```.-
help for ^orthog^                                      (statalist: 10 July 1998
> )
.-

Orthogonalize variables
-----------------------

^orthog^ [varlist] [weight] [^if^ exp] [^in^ range] ^,^
^g^enerate^(^newvarlist^)^ [ ^mat^rix^(^matname^)^ ^flo
> at^ ]

^aweight^s and ^fweight^s are allowed; see help @weights@.

Description
-----------

^orthog^ orthogonalizes "varlist" and creates a new set of orthogonal variables
"newvarlist" using a modified Gram-Schmidt procedure (Golub and Van Loan 1989).

The order of the variables in "varlist" determines the orthogonalization.
That is, if "varlist" is ^x1 x2 x3^, then the effect of the constant is first
removed from ^x1 x2 x3^, then ^x1^ is removed from ^x2^ and ^x3^, and then ^x2^
>  is
removed from ^x3^.  If "newvarlist" is ^q1 q2 q3^, we have

q1 = a10 + a11*x1
q2 = a20 + a21*x1 + a22*x2
q3 = a30 + a31*x1 + a32*x2 + a33*x3

where ^q1 q2 q3^ are orthogonal and aij are constants.

Options
-------

^generate(^newvarlist^)^ is not optional.  It creates new variables containing
the orthogonalized "varlist".  "newvarlist" must either contain exactly
the same number of variable names as "varlist" or be abbreviated using
either "newvar1-newvar#" or "newvar*".  See examples below.

^matrix(^matname^)^ creates a  m x m matrix called "matname" containing the
matrix R defined by X = QR, where X is the m x n matrix representation
of "varlist" and Q is the m x n matrix representation of "newvarlist"
(m = number of variables in "varlist" plus the constant; n = number of
observations).

^float^ specifies that the new variables be of type float.  The default is
double.

Warning
-------

With many variables, ^orthog^ will be slow.  Time required is proportional to
the square of the number of variables.

Examples
--------

. ^orthog x1 x2 x3, gen(u1 u2 u3)^
. ^orthog x1 x2 x3, gen(u1-u3)^
. ^orthog x1 x2 x3, gen(u*)^

. ^orthog x1 x2 x3, gen(u*) matrix(r)^
. ^orthog x*, gen(u*) mat(R) float^

The matrix R created by the ^matrix()^ option can be used to transform
coefficients from a regression:

. ^orthog x*, gen(u*) mat(R)^
. ^regress y u*^
. ^matrix bu = get(_b)^
. ^matrix invR = inv(R)^
. ^matrix b1 = bu*invR'^  [note that the transpose of invR is used]

. ^regress y x*^
. ^matrix b2 = get(_b)^

Then b1 and b2 will be the same.

The matrix R can also be used to recover X (original "varlist") from
Q (orthogonalized "newvarlist") one variable at a time:

. ^orthog price weight mpg, gen(upr uwei umpg) mat(R)^
. ^matrix c = R[.,"price"]^
. ^matrix c = c'^                      [^matrix score^ requires a row vector]
. ^matrix score double samepr = c^
. ^compare price samepr^

That is, the variable ^samepr^ is the same as the original ^price^.
This procedure can be performed as a check of the numerical soundness
of the orthogonalization.

Methods and formulas
--------------------

The X = QR orthogonalization is computed using a modified Gram-Schmidt
procedure (Golub and Van Loan 1989).

The columns of Q are orthogonal and R is upper triangular (actually R
is a permuted upper triangular matrix with row/column 1 interchanged
with row/column m so that the last row corresponds to the constant term).

Q is normalized so that

Q'WQ = NI

where W = diag(w1, w2,..., wn) with w1, w2,..., wn the weights (all 1
if weights not specified), and N is the sum of the weights.  If the
weights are ^aweight^s, they are first normalized so that N is the
number of observations.

Author
------

Bill Sribney
Stata Corporation
702 University Drive East
College Station, TX 77840
Phone: 409-696-4600
800-782-8272
Fax:   409-696-4601
email: tech_support@@stata.com

Reference
---------

Golub, G.H. and C.F. Van Loan. 1989.  Matrix Computations, 2nd ed.
Baltimore: Johns Hopkins University Press, pp. 218-219.

Also see
--------

Manual:  ^[R] orthpoly^
On-line:  help for @matrix@, @orthpoly@, @regress@
```