaussian curve.

The Variables Explained

x₀ (Center): The position of the peak.
η (Eta / Blend Fraction): The critical weight parameter (between 0 and 1) that tunes the shape.
γ & σ (Widths): The Lorentzian half-width and Gaussian standard deviation, respectively.
a & c: The amplitude and the baseline offset.

The Blending Factor (`η`)

η = 0: Yields a pure Gaussian core (fast decay).
η = 1: Yields a pure Lorentzian (heavier, slower-decaying wings).
Real-World Use: In X-ray diffraction (XRD), Raman, and NMR, peaks are rarely pure. Tuning η between 0 and 1 creates a hybrid that perfectly captures mixed instrumental and physical broadening.

The Computational Shortcut

Evaluating a true Voigt profile requires calculating the Faddeeva function, which can bog down fitting algorithms on large datasets. Because the Pseudo-Voigt replaces that with simple addition, it is incredibly fast to evaluate while reproducing the shape almost perfectly. Choose this model when speed and fitting robustness matter more than physical exactness.

References:
[1] Wikipedia, “Voigt profile” (pseudo-Voigt approximation). en.wikipedia.org/wiki/Voigt_profile
[2] P. Thompson, D. E. Cox, and J. B. Hastings, “Rietveld refinement of Debye–Scherrer synchrotron X-ray data from Al₂O₃,” J. Appl. Crystallogr. 20, 79–83 (1987). doi:10.1107/S0021889887087090

Voigt

This model (y = a·Re[w(z)]/√π + c) is the physically rigorous line shape obtained by convolving a Gaussian with a Lorentzian. It captures the reality of Woldemar Voigt's 1912 work: two different physical broadening mechanisms acting at the exact same time.

The Dual Broadening Parameters

σ (Gaussian Component): Represents classical Doppler broadening.
γ (Lorentzian Component): Represents lifetime, pressure, or collisional broadening.
w(z) (Faddeeva Function): Because the convolution has no elementary closed form, it is evaluated through the real part of this complex function, where z = ((x−x0) + i·γ)/(σ√2).

Estimating the Width (FWHM)

Because the shape is a convolution, the total width is not a simple addition of the two parts.
The Olivero–Longbothum Approximation:
f_V ≈ 0.5346·f_L + √(0.2166·f_L² + f_G²)
This formula combines the Lorentzian (f_L) and Gaussian (f_G) full widths to give you the total Voigt width with an accuracy of better than 0.1%.

The Cost of Physical Exactness

Evaluating the Faddeeva function is computationally expensive. You should only use the true Voigt profile for astronomical absorption lines and highly precise atomic spectroscopy where Doppler and pressure broadening both genuinely matter and physical exactness is mandatory. For general peak fitting, the Pseudo-Voigt is much faster and usually sufficient.

References:
[1] Wikipedia, “Voigt profile.” en.wikipedia.org/wiki/Voigt_profile
[2] W. Voigt, “Das Gesetz der Intensitätsverteilung innerhalb der Linien eines Gasspektrums,” Sitzungsber. Bayer. Akad. Wiss. 25, 603 (1912).

Asymmetric Lorentzian (`y = a / (1 + ((x−x0)/γ(x))²) + c`, γ side-dependent)

Lorentzian with different left and right widths — XRD line shapes with axial divergence, mass-spectrometer peaks with one-sided ion drift, ATR-FTIR absorbance with non-symmetric environment.

Sum of Two Gaussians (`y = a1·exp(−(x−μ1)²/(2σ1²)) + a2·exp(−(x−μ2)²/(2σ2²)) + c`)

Two-peak deconvolution — overlapping spectral lines, bimodal size histograms, mixture populations. AutoFit's across-seed tie-break resolves the two-peak labeling degeneracy automatically.

Sum of Two Lorentzians (`y = a1/(1+((x−x1)/γ1)²) + a2/(1+((x−x2)/γ2)²) + c`)

Two-peak deconvolution with Lorentzian line-shape — overlapping natural-linewidth spectral lines, RC-coupled resonance pairs in RF circuits.

Double Gaussian (Composite) (`y = a1·exp(−(x−μ1)²/(2σ1²)) + a2·exp(−(x−μ2)²/(2σ2²)) + c`)

Same form as Sum of Two Gaussians but with default bounds tuned for narrowly-spaced peaks — try this first when you know the two peaks are close together (within ~2σ).

Bi-Gaussian (Split Width) (`y = a·exp(−(x−μ)²/(2σ_L²))` for x<μ, `σ_R` otherwise)

Gaussian with different left and right widths — chromatogram peaks with column overload tailing, asymmetric instrument-response functions, sediment-grain-size histograms.

Log-Normal Peak (`y = a·exp(−[ln(x/μ)]²/[2σ²]) + c`)

Right-skewed bell in linear x (symmetric in ln x) — aerosol particle-size distributions, drug-residence-time, web-latency histograms. Defined only for x > 0.

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.9 Lognormal Distribution. itl.nist.gov/…/eda3669.htm
[2] Wikipedia, “Log-normal distribution.” en.wikipedia.org/wiki/Log-normal_distribution

Skew-Normal Distribution

This model (y = a·exp(−(x−μ)²/(2σ²))·[1 + erf(s·(x−μ)/(σ√2))] + c) takes an ordinary Gaussian and multiplies it by a normal cumulative term. Introduced by A. Azzalini in 1985, this elegant modification tilts the bell to one side while preserving its overall Gaussian character.

The Core Variables

μ & σ: The mean and standard deviation of the underlying Gaussian "core".
s (Shape / Skew): The critical multiplier inside the error function (erf) that dictates the tilt.
a & c: The amplitude and baseline offset.

Controlling the Tilt (`s`)

s = 0: The erf factor becomes constant, and the curve reduces exactly to a perfectly symmetric Gaussian.
s > 0: Creates positive skew, stretching the curve into a longer right tail.
s < 0: Creates negative skew, stretching the curve into a longer left tail.

Skew-Normal vs. EMG

Both models handle asymmetric peaks, but they represent different physical realities. Use the Skew-Normal when you need a clean, general-purpose handle on asymmetry. However, if the tail is specifically caused by a distinct physical process (like column retention in chromatography causing an exponential tailing effect), the Exponentially Modified Gaussian (EMG) is the more physically rigorous choice.

References:
[1] A. Azzalini, “A very brief introduction to the skew-normal distribution” — the originator’s own primer. azzalini.stat.unipd.it/SN/Intro/intro.html
[2] A. Azzalini, “A class of distributions which includes the normal ones,” Scand. J. Stat. 12, 171–178 (1985). jstor.org/stable/4615982

Exponentially Modified Gaussian (EMG)

This model (y = h·exp(−λ(x−μ) + σ²λ²/2)·Φ([x−μ]/σ − σλ) + c) is the result of convolving a Gaussian with a one-sided exponential. It preserves the symmetric bell of the normal curve but smears it out into a positively skewed shape with a heavy trailing tail.

The Core Variables

μ & σ: The center and width of the underlying Gaussian bell.
λ (Lambda / Decay Rate): Controls the tail. A small λ creates a long, slow-decaying tail. A large λ pulls the tail in, returning the curve toward a clean Gaussian.
h & c: The amplitude factor and baseline offset.

Physical Origins (Why use it?)

Chromatography: This is the standard chromatography peak shape. A Gaussian band-broadening process inside the column is "dragged out" by an exponential retention or kinetic delay.
Psychophysiology: Known in this field as the “ex-Gaussian,” it is the standard model for measuring stimulus-to-response reaction times.

Dealing with a Left-Sided Tail

The standard EMG equation assumes a peak with a Gaussian leading edge and an exponential trailing (right) tail. If your data exhibits negative skew (a heavy tail on the left, or leading edge), simply flip the sign of your x-axis data before fitting, and flip the fitted μ back afterward.

References:
[1] Wikipedia, “Exponentially modified Gaussian distribution.” en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
[2] E. Grushka, “Characterization of exponentially modified Gaussian peaks in chromatography,” Anal. Chem. 44, 1733–1738 (1972). doi:10.1021/ac60319a011

Exponential Gaussian Hybrid (EMG, alt form) (`y = a·exp(σ²λ²/2 − λ(x−μ))·Φ([x−μ]/σ − σλ) + c`)

Same as EMG with a different amplitude convention — pick whichever matches your chromatography software's parameter table.

Crystal Ball Function

This composite model (y = a·N(x; μ, σ) Gaussian core + power-law tail) takes its name from the Crystal Ball collaboration detector. It is the absolute standard signal-peak model in high-energy physics for capturing lossy processes that distort symmetrical peaks.

The Core Variables

μ & σ (Gaussian Core): The center and width of the main peak.
α (Alpha / Boundary): The critical transition point (measured in standard deviations from the center) where the Gaussian ends and the power-law begins.
n (Exponent): The exponent that dictates the decay of the power-law tail.

High-Energy Physics Origins

Bremsstrahlung: As electrons radiate photons that escape the calorimeter, energy is lost. This pulls events into a heavy, low-side tail that a plain Gaussian badly underestimates.
Shower Containment: Partial energy capture in detectors also contributes to this specific distortion.
Modern Usage: Used throughout invariant-mass spectra, including CMS and ATLAS Higgs and Z-boson analyses.

The Perfect Splice (Low-Side Tail)

Unlike the EMG or Skew-Normal models which smear the entire curve, the Crystal Ball function leaves the right side of the Gaussian completely intact to model a very specific low-side (left) tail. At the boundary α, the math explicitly matches both the value and the slope of the two pieces, ensuring a perfectly smooth splice between the Gaussian core and the power-law tail.

References:
[1] J. E. Gaiser, “Charmonium Spectroscopy from Radiative Decays of the J/ψ and ψ′,” Ph.D. thesis, SLAC-R-255 (1982) — full text. osti.gov/biblio/6237411
[2] Wikipedia, “Crystal Ball function.” en.wikipedia.org/wiki/Crystal_Ball_function

Pearson VII (`y = a·[1 + ((x−x0)/γ)²·(2^(1/m)−1)]^(−m) + c`)

Continuously tunable peak from Lorentzian (m=1) toward Gaussian (m→∞) — XRD line-shape fitting, NMR spectra, generalized resonance profiles. One extra parameter compared to Pseudo-Voigt with a different physical interpretation.

References
[1] Wikipedia, “Pearson distribution” (type VII). en.wikipedia.org/wiki/Pearson_distribution
[2] C. C. Craig, “A New Exposition and Chart for the Pearson System of Frequency Curves,” Ann. Math. Stat. 7, 16–28 (1936) — free (Project Euclid). doi:10.1214/aoms/1177732542
[3] M. M. Hall Jr. et al., “The approximation of symmetric X-ray peaks by Pearson type VII distributions,” J. Appl. Crystallogr. 10, 66–68 (1977). doi:10.1107/S0021889877012849

Sech-n Peak (`y = a / cosh(b·(x−x0))ⁿ + c`)

Soliton-style symmetric pulse with adjustable tail sharpness via exponent n — nonlinear-optics pulse profiles, smoothed kink solutions in field theory. Sharper drop than Lorentzian at large n.

References
[1] R. Paschotta, “sech²-shaped pulses,” RP Photonics Encyclopedia. rp-photonics.com/sech2_shaped_pulses.html
[2] Wikipedia, “Hyperbolic secant distribution.” en.wikipedia.org/wiki/Hyperbolic_secant_distribution

Derivative of Logistic (Bell) (`y = A·exp(−k(x−x0))/(1+exp(−k(x−x0)))² + C`)

Derivative of a sigmoid — symmetric bell lighter-tailed than Lorentzian and heavier than Gaussian. Useful when modeling the rate-of-change of a cumulative process (e.g., density of an underlying CDF).

Triangular Pulse (`y = a·max(0, 1 − |(x−x0)/w|) + c`)

Piecewise-linear symmetric bump with finite support — windowing functions in signal processing, idealized FIR response shapes, simple convolution kernels. Sharp corners make residuals non-smooth; use Gaussian for smoother bells.

Parabolic Hump (`y = a − b·(x − x0)²`)

Inverted-parabola peak — coarse summary of a smooth optimum near the apex. Three parameters, fast to fit; tails diverge from any real bell so use only over a narrow window around the peak.

Product of Gaussians (Multiplicative Peak) (`y = a·exp(−(x−m1)²/(2·s1²))·exp(−(x−m2)²/(2·s2²)) + c`)

Product of two Gaussians — algebraically a third Gaussian, but the parameters report two contributing sources separately (e.g., instrument response × source profile, spectral × spatial filters).

Fano Resonance

This model (y = a·(q·γ + (x−x0))² / (γ² + (x−x0)²) + c) describes an intrinsically asymmetric resonance characterized by a sharp peak paired with a distinct dip. Worked out by Ugo Fano in 1961, it arises whenever a discrete resonant state interferes with a continuum of background states.

The Core Variables

x₀ & γ: The resonant energy (center) and the spectral line width.
a & c: The amplitude and the baseline offset.
q (Fano Factor): The crucial shape parameter. It represents the ratio of the resonant transition amplitude to the background continuum amplitude.

Controlling the Asymmetry (`q`)

q ≈ 1: The two transition pathways are comparable in strength. This produces the most strongly asymmetric, distinctive Fano line (a massive peak adjacent to a deep dip).
q → ∞: The continuum background contribution vanishes, and the profile collapses back into a perfectly symmetric Lorentzian.

A Signature of Quantum Interference

Do not use this model simply because a peak looks lopsided (use the Skew-Normal or EMG for that). The Fano resonance is a highly specific physical signature. It is the definitive model for autoionizing atomic levels, plasmonic resonances, and quantum-interference line shapes in mesoscopic transport.

References:
[1] Wikipedia, “Fano resonance.” en.wikipedia.org/wiki/Fano_resonance
[2] U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124, 1866–1878 (1961). doi:10.1103/PhysRev.124.1866
[3] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010). doi:10.1103/RevModPhys.82.2257

9. Distribution PDFs and CDFs

Beta CDF (`y = a · I_x(b, c) + d`)

Regularized incomplete-beta function on x ∈ [0, 1] — cumulative Beta distribution for proportions, probabilities, [0,1]-bounded responses. Numerically heavy; uses finite-difference Jacobian.

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.17 Beta Distribution. itl.nist.gov/…/eda366h.htm
[2] Wolfram MathWorld, “Regularized Beta Function.” mathworld.wolfram.com/RegularizedBetaFunction.html
[3] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards (1964) — free full text. archive.org/details/AandS-mono600

Beta PDF (`y = a·x^(b−1)·(1−x)^(c−1) + d`)

Density form of the Beta distribution — versatile bell, U, or L-shape on [0, 1] depending on the two shape parameters. Use for histograms of proportions and Bayesian priors over probabilities.

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.17 Beta Distribution. itl.nist.gov/…/eda366h.htm
[2] Wolfram MathWorld, “Beta Distribution.” mathworld.wolfram.com/BetaDistribution.html

Weibull CDF (Cumulative Failure)

This model (y = a·(1 − exp(−((x − x0)/λ)^k)) + c) represents the cumulative form of the Weibull distribution. It traces the fraction of items that have failed (or events that have occurred) over time, making it the undisputed workhorse of reliability engineering, survival analysis, and wind-speed modeling.

The Core Variables

x₀ (Threshold): The starting time before which no failures occur.
λ (Lambda / Scale): The characteristic life parameter, dictating how stretched out the failure timeline is.
a & c: The curve rises from a baseline of c toward an ultimate ceiling of a + c.

The Shape Parameter (`k`) & The "Bathtub"

k < 1 (Infant Mortality): Decreasing hazard rate. Defective units fail early, and the survivors actually grow more reliable over time.
k = 1 (Random Failures): Constant hazard rate. The curve reduces to a simple exponential distribution where failures happen completely at random.
k > 1 (Wear-out): Increasing hazard rate. The classic aging regime where parts degrade and fail over time.

Diagnosing Failure Mechanisms

Reach for the Weibull CDF whenever you need a flexible time-to-event curve. Its greatest strength is diagnostic: rather than just giving you a "good fit," you can simply look at the resulting k parameter to immediately deduce which physical failure mechanism dominates your system.

References:
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.8 Weibull Distribution. itl.nist.gov/.../eda3668.htm
[2] W. Weibull, “A Statistical Distribution Function of Wide Applicability,” J. Appl. Mech. 18, 293–297 (1951). jhanley.biostat.mcgill.ca/.../Weibull-ASME-Paper-1951.pdf

Weibull PDF (3-parameter) (`y = a·(b/λ)·((x−x0)/λ)^(b−1)·exp(−((x−x0)/λ)^b) + c`)

Density form of the 3-parameter Weibull — failure-time histograms with a threshold x0, positively-supported asymmetric peaks in reliability and meteorology.

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.8 Weibull Distribution. itl.nist.gov/…/eda3668.htm
[2] W. Weibull, “A Statistical Distribution Function of Wide Applicability,” J. Appl. Mech. 18, 293–297 (1951) — the original paper (free PDF). jhanley.biostat.mcgill.ca/…/Weibull-ASME-Paper-1951.pdf

Normal CDF (Error Function)

This model (y = a·0.5·[1 + erf((x − μ)/(σ√2))] + b) represents the cumulative distribution of the normal (Gaussian) law. By using the error function, it forms a smooth, perfectly symmetric step that serves as the canonical model for threshold transitions.

The Core Variables

b (Baseline): The starting floor of the step.
a + b (Ceiling): The maximum plateau the curve reaches.
μ (Midpoint): The exact center of the transition where the step reaches half-height.
σ (Width): Controls the steepness or broadness of the transition.

Common Applications

Hardware & Sensors: Detector trigger curves and calibration sigmoids.
Biology & Medicine: The probit response function used extensively in dose–response modeling.
Psychophysics: Modeling perception thresholds and stimulus detection.

Normal CDF vs. Logistic CDF

Compared to the Logistic CDF, this curve is slightly steeper through the middle and lighter in the tails. Because they look almost identical near the center, it can be impossible to tell them apart by eye. Pro Tip: When in doubt, fit both models side-by-side and let the residuals in the extreme tails decide the winner.

References:
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution. itl.nist.gov/.../eda3661.htm
[2] NIST/SEMATECH e-Handbook, 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal. itl.nist.gov/.../eda3671.htm
[3] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards (1964) — free full text. archive.org/details/AandS-mono600

Log-Normal CDF (`y = a·0.5·[1 + erf((ln(x) − μ)/(σ√2))] + c`)

Cumulative log-normal — particle-size CDF, income distribution, web latency. Defined only for x > 0; reach for this when the underlying variable spans many orders of magnitude.

Gamma PDF

This model (y = a · x^(k−1) · exp(−x/θ) + c for x > 0) describes a right-skewed peak restricted strictly to the positive axis. Built from the product of a power term and an exponential decay, it is the mathematical standard for modeling total waiting times and queueing service durations.

The Core Variables

k (Shape): The critical parameter dictating the silhouette of the curve.
θ (Theta / Scale): Controls the stretch or spread of the distribution along the x-axis.
a & c: The amplitude scale and the baseline offset.

Controlling the Silhouette (`k`)

k = 1: The curve collapses into a simple exponential distribution, monotonically decreasing from a peak right at the origin.
k > 1: The peak (mode) is pushed away from zero, and the hump becomes progressively more symmetric as k increases.
Bayesian Context: Because of this shape flexibility, it is also the standard conjugate prior for rate parameters.

The Strictly Positive Constraint

Because the math relies on x > 0, you should only reach for the Gamma PDF when your data are strictly positive (like elapsed time, physical size, or a series of k independent reliability stages) and exhibit a single skewed peak. Feeding negative x values into this model will result in calculation errors.

References:
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.11 Gamma Distribution. itl.nist.gov/.../eda366b.htm
[2] Wikipedia, “Gamma distribution.” en.wikipedia.org/wiki/Gamma_distribution
[3] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards (1964). archive.org/details/AandS-mono600

Gamma CDF (`y = a · P(k, x/θ) + c`)

Regularized lower incomplete gamma — cumulative gamma distribution. Use for cumulative waiting-time data and as a flexible S-shape with controllable tail behavior.

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.11 Gamma Distribution. itl.nist.gov/…/eda366b.htm
[2] Wikipedia, “Incomplete gamma function.” en.wikipedia.org/wiki/Incomplete_gamma_function

Pareto CDF (`y = a·[1 − (x_min/x)^α] + c` for x ≥ x_min)

Power-law tail CDF — income / wealth concentration (the "80/20" rule), file-size distributions, city-size rank-frequency. Tail exponent α sets how slowly the distribution decays.

References
[1] Wolfram MathWorld, “Pareto Distribution.” mathworld.wolfram.com/ParetoDistribution.html

Inverse Gaussian / Wald PDF (`y = a·√(λ/(2π·x³))·exp(−λ(x−μ)²/(2μ²·x)) + c`)

Right-skewed positive distribution — first-passage times for Brownian motion with drift, neural reaction-time data, financial duration models. Sharper rise than log-normal at small x.

References
[1] M. C. K. Tweedie, “Statistical properties of inverse Gaussian distributions. I,” Ann. Math. Stat. 28, 362–377 (1957) — open access (Project Euclid). doi:10.1214/aoms/1177706964
[2] Wikipedia, “Inverse Gaussian distribution.” en.wikipedia.org/wiki/Inverse_Gaussian_distribution
[3] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., Wiley (1971).

Maxwell-Boltzmann (`y = a·x²·exp(−x²/(2σ²)) + c`)

Molecular-speed distribution from kinetic theory — gas-phase speed histograms, thermal-noise velocity distributions, plasma physics. Width parameter σ scales with √(kT/m).

References
[1] Wikipedia, “Maxwell–Boltzmann distribution.” en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution

GEV (PDF, right-skew) (`y = (a/σ)·(1 + ξ·(x−μ)/σ)^(−1−1/ξ)·exp(−(1+ξ·(x−μ)/σ)^(−1/ξ)) + c`)

Generalized Extreme Value density for block-maxima — annual flood heights, peak wind speeds, financial drawdown extremes. Shape ξ selects Gumbel (0), Fréchet (>0), or reversed-Weibull (<0) family.

References
[1] Wikipedia, “Generalized extreme value distribution.” en.wikipedia.org/wiki/Generalized_extreme_value_distribution
[2] A. F. Jenkinson, “The frequency distribution of the annual maximum (or minimum) values of meteorological elements,” Q. J. R. Meteorol. Soc. 81, 158–171 (1955). doi:10.1002/qj.49708134804

GEV (left-skew) (mirrored GEV form)

GEV PDF with the skew flipped — for minima rather than maxima, or right-truncated extremes (lowest temperatures recorded, minimum yield strength in a batch).

Gumbel PDF (`y = a·exp(−exp(−(x−μ)/β))·exp(−(x−μ)/β) + c`)

Density form of the Gumbel (Type-I extreme value) — the classical large-value asymptotic when the underlying tail is exponential-like (annual rainfall maxima, peak temperature).

References
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.16 Extreme Value Type I Distribution. itl.nist.gov/…/eda366g.htm
[2] Wikipedia, “Gumbel distribution.” en.wikipedia.org/wiki/Gumbel_distribution

Gumbel CDF (`y = a·exp(−exp(−(x−μ)/β)) + c`)

Gumbel cumulative — the S-shape from compounding extreme-value statistics. Use when reporting a probability of exceedance rather than a density.

Inverse Sigmoid (Probit / Ogive) (`y = a·Φ⁻¹((x−μ)/σ) + c`)

Inverse normal CDF — link function for probit regression, psychometric thresholds, item-response theory. Maps a probability onto an effective z-score; defined only for (x−μ)/σ ∈ (0,1).

References
[1] Wikipedia, “Probit.” en.wikipedia.org/wiki/Probit
[2] C. I. Bliss, “The method of probits,” Science 79, 38–39 (1934). doi:10.1126/science.79.2037.38

10. Piecewise, hinge, and ramp

Rectified Linear Unit (ReLU) (`y = a·max(0, x − x0) + b`)

Flat segment that hinges into a linear segment — yield onset, detection threshold, dead-zone followed by linear response. Sharp corner; LM convergence is slightly slower than smooth alternatives.

References
[1] Wikipedia, “Rectifier (neural networks)” — ReLU and variants. en.wikipedia.org/wiki/Rectifier_(neural_networks)

Leaky ReLU (`y = a·max(0, x − x0) + b·min(0, x − x0) + c`)

Two-slope hinge with distinct slopes above and below x0 — asymmetric piecewise-linear data, op-amp transfer function near saturation, soft-clipped audio response.

Piecewise Linear (Hinge) (`y = a·(x − x0) + intercept` for x < x0; `b·(x − x0) + intercept` for x ≥ x0)

Continuous two-segment hinge with both pieces passing through (x0, intercept) — change-point regression, V-shape stress responses, demand curve with kink at a price point. Continuous at the knee (no value jump).

Bilinear (Piecewise Linear, breakpoint) (`y = a1·x + b1` for x < x0; `y = a2·x + b2` for x ≥ x0)

Two independent linear segments meeting at x0 — broken-stick regression with a possible value discontinuity at the knee. Use over Piecewise Linear (Hinge) when the segments aren't constrained to meet.

Piecewise Constant 2-Step (`y = a` if x < x0; `b` if x ≥ x0)

Pure jump — calibration step-change, regime swap, binary trigger threshold. Use only when you really mean "constant on each side"; for a softer transition use Smoothed Heaviside Step.

Rectified Power Law (`y = a·max(0, x − x0)ᵇ + c`)

Power-law turn-on at a threshold — phonon density of states above a gap, photoresponse above a band edge, critical-phenomena order parameter above the transition.

Rectified Polynomial (degree n) (`y = a·max(0, x − x0)ⁿ + b`)

Generalized rectified power with integer or real exponent n — onset of higher-order responses, polynomial yield laws beyond a threshold.

Piecewise Rational (Bend / Hinge) (`y = (a1·x + b1)/(c1·x + d1)` for x < x0; rational right segment)

Two rational segments joined at a breakpoint — regime-switching saturation, control-system response with mode change, hysteresis branches. Heavy on parameters; use only when simpler piecewise fits visibly fail.

11. Periodic and oscillatory

Sinusoid (`y = A·sin(ω·x + φ) + C`)

Single-frequency sine — diurnal cycles, AC waveforms, simple vibrational data. AutoFit pre-seeds ω from a Lomb-Scargle periodogram so it works even on unevenly-sampled data. For two-frequency content use Sum of Sines.

Sum of Sines (2) (`y = A1·sin(ω1·x + φ1) + A2·sin(ω2·x + φ2) + C`)

Two-frequency oscillation — beat patterns, double-tone audio signals, fundamental + harmonic, ENSO-style climate cycles. Identifiability is poor when the two frequencies are within ~10% of each other.

Damped Sinusoid (`y = A·exp(−k·x)·sin(ω·x + φ) + C`)

Underdamped harmonic oscillator — RLC ringing, mechanical free-decay tests, pendulum with friction, ring-down spectroscopy. Reports decay rate k and angular frequency ω directly.

References
[1] MIT OpenCourseWare RES.8-009, “Introduction to Oscillations and Waves,” Lecture 4: Damped Oscillations. ocw.mit.edu/…/mitres_8_009su17_lec4.pdf

Linear Chirp (`y = A·sin(2π·(f0·x + (k/2)·x²) + φ) + C`)

Sinusoid whose instantaneous frequency sweeps linearly with time — radar / sonar pulses, FMCW range-finding, bat echolocation, NMR adiabatic-passage waveforms. Slope k is the chirp rate.

References
[1] Wikipedia, “Chirp.” en.wikipedia.org/wiki/Chirp

12. Dielectric and impedance spectroscopy

Real and imaginary parts of the complex permittivity ε(ω) for the standard relaxation models. Use these on broadband-dielectric- spectroscopy data, impedance Bode plots, or any frequency-domain response with a phase-lag — fit the real and imaginary parts side-by-side and compare the same parameters across both.

Debye Permittivity (Real) (`ε′(ω) = ε∞ + Δε / (1 + (ω·τ)²)`)

Real part of the single-relaxation-time Debye response — polar liquids well above their freezing point, dipolar gases. Reports the relaxation time τ directly; for distributed relaxations use Cole-Cole or Havriliak-Negami.

References
[1] Wikipedia, “Dielectric” (Debye relaxation). en.wikipedia.org/wiki/Dielectric
[2] P. Debye, Nobel Lecture (1936); P. Debye, Polar Molecules, Chemical Catalog Co. (1929). nobelprize.org/…/debye-lecture.pdf

Debye Permittivity (Imag) (`ε″(ω) = Δε · ω·τ / (1 + (ω·τ)²)`)

Imaginary (loss) part of Debye relaxation — single symmetric peak at ω·τ = 1. Use to extract the loss peak position and amplitude in dielectric spectra.

Cole-Cole Permittivity (Real)

This model (ε′(ω) = ε_∞ + Re[Δε / (1 + (iω·τ)^1−α)]) is a phenomenological extension of the Debye relaxation model for the frequency-dependent complex permittivity of a dielectric. Introduced in 1941, it was specifically designed to handle materials whose relaxation is not governed by a single time constant.

The Core Variables

ε_∞: The high-frequency permittivity limit.
Δε: The dielectric increment (the difference between the static and high-frequency permittivity).
τ (Tau): The characteristic relaxation time of the system.

The Broadening Exponent (`α`)

The Concept: This parameter (between 0 and 1) encodes a distribution of relaxation times rather than a single one.
α = 0: The expression collapses entirely to the standard, single-time Debye model.
α > 0: Symmetrically stretches the relaxation over a much wider span of the logarithmic frequency axis.

Modeling Disordered Dynamics

Pure crystals and simple liquids often follow the strict Debye model. You should upgrade to this Cole-Cole equation when dealing with systems that have disordered, dispersed dynamics. It is exceptionally well-suited for fitting impedance and dielectric data from biological tissues, polymers above their glass transition, and complex ionic solutions.

References:
[1] Wikipedia, “Cole–Cole equation.” en.wikipedia.org/wiki/Cole-Cole_equation
[2] S. Holm, “Time domain characterization of the Cole–Cole dielectric model,” J. Electr. Bioimpedance 11, 101–105 (2020) — open access. pmc.ncbi.nlm.nih.gov/articles/PMC7851980
[3] K. S. Cole and R. H. Cole, “Dispersion and Absorption in Dielectrics I,” J. Chem. Phys. 9, 341–351 (1941). doi:10.1063/1.1750906

Cole-Cole Permittivity (Imag) (`ε″(ω) = Im[Δε / (1 + (iω·τ)^(1−α))]`)

Imaginary part of Cole-Cole — symmetric loss peak broader than Debye, characteristic of disordered systems with overlapping relaxation modes.

References
[1] Wikipedia, “Cole–Cole equation.” en.wikipedia.org/wiki/Cole-Cole_equation
[2] S. Holm, “Time domain characterization of the Cole–Cole dielectric model,” J. Electr. Bioimpedance 11, 101–105 (2020) — open access. pmc.ncbi.nlm.nih.gov/articles/PMC7851980
[3] K. S. Cole and R. H. Cole, “Dispersion and Absorption in Dielectrics I,” J. Chem. Phys. 9, 341–351 (1941). doi:10.1063/1.1750906

Cole-Davidson Permittivity (Real) (`ε′(ω) = ε∞ + Re[Δε / (1 + iω·τ)^β]`)

Real part of the asymmetric (right-skewed) relaxation — glycerol and other glass-forming liquids, polymer dynamics where the high-frequency tail is power-law. β = 1 reduces to Debye.

References
[1] Wikipedia, “Havriliak–Negami relaxation” (Cole–Davidson is the α→0 case). en.wikipedia.org/wiki/Havriliak-Negami_relaxation
[2] D. W. Davidson and R. H. Cole, “Dielectric Relaxation in Glycerol, Propylene Glycol, and n-Propanol,” J. Chem. Phys. 19, 1484–1490 (1951). doi:10.1063/1.1748105

Cole-Davidson Permittivity (Imag) (`ε″(ω) = Im[Δε / (1 + iω·τ)^β]`)

Imaginary part of Cole-Davidson — asymmetric loss peak with extended high-frequency tail, diagnostic of glass-formers near T_g.

Havriliak-Negami Permittivity (Real) (`ε′(ω) = ε∞ + Re[Δε / (1 + (iω·τ)^(1−α))^β]`)

Real part of the most general single-relaxation model — combines Cole-Cole symmetric broadening (α) with Cole-Davidson asymmetry (β). Use when neither simpler model captures the shape.

References
[1] Wikipedia, “Havriliak–Negami relaxation.” en.wikipedia.org/wiki/Havriliak-Negami_relaxation
[2] S. Havriliak and S. Negami, “A complex plane representation of dielectric and mechanical relaxation processes in some polymers,” Polymer 8, 161–210 (1967). doi:10.1016/0032-3861(67)90021-3

Havriliak-Negami Permittivity (Imag) (`ε″(ω) = Im[Δε / (1 + (iω·τ)^(1−α))^β]`)

Imaginary part of Havriliak-Negami — the workhorse loss-spectrum fit for amorphous polymers, ionic liquids, biological membranes.

Lorentz Oscillator Permittivity (Real) (`ε′(ω) = ε∞ + Δε·(ω0² − ω²) / [(ω0² − ω²)² + (γω)²]`)

Real part of a damped harmonic-oscillator response — optical phonon resonances in crystals, infrared lattice absorption, classical bound-electron contribution to refractive index.

References
[1] J. Colton, “Lorentz Oscillator Model of the Dielectric Function,” Brigham Young University, Physics 581 notes. physics.byu.edu/…/lorentz-oscillator-model.pdf
[2] Wikipedia, “Lorentz oscillator model.” en.wikipedia.org/wiki/Lorentz_oscillator_model

Lorentz Oscillator Permittivity (Imag) (`ε″(ω) = Δε·γω / [(ω0² − ω²)² + (γω)²]`)

Imaginary part of the Lorentz oscillator — resonant absorption peak at ω = ω0 with width γ. Fit alongside the real part to extract oscillator strength.

Drude Permittivity Model

This model (ε′(ω) = ε_∞ − ω_p² / (ω² + γ²)) treats conduction electrons as a free, non-interacting gas driven by an electromagnetic field. Despite its simplicity, it remains the gold standard for describing the optical response of metals and doped semiconductors.

The Core Variables

ω_p (Plasma Frequency): The natural oscillation frequency of the electron density. It is directly tied to the carrier density.
γ (Damping Rate): The inverse of the mean time between electron collisions (γ = 1/τ).
ε_∞: The high-frequency background constant accounting for bound electrons.

Understanding Metallic Reflection

Sub-Plasma Frequency (ω < ω_p): The real permittivity is negative. This is the physical hallmark of metallic reflection.
Approaching Plasma Frequency: As ω rises toward ω_p, the real permittivity trends toward zero.

Diagnostic Utility

Use this model to fit the free-electron response of your material. A successful fit allows you to extract two fundamental physical constants directly from your data: the plasma frequency (revealing carrier density) and the damping rate (revealing scattering/collision frequency).

References:
[1] Wikipedia, “Drude model.” en.wikipedia.org/wiki/Drude_model
[2] P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306, 566–613 (1900). doi:10.1002/andp.19003060312

Drude Permittivity (Imag) (`ε″(ω) = ωp²·γ / [ω·(ω² + γ²)]`)

Imaginary part of Drude — monotonic loss diverging at low frequency, characteristic of free-carrier absorption in conductors.

References
[1] Wikipedia, “Drude model.” en.wikipedia.org/wiki/Drude_model
[2] P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306, 566–613 (1900). doi:10.1002/andp.19003060312

Modified Debye Loss + DC (`ε″(ω) = σ/(ε0·ω) + Δε·ω·τ / (1 + (ω·τ)²)`)

Debye dielectric loss plus a DC-conductivity contribution diverging at low frequency — electrolyte solutions, hydrated polymers, soil dielectric spectra where ionic conduction dominates the low-frequency tail.

13. Optics, antennas, and aperture diffraction

Sérsic Profile (Galactic Morphology)

This model (y = I₀·exp(−(x/x₀)^1/β) + c) is the standard astronomical tool for describing how a galaxy's surface brightness falls off with radius. Introduced by José Luis Sérsic, it generalized earlier laws to provide a single, flexible framework for everything from compact elliptical bulges to diffuse spiral disks.

The Core Variables

I₀ (Intensity): The central surface brightness.
x₀ (Radius): The characteristic scale length of the galaxy.
β (Sérsic Index n): The shape parameter that dictates how light is concentrated.
c (Background): The residual sky background level.

Decoding the Index (n)

n = 4: The de Vaucouleurs profile—typical for elliptical galaxies and the concentrated bulges of spiral galaxies.
n = 1: The exponential profile—typical for the disks of spiral galaxies.
n = 0.5: The profile reduces to a Gaussian distribution.

Diagnostic Utility for Astronomy

The best-fit Sérsic index strongly correlates with both the size and luminosity of a galaxy. Generally, larger and more massive systems favor higher n values (more centrally concentrated light with extended, diffuse wings). Use this model as your primary tool for modeling the structural components (bulges/disks) of galactic systems.

References:
[1] A. W. Graham and S. P. Driver, “A Concise Reference to (Projected) Sérsic R^(1/n) Quantities” (2005) — open preprint. arxiv.org/abs/astro-ph/0503176
[2] A. W. Graham and S. P. Driver, Publ. Astron. Soc. Aust. 22, 118–127 (2005). doi:10.1071/AS05001

Bessel Circular Aperture (Airy) (`y = a·[2·J1(b·(x−x0)) / (b·(x−x0))]² + c`)

Airy diffraction pattern — far-field intensity from a circular aperture, point-spread function of a diffraction-limited telescope or microscope. Central peak with concentric ring sidelobes; b sets the angular scale.

References
[1] Wikipedia, “Airy disk” (circular-aperture diffraction). en.wikipedia.org/wiki/Airy_disk
[2] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards (1964) — free full text. archive.org/details/AandS-mono600

Sinc Squared (`y = a·sinc²(b·(x − x0)) + c`)

Far-field intensity from a slit aperture or rectangular-pulse Fourier transform — single-slit diffraction, antenna array factor for a uniform line array, FIR filter side-lobe envelope.

References
[1] Wolfram MathWorld, “Sinc Function.” mathworld.wolfram.com/SincFunction.html

Sellmeier (3-term) (`n²(λ) = 1 + Σ Bi·λ² / (λ² − Ci)`, i=1..3)

Dispersion model for transparent optical glass — refractive index vs. wavelength for BK7, fused silica, and most optical-grade materials in their transparency window. The standard for ray-tracing software.

References
[1] R. Paschotta, “Sellmeier Formula,” RP Photonics Encyclopedia. rp-photonics.com/sellmeier_formula.html
[2] Wikipedia, “Sellmeier equation.” en.wikipedia.org/wiki/Sellmeier_equation

Cardioid Polar Pattern (`y = a·(1 + cos(x − x0))/2 + c`)

Directional sensitivity pattern of a first-order pressure-gradient microphone or end-fire dipole antenna — heart-shaped lobe with full sensitivity at x0 and a null on the opposite side.

References
[1] DPA Microphones, “Directional vs. Omnidirectional Microphones,” Mic University. dpamicrophones.com/mic-university/…
[2] Wolfram MathWorld, “Cardioid.” mathworld.wolfram.com/Cardioid.html

Uniform Linear Array Factor |AF|² (`U(θ) = a · [sin(N·kd·cos(θ)/2) / (N·kd·cos(θ)/2)]² + c`)

Normalized radiation intensity of an N-element broadside uniform linear array — main lobe at θ = π/2, sidelobes whose number and width are set by N, and inter-element spacing carried in kd (half-wavelength spacing → kd = π). Use to fit beamwidth, recover N, and read the matched array spacing from a measured pattern.

References
[1] S. V. Hum, “Antenna Arrays II,” University of Toronto ECE422 notes. waves.utoronto.ca/prof/svhum/ece422/notes/15-arrays2.pdf
[2] A. Ma, “Array Factor,” EM lecture notes. angms.science/doc/EM/EM_18_ArrayFactor.pdf
[3] N. K. Nikolova, “Antenna Arrays — Introduction (L13),” McMaster ECE. ece.mcmaster.ca/faculty/nikolova/…/L13_Arrays1.pdf

Planck Black-Body Radiation

This model (y = a · x³ / (exp(b·x) − 1) + c) describes the spectral radiance of a black body in thermal equilibrium. Famously, this result launched quantum theory when Max Planck assumed that energy was exchanged in discrete quanta—the foundation of the entire field.

The Core Variables

b (Temperature Parameter): Directly linked to the temperature (b = h / (k·T)). A larger b indicates a cooler source.
a (Amplitude): Rolls together the physical constants (2h/c²) and your system's specific calibration factor.
x: Represents the frequency.

Regimes of Radiation

Low-Frequency: The curve reduces to the classical Rayleigh–Jeans law.
High-Frequency: The curve transitions into the Wien tail.
Temperature Shift: As the source cools (larger b), the spectral peak shifts toward lower frequencies.

Diagnostic Utility

Use this model to fit thermal continuum spectra—such as stellar output, the cosmic microwave background, or thermal-infrared source calibrations. By extracting the temperature from your fitted parameter b, you can characterize the thermal state of a wide range of physical and astronomical systems.

Historical & Physical Context:
The Planck law resolved the “ultraviolet catastrophe”—the prediction by classical physics that an ideal black body at thermal equilibrium would emit infinite energy at short wavelengths. By introducing the quantization of energy (E = hν), Planck successfully bridged the gap between the Rayleigh–Jeans law at low frequencies and the Wien approximation at high frequencies. Today, this model is essential not just for calibration, but for identifying thermal deviations in experimental data caused by non-ideal black bodies (emissivity < 1).

References:
[1] Wikipedia, “Planck’s law.” en.wikipedia.org/wiki/Planck’s_law
[2] M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Ann. Phys. 309, 553–563 (1901). doi:10.1002/andp.19013090310

14. DSP filter responses and pulse shapes

Butterworth Lowpass |H|² (`|H(ω)|² = a / (1 + (ω/ωc)^(2n)) + c`)

First described by the British engineer Stephen Butterworth in his 1930 paper “On the Theory of Filter Amplifiers,” this filter is designed to be “maximally flat” in the passband — its magnitude-squared response has as many vanishing derivatives at zero frequency as the order allows, so there are no ripples before the roll-off begins. The response is |H(ω)|² = a/(1 + (ω/ω_c)²ⁿ) + c, where ω_c is the cutoff frequency — the half-power point at which the power drops to half its passband value (about −3 dB, a voltage gain near 1/√2 ≈ 0.7071) — and the order n sets the steepness of the transition into the stopband, higher n giving a sharper roll-off. The amplitude a scales the passband level and c accounts for a noise floor. It is the standard analog and digital lowpass family; fit measured frequency-response magnitude data to recover the cutoff ω_c and effective order n.

References
[1] S. Butterworth, “On the Theory of Filter Amplifiers,” Experimental Wireless & the Wireless Engineer 7, 536–541 (1930) — full-text scan. worldradiohistory.com/…/Wireless-Engineer-1930-10.pdf
[2] Wikipedia, “Butterworth filter.” en.wikipedia.org/wiki/Butterworth_filter

Butterworth Highpass |H|² (`|H(ω)|² = a·(ω/ωc)^(2n) / (1 + (ω/ωc)^(2n)) + c`)

Magnitude-squared response of an n-th-order Butterworth highpass — DC blockers, AC-coupling characterization, rumble filters. Complementary to the lowpass form.

Bandpass Biquad / Peak EQ |H|² (`|H(ω)|²` = standard RBJ bandpass / peaking EQ form)

Magnitude-squared of a single second-order section configured as a bandpass or peaking EQ — fit measured audio-EQ curves, vibration-isolation transmissibility, mechanical resonance Bode magnitude. Reports center frequency, Q, and gain.

References
[1] R. Bristow-Johnson, “Cookbook formulae for audio EQ biquad filter coefficients” (Audio EQ Cookbook), W3C. w3.org/TR/audio-eq-cookbook/

Notch Biquad |H|² (`|H(ω)|²` = standard RBJ notch form)

Magnitude-squared of a second-order notch — 50/60 Hz hum rejection, line-frequency removal in ECG/EEG, narrow-band interference suppression. Reports notch frequency and depth.

References
[1] R. Bristow-Johnson, “Cookbook formulae for audio EQ biquad filter coefficients” (Audio EQ Cookbook), W3C. w3.org/TR/audio-eq-cookbook/

Raised-Cosine Pulse (`y` = standard root-raised-cosine pulse shape)

Pulse-shape used in QAM / PSK digital communications to limit inter-symbol interference — fit captured eye-diagram averages or measured matched-filter outputs. Roll-off factor α trades bandwidth for transition sharpness.

References
[1] J. M. Pauly, “Line Codes and Pulse Shaping,” EE179 Lecture 15, Stanford University. web.stanford.edu/class/ee179/lectures/notes15.pdf
[2] Wikipedia, “Raised-cosine filter.” en.wikipedia.org/wiki/Raised-cosine_filter

Soft-Knee Compressor (`y` = output dB vs. input dB across a soft-knee compressor curve)

Smooth piecewise mapping for audio dynamics processors — linear below threshold, compressed above, with a smooth transition over a fitted knee width. Reports threshold, ratio, and knee width.

15. Chemical, material, and reaction kinetics

Arrhenius equation Wikipedia

This model (y = A·exp(−E_a/(R·T)) + C) captures how reaction rates increase with temperature. Proposed by Svante Arrhenius in 1889, it models the fraction of molecular encounters that possess enough kinetic energy to overcome a specific reaction barrier.

The Core Variables

E_a (Activation Energy): The height of the energy barrier.
A (Pre-exponential Factor): The "frequency factor" representing how often successful molecular collisions occur.
T: Absolute temperature.
R: The ideal gas constant.

The Arrhenius Plot

Linearization: By taking the natural log, we get ln(y) = ln(A) - (E_a/R)·(1/T).
Diagnostic Tool: Plotting ln(y) against 1/T creates a straight line. The slope is equal to -E_a/R, allowing you to read the activation energy directly.

Important Limitations

The assumption that A and E_a are temperature-independent usually only holds over a narrow temperature window. Warning: Be careful with data measurement—because you are plotting against 1/T, even small errors in your temperature readings are significantly magnified at high values of 1/T.

Physical Context:
Beyond chemical kinetics, this same exponential dependence is the standard way to model diffusion coefficients, material viscosity, electrical conductivity, and even accelerated-life reliability testing. Whenever a physical process is constrained by an energy barrier, the Arrhenius form is the natural mathematical framework to describe it.

References:
[1] NIST/SEMATECH e-Handbook of Statistical Methods, 8.1.5.1 Arrhenius. itl.nist.gov/…/apr151.htm

Eyring Rate Equation (Transition State Theory)

This model (y = (k_B·T/h)·exp(−ΔG^‡/(R·T)) + c) is derived from Transition State Theory (TST). It describes the rate of a chemical reaction by considering the high-energy "activated complex" that exists at the top of the energy barrier.

The Core Variables

ΔG^‡ (Activation Free Energy): The difference between the reactants and the transition state.
k_B / h: Boltzmann’s constant and Planck’s constant.
T & R: Absolute temperature and the ideal gas constant.

Thermodynamics of the Barrier

The Eyring Plot: By plotting ln(k/T) vs 1/T, the data yields a straight line.
Slope: Gives you −ΔH^‡/R (activation enthalpy).
Intercept: Gives you ΔS^‡/R (activation entropy).

Why go beyond Arrhenius?

Arrhenius is purely empirical; the Eyring equation is rooted in statistical mechanics. Use Eyring when you need to know *why* a reaction is slow—is it because the energy barrier is too high (enthalpy), or because the transition state is highly ordered and unlikely to form (entropy)? This is essential for organometallic and enzymatic research.

Deep Dive:
Unlike the Arrhenius model, which assumes a simple collision frequency, the Eyring approach models the system as a quasi-equilibrium between reactants and the activated complex. Because it is based on fundamental principles of statistical mechanics, it is robust across gas, liquid, and solution phases, making it the standard for complex kinetics where molecular orientation and phase interactions play a role in the reaction rate.

References:
[1] Chemistry LibreTexts, “Eyring equation.” chem.libretexts.org/Eyring_equation
[2] H. Eyring, “The Activated Complex in Chemical Reactions,” J. Chem. Phys. 3, 107–115 (1935). doi:10.1063/1.1749604

Vogel-Fulcher-Tammann (VFT) (`y = A·exp(B/(T − T0)) + c`)

Super-Arrhenius viscosity / relaxation-time divergence near the glass transition — silicate glasses, polymer melts, supercooled liquids, ionic-liquid conductivity. Diverges at the Vogel temperature T0.

References
[1] Wikipedia, “Vogel–Fulcher–Tammann equation.” en.wikipedia.org/wiki/Vogel-Fulcher-Tammann_equation
[2] G. S. Fulcher, “Analysis of Recent Measurements of the Viscosity of Glasses,” J. Am. Ceram. Soc. 8, 339–355 (1925). doi:10.1111/j.1151-2916.1925.tb16731.x

Williams-Landel-Ferry (WLF) (`log(aT) = −C1·(T − T_ref) / (C2 + T − T_ref)`)

Time-temperature superposition shift factor for polymers above T_g — viscoelastic master-curve construction, dynamic mechanical analysis, rheology of amorphous solids.

References
[1] ScienceDirect Topics, “Williams–Landel–Ferry Equation” — free overview (free-volume theory, time–temperature superposition). sciencedirect.com/topics/engineering/williams-landel-ferry-equation
[2] M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc. 77, 3701–3707 (1955). doi:10.1021/ja01619a008

Avrami / JMAK (`y = 1 − exp(−(k·t)ⁿ)`)

Johnson-Mehl-Avrami-Kolmogorov isothermal phase-transformation kinetics — crystallization of polymers and metals, devitrification of glasses, nucleation-and-growth precipitation in alloys. Avrami exponent n diagnoses the nucleation/growth dimensionality.

References
[1] ScienceDirect Topics, “Avrami Equation” — free overview of JMAK transformation kinetics. sciencedirect.com/topics/engineering/avrami-equation
[2] M. Avrami, “Kinetics of Phase Change. I. General Theory,” J. Chem. Phys. 7, 1103–1112 (1939). doi:10.1063/1.1750380

Butler-Volmer (`y = i0·[exp(α·F·η/(R·T)) − exp(−(1−α)·F·η/(R·T))]`)

Electrode kinetics — current density vs. overpotential at a metal-electrolyte interface, fundamental to battery, fuel-cell, and corrosion analysis. Reports exchange current density i0 and transfer coefficient α.

References
[1] M. Z. Bazant, “Butler–Volmer equation,” MIT 10.626 Lecture 13, MIT OpenCourseWare (2014). ocw.mit.edu/…/S11lec13.pdf
[2] R. Guidelli et al., “Defining the transfer coefficient in electrochemistry (IUPAC Technical Report),” Pure Appl. Chem. 86, 245–258 (2014) — open access. doi:10.1515/pac-2014-5026

Van der Waals Isotherm

This model (P = R·T / (V − b) − a / V²) was proposed by Johannes Diderik van der Waals in 1873 to repair the ideal-gas law. It accounts for the non-ideal behavior of real fluids by introducing corrections for finite molecular size and intermolecular attraction.

Molecular Corrections

a (Attractive Force): The term −a/V² lowers the observed pressure due to intermolecular attraction.
b (Excluded Volume): Replaces V with V − b to account for the physical space occupied by the molecules themselves.
T & R: Absolute temperature and the ideal gas constant.

The Critical Point

The Inflection Point: Van der Waals observed that the critical isotherm has a horizontal slope at the critical point.
Law of Corresponding States: This mathematical feature allows a and b to be determined directly from a substance’s critical pressure, volume, and temperature.

Important Constraint

Note the mathematical pole at V = b, where the pressure diverges toward infinity. When performing curve fitting, ensure your input data stays safely above this excluded volume; otherwise, the fit will encounter a singularity and fail to converge.

Physics Context:
While the ideal gas law assumes molecules are dimensionless points that never interact, the Van der Waals equation treats them as real-world objects. By adding these two simple parameters, it successfully reproduces gas-liquid phase behavior, making it the bedrock of real-fluid thermodynamics and the starting point for more complex models of phase transitions.

References:
[1] J. D. van der Waals, “The Equation of State for Gases and Liquids,” Nobel Lecture (1910). nobelprize.org/prizes/physics/1910/waals/lecture/
[2] Wikipedia, “Van der Waals equation.” en.wikipedia.org/wiki/Van_der_Waals_equation

Argus

y = a·(x − x0)·√(x_max − x)·... — Argus-collaboration two-body invariant-mass background.

Threshold-shaped background for two-body decay invariant-mass spectra — endpoint behavior near a kinematic cutoff in B-meson and charm-meson studies. Standard in HEP background fits.

16. NIST StRD reference equations

These 17 entries reproduce the problems from the NIST Statistical Reference Datasets — the canonical hard cases for nonlinear least squares. The solver reaches NIST's certified parameter values to at least 4 digits on every problem and 8 digits on most. Use them when your data matches a NIST problem's functional form, or as a solver benchmark.

Misra1a (`y = b1·(1 − exp(−b2·x))`)

Saturating-exponential rise — dental-tissue absorption; first of the four Misra forms. Also matches BoxBOD biochemical-oxygen-demand data.

References
[1] NIST StRD Nonlinear Regression, “Misra1a” — monomolecular adsorption (D. Misra, NIST 1978). itl.nist.gov/div898/strd/nls/data/misra1a.shtml

Misra1b (`y = b1·(1 − (1 + b2·x/2)⁻²)`)

Saturation with a squared rational denominator — slightly sharper foot than Misra1a.

References
[1] NIST StRD Nonlinear Regression, “Misra1b” — monomolecular adsorption (D. Misra, NIST 1978). itl.nist.gov/div898/strd/nls/data/misra1b.shtml

Misra1c (`y = b1·(1 − (1 + 2·b2·x)^(−1/2))`)

Saturation with a square-root denominator — slowest approach to plateau in the Misra family.

References
[1] NIST StRD Nonlinear Regression, “Misra1c” — monomolecular adsorption (D. Misra, NIST 1978). itl.nist.gov/div898/strd/nls/data/misra1c.shtml

Misra1d (`y = b1·b2·x / (1 + b2·x)`)

Michaelis–Menten / Langmuir in Misra naming — same saturation curve, fastest approach to plateau in the family.

References
[1] NIST StRD Nonlinear Regression, “Misra1d” — monomolecular adsorption (D. Misra, NIST 1978). itl.nist.gov/div898/strd/nls/data/misra1d.shtml

Chwirut (`y = exp(−b1·x) / (b2 + b3·x)`)

Ultrasonic-response decay — exponential numerator over rational denominator. Single entry covers both NIST Chwirut1 and Chwirut2 with bounds spanning their certified ranges.

References
[1] NIST StRD Nonlinear Regression, “Chwirut1” — ultrasonic calibration study (D. Chwirut, NIST). itl.nist.gov/div898/strd/nls/data/chwirut1.shtml

Lanczos (`y = b1·exp(−b2·x) + b3·exp(−b4·x) + b5·exp(−b6·x)`)

Sum of three exponentials — Lanczos's notorious triple-decay test. Severe rate-constant labeling degeneracy resolved deterministically by the AutoFit tie-break.

References
[1] NIST StRD Nonlinear Regression, “Lanczos1” — generated three-exponential data (C. Lanczos). itl.nist.gov/div898/strd/nls/data/lanczos1.shtml

Gauss (`y = b1·exp(−b2·x) + b3·exp(−(x−b4)²/b5²) + b6·exp(−(x−b7)²/b8²)`)

Exponential background plus two Gaussian peaks — NIST Gauss1, Gauss2, Gauss3 share this 8-parameter form. The two-Gaussian basin swap is the hardest local-minimum trap in StRD.

References
[1] NIST StRD Nonlinear Regression, “Gauss1” — two Gaussians on an exponential baseline (B. Rust, NIST). itl.nist.gov/div898/strd/nls/data/gauss1.shtml

DanWood (`y = b1·x^b2`)

Two-parameter power law — Daniel & Wood's radiant-energy temperature scaling. Simplest NIST problem but sensitive to initial guess.

References
[1] NIST StRD Nonlinear Regression, “DanWood” — radiated energy vs. temperature (Daniel & Wood). itl.nist.gov/div898/strd/nls/data/daniel_wood.shtml

Kirby2 (`y = (b1 + b2·x + b3·x²) / (1 + b4·x + b5·x²)`)

Quadratic-over-quadratic rational — quartz-crystal frequency calibration. Identifiable but with strong parameter correlations.

References
[1] NIST StRD Nonlinear Regression, “Kirby2” — scanning electron microscope calibration (R. Kirby, NIST). itl.nist.gov/div898/strd/nls/data/kirby2.shtml

Hahn1 (`y = (b1 + b2·x + b3·x² + b4·x³) / (1 + b5·x + b6·x² + b7·x³)`)

Cubic-over-cubic rational — copper thermal-expansion (Hahn1) and electron-mobility (Thurber) calibration. Single catalog entry; bounds widened to the union of both certified ranges.

References
[1] NIST StRD Nonlinear Regression, “Hahn1” — thermal expansion of copper (T. Hahn, NIST). itl.nist.gov/div898/strd/nls/data/hahn1.shtml

MGH09 (`y = b1·(x² + b2·x) / (x² + b3·x + b4)`)

Kowalik rational enzyme-kinetic model — More-Garbow-Hillstrom test problem 9. The SSR surface has a deep non-axial valley.

References
[1] NIST StRD Nonlinear Regression, “MGH09” — Kowalik & Osborne problem. itl.nist.gov/div898/strd/nls/data/mgh09.shtml
[2] J. J. More, B. S. Garbow, and K. E. Hillstrom, “Testing Unconstrained Optimization Software,” ACM Trans. Math. Softw. 7, 17–41 (1981). doi:10.1145/355934.355936

MGH10 (`y = b1·exp(b2 / (x + b3))`)

Meyer's exponential-in-reciprocal-x — Arrhenius-style temperature dependence in MGH form. Famous for its long narrow SSR valley.

References
[1] NIST StRD Nonlinear Regression, “MGH10” — Meyer thermistor problem. itl.nist.gov/div898/strd/nls/data/mgh10.shtml
[2] J. J. More, B. S. Garbow, and K. E. Hillstrom, “Testing Unconstrained Optimization Software,” ACM Trans. Math. Softw. 7, 17–41 (1981). doi:10.1145/355934.355936

Rat43 (`y = b1 / (1 + exp(b2 − b3·x))^(1/b4)`)

Asymmetric Richards-style sigmoid — Ratkowsky's growth-curve benchmark. Same family as Generalized Logistic.

References
[1] NIST StRD Nonlinear Regression, “Ratkowsky3 (Rat43)” — sigmoidal growth (D. A. Ratkowsky). itl.nist.gov/div898/strd/nls/data/ratkowsky3.shtml

Eckerle4 (`y = (b1/b2)·exp(−½·((x − b3)/b2)²)`)

Gaussian peak with prefactor coupled to width — circular-interferometer fringe profile. Coupling makes the amplitude/width seed pair matter.

References
[1] NIST StRD Nonlinear Regression, “Eckerle4” — circular interference transmittance (K. Eckerle, NIST). itl.nist.gov/div898/strd/nls/data/eckerle4.shtml

Bennett5 (`y = b1·(b2 + x)^(−1/b3)`)

Superconducting-magnet relaxation — shifted power law with reciprocal exponent. Classic NIST hard problem; certified parameters live in a narrow ravine.

References
[1] NIST StRD Nonlinear Regression, “Bennett5” — superconductivity magnetization (L. Bennett, NIST). itl.nist.gov/div898/strd/nls/data/bennett5.shtml

Roszman1 (`y = b1 − b2·x − arctan(b3/(x − b4))/π`)

Quantum-state population vs. external-field response — linear trend plus arctan resonance. Sensitive to the seed for resonance center b4.

References
[1] NIST StRD Nonlinear Regression, “Roszman1” — quantum defects in iodine atoms (L. Roszman, NIST). itl.nist.gov/div898/strd/nls/data/roszman1.shtml

ENSO (`y = b1 + b2·cos(2πx/12) + b3·sin(2πx/12) + b5·cos(2πx/b4) + b6·sin(2πx/b4) + b8·cos(2πx/b7) + b9·sin(2πx/b7)`)

El Niño / Southern Oscillation pressure-difference time series — annual cycle (period 12 months, fixed) plus two free-period cycles. Hardest parameter trade-off in StRD after the Gauss basin swap.

References
[1] NIST StRD Nonlinear Regression, “ENSO” — El Niño atmospheric pressure differences (Kahaner, Moler & Nash). itl.nist.gov/div898/strd/nls/data/enso.shtml

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Model Reference — 100+ Scientific Equations

1. Linear and polynomial baselines

Linear (y = a·x + b)

Quadratic (y = a·x^2 + b·x + c)

Cubic (y = a·x^3 + b·x^2 + c·x + d)

Polynomial degree 4 (y = a·x^4 + b·x^3 + c·x^2 + d·x + e)

2. Rational functions

Reciprocal (y = a/(x + b) + c)

Rational (1/1) (y = (a·x + b) / (c·x + d))

3. Exponential growth and decay

Exponential (y = a·exp(b·x) + c)

Exponential Decay + Linear (y = a·exp(b·x) + c·x + d)

Exponential Rise (to baseline) (y = A·(1 − exp(−k·(x−x0))) + C)

Leaky Integrator (y = A·(1 − exp(−(x−x0)/τ)) + B)

Sum of Two Exponentials (y = a1·exp(b1·x) + a2·exp(b2·x) + c)

Sum of Two Exponentials (Decay) (y = a1·exp(−k1·x) + a2·exp(−k2·x) + c)

Sum of Two Exponentials (Rise) (y = a1·(1 − exp(−k1·x)) + a2·(1 − exp(−k2·x)) + c)

Sum of Three Exponentials (y = a1·exp(b1·x) + a2·exp(b2·x) + a3·exp(b3·x) + c)

Polynomial × Exponential (y = (a·x² + b·x + c)·exp(d·x) + e)

Stretched Exponential (y = a·exp(−|(x−x0)/b|ᶜ) + d)

Double Exponential Difference (y = a·exp(−k1·x) − b·exp(−k2·x) + c)

4. Power-law and allometric

The Variables Explained

Common Applications

Diagnostic Caution

Shifted Power Law (y = a·(x − x0)ᵇ + c)

Inverse Quadratic (y = a/(x − x0)² + c)

Hyperbolic Decay (y = a/(b + x) + c)

Double Power + Constant (y = a0·xᵉ⁰ + a1·xᵉ¹ + c)

Sum of Three Powers (y = a0·xᵉ⁰ + a1·xᵉ¹ + a2·xᵉ² + c)

5. Saturation, binding, and approach-to-limit

Logarithmic (y = a·ln(b·x) + c)

Saturating Hyperbola (y = a·x / (b + x) + c)

The Variables Explained

Two Contexts, Identical Algebra

When to use this model

Michaelis-Menten with Inhibition (y = Vmax·x / (Km + x + x²/Ki))

Exponential-Sigmoid Saturation (y = L·(1−exp(−k·(x−x0))) / (1 + exp(−b(x−w))) + C)

Double Michaelis–Menten (y = V1·x/(K1 + x) + V2·x/(K2 + x) + C)

Exponential-Linear Crossover (y = a·(1−exp(−b(x−x0))) then linear)

BET Isotherm (y = (a·K·x) / [(1 − x)(1 + (K − 1)·x)] + c)

The Variables Explained

Interpreting the Hill Coefficient (n)

Relationship to 4PL

Hill Inhibition (y = y0 + (ymax − y0) / (1 + (x/c)^n))

6. Symmetric sigmoid / S-curve

The Variables Explained

Key Assumptions

Which S-Curve should you choose?

Smoothed Heaviside Step (k=100) (y = a/(1+exp(−100·(x−x0))) + b)

Sigmoid + Linear (y = L/(1+exp(−k·(x−x0))) + a·x + b)

Hyperbolic Tangent (y = a·tanh(b·(x−x0)) + c)

Arctangent Step (y = a·arctan(b·(x−x0)) + c)

Generalized Lorentzian (Power Bell) (y = a / (1 + |b·(x − x0)|ⁿ) + c)

Softplus (Smooth Rectifier) (y = a·ln(1 + exp(b·(x−x0))) + c)

7. Asymmetric and generalized growth (Richards, Gompertz, Hill)

The 5 Parameters Explained

The Power of Asymmetry

The Cost of Flexibility (Overfitting Risk)

The 4 Parameters Explained

Why use a 4PL Constraint?

Best Practice: 4PL vs 5PL

The Core Variables

Fixed Asymmetry

Gompertz vs. Richards (5PL)

Logistic-Hill (4PL) (y = d + (a − d) / (1 + (x/c)ᵇ))

Logistic (5PL) (y = d + (a − d) / (1 + (x/c)ᵇ)ˢ)

Logistic (5PL, skewed) (y = d + (a − d) / (1 + (x/c)ᵇ)ᵍ)

Baranyi (lag-growth) (y = y0 + (L − y0) / (1 + exp(−k·(x − D))))

Logistic + Exponential Tail (y = L/(1+exp(−k·(x−x0))) + A·exp(b·x) + C)

Logistic + Exponential Mixture (y = L/(1 + exp(−k·(x − x0))) + A·exp(−b·x) + c)

Generalized Logistic + Exponential Tail (y = A/(1+exp(−k(x−x0)))ᵛ + B·exp(−λ(x−t0)) + c)

Logistic + Linear Drift (y = a/(1 + exp(−b·(x−x0))) + m·x + c)

Logistic Hazard (y = 1 − [1/(1 + exp(k·(x − x0)))]ᵐ)

8. Peaks, bumps, and bells

Linear (`y = a·x + b`)

Quadratic (`y = a·x^2 + b·x + c`)

Cubic (`y = a·x^3 + b·x^2 + c·x + d`)

Polynomial degree 4 (`y = a·x^4 + b·x^3 + c·x^2 + d·x + e`)

Reciprocal (`y = a/(x + b) + c`)

Rational (1/1) (`y = (a·x + b) / (c·x + d)`)

Exponential (`y = a·exp(b·x) + c`)

Exponential Decay + Linear (`y = a·exp(b·x) + c·x + d`)

Exponential Rise (to baseline) (`y = A·(1 − exp(−k·(x−x0))) + C`)

Leaky Integrator (`y = A·(1 − exp(−(x−x0)/τ)) + B`)

Sum of Two Exponentials (`y = a1·exp(b1·x) + a2·exp(b2·x) + c`)

Sum of Two Exponentials (Decay) (`y = a1·exp(−k1·x) + a2·exp(−k2·x) + c`)

Sum of Two Exponentials (Rise) (`y = a1·(1 − exp(−k1·x)) + a2·(1 − exp(−k2·x)) + c`)

Sum of Three Exponentials (`y = a1·exp(b1·x) + a2·exp(b2·x) + a3·exp(b3·x) + c`)

Polynomial × Exponential (`y = (a·x² + b·x + c)·exp(d·x) + e`)

Stretched Exponential (`y = a·exp(−|(x−x0)/b|ᶜ) + d`)

Double Exponential Difference (`y = a·exp(−k1·x) − b·exp(−k2·x) + c`)

Shifted Power Law (`y = a·(x − x0)ᵇ + c`)

Inverse Quadratic (`y = a/(x − x0)² + c`)

Hyperbolic Decay (`y = a/(b + x) + c`)

Double Power + Constant (`y = a0·xᵉ⁰ + a1·xᵉ¹ + c`)

Sum of Three Powers (`y = a0·xᵉ⁰ + a1·xᵉ¹ + a2·xᵉ² + c`)

Logarithmic (`y = a·ln(b·x) + c`)

Saturating Hyperbola (`y = a·x / (b + x) + c`)

Michaelis-Menten with Inhibition (`y = V_max·x / (K_m + x + x²/K_i)`)

Exponential-Sigmoid Saturation (`y = L·(1−exp(−k·(x−x0))) / (1 + exp(−b(x−w))) + C`)

Double Michaelis–Menten (`y = V1·x/(K1 + x) + V2·x/(K2 + x) + C`)

Exponential-Linear Crossover (`y = a·(1−exp(−b(x−x0)))` then linear)

BET Isotherm (`y = (a·K·x) / [(1 − x)(1 + (K − 1)·x)] + c`)

Interpreting the Hill Coefficient (`n`)

Hill Inhibition (`y = y0 + (y_max − y0) / (1 + (x/c)^n)`)

Smoothed Heaviside Step (k=100) (`y = a/(1+exp(−100·(x−x0))) + b`)

Sigmoid + Linear (`y = L/(1+exp(−k·(x−x0))) + a·x + b`)

Hyperbolic Tangent (`y = a·tanh(b·(x−x0)) + c`)

Arctangent Step (`y = a·arctan(b·(x−x0)) + c`)

Generalized Lorentzian (Power Bell) (`y = a / (1 + |b·(x − x0)|ⁿ) + c`)

Softplus (Smooth Rectifier) (`y = a·ln(1 + exp(b·(x−x0))) + c`)

Logistic-Hill (4PL) (`y = d + (a − d) / (1 + (x/c)ᵇ)`)

Logistic (5PL) (`y = d + (a − d) / (1 + (x/c)ᵇ)ˢ`)

Logistic (5PL, skewed) (`y = d + (a − d) / (1 + (x/c)ᵇ)ᵍ`)

Baranyi (lag-growth) (`y = y0 + (L − y0) / (1 + exp(−k·(x − D)))`)

Logistic + Exponential Tail (`y = L/(1+exp(−k·(x−x0))) + A·exp(b·x) + C`)

Logistic + Exponential Mixture (`y = L/(1 + exp(−k·(x − x0))) + A·exp(−b·x) + c`)

Generalized Logistic + Exponential Tail (`y = A/(1+exp(−k(x−x0)))ᵛ + B·exp(−λ(x−t0)) + c`)

Logistic + Linear Drift (`y = a/(1 + exp(−b·(x−x0))) + m·x + c`)

Logistic Hazard (`y = 1 − [1/(1 + exp(k·(x − x0)))]ᵐ`)

The FWHM Rule (Estimating `σ`)

The Blending Factor (`η`)

Asymmetric Lorentzian (`y = a / (1 + ((x−x0)/γ(x))²) + c`, γ side-dependent)

Sum of Two Gaussians (`y = a1·exp(−(x−μ1)²/(2σ1²)) + a2·exp(−(x−μ2)²/(2σ2²)) + c`)

Sum of Two Lorentzians (`y = a1/(1+((x−x1)/γ1)²) + a2/(1+((x−x2)/γ2)²) + c`)

Double Gaussian (Composite) (`y = a1·exp(−(x−μ1)²/(2σ1²)) + a2·exp(−(x−μ2)²/(2σ2²)) + c`)

Bi-Gaussian (Split Width) (`y = a·exp(−(x−μ)²/(2σ_L²))` for x<μ, `σ_R` otherwise)

Log-Normal Peak (`y = a·exp(−[ln(x/μ)]²/[2σ²]) + c`)

Controlling the Tilt (`s`)

Exponential Gaussian Hybrid (EMG, alt form) (`y = a·exp(σ²λ²/2 − λ(x−μ))·Φ([x−μ]/σ − σλ) + c`)

Pearson VII (`y = a·[1 + ((x−x0)/γ)²·(2^(1/m)−1)]^(−m) + c`)

Sech-n Peak (`y = a / cosh(b·(x−x0))ⁿ + c`)

Derivative of Logistic (Bell) (`y = A·exp(−k(x−x0))/(1+exp(−k(x−x0)))² + C`)

Triangular Pulse (`y = a·max(0, 1 − |(x−x0)/w|) + c`)

Parabolic Hump (`y = a − b·(x − x0)²`)

Product of Gaussians (Multiplicative Peak) (`y = a·exp(−(x−m1)²/(2·s1²))·exp(−(x−m2)²/(2·s2²)) + c`)

Controlling the Asymmetry (`q`)

Beta CDF (`y = a · I_x(b, c) + d`)

Beta PDF (`y = a·x^(b−1)·(1−x)^(c−1) + d`)

The Shape Parameter (`k`) & The "Bathtub"

Weibull PDF (3-parameter) (`y = a·(b/λ)·((x−x0)/λ)^(b−1)·exp(−((x−x0)/λ)^b) + c`)

Log-Normal CDF (`y = a·0.5·[1 + erf((ln(x) − μ)/(σ√2))] + c`)

Controlling the Silhouette (`k`)

Gamma CDF (`y = a · P(k, x/θ) + c`)

Pareto CDF (`y = a·[1 − (x_min/x)^α] + c` for x ≥ x_min)

Inverse Gaussian / Wald PDF (`y = a·√(λ/(2π·x³))·exp(−λ(x−μ)²/(2μ²·x)) + c`)

Maxwell-Boltzmann (`y = a·x²·exp(−x²/(2σ²)) + c`)

GEV (PDF, right-skew) (`y = (a/σ)·(1 + ξ·(x−μ)/σ)^(−1−1/ξ)·exp(−(1+ξ·(x−μ)/σ)^(−1/ξ)) + c`)

Gumbel PDF (`y = a·exp(−exp(−(x−μ)/β))·exp(−(x−μ)/β) + c`)

Gumbel CDF (`y = a·exp(−exp(−(x−μ)/β)) + c`)

Inverse Sigmoid (Probit / Ogive) (`y = a·Φ⁻¹((x−μ)/σ) + c`)

Rectified Linear Unit (ReLU) (`y = a·max(0, x − x0) + b`)

Leaky ReLU (`y = a·max(0, x − x0) + b·min(0, x − x0) + c`)

Piecewise Linear (Hinge) (`y = a·(x − x0) + intercept` for x < x0; `b·(x − x0) + intercept` for x ≥ x0)