The problem of fitting a polynomial to a set of observational
data so that the sum of the squared residuals is a
minimum has been frequently investigated. A.C. Aitken, in
an appendix to his paper, "On the Graduation of Data by the
Orthogonal Polynomials of Least Squares ", (Proc. Roy. Soc.
Edin. Vol. LIII (1933) pp. 77 -78,) provides a list of the
more important papers on this subject. Tchebychef and Gram
were the first to expound, more than fifty years ago, the
method of fitting my means of orthogonal polynomials. Their
work has been followed up in more recent times by several
writers, including Jordan, R.A. Fisher and A.C. Aitken.
W.F. Sheppard and C.W.M. Sherriff develop the equivalent
method of linear combination of data. However, in the latter
method, the fitted value of the central observation only is
considered in detail, and an odd number of data is therefore
It has been shown by W.F. Sheppard, and recently, with
much more conciseness by G.J. Lidstone that the methods
of Least Square Fitting and Linear Combination of Minimal
Reduction Co-efficient lead to identical results.
It is proposed, in the following investigation, to express
all the fitted values as linear combinations of the observed
values. The data considered are either odd or even in number,
equidistant, unweighted and uncorrelated.
In Chapter I we shall investigate the form of the matrix and its properties, and shall give examples of its construction
and practical use.
The corresponding matrices
of co-efficients, obtained in this way, of lower and lower order, are discussed in Chapter II, and appropriate examples
given. The properties of these matrices are very similar
to, and often identical with those of the matrix C
Chapter III contains alternative methods of fitting, with
simple checks on accuracy of working. Examples of each
method are given.
The appendix consists of tables of the numerical values
of the matrices connecting observed and fitted values, for
numbers of data equal to 4, 5, 6, 7 - - - 15, and for
fitted polynomials of degree 0, 1, 2, 3, 4, 5, together
with the corresponding matrices connecting the differences.
There is also a short bibliography.
Methods of fitting involving the matrices discussed
here are particularly suitable for rapid calculation with a
machine. Indeed the use of a machine is taken for granted.
The same example is used throughout to simplify the
comparison of the various methods.