Texas Instruments Titanium TI-89, User Manual

Page: 920 / 977

Appendix B: Technical Reference 917
Most of the regressions use non-linear recursive least-squares techniques
to optimize the following cost function, which is the sum of the squares of
the residual errors:
[ ]
J residualExpression
i
N
=
=
∑
1
2
where: residualExpression is in terms of x
i
and y
i
x
i
is the independent variable list
y
i
is the dependent variable list
N is the dimension of the lists
This technique attempts to recursively estimate the constants in the model
expression to make
J as small as possible.
For example, y=a sin(bx+c)+d is the model equation for SinReg . So its
residual expression is:
a sin(bx
i
+c)+d
ì
y
i
For SinReg , therefore, the least-squares algorithm finds the constants a, b,
c, and d that minimize the function:
[ ]
J a bx c d y
i i
i
N
= + + −
=
∑
sin ( )
2
1
Regression Description
CubicReg Uses the least-squares algorithm to fit the third-order
polynomial:
y=ax
3
+ bx
2
+ cx+d
For four data points, the equation is a polynomial fit; for
five or more, it is a polynomial regression. At least four
data points are required.
ExpReg Uses the least-squares algorithm and transformed values
x and ln(y) to fit the model equation:
y=ab
x
LinReg Uses the least-squares algorithm to fit the model equation:
y=ax+b
where a is the slope and b is the y-intercept.
Regression Formulas
This section describes how the statistical regressions are
calculated.
Least-Squares
Algorithm
Regressions