NNLS LawsonHanson 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

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Flowchart
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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 PNULL

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Flowchart
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Stopping condition

• Start of the main loop
• Dual vector
• Stopping condition
• set Z is empty or

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Flowchart
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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

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Flowchart
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Compute subproblem

• Start of the inner loop
• Subproblem
• where column j of Ep

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Flowchart
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Nonnegativity conditions

• If z satisfies nonnegativity conditions then we
  set xz and jump to stopping condition
• else continue

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Flowchart
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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.

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Testing the algorithm

• Ex. Values of polynomial
• are calculated at points x1,2,3,4 with fixed
  p1 and p2.
• Columns of E hold the values of polynomial y(x)x
  and polynomial at points
  x1,2,3,4.
• Values of p1 and p2 are estimated with NNLS.

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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 vector0.1 0.5
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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
vector0.1 0.5 0.13
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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
vector0.1 0.5 0.13 0
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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 vector0.1
7.26423e-16 0.13
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• Kaisa Sederholm Turku PET Centre Modelling
  report TPCMOD0020 2003-05-23