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Curve Model Reference – Uses, Equations, and Guidance

Basic and Widely-Applicable Models

Linear (y = a·x + b)

   Linear models describe proportional relationships between x and y. Use this for data that trends steadily in one direction, such as calibration curves or simple first-order kinetics. Results are intuitive and easy to interpret. This is the standard starting point for quantitative data analysis.

Quadratic (y = a·x² + b·x + c)

   Use a quadratic model to fit data that forms a single arch or valley—like ballistics, standard parabolas, or certain efficiency curves. The equation captures one turning point (a maximum or minimum). It describes acceleration, deceleration, or symmetrical crest/dip behaviors. Can be a better fit than linear for data with one clear bend.

Cubic (y = a·x³ + b·x² + c·x + d)

   A cubic fit models data with one inflection point and up to two bends, such as S-curves, reaction rate transitions, or certain titration curves. It is suitable when the dataset crosses through an S-shape or has asymmetric features. Use it when quadratic does not capture an extra "twist" in your trend. Cubic fits are essential for more complex, smoothly varying processes.

Polynomial (Degree 4) (y = a·x⁴ + b·x³ + c·x² + d·x + e)

   Quartic polynomials are capable of fitting complex, wavy curves with up to three bends. Use with caution because overfitting can capture noise rather than true patterns, but justified for phenomena with multiple peaks or dips. This model is often used in exploratory analysis where the true data-generating process is unknown. Utilize only when lower-degree polynomials clearly cannot describe your data.

Growth, Decay, and Scaling Models

Exponential (y = a·exp(b·x) + c)

   The exponential function fits processes with accelerating (or decelerating) change, such as population growth, radioactive decay, or chemical kinetics. Use it when changes in y are proportional to the current value of y. Recognize from data that grows or shrinks faster over time. Log-transforming your data (ln(y) vs. x) can help confirm appropriateness.

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

   Combines one fast exponential component with a steady linear background. Useful in sensor drift, fluorescence decay with slow baseline, or physical systems showing rapid initial change then slow drift. Use this to capture "fast drop plus slow background" features. Provides better fits than simple exponential or linear alone for many real systems.

Power Law (y = a·xᵇ + c)

   Power law models describe scaling relationships where change in y depends on x raised to an exponent—typical in allometry, city size, or frequency distributions. Use this model when log-log plots of your data are straight lines. These equations do not saturate; they often appear in biological, ecological, and physical scaling. Good for modeling systems with broad, scale-free behaviors.

General Power Law (shifted) (y = a·(x–x₀)ᵇ + c)

   The shifted power law allows the scaling relationship to start at an arbitrary x₀, not the origin. This is useful when the effect begins after a threshold or delay, for example in stress-strain onset or physical/biological activation thresholds. The shift increases flexibility for onset/location, making this fit preferable for data with a lag. Use when a regular power law does not match your data's onset location.

Logarithmic (y = a·ln(b·x) + c)

   Applies when y rises quickly for small x and then saturates or slows for larger x. Typical in diminishing returns, learning curves, or chemical titration responses. Use this model if increasing x gives ever smaller increments in y. Logarithmic fits are suitable for phenomena where growth slows down over time.

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

   Models processes that approach a plateau from below, such as charging a capacitor or population growth to a fixed upper limit. The function starts at baseline and curves upward, leveling off at high x. Use this for systems with rapid start that slow near a ceiling. Indicates saturation or limiting behavior common in engineering and biology.

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

   Captures data with both curved (polynomial) and exponentially-increasing or -decreasing trends. Used in advanced signal processing, growth/decay superimposed with curvature, or certain optical and physical models. Lets you describe shapes like "exponentially-warped bells" or steeper-than-exponential changes. Use if simple exponential or polynomial cannot capture the curvature you observe.

Arrhenius Equation (y = A·exp(–Ea/(R·T)) + C)

   Used in chemistry, physics, and materials science for temperature-dependence of reaction rates or conductivity. The model relates observable rates to inverse temperature, allowing extraction of activation energy Ea. Use for systems where physical theory or literature suggests Arrhenius dependence. Plotting ln(y) vs 1/T should yield a straight line if appropriate.

Exponential-Pareto (ExpPower) (y = a·x^{-b}·exp(–λx) + c)

   Models signals or distributions with a "heavy" (power law) tail that is ultimately cut off by an exponential decay. Used for reliability, financial tails, returns, or hazard rates in physics. If your log-log plot rolls over (cuts off) at high x, consider this model. Useful where pure power-law or pure exponential is either too shallow or too sharp.

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

   Useful in time decay phenomena like dielectric relaxation, battery chemistry, glassy systems, and certain lifetimes. The model describes broader or narrower than normal exponential decay, controlled by the exponent c. Choose this when normal exponential decay is too sharp or too broad for your data. Also called the Kohlrausch function.

Saturation and Threshold Models

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

   Fits data that falls rapidly at first and then flattens out, common in enzyme kinetics, electronic resistance in parallel, and competitive inhibition. Recognize this when y starts large for small x and then approaches a finite baseline at high x. Also appears in environmental or biological decay with limiting background. Good for quick diagnostic fits for saturating inverse relationships.

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

   This is a classic Michaelis-Menten model, commonly seen in enzyme kinetics, adsorption, and growth-limiting processes. The function is zero at x = 0, rises steeply, then levels off as x increases. Use for saturating response curves where output is limited by a process capacity. Robust, interpretable kinetic parameters.

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

   Models decay processes that never quite reach zero but instead level at a background value c. Appears in environmental concentration decay, signal dilution, or long-tailed kinetic processes. Use when the decrease slows substantially at high x, revealing an offset baseline. Related to classic reciprocal forms with additional flexibility.

Langmuir Isotherm (y = (Qmax·K·x)/(1 + K·x) + c)

   Describes surface adsorption to a fixed number of sites, especially in chemistry and environmental science. Useful for fitting chemical and biological data where binding is limited by site saturation. Recognize by the curve’s rapid initial rise and eventual plateau as capacity is reached. Key model for interfacial and sorption science.

Monod Growth (y = umax·x / (Ks + x) + y0)

   Generalizes the hyperbolic response to include baseline offset (y0), commonly used for microbial, bacterial, and ecological growth under limiting nutrient. Useful when growth increments are proportional to available resources. Plateau is achieved at high x (nutrient). Suitable for biological systems with limiting substrate/food.

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

   Used in neuroscience (LIF neuron models), electronics (RC circuits), and other systems with a saturating approach to steady-state. The curve rises quickly, then levels off. τ (tau) determines the rate of rise. Use to describe processes that accumulate then leak to a constant value over time or input.

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

   Models enzyme kinetics with substrate inhibition; as substrate increases, rate initially rises, then falls. Used in toxicological, metabolic, and biochemical processes exhibiting self-inhibition. Shows characteristic peak and downturn at high substrate. Recognize when simple Michaelis-Menten is not sufficient.

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

   Captures processes with two distinct saturating responses, such as dual-pathway metabolism, multi-site enzyme systems, or overlapping uptake mechanisms. Useful for complicated experimental systems where a single hyperbola underestimates the observed plateau. Use when one Michaelis-Menten curve is insufficient. Provides more realistic modeling of complex biological or chemical pathways.

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

   Fits data with abrupt step changes between two levels. Useful for switching devices, 2-level transitions, or clear-cut “on/off” signals. Use as a simple discontinuous step detector. Typical in digital circuits or binary-state responses.

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

   Models data with two linear segments joined at a breakpoint (x₀). Good for thresholded or phase change behaviors, such as stress-strain in materials or economic demand with price thresholds. Identifies a change in slope within the dataset. Apply to “elbow-shaped” plots or piecewise physical effects.

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

   Similar to bilinear, but uses one slope before and a different slope after a single breakpoint. Suitable for piecewise linear transitions showing a “kink” or hinge, especially when smoothness at the joint is not required. Use for processes that behave differently above and below a certain threshold. Applies in mechanical, electrical, or economic stepwise systems.

Piecewise Rational (Hinge) (y = (a₁·x + b₁)/(c₁·x + d₁), x < x₀; (a₂·x + b₂)/(c₂·x + d₂), x ≥ x₀)

   Fits data using two separate rational functions joined at a bend or hinge. Useful for modeling physical processes with regime changes, such as catalysis before and after saturation, or technological processes with a transition. Provides flexibility in both slopes and curvatures before and after the break point. Strong fit for complicated real-world transitions.

Step (Heaviside) (y = a·H(x–x₀) + b)

   Used for sudden threshold changes or switching behavior, where output jumps sharply at a critical value. Useful for activation/inactivation, digital switching, or step function responses. Recognize in data as a sharp vertical step. No curve—just an abrupt jump.

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

   Describes systems where nothing happens until a threshold, then output increases linearly. Common in neural networks, input activation, and clipped data. Recognize in data that is flat for low x, then rises. Simple, interpretable, often used as a building block in more complex systems.

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

   Generalizes the ReLU by allowing a small, nonzero slope below the threshold (the “leak”). Used in advanced neural network modeling, rectified and asymmetric signals, and electronic threshold circuits. Choose this when you see a slope rather than a flat region for low x, or to avoid “dead” zones. Useful for fitting electronics, activation, and robust regression.

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

   Provides a smooth transition from flat to linear, unlike ReLU or step models. Used in machine learning, psychophysics, or models requiring differentiable thresholds. Good when the “switch-on” is gradual, not abrupt. Helps avoid non-differentiability in systems modeled computationally.

Hockey Stick (Step+Ramp) (y = b for x < x₀; a·(x–x₀) + b for x ≥ x₀)

   Fits data that sits at a low plateau and then, past a critical value, increases linearly. Common in climate analysis, material stress, or finance where “break” initiates rapid change. Choose for “flat, then ramp” patterns. Useful for modeling responses only triggered after a threshold.

Sigmoidal and Logistic Models

Logistic (Sigmoid) (y = L / (1 + exp(–k·(x–x₀))))

   The standard S-shaped curve, capturing slow start, rapid acceleration, and a plateau. Used for population growth, epidemics, and switch-like transitions. These fits have hard upper and lower limits. Good for modeling binary transitions in continuous systems.

3-Parameter Logistic Growth (y = a / (1 + exp(–k·(x–x₀))))

   Similar to the logistic sigmoid, but with simpler parameterization (no baseline offset). Useful for population and uptake processes where growth starts slow, accelerates, then saturates. Use when data cleanly “S’s” upward without need for background level. Can be interpreted in terms of carrying capacity and intrinsic growth rate.

Generalized Logistic (Richards) (y = A + (K–A) / (1 + Q·exp(–B(x–vx)))1/ν)

   Adds asymmetry and shape flexibility to the regular sigmoid. Captures transitions that are not symmetrical—such as real-world population changes, epidemics, or growth with slow acceleration and fast fall (or vice versa). Adjusts rise and fall rates independently for greater realism. Use when classic logistic models miss the real transition shape.

Five-Parameter Logistic (5PL, Asym-Sigmoid) (y = d + (a–d)/[(1 + (x/c)^b)^s])

   A highly flexible sigmoidal model allowing asymmetrical (skewed) transitions and varying slope shapes. Used in dose-response modeling, immunoassays, and advanced bioassay analysis. Adjusts midpoints, asymptotes, slope, and transition skewness independently. Choose when your S-curve is not symmetric or standard four-parameter logistic misses essential features.

Log-Logistic (4PL) (y = d + (a–d)/[1 + (x/c)^b])

   Another common four-parameter S-curve, often used in dose-response curves and bioassays. Good for systems where responses scale with the logarithm of dose or input. Features variable asymmetry and flexible steepness for sigmoidal curves. Often fits biological and chemical quantitation data well.

Boltzmann Sigmoid (y = A₁ + (A₂–A₁)/[1+exp((x–x₀)/dx)])

   Used for sharp, finite transitions between two levels, common in electronic switching, physical transitions, or phase change processes. Controls upper and lower asymptotes, midpoint, and slope width. Ideal for “finite window” transitions instead of infinitely gradual changes. Often applied in spectroscopy, conductivity, and critical temperature phenomena.

Generalized Logistic (Verhulst-Richards, shape v) (y = K / (1 + Q·exp(–B(x–x₀)))1/v)

   Another form of the generalized logistic, parameterized for population biology and growth contexts. The “v” parameter tunes asymmetry; other parameters tune slope and inflection position. Lets you model skew in both rise and fall. Often used to capture real biological or material transitions that do not follow a simple S-curve.

Logistic (Carrying Cap Offset) (y = K / (1 + Q·exp(–r(x–x₀))) + C)

   Extends the regular logistic to include a baseline (offset). Good for systems that do not fall to zero at low input. Use this when your observed data starts and ends at different nonzero levels. Useful in population modeling, technological adoption, and growth with pre-existing background.

Logistic w/ Lag (y = L / (1 + exp(–k(x–x₀–D))))

   Adds a lag delay to the logistic function, shifting the start of the S-curve. Appropriate for processes that do not begin immediately, such as population growth after gestation, diffusion delays, or cell culture lag phases. The “D” parameter gives the delay before growth initiates. Use to model delayed response systems.

Hill (4-param & Baseline) (y = d + (a–d)/[1 + (x/c)^b], y = y₀ + (ymax–y₀)/(1 + (c/x)^n))

   The Hill equation describes sigmoidal dose-response, binding, or transport curves with two asymptotes. Used in pharmacology, toxicology, and PCR quantitation. The “baseline” form fits data with nonzero minimal response. Use when you require a flexible sigmoidal model for an experimental system with adjustable starting and ending levels.

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

   The inhibition form of the Hill equation is specialized for competitive binding, antagonistic responses, or negative feedback. It models systems where increased x reduces y towards a baseline value. Used in biochemistry and drug tolerance. Employ when the typical response curve “falls” instead of “rises.”

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

   Provides an S-shaped curve with more gradual tails compared to the logistic. Used for neural network activations, physics crossovers, or systems with soft thresholds. Can model smooth, non-asymptotic transitions between two states. Use when sigmoid is too sharp or the process “softens out” at the sides.

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

   Produces a soft S-shaped transition with even slower tails, used in statistical transitions, fuzzy logic, and soft switching. The arctangent form never reaches fixed upper/lower limits. Useful for modeling gradual transitions with “long wings.” Common in psychometric or soft probability models.

Beta CDF (y = a·Iₓ(b, c) + d)

   Models transitions with controlled asymmetry and span, confined to a domain [0,1]. Used for probability, nonlinear statistics, bounded growth processes, or shape-controlled biological transitions. Adjusts both slope and asymmetry via two shape parameters. Good for bounded S-curve data.

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

   Captures data that follows a flexible S-curve with an initial or trailing exponential component. Used in epidemiology, hybrid growth processes, and advanced physical systems where initial fast rise or slow decay are present. Provides maximal flexibility to account for multiple regimes in a single smooth model. Use when regular sigmoid or exponentials miss secondary dynamic features.

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

   Adds an exponential tail or baseline alongside a logistic S-curve. Useful for dose-response or signal detection with leakage, background, or ongoing exponential trend. Allows for lagging or persisting response beyond the main transition. Choose when regular logistic fits systematically miss a slow tail.

Power Sigmoid (y = a/(1+|b(x–x₀)|ⁿ) + c)

   Provides very flexible compression curves, commonly used for psychophysical scaling, economics, or control surfaces. Allows independent adjustment of width, midpoint, and the “sharpness” of transition (n). Use when neither a standard sigmoid nor hyperbolic function matches your curve’s compression. Good for non-standard inputs and perceptual phenomena.

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

   Used to model cumulative phenomena, survival times, or response latencies where underlying events are log-normally distributed. The curve starts at the minimum, rises sigmoidally, and plateaus. Useful in dose-tolerance, population adoption, or time-to-failure data. Fits left-skewed transitions well.

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

   This is the inverse of the cumulative distribution function (CDF) of the normal distribution. Used in psychometrics, statistics, and bioassays where probability must be mapped back to stimulus level. Fits ogive-shaped response data or maps percentiles to input. Use for reverse S-curves or threshold backcalculation.

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

   Combines a sigmoidal transition with an added linear drift or background. Useful when the main curve saturates but the baseline still changes with x. Common in finance, p-value corrections, or chemical processes with time-varying backgrounds. Apply when your S-curve isn’t strictly flat at ends.

Logistic with Lower Asymptote (y = L / (1+exp(–k(x–x₀))) + ylo)

   Like the classic logistic, but the lower limit (asymptote) is not fixed at zero. Fits dose–response or adoption data with a persistent minimum value. Used in psychometrics or drug response where there’s always baseline signal. Good if the lower tail never truly vanishes.

Logistic Threshold (Window) (y = a/(1+exp(–b(x–x₀))) + c)

   Describes the entry into a new regime at a threshold value with saturated limits on both sides. Used in neural firing rates, technology adoption, or any system with thresholded activation within a window. Flexible slope and baseline. Appropriate for S-shaped transitions centered anywhere along the x-axis.

Cumulative Logistic (Fraction Max) (y = N/(1 + exp(–k·(x–x₀))))

   Common in epidemiology for total outbreak fractions or Reed-Frost models. Used for cumulative counts, fraction-of-population processes, or technology deployment analysis. When total output has a clear maximum, use this form. Matches data that rises rapidly, slows, and then plateaus at an upper fraction.

Richards Growth (y = K/[1 + ((K–y₀)/y₀)·exp(–r·(x–x₀))]1/ν)

   The most general sigmoid, it controls initial value, inflection, growth rate, and shape symmetry independently. Used for advanced fitting of population, adoption, or logistic-like data that cannot be fit by any simpler sigmoid. Use when skew or multi-phase S-curve is observed. Especially common in biological or multi-phase growth systems.

Peak, Mixture, and Distribution Models

Gaussian (Normal) (y = a·exp(–((x–μ)²/(2σ²))) + c)

   The classic symmetric bell curve, fundamental to probability and experimental error modeling. Used for peak fitting, error distributions, and random noise characterization. Standard in signal processing and statistics. Recognize by its symmetry and rapid decay.

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

   Fits data with two overlapping or closely spaced peaks, such as mixed chemical species, population bimodality, or spectra. Resolves mixed or compound distributions in noisy data. Useful for quantifying two populations or doubly-peaked signals. Choose when a single Gaussian cannot fit all observed features.

Sum of Two Lorentzians (y = a₁/(1+((x–x₁)/g₁)²) + a₂/(1+((x–x₂)/g₂)²) + c)

   Fits two closely spaced or overlapping Lorentzian-shaped peaks, typical in spectroscopy, Raman, or resonance data. Lorentzians have sharp narrow peaks with broad tails. Use this for strong resonance, damped oscillation, or unresolved spectral lines. Provides extra flexibility over single-peak fits.

Product of Gaussians (y = a·exp(–((x–m₁)²/(2s₁²)))·exp(–((x–m₂)²/(2s₂²))) + c)

   Captures the overlap (or intersection) of two populations or contributing sources with Gaussian spread. The peak is reduced and narrowed compared to a sum of Gaussians. Use when only the region shared by two effects should be modeled, such as population overlap or joint probabilities. Strong for modeling co-occurrence or intersecting distributions.

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

   Generates asymmetric, right- or left-skewed peak shapes, common in chromatography, fluorescence, or detection signals. Models situations where a rapid onset or decay adds asymmetry to the main (symmetric) peak. Useful for any peak that is not well described by a symmetric curve. Use when you observe a “tail” on one side of a peak.

Skew-Normal (y = a·exp(–((x–μ)²/(2σ²)))·[1+erf(s(x–μ)/(σ√2))] + c)

   This model accounts for peaks that are asymmetric and have heavier weight on one side. Used in finance, psychology, and physiology where distributions are not perfectly symmetric. Detectable by unequal “shoulders” or tails in a measured peak. More realistic for natural systems than the pure Gaussian when asymmetry is observed.

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

   Models data where peaks skew to one side, as in particle size distribution, economics, or lifetime data. The log-normal distribution is useful for phenomena that are only positive and have a long tail. Appears when underlying processes multiply or grow logarithmically. Appropriate for distributions of biological size, financial returns, or time-to-failure.

Gumbel Distribution (y = a·exp(–exp(–(x–μ)/β)) + c)

   Used for modeling extreme value statistics, particularly for the distribution of maxima (or minima) of datasets; often used in flood, burst, or stress event analysis. Gumbel curves appear right-skewed and are typical where a process is limited by rare, extreme events. Apply when analyzing event size distributions (e.g., rainfall, loads, or maxima). Choose for systems where maxima (or minima) are key values.

GEV Distribution (y = a·exp(–(1+ξ·(x–μ)/σ)–1/ξ) + c)

   The Generalized Extreme Value (GEV) distribution models maxima or minima with tunable tail shape. Used in hydrology, risk assessment, or meteorology. Greater shape flexibility than Gumbel. Suitable when you need a model for the most extreme observations.

Five-Parameter Composite (Double Gaussian) (y = a₁·exp(–(x–μ₁)²/(2σ₁²)) + a₂·exp(–(x–μ₂)²/(2σ₂²)) + c)

   Extends the sum-of-Gaussians model to more robustly handle noisy, separated, or ill-posed double peaks. Designed for material science, spectroscopy, or any domain with partially resolved, asymmetric, or mixed populations. Choose this to separate closely spaced or challenging-to-resolve mixed distributions. Use for population analysis, overlapped events, or spectral peak separation.

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

   Represents a symmetric, bell-like peak that is the derivative of a sigmoid curve. Used to fit transition rate peaks, response rate maxima, and smooth “bell” transitions in logistic-regulated systems. Provides a close match to bell-shaped responses in systems with logistic control. Recognizable by moderate peak width and symmetry.

Weibull PDF (y = a·(b/λ)·((x–x₀)/λ)b–1·exp(–((x–x₀)/λ)b) + c)

   Used to fit lifetimes, failure times, reliability engineering data, or ecological abundance. The Weibull distribution is asymmetric with a tunable shape parameter. Appears in failure analysis, reliability, hydrology, or ecology. Select for positive, skewed data sets that decrease after a peak.

Double Sided Sinh (y = a / (cosh(b(x–x₀)))ⁿ + c)

   Produces symmetric, “hill-shaped” curves with variable tail weight. Used for symmetric events or responses with heavier than normal or lighter than normal decay. Applicable to growth/decay systems with symmetric properties. Recognizable by a flat top and steep tails.

Triangular Pulse (“hat” function) (y = a·max(0, 1–|(x–x₀)/w|) + c)

   Models symmetric, sharp events with finite width, such as laser pulses, switching events, or square events with finite rise/fall. Use for short, localized, and symmetric signals. The curve rises, peaks, then falls sharply back to baseline. Often used as a simple idealization for rectangular or “blip” signals.

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

   Models data that rises only after a threshold (x₀) with shape controlled by power n. Applies to spike shapes, growing signals, or phenomena that start at a critical value. Useful for decay/plateau synthesis or when only the right tail of a polynomial is present. Select for highly right-skewed, thresholded signals.

Periodic and Oscillatory Models

Sinusoid (y = A·sin(ωx + φ) + C)

   Applies to periodic, repeating signals—common in physics (waves), engineering (signals), and biology (cycles). Use when data shows regular up/down trends. Model provides information about amplitude, frequency, and phase. Suitable for any system with oscillatory or cyclic response.

Sum of Sines (2) (y = A₁·sin(ω₁x + φ₁) + A₂·sin(ω₂x + φ₂) + C)

   Fits data where several frequencies contribute, resulting in beating or complex periodicity (e.g., acoustics, composite signals). The sum of two sinusoids can represent interference or harmonic content. Look for nested or “beating” waves in data. Analyze mixed oscillatory phenomena across science and engineering.

Damped Oscillator (y = A·exp(–k·x)·sin(ω·x + φ) + C)

   Used for oscillatory systems that lose amplitude, such as damped springs, RLC circuits, or signal decay. Captures both the exponential envelope and oscillation. Important for dynamic systems with dissipation. Use when your periodic data “rings down” with time.

Damped Cosine (y = A·exp(–k·x)·cos(ω·x + φ) + C)

   Models damped oscillations with a cosine base, often used in NMR, EPR, signal processing, or resonance decay measurements. Identifies frequency, decay rate, and phase of periodic signals that extinguish over time. Select for symmetric oscillations that decrease predictably. Ideal for analysis of free induction decay and resonance.

Specialized and Composite Models

Sersic Profile (y = I₀·exp(–(x/x₀)1/β) + c)

   Widely used in astrophysics to model light profiles of galaxies, but also relevant for optical/EM signals. Tailors curve sharpness via beta—neither pure exponential nor Gaussian. Use when analyzing spatial distributions with sharp or flat cores. Fits a wide range of radial decay phenomena.

Generalized Gompertz (y = a·exp(–b·exp(–c(x–x₀)q)) + d)

   Extends the Gompertz growth model to allow variable curvature and inflection by exponent q. Applicable to time-dependent biological, social, or technical systems with nonstandard acceleration profiles. Turn to this model for population, tech, or ecological systems that deviate from standard sigmoid transitions. Controls early, mid, and late phase growth flexibly.

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

   Fits data that shows both a polynomial (curving) trend and exponential growth or decay. This can appear in population modeling, signal decay with underlying trend, or multiphase chemical reactions. Use for advanced fitting where pure exponential or polynomial models leave systematic residuals. Robust for signals with non-uniform warping of exponential trends.

Piecewise Rational (Bend/Hinge) (y = [a₁·x + b₁]/[c₁·x + d₁] for x < x₀; [a₂·x + b₂]/[c₂·x + d₂] for x ≥ x₀)

   Describes systems with different behavior or curvatures before and after a breakpoint, such as regime changes or process limits. More flexible than regular piecewise linear models. Useful in chemical, ecological, or engineering processes that switch mechanisms. Pick this for difficult-to-fit “kinked” curve data.

Expo–Linear Crossover (Saturating Expo) (y = a(1–exp(–b·(x–x₀))), x < xp; a+b·(x–xp)–a, x ≥ xp)

   Models systems that rise quickly, then switch to a steady linear increase (or plateau) beyond a switch point. Used in signal acquisition, switching, or growth that reaches capacity then is limited externally. Use when initial curve transitions to linear behavior. Highlights processes with an initial exponential/or sigmoidal approach, then a "hand-off" to new behavior.

Bilinear (Piecewise Linear, breakpoint)

   See above in Threshold models; it also appears here because it bridges segmented and composite model classes.

Parabolic Hump (Quadratic Peak) (y = a – b·(x–x₀)² + c)

   Quickly fits symmetric, parabola-shaped peaks, especially for short or narrow signals. Useful for chromatograms, local maxima, or process events with quadratic rise and fall. Best for quick, rough fits where broader or symmetric behavior is clear and not extended far from the peak. Not suitable for strongly skewed or long-tailed data.

Beta PDF (y = a·x^{b–1}·(1–x)^{c–1} + d)

   Models bounded, asymmetric “hump” distributions for data between 0 and 1, such as proportions, probabilities, or normalized units. Useful in statistics, ecology, and reliability, especially for modeling transition, risk, or frequency of rare events. Adjustable via two shape parameters, making it model bell, U, or L-shaped data over unit interval. Use for 0–1 bounded histograms or frequency data.

Other and Hybrid Models

Cumulative Normal (erf) S-curve (y = a*0.5*(1+erf((x-μ)/(σ√2))) + b)

   An S-curve modeled after the cumulative distribution function of the normal (Gaussian) distribution. Suitable for error modeling, transition probability, or percent-within-threshold data. Used for cumulative response, coverage, or tolerance curves in physics, bioassay, and statistics. Recognize from smooth, sigmoidal transition with symmetric midrise.

Log-Logistic (Type III) (y = a / (1 + (x/x₀)^b ) + c)

   Models survival, failure rates, and financial/insurance heavy-tailed data where cumulative curves have a sigmoidal/non-symmetric S-shape. Explains “slow start, faster-than-logistic rise, then long tail” patterns. Used in reliability, economics, and lifespan analysis. Fits positive-valued, skewed data with non-exponential decay.