1. Multiple linear regression
dependence on more than one variable
e.g. dependence of runoff volume on soil type and land cover

2. With two independent variables, get a surface

3. Much like polynomial regression
Sum of squared residuals

4. Rearrange to get
Very much like normal equations for polynomial regression

5. Once again, solve by any matrix method
Cholesky is appropriate - symmetric and positive definite

6. Example: Strength of concrete depends on cure time and cement/water ratio

7. Samples

9. Solve by Cholesky decomposition
Backsubstitution

10. General least squares
Given
z are functions, e.g

11. Can express as
and define Sr

12. As usual, take partials to minimize
lead to matrix equations
Solve this for [a]
Cholesky
LU or Gauss elimination
Matrix inverse

13. Confidence intervals
If we say the elements of are
then

14. Use Excel to get t-distribution
TINV(a,n-2)

15. Nonlinear regression
Use Taylor series expansion to linearize original equation - Gauss Newton approach
Let model be
Where f is a nonlinear function of x
are one of a set of n observations

16. Use Taylor series for f, and chop
j - initial guess
j+1 - improved guess

17. Plug the Taylor series into original equation

18. Given all n equations
Set up matrix equation

19. Where

20. Using same least squares approach (minimizing sum of squares of residuals E)
Get
from
Now change
with
and do
again until convergence is reached

21. Example:
n=14

22. Model it with

23. Choose an initial a0=1, a1=-1
Matlab demo