Many thanks to all who have answered my question.
I asked for a definition/explanation for splines.
For others who might be interested, the replies are included below:
(sorry for the delay; it was due to a problem with my Hotmail account)
From: Matthew Coates <[log in to unmask]>
If I remember rightly, the general term "spline" is a contraction of split
line interpolation. This means that, rather than fitting a single function
through a set of points, a series of functions are plotted through adjacent
sets of two or more points.
This might be as simple as two straight lines which cross at some point in
the middle of the data set. More usually, a separate of cubic function is
fitted to each pair of adjacent points to generate a smooth curve which
passes through every point in the data set. This is achieved (as we know
it is not possible to fit a cubic to just two points) by constraining each
segment so that the instantaneous slope is the same as that in the adjacent
segment at the point where they meet. Clever stuff.
The STATISTICA electronic manual says this:
[
Spline Fitting
It can be demonstrated that curves of any complexity can be described by a
sequence of segments defined as polynomials. In practice, most real-life
curves can be reliably approximated by a sequence of third-order (cubic)
polynomials.
For bivariate data sets (correlations that involve two variables), to
determine the curve, the spline procedure solves cubic equations for every
point at a regular interval (for more information on cubic interpolation,
see de Boor, 1978; Johnson and Ries, 1982; Dahlquest and Bjorck, 1976; and
Gerald and Wheatley, 1989).
A minimum of at least 3 data points are necessary to perform a spline fit;
the algorithm ignores overlapping data points.
]
Respect
Matt Coates
From: "Dr. Roger Brown" <[log in to unmask]>
Definition: A smoothly joined piecewise polynomial of degree n.
Spline smoothing fits a curved line through every point in the
plot such that the curve is smooth everywhere. For example, a
popular spline is the cubic spline:
Y = a + bX + cX^2 + dX^3
where a is a constant, and b, c and d are coefficients. See Brodlie
(1980) for more details.
Reference
Brodlie, K. W. (1980). A review of methods for curve and function
drawing. In K. W. Brodlie (Ed.), Mathematical Methods in Computer
Graphics and Design. London: Academic Press, Inc.
R. Brown
From: Jason Hiscox <[log in to unmask]>
Dear Blue
Depends what you mean - in the most generic meaning splines are smoothed
curves
to fit through a series of points (usually by fitting an nth degree
polynomial
function to n points, dropping an end point and adding a point at the other
end.
The differing polynomials need to satisfy criteria about the values of 1st,
2nd
etc derivatives at each data point being the same)
Hope this helps
Jason
From: Tim Cole <[log in to unmask]>
A spline curve is a smooth curve made up by joining together sections of
polynomial curves, e.g. quadratic or cubic curves. Each section of the
curve is chosen to have the same position and slope (and possibly the
second derivative) as the next section, so the two sections join together
smoothly.
Typically the spline curve is fitted to data as a form of regression curve.
The points where the sections join (called nodes) can be specified in
advance, or alternatively for the class of natural cubic splines the nodes
are at every point on the x scale where there is a data point.
A good book about cubic splines in the context of penalised likelihood,
i.e. where curves are fitted by maximum likelihood but penalised by the
roughness of the curve, is: Green PJ, Silverman BW. Nonparametric
regression and generalized linear models. London: Chapman & Hall, 1994.
Tim Cole
From: Jay Warner <[log in to unmask]>
consider a series of (x,y) points, running along more or less in the x
direction, roughly const. y value. You can draw a line between any
adjacent pair of points. The connect-the-dots sequence will be a set of
straight lines. I can say that
a) the y(x(i)) for each x(i) is the same, and equal to the point
value, for each line at x(i).
b) the slope, dy/dx at each x(i) is not the same for each of the two
lines at x(i).
The line is a linear form: y = b(o) + b(1)*x, where b(o) and b(1) are
different for each line.
Now I connect the points, only I state that
a) the y(x(i)) for each x(i) is the same, and equal to the point
value, for each line at x(i).
b) the slope, dy/dx at each x(i) must also be equal for each of the
two lines at x(i).
To get this, I must express the line as a curve,
y = b(o) + b(1)*x + b(2)*x^2
and adjust b(2), b(1) and b(o), until conditions a and b above are met.
I trust you can see a set of simultaneous equations here someplace.
You can do the spline across more than 2 points, say 3 or 7 or whatever.
You can do the spline conditions with more derivatives. Often the 2nd
derivative must be equal, too.
See why software is nice?!
Also, we usually take a curve between points to indicate a prediction of
y, given an intermediate x. But note that the coefficients of the
curve(s) do not necessarily relate to any physical principle/mechanism.
Translation: It looks pretty, but doesn't mean much.
If you went back and collected the data again, would you get about the
same shape, or something else?
good luck,
Jay
From: Paul Seed <[log in to unmask]>
Splines are smooth curves, drawn according to a formula.
The shape of the curve is set by a limited number of parameters
(fixed values). The parameters changes at a linited number of fixed
points known as knots. By carefully chosing the position of the knots and
defining the way in which the parameters change, the curve is smooth
(no discontinuity event to (say) 2nd or 3rd derivatives.)
The commonest form is the cubic spline, which is used in designing roads.
A cubic spline can be fitted to a set of points rather in the manner of a
straight line
regression. Parameters are set so that the spline passes as near to all the
points as possible, (for some suitable defintion of "near".) The more
knots, the more changes in slope, curvature & direction, so the more wiggly
the spline.
Cubic splines have the big advantage that what hapepns at one extreme is not
affected by odd values at the other extreme. They are often used as
smoothing curves to describe the underlying shape of a relationship,
and to check for a possible linear fit: if the spline roughly straight, a
straight line might be good enough.
Hope this helps.
From: rui manuel carreira rodrigues <[log in to unmask]>
Hello
consider a partition a=to < t1 < ... < tn=b of some real interval [a,b].
Spline function is a function defined on [a,b] with the same expression in
each subinterval of the partition (for example a cubic polinomial) and with
the maximum smoothness possible in [a,b]. Spline function also satisfy some
boundary and interpolation conditions.
Some example are:
1)Polinomial spline (which includes the well known cubic spline: locally a
cubic polinomial and with continuous derivatives up till the second
derivative)
2)Generalized splines
I hope I´ve been of some help
Bye
Rui
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