Chemometrics

The most com­monly used func­tion to cal­cu­late the matrix rank is prob­a­bly the one in the MATLAB soft­ware pack­age [1]. It is based on the cal­cu­la­tion of sin­gu­lar val­ues of the matrix A, and the num­ber of sin­gu­lar val­ues that are greater than some tol­er­ance defines the rank. The algo­rithm is sim­ple: r = svd(A); …

The numer­i­cal esti­mate of matrix rank Read More »

Dur­ing 2011 year, I have tried many dif­fer­ent ways of mak­ing a PLS algo­rithm (PLS1) as sta­ble pos­si­ble. The goal was to make it as numer­i­cally good as the SVD imple­mented in MATLAB. The pseudocode that solved the PLS1 case for one y‑varable is shown below and it includes the addi­tion of null vec­tors to …

A numer­i­cally sta­ble PLS algo­rithm Read More »

There are many meth­ods to obtain null vec­tors to a given set of vec­tors. One of the numer­i­cally best meth­ods is imple­mented in MATLAB, based on sin­gu­lar value decom­po­si­tion, SVD. Null vec­tors can be used for many dif­fer­ent things; one of them is to remove prop­er­ties that are ore orthog­o­nal to another, given prop­erty. Such …

Numer­i­cally good null spaces Read More »

Please note that this is not regard­ing the Lanc­zos or the PLS bidi­ag­o­nal­iza­tion algo­rithm. This is about the House­holder bidi­ag­o­nal­iza­tion algo­rithm. The dif­fer­ence is huge. For exam­ple, there are no y‑variables in the House­holder decom­po­si­tion. It is a decom­po­si­tion of X that results in a fac­tor­iza­tion with a bidi­ag­o­nal matrix in the mid­dle, sim­i­lar to …

A fast House­holder bidi­ag­o­nal­iza­tion algo­rithm Read More »

I got a request to send the code of the nine algo­rithms that were com­pared in “A com­par­i­son of nine PLS1 algo­rithms” and pub­lished in the Jour­nal of Chemo­met­rics 2009; 23: 518–529 and led to the Kowal­ski price 2010 for best the­o­ret­i­cal paper. Then I thought it would be even bet­ter if they would be …

Nine algo­rithms Read More »