1. NNLS (Lawson-Hanson) method in linearized models

2. LSI & NNLS LSI = Least square with linear equality constraints NNLS = nonnegative least square

3. Flowchart

4. 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

5. Flowchart

6. Stopping condition Start of the main loop Dual vector Stopping condition: set Z is empty or

7. Flowchart

8. 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

9. Flowchart

10. Compute subproblem Start of the inner loop Subproblem where column j of E p

11. Flowchart

12. Nonnegativity conditions If z satisfies nonnegativity conditions then we set x:=z and jump to stopping condition else continue

13. Flowchart

14. 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.

15. Testing the algorithm Ex. Values of polynomial are calculated at points x=1,2,3,4 with fixed p 1 and p 2. Columns of E hold the values of polynomial y(x)=x and polynomial at points x=1,2,3,4. Values of p 1 and p 2 are estimated with NNLS.

16. nnls_test 0.1 (c) 2003 by Turku PET Centre Matrix E: 1 2 4 3 9 4 16 Vector f: 0.6 2.2 4.8 8.4 Result vector:0.1 0.5

17. 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

18. nnls_test 0.1 (c) 2003 by Turku PET Centre Matrix E: 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

19. 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

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