The curve fitting platform allows you to select from a library of model types.
In this post I take a closer look at the different model types that are available to support curve fitting.
Each of the model categories contain a variety of models with differing numbers of parameters:
Polynomial Models
If you use linear regression (standard least squares) you will be familiar with this type of model:
[latex size=”1″]y=\ \beta_0+\ \beta_1\ x+\ \beta_2\ x^2+\ \beta_3\ x^3\ \ + [/latex] . . .
Whilst gradient descent algorithms can be used to estimate these parameters, the primary role of curve fitting is to fit parameters that form part of a nonlinear equation – typically representing some mechanistic model relating to a scientific application. All other model types fall into this category of nonlinear models.
Sigmoid / Logistic Curves
The basic sigmoid function takes the following form:
[latex size=”1″]y = [/latex] [latex size=”2″]\frac{1}{1 + e^{-x}}[/latex]
It characterises the case where an unbounded x variable is transformed into a y variable that is contained within a range 0 to 1. It is therefore particularly useful for modelling a response that represents a proportion.
The logistic function introduces one or more parameters to generalise the behaviour of this S-curve. For example, a parameter can be introduced to control the growth rate:
[latex size=”1″]y = [/latex] [latex size=”2″]\frac{1}{1 + e^{-a \cdot x}}[/latex]
The curve has a point of inflection at x=0. The introduction of a second parameter allows the location of this inflection point to be adjusted:
[latex size=”1″]y = [/latex] [latex size=”2″]\frac{1}{1 + e^{-a \cdot (x-b)}}[/latex]
This is the formula for the Logistic 2P model.
Whilst y is a continuous response, these types of model are often used to model a binary outcome (0 or 1). In this case the y value is interpreted as the probability of an outcome of 1 given a specified value of x.
The Logistic 3P model introduces a third parameter allowing the curve to have an upper asymptote other than 1:
[latex size=”1″]y = [/latex] [latex size=”2″]\frac{c}{1 + e^{-a \cdot (x-b)}}[/latex]
And the Logistic 4P model provides a description of both upper and lower asymptotes with parameters c and d:
[latex size=”1″]y = d + [/latex] [latex size=”2″]\frac{c-d}{1 + e^{-a \cdot (x-b)}}[/latex]
There is also a Logistic 5P model that allows the curve to be asymmetric about the inflection point:
[latex size=”1″]y = d + [/latex] [latex size=”2″]\frac{c-d}{[{1 + e^{-a \cdot (x-b)}}]^f}[/latex]
Sigmoid/Probit Curves
The logistic functions described earlier typically represent the case where the response is derived from the probability of a binary outcome. Alternatively, we can model the S-curve on the basis that it represents the cumulative distribution function Φ of a Normal distribution:
[latex size=”1″]y = \Phi [/latex][latex size=”2″](\frac{x-b}{a})[/latex]
where the parameter a represents the growth rate and b is the point of inflection. This is the Probit 2P model.
The Probit 4P model introduces parameters to control the lower and upper asymptotes:
[latex size=”1″]y = c+(d-c)\cdot\Phi [/latex][latex size=”2″](\frac{x-b}{a})[/latex]
Sigmoid/Gompertz Curves
The 5-parameter logistic model describes an S-shaped curve that is asymmetric about the inflection point. A Gompertz curve can be considered to be a special case of this model. As described in Wikipedia the model was first proposed as a description of human mortality.
[latex size=”2″]y = a\cdot e^{-e^{-b(x-c)}} [/latex]
A four-parameter model is also available that provides parameters for both lower and upper asymptotes.
Sigmoid/Weibull Growth Curve
Another S-shaped curve is the Weibull Growth model, often used in reliability engineering:
[latex size=”2″]y = a[1-e^{-(x/c)^b} ][/latex]
Where a is the upper asymptote, b is the growth rate, and c is the inflection point.
Exponential Growth and Decay
Exponential 2P is the basic exponential model:
[latex size=”2″]y = b\cdot e^{\lambda \cdot x}[/latex]
The parameter b is a scaling parameter and λ represents the growth rate. If λ is negative, then it represents the rate of decay.
The Exponential 3P model adds an additive term to control the asymptote of the curve:
[latex size=”2″]y = a+b\cdot e^{\lambda \cdot x}[/latex]
An alternative parameterisation is the mechanistic growth model:
[latex size=”2″]y = a(1-b\cdot e^{\lambda \cdot x})[/latex]
JMP also supports bi-exponential models. These models are the sum of two exponentials and appear as 4-parameter and 5-parameter models:
[latex size=”2″]y = a\cdot e^{-b \cdot x}+c\cdot e^{-d \cdot x}[/latex]
[latex size=”2″]y = a+b\cdot e^{-c \cdot x}+d\cdot e^{-e \cdot x}[/latex]
Cell Growth
Growth of cells in a bioreactor can be characterised by a number of phases:
JMP’s Cell Growth 4P model takes the form:
[latex size=”2″]y = \frac{ab}{(a-b) e^{-c \cdot x}+b e^{d \cdot x}}[/latex]
where:
a = peak value if mortality rate is zero
b = response at time zero
c = cell division rate
d = cell mortality rate
Peak Models
The bell-shaped curve associated with a Normal distribution is more generically described by a Gaussian function of the form:
[latex size=”2″]y = ae^{\frac{(x-b)^2}{2c^2}}[/latex]
The Lorentzian curve is superficially similar to the Gaussian bell-shape, but has heavier tails:
[latex size=”2″]y = \frac{ab^2}{(x-c)^2+b^2}[/latex]
Peak curves are used, for example, to model spectroscopic peaks.
For both models the parameter a corresponds to the maximum value of the peak; b represents the growth rate, and c is the critical point: the value of x where the curve reaches its maximum value.
Pharmacokinetic Models
Pharmocokinetic models seek to describe the kinetics of a drug once it has been administered into the body. The One Compartment Oral Dose model has the following parameterisation:
[latex size=”2″]y = \frac{abc}{c-b}(e^{-b\cdot x}-e^{-c\cdot x})[/latex]
where:
a = area under the curve
b = elimination rate
c = absorption rate
JMP also supports a Two Compartment IV Bolus Dose model, but that is beyond my latex skills!
Michaelis-Menten Model
Named after the biochemists Leonor Michaelis (1875-1949) and Maud Menten (1879-1960), this model is used to describe enzyme kinetics:
[latex size=”2″]y = \frac{ax}{b+x}[/latex]
The parameter a represents the maximum reaction rate (in literature often referred to as Vmax), and the b parameter (in literature often referred to as the Michaelis constant Km) is the value of x such that the response is half Vmax ; it is an inverse measure of the substrates affinity for the enzyme.
And There Is More
We are not limited to selecting from a pre-defined library of curve types. Any nonlinear function can be expressed as a column formula and fitted using the Nonlinear Platform. In fact it is one of my most frequently used platforms. But that is a topic for another day.