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Zlamal__Jakub

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12-21-2016
10:30 PM

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Least squares - estimation of fit error

I am using routines DGEQPF, DORMQR and DTRSV (according to example in dgeqpfx.f) and it works well.

I also need to calculate estimation of errors of fitted parameters but I did not find any example.

Can someone give me advice how to calculate it?

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mecej4

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12-22-2016
05:46 AM

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The topic you ask about involves a combination of linear algebra, linear least squares and statistics, but it is not difficult to put the pieces together.

Let *c*_{0} be the "regression coefficients" or "fitted parameters" that you have found, and let *R* be the upper triangular matrix obtained from the Q-R factorization of the regressor matrix. The "confidence interval" for *c*, the true but unknown value of the regression coefficients, is given by *c*_{0} - *δ*_{c} < *c* < *c*_{0} + *δ*_{c}, where

* δ*_{c} = *t _{ν,α}*

*t* is the Student-*t* variable, with tail area *α*/2 and degrees-of-freedom *ν = **n - m* of the Student-*t* PDF,

*n* = number of observations,

*m* = number of regression coefficients,

*y*, *y _{f}* are the observed and fitted values of the dependent variable.

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mecej4

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12-22-2016
05:46 AM

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The topic you ask about involves a combination of linear algebra, linear least squares and statistics, but it is not difficult to put the pieces together.

Let *c*_{0} be the "regression coefficients" or "fitted parameters" that you have found, and let *R* be the upper triangular matrix obtained from the Q-R factorization of the regressor matrix. The "confidence interval" for *c*, the true but unknown value of the regression coefficients, is given by *c*_{0} - *δ*_{c} < *c* < *c*_{0} + *δ*_{c}, where

* δ*_{c} = *t _{ν,α}*

*t* is the Student-*t* variable, with tail area *α*/2 and degrees-of-freedom *ν = **n - m* of the Student-*t* PDF,

*n* = number of observations,

*m* = number of regression coefficients,

*y*, *y _{f}* are the observed and fitted values of the dependent variable.

Zlamal__Jakub

New Contributor III

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12-22-2016
06:29 AM

56 Views

Thank You very much. I did not found this equation for QR factorisation.

I have one another question. Routine DGEQPF permutes columns of orignial matrix. Then I assume that the values \delta_c will be permuted so. Is it true?

mecej4

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12-22-2016
09:38 AM

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Whether pivoting is used or not is a detail at the programming level, and depends on which Lapack routine is used for performing the QR decomposition.

If you use DGEQRF, pivoting is not used and the question goes away.

Please note that DGEQPF is now deprecated; use DGEQP3 instead. If pivoting is used, the permutation vector does need to be used in reporting *δ*_{c}. However, only the diagonal of *R ^{T}R* needs to be permuted, so doing so is not difficult, although it can appear tricky to do so at first look.

Zlamal__Jakub

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12-23-2016
11:25 AM

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Thanks for advice, I will try.

Zlamal__Jakub

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01-01-2017
07:31 AM

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I have finished my routine calculating LSQ fit and also standard deviations of fitted parameters. It is not well memory optimised, but seems to work.

subroutine DSOLVEQRErrorEstim_MKL(A,B,X,Xerr,m,n,tol,info) implicit none integer*4,intent(in) :: m,n ! m - number of equations, n - number of unknowns real*8,intent(in) :: A(m,n) ! matrix A (A*X=B) real*8,intent(inout) :: B(m,1) ! righthand side real*8,intent(out) :: X(n,1) ! fitted parameters real*8, intent(out) :: Xerr(n,1) ! standard deviations of fitted parameters real*8, intent(in) :: tol ! Choose TOL to reflect the relative accuracy of the input data integer*4, intent(inout) :: info ! result of MKL routines 0 if all is ok INTEGER MMAX, NMAX, LDA, LDB, LDX, NRHMAX, LWORK real*8,parameter :: ZERO=0.0D0 INTEGER I, IFAIL, J, K, NRHS real*8 TAU(N),WORK(64*n) DOUBLE PRECISION RWORK(2*N) INTEGER JPVT(N),ipiv(n) CHARACTER CLABS(1), RLABS(1) real*8 R(n,n) ! factorised matrix R real*8 Acopy(m,n),Bcopy(m) ! copy of input matrices real*8 Bout(m) real*8 err,sigma lda=m ldb=m nrhs=1 lwork=64*n ! get a copy of input matrices to evaluate deviations Acopy=A Bcopy(1:m)=B(1:m,1) ! Initialize JPVT to be zero so that all columns are free JPVT=0 ! Compute the QR factorization of A CALL DGEQP3(M,N,A,LDA,JPVT,TAU,WORK,lwork,INFO) if (info/=0) then return endif ! copy R for evaluation of deviations R=0.D0 do i=1,n ! row R(1:i,i)=A(1:i,i) enddo call dtrmm('Left','Upper','Transpose','Not diagonal',n,n,1.D0,A,m,R,n) ! in R is not R^T*R ! Determine which columns of R to use DO 20 K = 1, N IF (ABS(A(K,K)).LE.TOL*ABS(A(1,1))) GO TO 40 20 CONTINUE ! Compute C = (Q**H)*B, storing the result in B 40 K = K - 1 CALL DORMQR('Left','Transpose',M,NRHS,N,A,LDA,TAU,B,LDB,WORK,LWORK,INFO) if (info/=0) then return endif ! Compute least-squares solution by backsubstitution in R*B = C CALL DTRSV('Upper','No transpose','Non-Unit',K,A,LDA,B(1,1),1) ! Set the unused elements of the I-th solution vector to zero B(K+1:N,1)=ZERO 60 CONTINUE ! Unscramble the least-squares solution stored in B DO 100 I = 1, N X(JPVT(I),1) = B(I,1) 100 CONTINUE ! now estimate errors ! we need inverse of R ! DGETRF computes an LU factorization of a general M-by-N matrix A ! using partial pivoting with row interchanges. call DGETRF(n, n, R, n, ipiv, info) if (info /= 0) then !stop 'Matrix is numerically singular!' return end if ! DGETRI computes the inverse of a matrix using the LU factorization ! computed by DGETRF. call DGETRI(n, R, n, ipiv, work, n, info) if (info/=0) then return endif ! in R is now (R^T*R)^(-1) ! evaluate A*X call DGEMM('None','None',m,1,n,1.D0,Acopy,m,X,n,0.D0,Bout,m) ! calculate error of fit Bout=Bout-Bcopy err=dot_product(Bout,Bout) sigma=dsqrt(err/(m-n)) ! Unscramble the R DO I = 1, N Xerr(JPVT(I),1) = DSQRT(R(I,I))*sigma enddo return end subroutine

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