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help for ^orthpoly^                                              (STB-25:  sg37)
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Create orthogonal polynomials
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	^orthpoly^ varname [weight] [^if^ exp] [^in^ range] ^,^
			{ ^g^enerate^(^varlist^)^ ^p^oly^(^matname^)^ } [ ^d^egree^(^#^)^ ]


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


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

^orthpoly^ computes orthogonal polynomials for a variable "varname".
Either ^generate()^ or ^poly()^ or both must be specified.


Options
-------

^degree(^#^)^ specifies the highest degree polynomial to include.  Orthogonal 
    polynomials of degree 1, 2,..., d = # are computed.  Default is 1.

^generate(^varlist^)^ creates d new variables (of type double) containing
    orthogonal polynomials of degree 1, 2,..., d evaluated at "varname".
    The varlist must either contain exactly d new variable names or be
    abbreviated using either "newvar1-newvard" or "newvar*".  For both
    styles of abbreviation, new variables newvar1, newvar2,..., newvard
    are generated.

^poly(^matname^)^ creates a (d+1)x(d+1) matrix called "matname" containing the
    coefficients of the orthogonal polynomials.  The orthogonal polynomial of
    degree i is

        matname[i, d+1] + matname[i, 1]*varname + matname[i, 2]*varname^^2
	+ ... + matname[i, i]*varname^^i

    Note: The coefficients corresponding to the constant term are placed in
    the last column of the matrix.


Examples
--------

 . ^orthpoly x, deg(3) gen(p1 p2 p3)^
 . ^orthpoly x, deg(3) gen(p1-p3)^
 . ^orthpoly x, deg(3) gen(p*)^
 . ^orthpoly x, deg(3) poly(A)^
 . ^orthpoly x, deg(3) gen(p*) poly(A)^

Suppose x1=x, x2=x^^2, x3=x^^3, and x4=x^^4 are natural polynomials and px1, px2,
px3, and px4 are the corresponding orthogonal polynomials.  If we compute

 . ^orthpoly x, deg(4) gen(px*) poly(P)^
 . ^regress y px1-px4^
 . ^matrix bp = get(_b)^
 . ^matrix b1 = bp*P^

 . ^regress y x1-x4^
 . ^matrix b2 = get(_b)^

then b1 and b2 will be the same.  (This is the rationale for the ordering of
the rows and columns of P.)


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

The orthogonal polynomials are computed using the Christoffel-Darboux
recurrence formula.  They are normalized so that X'DX = NI, where
D = 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


Also see
--------

    STB:  STB-25  sg37