NON-LINEAR REGRESSION
(Statistics of Two Variables)
Many relationships between two variables follow patterns that are NOT linear.
Non-linear regression is an analytic technique for determining a curve of best
fit for data from such relationships.
The correlation of determination, r2, is a measure of how closely a curve fits
the data. It can be likened to the correlation coefficient, r, in linear regression.
THE CORRELATION OF DETERMINATION:
r
where
2

( y est  y) 2
( y  y) 2
y
is the mean value
yest is the estimated value for a given value of x
y is the actual value for a given value of x
The coefficient of determination can have values between 0 and 1, where 1 is a
perfect fit and 0 is a poor fit. The curve of best fit will be the one that has the
highest value for r2.
EXPONENTIAL and POWER REGRESSION
Exponential regressions produce equations with the form y = abx .
Power regressions produce equations with the form y = axb.
Example 
Time (h)
Population
A laboratory technician monitors the growth of a bacterial culture by
scanning it every hour and estimating the number of bacteria.
1
10
2
21
3
43
4
82
5
168
6
320
7
475
Using the graphing calculator, compare the data using linear, exponential and power
regression and determine the equation of the curve of best fit.
LinReg(ax+b)
PwrReg
ExpReg
y=ax + b
y=a*x^b
y=a*b^x
a=
a=
a=
b=
b=
b=
r2 =
r2 =
r2 =
Which regression gives a better coefficient of determination? ____________________
 ____________________________________ is the curve of best fit.
POLYNOMIAL REGRESSION (Quadratic or Cubic)
Polynomial regressions produce equations with the form y = ax2 + bx + c (quadratic)
or y = ax3 + bx2 + cx + d (cubic).
Example 
The laboratory technician takes further measurements of the bacterial
culture in example :
1
Time (h)
Population 10
2
21
3
43
4
82
5
6
7
8
9
10
11
12
13
14
168 320 475 630 775 830 980 1105 1215 1410
Using the graphing calculator, compare the data using exponential, power, and
polynomial regression.
ExpReg
PwrReg
QuadReg
CubicReg
y=a*b^x
y=a*x^b
y=ax2+bx+c
y=ax3+bx2+cx+d
a=
a=
a=
a=
b=
b=
b=
b=
r2 =
r2 =
c=
c=
r2 =
d=
r2 =
 _____________________________________________ is the curve of best fit.
Homework: p.191–193 #1, 2, 4, 5, 7
use graphing calculator