A Teaching Tool for Nonlinear Regression: Visual Fit

Figures & data

Table 1. List of Functions and Parameterizations Used in Visual Fit

Function	Parameterization	Description of Parameters
Linear	$f\left(x\right)=a+bx$	a - y-intercept b - Slope of the line
	$f(x)\text{}\text{}=\text{}a+b(x-\overline{x})$	a - Height of the function at $x=\overline{x}$ b - Slope of the line
Quadratic	$f(x)\text{}\text{}=\text{}a+b(x-\overline{x})+c{(x-\overline{x})}^2$	a - Height of the function at $x=\overline{x}$ b - Slope of the function at $x=\overline{x}$ c - Quadratic effect
Exponential	$f(x)\text{}\text{}=\text{}a(1-e^{-bx})$	a - Maximum height of the function b - Rate of approach towards a
	$f(x)\text{}\text{}=\text{}a(1-2^{-x/xh})$	a - Maximum height of the function xh - Value of x at which the function equals a/2
	$f(x)\text{}\text{}=\text{}a(1-e^{-dx/a})$	a - Maximum height of the function d - Slope of the function at origin
Exponential with Intercept	$f(x)\text{}\text{}=\text{}a(1-e^{-bx})+c$	a - Maximum height (from c) of the function b - Rate of approach towards a c - y-intercept
	$f(x)\text{}\text{}=\text{}a(1-2^{-x/xh})+c$	a - Maximum height (from c) of the function xh - Value of x at which the function equals a/2+c c - y-intercept
	$f(x)\text{}\text{}=\text{}a(1-e^{-dx/a})+c$	a - Maximum height (from c) of the function d - Slope of the function at origin c - y-intercept
Logistic	$f(x)\text{}\text{}=\text{}\frac{a}{1+e^{\frac{4m}{a}(x_h-x)}}$	a - Maximum height of the function xh - Value of x at which the function equals a/2 m - Slope of function at ^x ^= xh

Figure 1. Horizontal Reference Lines. Reference lines are drawn on the graph to help convey the meaning of the parameters.

Figure 2. Vertical Reference Lines. The vertical reference line indicates the distance represented by parameter a.

Figure 3. Representation of a Complicated Parameter. In this model, f(x) = a(1 - e-bx), the parameter b is described as a ratio between the lengths of the two pink lines.

Table 2. List of Java Classes Comprising Visual Fit

Class name	Description
VisualFit	Main body of program. Handles most user actions.
Function	Performs all of the actions specific for the data, the trial fit, and each regression model.
Graph	Manages the scatterplot graph. Also controls the mouse interface with parameter box controls.
ControlPanel	Manages the control panel next to the graph.
Convert	Converts back and forth between the graph’s continuous coordinates and the screen’s pixel coordinates.
Format	Controls number formatting (written by Cay Horstmann, Sun Microsystems, Inc.).
MessageArea	Controls the message panel on the side.
SSBar	Calculates and draws the sum of squares bar.
ArrayOp	A set of functions used for manipulating arrays.
VisualFitFrame	A frame that houses the applet in VisualFit. Only used when program is run as a stand-alone.
Generate	Performs all of the necessary random number generation.
Sort	Performs all of the necessary data sorting.

Figure 4. Start of the Visual Fit Applet.

Figure 5. Changing Model Parameters. Parameters may be controlled by dragging their corresponding parameter box controls.

Figure 6. Comparison With Least Squares. Users’ estimated parameters can be compared with least squares estimates.

Figure 7. A Dataset With an Outlier. Outliers and other irregularities may be generated.