Solving linear models

x

y xbby10

The two-parameter linear model

0b

xy )0(

0yb

xyb

1

Linear / statistically linear

• Linear model = fit a straight line

• Statistically linear = linear in the parameters Ex.

2

210xbxbby

xbby ln10

)2sin()sin(210

xbxbby

Residual term

x

y

xbby10

modelling methods

minimize

• least square sum

• sum of residuals

• minmax

N

ii

0

2

N

ii

0

ini

,...,1max

Solving least square problems

Ex.

• Derivation of the object function

• QR decomposition of E

• Singular value decomposition (SVD) of E

• Nonnegative least square algorithm (NNLS)

n

ii

fEx1

2minminmin

Singular value decomposition

• E=USV, U and V are orthogonal and S is a diagonal matrix

• We get x=Vp

• Approximately as fast as e. g. NNLS

fUggSpfEx T ,

1. Ruskeepää, H.: Mallintamisen perusteet

2. Lawson, C. L., Hanson, R.J.: Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1974