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

With two independent variables, get a surface

Much like polynomial regression Sum of squared residuals

Rearrange to get Very much like normal equations for polynomial regression

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

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

Samples

Solve by Cholesky decomposition Backsubstitution

General least squares Given z are functions, e.g

Can express as and define Sr

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

Confidence intervals If we say the elements of are then

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

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

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

Plug the Taylor series into original equation

Given all n equations Set up matrix equation

Where

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

Example: n=14

Model it with

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