Curve fitting

From Exampleproblems

Curve fitting is finding a curve which matches a series of data points and possibly other constraints. This section is an introduction to both interpolation (where an exact fit to constraints is expected) and curve fitting/regression analysis (where an approximate fit is permitted).

1 Fitting data points to lines and polynomial curves
2 Fitting data points to other curves
3 Application to surfaces
4 See also
5 External links

Fitting data points to lines and polynomial curves

Let us start with a first degree polynomial equation:

y = a x + b .

This is a line with slope a (as opposed to the usual slope m for a line in slope/intercept form, or y = mx + b) . It is known that a line will connect any two points. So, a first degree polynomial equation is an exact fit for two points.

If we increase the order of the equation to a second degree polynomial, we get:

y = a x 2 + b x + c

This will exactly fit three points.

If we increase the order of the equation to a third degree polynomial, we get:

y = a x 3 + b x 2 + c x + d .

This will exactly fit four points.

A more general statement would be to say it will exactly fit four constraints. Each constraint can be a point, angle, or curvature (which is the reciprocal of the radius, or 1/R). Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions. Identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as "the change in the rate of curvature", could also be added. This, for example, would be useful in highway cloverleaf design to understand the forces applied to a car, as it follows the cloverleaf, and to set reasonable speed limits, accordingly.

Bearing this is mind, the first degree polynomial equation could also be an exact fit for a single point and an angle while the third degree polynomial equation could also be an exact fit for two points, an angle constraint, and a curvature constraint. Many other combos of constraints are possible for these and for higher order polynomial equations.

If we have more than n + 1 constraints (n being the degree of the polynomial), we can still run the polynomial curve through those constraints. However, we are not guaranteed to get an exact fit through all the constraints (but we might, for example, in the case of three collinear points exactly fitting a first degree polynomial). In general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations.

Now, one might wonder why we would ever want to get an approximate fit when we could just increase the degree of the polynomial equation and get an exact match. There are several reasons:

Just because an exact match exists doesn't necessarily mean we can find it. Depending on the algorithm used, you may very well get a divergent case where the exact fit can't be calculated, or it might take too much CPU time to find the solution and you'll end up having to accept an approximate solution, anyway.

One may actually prefer the effect of averaging out questionable data points in a sample, rather than distorting the curve to fit them exactly.

High order polynomials can be highly oscillatory. If one runs a curve through two points A and B, one would expect the curve to run somewhat near the midpoint of A and B, as well. This may not happen with high-order polynomial curves, they may even have values that are very large in positive or negative magnitude. With low-order polynomials, the curve is more likely to fall near the midpoint (it's even guaranteed to exactly run through the midpoint on a first degree polynomial).

Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of ogee/inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to negative. You could also say this is where it transitions from "holding water" to "shedding water". Note that it is only "possible" that high order polynomials will be lumpy, they could also be smooth, but there is no guarantee, unlike with low order polynomial curves. A fifteenth degree polynomial could have, at most, thirteen inflection points, but could also have twelve, eleven, or any number down to zero.

Now that we have talked about using a degree too low for an exact fit, let's also discuss what happens if the degree of the polynomial curve is higher than needed for an exact fit. This is bad for all the reasons listed previously for high order polynomials, but also leads to a case where there are an infinite number of solutions. For example, a first degree polynomial (a line) constrained by only a single point, instead of the usual two, would give us in infinite number of solutions. This brings up the problem of how to compare and choose just one solution, which can be a problem for software and for humans, as well. So, it's usually best to choose as low a degree as possible for an exact match on all constraints, and perhaps an even lower degree, if an approximate fit is acceptable.

Fitting data points to other curves

Other types of curves, such as conic sections (circular, elliptical, parabolic, and hyperbolic arcs) or trigonometric functions (such as sine and cosine), may also be used, in certain cases. For example, trajectories of objects under the influence of gravity follow a parabolic path, when air resistance is ignored. Hence, matching trajectory data points to a parabolic curve would make sense. Tides follow sinusoidal patterns, hence tidal data points should be matched to a sine wave, or the sum of two sine waves of different periods, if the effects of the Moon and Sun are both considered.

Application to surfaces

Note that while this discussion was in terms of 2D curves, much of this logic also extends to 3D surfaces, each patch of which is defined by a net of curves in two parametric directions, typically called u and v. A surface may be composed of one or more surface patches in each direction.