Appendix: Built-In Nonlinear Regression Models

Nonlinear regression models come in all shapes and sizes – in CurveExpert, an attempt has been made to include as many models as possible, while at the same time providing relevant models to a wide range of applications. Also, keep in mind that if the model that you desire is not built-in to CurveExpert, you may define your own using the Define User Models facility. In CurveExpert, the nonlinear models has been divided into families based on their characteristic behavior. These families and their members are enumerated below:

Exponential Family

Exponential models have the exponential or logarithmic functions involved. They are generally convex or concave curves, but some models in this group are able to have an inflection point and a maximum or minimum.

• Exponential y=a*exp(b*x)
• Modified Exponential y = a*exp(b/x)
• Logarithm y = a+b*ln(x)
• Reciprocal Logarithm y = 1/(a+b*ln(x))
• Vapor Pressure Model y = exp(a+b/x+c*ln(x))

Power Family

The Power Family involves raising one or more parameters to the power of the independent variable, or raising the dependent variable to the power of a given parameter. This family is generally a set of convex or concave curves with no inflection points or maxima/minima.

• Power Fit y = a*x^b
• Modified Power y = a*b^x
• Shifted Power y = a*(x-b)^c
• Geometric y = a*x^(b*x)
• Modified Geometric y = a*x^(b/x)
• Root Fit y = a^(1/x)
• Hoerl Model y = a*(b^x)*(x^c)
• Modified Hoerl Model y = a*b^(1/x)*(x^c)

Yield-Density Models

The yield-density models are widely used, especially in agricultural applications. These models historically have been used to model the relationship between the yield of a crop and the spacing or density or planting. Essentially two types of response are observed in practice: the “asymptotic” and “parabolic” yield-density relations. If the response is such that as density (x) increases, but the yield (y) approaches a fixed value, the relationship is asymptotic. If the response is such that there is a distinct optimum as the density increases, the relationship is parabolic. Of course, these types of relationships occur commonly in other scientific areas; therefore, this family of models is very useful.

• Reciprocal Model y = 1 / (a + bx)
• Reciprocal Quadratic y = 1 / (a + bx + cx^2)
• Bleasdale Model y = (a + bx) ^ (-1/c)
• Harris Model y = 1 / (a + bx^c)

Growth Family

Growth models are characterized by a monotonic growth from some fixed value to an asymptote. These models are most common the engineering sciences.

• Exponential Association (2) y = a*(1-exp(-bx))
• Exponential Association (3) y = a*(b-exp(-cx))
• Saturation Growth y = ax / (b + x)

Sigmoidal Family

Processes producing sigmoidal or “S-shaped” growth curves are common in a wide variety of applications such as biology, engineering, agriculture, and economics. These curves start at a fixed point and increase their growth rate monotonically to reach an inflection point. After this, the growth rate approaches a final value asymptotically. This family is actually a subset of the Growth Family, but are separated in CurveExpert because of their distinctive behavior.

• Gompertz Model y = a * exp (-exp(b - cx))
• Logistic Model y = a / (1 + exp (b - cx))
• Richards Model y = a / (1 + exp(b - cx))^(1/d)
• MMF Model y = (ab + cx^d)/(b + x^d)
• Weibull Model y = a - b*exp(-cx^d)

Miscellaneous Family

As with many things in life, some things just don’t fit into nice categories. The miscellaneous family is the one in which these “different” nonlinear regression models live.

• Sinusoidal Fit y = a + b*cos(c*x + d)
• Gaussian Model y = a*exp((-(x - b)^2)/(2*c^2))
• Hyperbolic Fit y = a + b/x
• Heat-Capacity Model y = a + bx + c/x^2
• Rational Function y = (a + bx) / (1 + cx + dx^2)