solving linear models. x y the two-parameter linear model
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Solving linear models
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x
y xbby10
The two-parameter linear model
0b
xy )0(
0yb
xyb
1
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Linear / statistically linear
• Linear model = fit a straight line
• Statistically linear = linear in the parameters Ex.
2
210xbxbby
xbby ln10
)2sin()sin(210
xbxbby
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Residual term
x
y
xbby10
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modelling methods
minimize
• least square sum
• sum of residuals
• minmax
N
ii
0
2
N
ii
0
ini
,...,1max
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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
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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 ,
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1. Ruskeepää, H.: Mallintamisen perusteet
2. Lawson, C. L., Hanson, R.J.: Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1974