# NNLS (Lawson-Hanson) method in linearized models - PowerPoint PPT Presentation

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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

### 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