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Predefined Nonlinear Curves

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

The table below lists the pre-defined non-linear curve types:

Class	Description
ExponentialCurve	A sum of exponential terms ae^b.
FourParameterLogisticCurve	The 4-parameter logistic function.
FiveParameterLogisticCurve	The 5-parameter logistic function with asymmetry factor.
GaussianCurve	A 4-parameter Gaussian ('Bell') curve.
LorentzCurve	A 4-parameter Lorentz peak curve.
RationalCurve	A quotient of two polynomials.
SineCurve	A 4-parameter periodic curve.

Table 1. Predefined nonlinear curves

These classes reside in the Extreme.Mathematics.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.

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 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.

Four Parameter Logistic Curve

A logistic curve is commonly used to represent growth processes.

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

Index	Property	Description
0	InitialValue	The value of the curve as x goes to zero.
1	FinalValue	The value of the curve as x goes to infinity.
2	Center	The halfway point in the transition from `InitialValue` to `FinalValue`.
3	HillSlope	The Hill slope of the curve.

Table 2. 4-parameter logistic curve parameters

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

Index	Property	Description
0	InitialValue	The value of the curve as x goes to zero.
1	FinalValue	The value of the curve as x goes to infinity.
2	Center	The halfway point in the transition from `InitialValue` to `FinalValue`.
3	HillSlope	The Hill slope of the curve.
4	AsymmetryFactor	The asymmetry factor.

Table 3. 5-parameter logistic curve parameters

Gaussian Curve

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

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

Index	Property	Description
0	Offset	The value of the curve as x goes to infinity.
1	Center	The location of the peak.
2	Width	The width of the peak halfway between the baseline and the top of the peak.
3	Area	The area between the curve and a horizontal line at y = `Offset`.

Table 4. Gaussian curve parameters

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.

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

Index	Property	Description
0	Offset	The value of the curve as x goes to infinity.
1	Center	The location of the peak.
2	Width	The width of the peak halfway between the baseline and the top of the peak.
3	Area	The area between the curve and a horizontal line at y = `Offset`.

Table 5. Lorentz curve parameters

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

Sine Curve

A sine curve represents a periodic function.

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

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

Table 5. Sine curve parameters