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Curve Fitting Basics
Curve fitting is the task of finding relationships, if any, that exist between a quantitative response and one or more explanatory variables, primarily to predict or obtain insights into the mechanisms responsible for the observed responses. If the plot of the experimental data suggests a relationship, it is curve fitting's endeavor to express this relationship with a model in the form of a function.
The set of experimental observations can often be characterized by fitting the data to a model dependent upon unknown adjustable coefficients. The model can be a straight line, or perhaps an nth order polynomial, where its coefficients serve simply to represent the discrete data in a continuous fashion for the purpose of interpolation.
On the other hand, the function's coefficients may have some theoretical relationship to a physical, biochemical or physiological process. For example; the maximum physiological effect of a drug in a sigmoidal response; the rate coefficient in a kinetic equation describing a chemical reaction; or the rate of growth in a biological phenomena as described by an exponential curve.

How SlideWrite's curve fitting options work.

SlideWrite offers two types of curve fitting so you can get results quickly and easily.
 •Quick curve fitting – automatically fits the data with a curve of a standard form.
 •Advanced curve fitting – allows a choice of linear, non–linear and user–defined curve fits plus the ability to tailor the fit.
Quick Curve Fitting is available in the Chart – Curves, Equations and Error Bars menu. These types of fits are most useful for identifying general trends within the data. The spline fit can be used to draw a smooth curve that passes through each point. When you select a Curve Fit, SlideWrite derives an approximating function using a least squares regression.
The curve fits available are:
 •Linear Y = a1 X + a0
 •Exponential Y = a0 e ^ (a1 X)
 •Log Y = a0 + a1 ln x
 •Power Y = a0 X ^ (a1)
 •Polynomial Y = a0 + a1 X + a2 X^2 +... + a6 X^6
 •Spline (no coefficients reported; requires at least 4 data points)
Advanced Curve Fitting is available using the Math – Curve Fitter option. The Curve Fitter uses statistical methods to determine the "best" equation which fits a set of observations. Both linear and nonlinear algorithms are employed to enable you to describe a statistical relationship between graph series. The Curve Fitter can fit both 2–dimensional (X,Y) and 3–dimensional (X,Y,Z) data. The mode of the Curve Fitter (2D or 3D) will depend on the current Graph Type selected under Chart.
 •2-D Mode – Choose a linear, nonlinear, or user–defined fit of the form y=f(x).
 •3-D Mode – Choose a user defined fit of the form z=f(x,y).

Nonlinear Curve Fits in 3 Easy Steps

SlideWrite can fit a complex, nonlinear curve to your data quickly and easily.
It takes just minutes to:
  1. Set up a curve fit for your data set
  2. Calculate it and
  3. Plot it on your graph.
Let us show you this easy 3-step process:
1. Enter or import your data and select the Curve Fitter option from the Math menu


2. Next select the Nonlinear Fit option

.. and then choose your nonlinear fitting functions to be tried.
   SlideWrite will calculate the best fit for you automatically.


3. Then you can view the fit statistics.

...save the fit equation for graphing and

...plot your results.

Choosing a Nonlinear Function

Many users of SlideWrite have inquired about the general shape of the built-in nonlinear functions supported by SlideWrite's CurveFitter. Here is a brief overview of the different functions available in SlideWrite. Remember that you can also enter your own functions or modify an existing function to get a tailor-made fit. Just select the User-defined Fit Option in the Curvefitter to enter your own custom equation and starting coefficients.
S-shaped transition functions:

There are four transition functions: Sigmoidal, Logistic Dose Response, Cumulative, and a pH-Activity function.

The four coefficients in the Sigmoidal, Logistic Dose Response,and Cumulative functions govern the y-offset, height, center, and width of the S-shape. When viewed with equal widths at 25% and 75% of total height, the differences in the shapes of the curves becomes apparent.

The Logistic Dose Response is an asymmetric transition function. The Cumulative function has the steepest overall transition, followed by the Sigmoidal and the Logistic Dose Response.

The pH-Activity function addresses the dissociation of a weak acid, and has a transition that is both sharp and symmetrical. The three coefficients in this function represent y at low pH, y at high pH and the log of the dissociation constant.
Waveform Functions - Sine and Sine2:

Sine2 function, with the same coefficients as a Sine function, is always positive with twice the frequency of the Sine function.
Peak Functions:

There are four peak functions: Logistic, Erfc, Gaussian, Lorentzian, and Log-Normal.

All but the Log-Normal function are symmetrical in nature. When viewed with equal widths at 50% of the height, we can see that the Lorentzian function has the widest tail, followed by Logistic, then Gaussian, and finally Erfc. The asymmetry of the Log-Normal function is obvious.
Hyperbolic, Exponential and Power Functions:

Photosynthesis Rate and I-Site Ligand/Michaelis-Menten are simply different expressins of the rectangular hyperbola.

The more common exponential decay, exponential growth, power functions are also shown here.

Note: For all functions, the sign of certain coefficients can alter the trend of the curve. For example, the sigmoidal curve shown above has an equation of the form:
Y=a0+al/(1+exp(-(x-a2)/a3))

If we change the sign of coefficient a3, we will see a curve which decreases from left to right.