Plotted data usually will not fit a linear model i.e. it is obvious when you plot your data on a scatter plot that a straight line is NOT the curve to draw.

For example: bacteria growing on a slice of bread ; bacteria grows at an exponential rate, so a scatter plot of time (x-axis) versus the number of bacteria on the bread (y-axis) would look as though an exponential curve should be drawn through the data points.



Non-linear regression: technique of finding a curve of best fit for data that is not described by a straight line.

Model: the name given to curves of best fit

Here are different models that can be fitted to non-linear data:

Model	Equation	Example
Quadratic	Y = ax² + bx + c	Distance fallen under influence of gravity
Cubic	Y = ax³ +bx² + cx + d
Exponential	Y = ab^x , a,b+constant Y = ae^x , e=2.718, a=constant	Bacterial growth
Power	Y = ax^b, a,b=constant

- For the curve above, you would apply an exponential curve: y= ae^x, where e = 2.718

- Now you must find the values of a and x (this has to be a guess)

- After "guessing" the values of a and x, you will then calculate how good your exponential function applies to your data points by using this equation:





- To calculate r², you would have to use ALL of your data's y-values. and for each of those y-values, you would include a Yfrom your equation

- So, it would definitely be best, if before calculating r², to conduct a table like this as to make thing much easier for you:



- r² has values from 0 to 1
- If the value is 1, your model is a perfect match with the data
- if the value is 0, your model is terrible and you must try again!
- so basically, the closer r is to 1, the better your model is

- r² is used for all types of curves: straight lines, quadratic, cubic, power, or even exponential
- do not be mistaken!! r is only for straight lines!!


LETS DO AN EXAMPLE:

The following table lists the number of bacteria produced from a single bacteria

T (hours)	Y (bacteria)
1	2
2	25
3	80
4	200
5	375

**By observing the scatter plot, it is believed that the
curve of best fit for this data is y= 3t. It will be
presumed that a value of r² > 0.7 will indicate
a strong correlation**

**** Is the chosen curve a good choice?
Prove it!



HERE IS THE SCATTER PLOT:




ANSWER:

In order to determine if the curve is a good choice, we must calculate r². We must find the different values that are in the equation.
























Overall, the curve was a good choice since r^2 > 0.7