Predefined Nonlinear Curves

Defining a nonlinear curve and calculating the partial derivatives can get quite complicated. For this reason, Numerics.NET provides a series of predefined nonlinear curves. Each class implements the ValueAt, SlopeAt, and FillPartialDerivatives methods.

The table below lists the predefined non-linear curve types, organized by their primary applications:

Predefined nonlinear curves

Category	Class	Description
Oscillatory & Periodic	SineCurve	A 4-parameter periodic sine curve.
	DampedSineCurve	A 5-parameter damped sinusoidal curve for oscillatory decay.
Peak / Line-Shape Models	GaussianCurve	A 4-parameter Gaussian 'bell' curve for symmetric peaks.
	LorentzCurve	A 4-parameter Lorentzian peak curve.
	LogNormalCurve	A 4-parameter log-normal distribution for skewed peaks.
	ExponentiallyModifiedGaussianCurve	A 5-parameter curve for modeling skewed peaks in chromatography.
	PseudoVoigtCurve	A 5-parameter mixed Gaussian-Lorentzian curve for spectroscopy.
Logistic & Dose–Response (Sigmoids)	FourParameterLogisticCurve	The 4-parameter logistic function for dose-response curves.
	FiveParameterLogisticCurve	The 5-parameter logistic function with asymmetry factor.
	HillEquationCurve	A 3-parameter curve for cooperative binding.
	DoubleSigmoidCurve	A 6-parameter curve for processes that rise then fall.
Growth / Survival (CDF-like Sigmoids)	GompertzCurve	A 4-parameter asymmetric growth curve.
	RichardsCurve	A 5-parameter flexible growth curve.
	NormalCdfCurve	A 4-parameter normal CDF curve for step-like transitions.
	WeibullCdfCurve	A 4-parameter cumulative Weibull distribution for reliability analysis.
Saturation & Kinetics	MichaelisMentenCurve	A 2-parameter curve for enzyme kinetics and saturation.
Monotonic Scaling & Decay Laws	ExponentialCurve	A sum of exponential terms.
	PowerCurve	A 3-parameter power-law curve for scaling phenomena.
	RationalCurve	A quotient of two polynomials.

These classes reside in the Numerics.NET.Curves.Nonlinear namespace. A more detailed description of each curve type is given below. In the formulas, a parameter a_i represents the parameter with index i in the curve's Parameters collection. For some curves, the individual parameters have a specific meaning, and the class implementing the curve has properties with descriptive names for these parameters. These properties are also listed where available.

Sine Curve

A sine curve represents a periodic function.

The equation is:

f (x; y_{0}, x_{0}, T, A) = y_{0} + A \sin (\frac{π (x - x_{0})}{T})

A sine curve is implemented by the SineCurve class and takes four parameters:

Sine curve parameters

Index	Parameter	Property	Description
0	y₀	Offset	The mean value of the curve. The curve oscillates around this value.
1	x₀	Center	The starting point of the oscillation.
2	T	Period	The period of the curve.
3	A	Amplitude	The amplitude of the curve.

Damped Sine Curve

A damped sine wave curve combines oscillatory behavior with exponential decay.

The equation is:

f (x; A, k, T, ϕ, C) = A e^{- k x} \sin (\frac{2 π x}{T} + ϕ) + C

A damped sine wave curve is implemented by the DampedSineCurve class and takes five parameters:

Damped sine wave curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Initial amplitude (amplitude at x = 0).
1	k	DampingCoefficient	Damping coefficient (rate of exponential decay).
2	T	Period	Period (determines oscillation rate).
3	$ϕ$	PhaseOffset	Phase offset (horizontal shift of the oscillation).
4	C	Offset	Baseline offset (equilibrium position).

Gaussian Curve

A Gaussian curve is one type of curve that is used to represent functions with a peak shape:

The equation is:

f (x; y_{0}, μ, w, A) = y_{0} + \frac{A \sqrt{2}}{w \sqrt{π}} \exp (- 2 {(\frac{x - μ}{w})}^{2})

A Gaussian curve is implemented by the GaussianCurve class and takes four parameters:

Gaussian curve parameters

Index	Parameter	Property	Description
0	y₀	Offset	The offset (value as x goes to infinity).
1	$μ$	Center	The location of the peak.
2	w	Width	The width of the peak halfway between the baseline and the top of the peak.
3	A	Area	The area between the curve and a horizontal line at y = Offset.

In addition, the GaussianCurve class has an Amplitude property that returns the difference between the function value at the peak and the baseline (Offset).

Lorentz Curve

A Lorentz curve is used to represent functions with a peak shape. The peak is sharper than the Gaussian curve.

The equation is:

f (x; y_{0}, μ, w, A) = y_{0} + \frac{2 A}{π w} \frac{1}{1 + 4 {(\frac{x - μ}{w})}^{2}}

A Lorentz curve is implemented by the LorentzCurve class and takes four parameters:

Lorentz curve parameters

Index	Parameter	Property	Description
0	y₀	Offset	The offset (value as x goes to infinity).
1	$μ$	Center	The location of the peak.
2	w	Width	The width of the peak halfway between the baseline and the top of the peak.
3	A	Area	The area between the curve and a horizontal line at y = Offset.

In addition, the LorentzCurve class has an Amplitude property that returns the difference between the function value at the peak and the baseline (Offset).

Log-Normal Curve

A log-normal curve represents a right-skewed distribution commonly used in many fields. It is the probability density function of a log-normal distribution.

The equation is:

f (x; A, μ, σ, C) = \frac{A}{x σ \sqrt{2 π}} \exp (- \frac{(\ln x - μ)^{2}}{2 σ^{2}}) + C

A log-normal curve is implemented by the LogNormalCurve class and takes four parameters:

Log-normal curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Amplitude or scaling factor.
1	$μ$	Median	Location parameter (median, mean of the logarithm).
2	$σ$	Scale	Scale parameter (standard deviation of the logarithm).
3	C	Offset	Offset offset.

Exponentially Modified Gaussian Curve

An Exponentially Modified Gaussian curve models skewed peaks commonly seen in chromatography.

The equation is:

f (x; A, μ, σ, τ, C) = \frac{A}{2 τ} \exp (\frac{σ^{2}}{2 τ^{2}} - \frac{x - μ}{τ}) erfc (\frac{1}{\sqrt{2}} [\frac{σ}{τ} - \frac{x - μ}{σ}]) + C

An Exponentially Modified Gaussian curve is implemented by the ExponentiallyModifiedGaussianCurve class and takes five parameters:

Exponentially modified Gaussian curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Peak amplitude (total area under the curve).
1	$μ$	Center	Location parameter (relates to peak center).
2	$σ$	Width	Width parameter (Gaussian standard deviation).
3	$τ$	DecayTime	Exponential relaxation time (controls tailing).
4	C	Offset	Offset offset.

Pseudo-Voigt Curve

A Pseudo-Voigt curve is a linear combination of Gaussian and Lorentzian profiles used in spectroscopy.

The equation is:

f (x; A, μ, w, η, C) = A [(1 - η) G (x; μ, w) + η L (x; μ, w)] + C

where G is a normalized Gaussian and L is a normalized Lorentzian. A Pseudo-Voigt curve is implemented by the PseudoVoigtCurve class and takes five parameters:

Pseudo-Voigt curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Peak amplitude (total area under the peak).
1	$μ$	Center	Peak center (position of maximum).
2	w	Width	Peak width (full width at half maximum).
3	$η$	MixingParameter	Mixing parameter (0 = pure Gaussian, 1 = pure Lorentzian).
4	C	Offset	Offset offset.

Four Parameter Logistic Curve

A logistic curve is commonly used to represent growth processes and dose-response relationships.

The equation is:

f (x; A_{1}, A_{2}, x_{0}, p) = A_{2} + \frac{A_{1} - A_{2}}{1 + (x / x_{0})^{p}}

A logistic curve is implemented by the FourParameterLogisticCurve class and has four parameters:

4-parameter logistic curve parameters

Index	Parameter	Property	Description
0	A₁	InitialValue	The upper asymptote (value as x approaches zero).
1	A₂	FinalValue	The lower asymptote (value as x approaches infinity).
2	x₀	Center	The inflection point (halfway point in the transition from InitialValue to FinalValue).
3	p	HillSlope	The Hill slope (steepness parameter) of the curve.

Five Parameter Logistic Curve

The four-parameter logistic curve is symmetrical in that the change from the initial value and the change towards the final value occur at roughly the same rate. When the data is not symmetrical in this way, the fitted curve may not be a good fit. The five-parameter logistic curve adds an asymmetry parameter that remedies this situation.

The equation is:

f (x; A_{1}, A_{2}, x_{0}, p, s) = A_{2} + \frac{A_{1} - A_{2}}{(1 + (x / x_{0})^{p})^{s}}

The 5 parameter logistic curve is implemented by the FiveParameterLogisticCurve class and has five parameters:

5-parameter logistic curve parameters

Index	Parameter	Property	Description
0	A₁	InitialValue	The upper asymptote (value as x approaches zero).
1	A₂	FinalValue	The lower asymptote (value as x approaches infinity).
2	x₀	Center	The inflection point (halfway point in the transition from InitialValue to FinalValue).
3	p	HillSlope	The Hill slope (steepness parameter).
4	s	AsymmetryFactor	The asymmetry factor (s = 1 reduces to 4PL).

Hill Curve

A Hill curve models cooperative binding and sigmoidal dose-response relationships.

The equation is:

f (x; V_{m a x}, K, n) = \frac{V_{m a x} \cdot x^{n}}{K^{n} + x^{n}}

A Hill curve is implemented by the HillEquationCurve class and takes three parameters:

Hill curve parameters

Index	Parameter	Property	Description
0	V_max	VMax	Maximum response (asymptotic value as x → ∞).
1	K	K	Half-maximal concentration (x value at which y = V_max/2).
2	n	HillCoefficient	Hill coefficient (cooperativity parameter).

Double Sigmoid Curve

A double sigmoid curve models processes that rise and then fall, creating a bell-shaped response.

The equation is:

f (x; A, k_{1}, x_{1}, k_{2}, x_{2}, C) = A [\frac{1}{1 + e^{- k_{1} (x - x_{1})}}] [1 - \frac{1}{1 + e^{- k_{2} (x - x_{2})}}] + C

A double sigmoid curve is implemented by the DoubleSigmoidCurve class and takes six parameters:

Double sigmoid curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Maximum amplitude (peak height above baseline).
1	k₁	RisingHillSlope	Hill slope of the rising phase.
2	x₁	RisingCenter	Midpoint of the rising phase.
3	k₂	FallingHillSlope	Hill slope of the falling phase.
4	x₂	FallingCenter	Midpoint of the falling phase.
5	C	Offset	Offset offset.

Gompertz Curve

A Gompertz curve models asymmetric growth processes.

The equation is:

f (x; A, b, c, C) = A \exp (- b e^{- c x}) + C

A Gompertz curve is implemented by the GompertzCurve class and takes four parameters:

Gompertz curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Asymptotic amplitude (maximum value minus baseline).
1	b	Displacement	Displacement along the x-axis.
2	c	GrowthRate	Growth rate (controls the steepness).
3	C	Offset	Offset offset (lower asymptote).

Richards Curve

A Richards curve provides flexible modeling of growth processes with adjustable asymmetry.

The equation is:

f (x; A, ν, k, x_{0}, C) = A {(1 + ν e^{- k (x - x_{0})})}^{- 1 / ν} + C

A Richards curve is implemented by the RichardsCurve class and takes five parameters:

Richards curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Upper asymptote (maximum value minus baseline).
1	$ν$	Shape	Shape parameter (controls asymmetry).
2	k	GrowthRate	Growth rate parameter.
3	x₀	InflectionPoint	Inflection point location.
4	C	Offset	Offset offset (lower asymptote).

Normal CDF Curve

A normal CDF curve models smooth step-like transitions. It is the cumulative distribution function of the normal distribution.

The equation is:

f (x; A, μ, σ, C) = A \cdot \erf (\frac{x - μ}{σ \sqrt{2}}) + C

A normal CDF curve is implemented by the NormalCdfCurve class and takes four parameters:

Error function curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Amplitude (half the total change from minimum to maximum).
1	$μ$	Center	Center or inflection point (location of the transition).
2	$σ$	Width	Width parameter (controls the steepness of the transition).
3	C	Offset	Offset offset (vertical shift).

Weibull CDF Curve

A Weibull CDF curve models reliability and survival data. It is the cumulative distribution function of the Weibull distribution.

The equation is:

f (x; A, λ, k, C) = A (1 - e^{- (x / λ)^{k}}) + C

A Weibull CDF curve is implemented by the WeibullCdfCurve class and takes four parameters:

Weibull CDF curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Amplitude (total change from baseline to asymptote).
1	$λ$	Scale	Scale parameter (characteristic life or 63.2nd percentile).
2	k	Shape	Shape parameter (determines failure rate behavior).
3	C	Offset	Offset offset.

Michaelis-Menten Curve

A Michaelis-Menten curve models enzyme kinetics and saturation processes.

The equation is:

f (x; V_{m a x}, K_{m}) = \frac{V_{m a x} \cdot x}{K_{m} + x}

A Michaelis-Menten curve is implemented by the MichaelisMentenCurve class and takes two parameters:

Michaelis-Menten curve parameters

Index	Parameter	Property	Description
0	V_max	VMax	Maximum reaction rate (asymptotic value as x → ∞).
1	K_m	Km	Michaelis constant (substrate concentration at half-maximum rate).

Exponential Curves

An exponential curve is a curve that involves one or more exponential functions e^ax. There is immense variety in the exact definition of different exponential curves. The ExponentialCurve class represents a curve that is the sum of one or more exponential terms:

The equation is:

f (x; a_{0}, b_{0}, a_{1}, b_{1}, \dots, a_{n - 1}, b_{n - 1}) = \sum_{i = 0}^{n - 1} a_{i} e^{b_{i} x}

The constructor takes one integer parameter that specifies the number of terms in the curve. The curve has 2n parameters, where n is the number of terms.

Power Curve

A power curve models scaling relationships and self-similar phenomena.

The equation is:

f (x; A, α, C) = A x^{- α} + C

A power curve is implemented by the PowerCurve class and takes three parameters:

Power curve parameters

Index	Parameter	Property	Description
0	A	Amplitude	Amplitude or scaling coefficient.
1	$α$	Exponent	Power-law exponent (often positive for decay).
2	C	Offset	Offset offset.

Rational Curves

A rational curve is a curve that is the quotient of two polynomials. Multiplying both polynomials by a constant gives the same curve. To resolve this ambiguity, the constant term of the polynomial in the denominator is set to 1. A rational curve therefore has n+d+1 parameters, where n is the degree of the polynomial in the numerator, and d is the degree of the polynomial in the denominator:

The equation is:

f (x; a_{0}, a_{1}, \dots, a_{n}, b_{1}, b_{2}, \dots, b_{d}) = \frac{a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}}{1 + b_{1} x + b_{2} x^{2} + \dots + b_{d} x^{d}}

The RationalCurve class implements this type of curve. The constructor takes two integer parameters: the degree of the polynomial in the numerator, and the degree of the polynomial in the denominator.