NNLS (Lawson-Hanson) method in linearized models

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## NNLS (Lawson-Hanson) method in linearized models

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**LSI & NNLS**• LSI = Least square with linear equality constraints • NNLS = nonnegative least square**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**Stopping condition**• Start of the main loop • Dual vector • Stopping condition: set Z is empty or**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**Compute subproblem**• Start of the inner loop • Subproblem where column j of Ep**Nonnegativity conditions**• If z satisfies nonnegativity conditions then we set x:=z and jump to stopping condition • else continue**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