Nnls lawson hanson method in linearized models
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NNLS (Lawson-Hanson) method in linearized models. LSI & NNLS. LSI = Least square with linear equality constraints NNLS = nonnegative least square . Flowchart. Initial conditions. Sets Z and P Variables indexed in the set Z are held at value zero

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NNLS (Lawson-Hanson) method in linearized models

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NNLS (Lawson-Hanson) method in linearized models


LSI & NNLS

  • LSI = Least square with linear equality constraints

  • NNLS = nonnegative least square


Flowchart


Initial conditions

  • Sets Z and P

  • Variables indexed in the set Z are held at value zero

  • Variables indexed in the set P are free to take values different from zero

  • Initially and P:=NULL


Flowchart


Stopping condition

  • Start of the main loop

  • Dual vector

  • Stopping condition:

    set Z is empty or


Flowchart


Manipulate indexes

  • Based on dual vector, one parameter indexed in Z is chosen to be estimated

  • Index of this parameter is moved from set Z to set P


Flowchart


Compute subproblem

  • Start of the inner loop

  • Subproblem

    where column j of Ep


Flowchart


Nonnegativity conditions

  • If z satisfies nonnegativity conditions then we set x:=z and jump to stopping condition

  • else continue


Flowchart


Manipulating the solution

  • x is moved towards z so that every parameter estimate stays positive. Indexes of estimates that are zero are moved from P to Z. The new subproblem is solved.


Testing the algorithm

  • Ex. Values of polynomial

    are calculated at points x=1,2,3,4 with fixed p1 and p2.

  • Columns of E hold the values of polynomial y(x)=x and polynomial at points x=1,2,3,4.

  • Values of p1and p2 are estimated with NNLS.


nnls_test 0.1 (c) 2003 by Turku PET Centre

Matrix E:

1 1

2 4

3 9

4 16

Vector f:

0.6 2.2 4.8 8.4

Result vector:0.1 0.5


nnls_test 0.1 (c) 2003 by Turku PET Centre

Matrix E:

1 1 1

2 4 8

3 9 27

4 16 64

Vector f:

0.73 3.24 8.31 16.72

Result vector:0.1 0.5 0.13


nnls_test 0.1 (c) 2003 by Turku PET Centre

Matrix E:

1 1 1 1

2 4 8 16

3 9 27 81

4 16 64 256

Vector f:

0.73 3.24 8.31 16.72

Result vector:0.1 0.5 0.13 0


nnls_test 0.1 (c) 2003 by Turku PET Centre

Matrix E:

1 1 1

2 4 8

3 9 27

4 16 64

Vector f:

0.23 1.24 3.81 8.72

Result vector:0.1 7.26423e-16 0.13


  • Kaisa Sederholm: Turku PET Centre Modelling report TPCMOD0020 2003-05-23


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