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From Chris Nix <chris....@gmail.com>
Subject Re: [math] Pivoting in QR decomposition
Date Sat, 23 Jul 2011 19:45:43 GMT
We can do this with the current Householder reflection implementation,
except instead of just obtaining reflections from columns in sequence across
the input matrix, we select the column with the greatest L2-norm at each
iteration.  The resulting permutation matrix is thus built and can be
returned later with a getP() method.  A pleasing by-product is that the
resulting R matrix is 'rank-revealing', allowing for a quicker getRank()
than currently exists in the SingularValueDecompositionImpl class.

Does it make sense to extend the current QRDecomposition interface to one
for rank-revealing QR decompositions that have the existing methods, plus a
getP() and getRank()?  The implementing class could extend the current
QRDecompositionImpl class to save reproducing code, at the cost of opening
up some private variables and methods, and the solver.

I'll open a JIRA issue.

Chris.

On 23 July 2011 18:44, Greg Sterijevski <gsterijevski@gmail.com> wrote:

> Chris, you had an algorithm in mind? -Greg
>
> On Sat, Jul 23, 2011 at 11:29 AM, Phil Steitz <phil.steitz@gmail.com>
> wrote:
>
> > On 7/22/11 11:40 AM, Greg Sterijevski wrote:
> > > Sorry,
> > >
> > > public ConstrainedOLSMultipleRegression extends OLSMultipleRegression{}
> > >
> > > should read:
> > >
> > > public ConstrainedOLSMultipleRegression extends OLSMultipleRegression{
> > >             @Override
> > >     public void newSampleData(double[] data, double[][] coeff, double[]
> > rhs,
> > > int nob, int nvars) {
> > >        adjustData( data,  coeff, rhs);
> > >        super.newSampleData(data, nobs, nvars);
> > >         qr = new QRDecompositionImpl(X);
> > >     }
> > >
> > >> }
> > > The data would be transformed on the way in, and everything else would
> > > remain the same...
> >
> > Got it.  Sounds good.  Patch away...
> >
> > Couple of things to keep in mind:
> >
> > 0) We may want to dispense with the QRDecomposition interface
> > altogther.  If we keep it, we should trim it down to what is common
> > and meaningfully implemented in all impls.  So both the Householder
> > and permutation getters are dropped.  If you need a pivoting impl,
> > go a head and code it up and we can reassess the interface.
> >
> > 1) We should be aiming to standardize the regression API.  Lets pick
> > up the other thread on regression refactoring.  Before hacking too
> > much more on OLS, lets refine and retrofit the new regression API on
> > this class.
> >
> > Phil
> > >
> > >
> > > On Fri, Jul 22, 2011 at 1:37 PM, Greg Sterijevski <
> > gsterijevski@gmail.com>wrote:
> > >
> > >> On the need for pivoting:
> > >>
> > >> Here is my first approach for changing OLSMultipleRegression to do
> > >> constrained estimation:
> > >>
> > >>     public double[] calculateBeta(double[][] coeff, double[] rhs) {
> > >>         if (rhs.length != coeff.length) {
> > >>             throw new IllegalArgumentException("");
> > >>         }
> > >>         for (double[] rest : coeff) {
> > >>             if (rest.length != this.X.getColumnDimension()) {
> > >>                 throw new IllegalArgumentException("");
> > >>             }
> > >>         }
> > >>         RealMatrix Coeff = new Array2DRowRealMatrix(coeff, false);
> > >>         RealVector rhsVec = new ArrayRealVector(rhs);
> > >>         QRDecomposition coeffQRd = new
> > >> QRDecompositionImpl(Coeff.transpose());
> > >>         RealMatrix Qcoeff = coeffQRd.getQ();
> > >>         RealMatrix R = X.multiply(Qcoeff);
> > >>
> > >>         final int nvars = X.getColumnDimension();
> > >>         final int nobs = X.getRowDimension();
> > >>         final int ncons = coeff.length;
> > >>
> > >>         RealMatrix R2 = R.getSubMatrix(
> > >>                 0, nobs - 1, ncons, nvars - 1);
> > >>
> > >>         RealMatrix R1 = R.getSubMatrix(
> > >>                 0, nobs - 1, 0, ncons - 1);
> > >>
> > >>         RealVector gamma = rhsVec.copy();
> > >>
> > >>         RealMatrix coeffR = coeffQRd.getR().getSubMatrix(
> > >>                 0, ncons - 1, 0, ncons - 1);
> > >>
> > >>         MatrixUtils.solveLowerTriangularSystem(coeffR.transpose(),
> > gamma);
> > >>
> > >>         RealVector gammPrime = Y.subtract(R1.operate(gamma));
> > >>
> > >>         QRDecomposition qr2 = new QRDecompositionImpl(R2);
> > >>
> > >>         RealVector constrainedSolution =
> > >> (qr2.getSolver().solve(gammPrime));
> > >>
> > >>         RealVector stackedVector =
> > >>                 new ArrayRealVector(
> > >>                 gamma.toArray(),
> > >>                 constrainedSolution.toArray());
> > >>
> > >>         stackedVector = Qcoeff.operate(stackedVector);
> > >>
> > >>         return stackedVector.toArray();
> > >>     }
> > >>
> > >> This approach is based on Dongarra et al:
> > >>
> > >> LAPACK Working Note
> > >> Generalized QR Factorization and its Applications
> > >> Work in Progress
> > >> E. Anderson, Z. Bai and J. Dongarra
> > >> December 9, 1991
> > >> August 9, 1994
> > >>
> > >> There is nothing terrible about this approach, the coding is not
> > finished
> > >> and tidy, but its a work in progress.
> > >>
> > >> I am also aware of second approach. I do not have a cite for it, I
> think
> > I
> > >> may have derived it myself, but it would not surprise me if it is in
> > some
> > >> textbook somewhere... That second approach takes the QR decomposition
> of
> > the
> > >> coefficient matrix and calculates adjustment matrices for the design
> > matrix
> > >> and dependent vector. The problem is that I need to reorganize the
> > design
> > >> matrix by the pivots of the QR decomposition. Once I have the
> adjustment
> > >> matrices, everything should proceed as in the case of an unconstrained
> > >> estimation. I like the idea that if we transform the data, everything
> > works
> > >> the same way.
> > >>
> > >> Since then the ConstrainedOLSMultipleRegression class looks like:
> > >> public ConstrainedOLSMultipleRegression extends OLSMultipleRegression{
> > >>
> > >> }
> > >>
> > >>
> > >> As for the fact that the QRDecompositionImpl reflects its interface.
> We
> > >> should probably add the functions:
> > >>  public int[] getPivots();
> > >>  public boolean isPivotting();
> > >>
> > >> to the interface. As Christopher pointed out, if the current
> > decomposition
> > >> is non pivoting, its pivot record is the canonical one,
> {0,1,2,...,n-1}.
> > >>
> > >> -Greg
> > >>
> > >>
> > >>
> > >>
> > >>
> >
> >
> > ---------------------------------------------------------------------
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> >
> >
>

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