ORTHPOLYNOMIAL procedure

Calculates orthogonal polynomials (P.W. Lane).

Options

`MAXDEGREE` = scalar	Maximum degree of polynomial to be calculated; default is the number of identifiers in the pointer specified by the `POLYNOMIAL` parameter
`WEIGHTS` = variate	Weights to be used in orthogonalization; default `*` gives an equal weight to each unit

Parameters

`X` = variates	Values from which to calculate the polynomials; no default – this parameter must be set
`POLYNOMIAL` = pointers	Identifiers of variates to store results; no default – this parameter must be set

Description

Polynomials of low degree can be fitted by ordinary linear regression, estimating effects of terms X, X**2, X**3, and so on for a variate X. However, it is sometimes preferable to arrange that successive polynomial terms are orthogonal to each other; certainly, there are likely to be numerical problems with polynomials of degree five or more, if they are not orthogonal. ORTHPOLYNOMIAL calculates orthogonal polynomials up to a specified maximum degree from a given variate. The orthogonalization can be weighted by specifying a variate of weights.

Options: MAXDEGREE, WEIGHTS. Parameters: X, POLYNOMIAL.

Method

Successive formation of polynomials, starting with p₁ = x – mean(x), ensuring orthogonality of p_i with p₁ …p_i–₁; that is:

∑ ( weight × p_i × p_j ) = 0

Action with `RESTRICT`

A variate in the X parameter can be restricted: the restriction is transferred to the calculated polynomials, and to the weight variate if specified.

Example

CAPTION  'ORTHPOLYNOMIAL example',!t(\
         'The example models changes of population with time by fitting a',\  
         'quartic regression model; future populations are predicted by',\    
         'including future dates with weight zero. Thus population figures',\ 
         '(variate pop) are available for dates 1811, 1821 ... 1931; for',\   
         'dates 1941, 1951 ... 1991 population figures are given as missing',\
         'values * and weights w as 0.'); STYLE=meta,plain
VARIATE  [VALUES=1811, 1821 ... 1991] year
&        [VALUES=13(1), 6(0)] w
&        [VALUES=10.16, 12.00, 13.90, 15.91, 17.93, 20.07, 22.71,\ 
                 25.97, 29.00, 32.53, 36.07, 37.89, 39.95, 6(*)] pop
PRINT    'Get (weighted) orthogonal polynomials up to degree 4.'
ORTHPOLYNOMIAL [MAX=4; WEIGHT=w] year; POLY=p2
PRINT    year,w,pop,p2[]; FIELD=5,2,6,4(12); DEC=0,0,2,4(0)
PRINT    'Fit quartic model: correlations should be zero.'
MODEL    [WEIGHT=w] pop
FIT      [PRINT=model,summary,estimates,correlation,fitted] p2[]
PRINT    'Display dangers of extrapolation!'
RKEEP    FITTED=fitted
GRAPH    [NROWS=21; NCOLUMNS=61] fitted,pop; year; METHOD=curve,point
PRINT    'Compare with non-orthogonal polynomials.'
CALC     year2,year3,year4 = year,year2,year3 * year
FIT      year,year2,year3,year4

Updated on March 6, 2019

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