.. _modelsbuiltin:

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 <custommodels.html>`_ 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)