Curve fitting

# Curve fitting

Discussion

Encyclopedia
Curve fitting is the process of constructing a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

, or mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

, that has the best fit to a series of data
Data
The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

points, possibly subject to constraints. Curve fitting can involve either interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

, where an exact fit to the data is required, or smoothing
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. Many different algorithms are used in smoothing...

, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

, which focuses more on questions of statistical inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...

such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation
Extrapolation
In mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...

refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

### Fitting lines and polynomial curves to data points

Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

equation:

This is a line with slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

a. We know that a line will connect any two points. So, a first degree polynomial equation is an exact fit through any two points with distinct x coordinates.

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

This will exactly fit a simple curve to three points.

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

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
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

, or curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

(which is the reciprocal of the radius of an osculating circle
Osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...

). 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
Spline (mathematics)
In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...

. 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
Cloverleaf interchange
A cloverleaf interchange is a two-level interchange in which left turns, reverse direction in left-driving regions, are handled by ramp roads...

design to understand the rate of change of the forces applied to a car (see jerk), as it follows the cloverleaf, and to set reasonable speed limits, accordingly.

Bearing this in 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 combinations 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. An exact fit to all constraints is not certain (but might happen, for example, in the case of a first degree polynomial exactly fitting three collinear points). In general, however, some method is then needed to evaluate each approximation. The least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

method is one way to compare the deviations.

Now, you 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:
• Even if an exact match exists, it does not necessarily follow that we can find it. Depending on the algorithm used, we may encounter a divergent case, where the exact fit cannot be calculated, or it might take too much computer time to find the solution. Either way, you might end up having to accept an approximate solution.

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

• Runge's phenomenon
Runge's phenomenon
In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...

: high order polynomials can be highly oscillatory. If we run a curve through two points A and B, we 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
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

. 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 point
Inflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...

s 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. We can 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 of this, 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 an 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. For this reason, it is 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.

For more details, Polynomial interpolation
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...

.

### Fitting other curves to data points

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.

### Algebraic fit versus geometric fit for curves

For algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical (y-axis) displacement of a point from the curve (e.g., ordinary least squares
Ordinary least squares
In statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...

). However for graphical and image applications geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include both axes of displacement of a point from the curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result.

### Fitting a circle by geometric fit

Coope approaches the problem of trying to find the best visual fit of circle to a set of 2D data points. The method elegantly transforms the ordinarily non-linear problem into a linear problem that can be solved without using iterative numerical methods, and is hence an order of magnitude faster than previous techniques.

### Fitting an ellipse by geometric fit

The above technique is extended to general ellipses by adding a non-linear step, resulting in a method that is fast, yet finds visually pleasing ellipses of arbitrary orientation and displacement.

### 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.

For more details, see the computer representation of surfaces
Computer representation of surfaces
In technical applications of 3D computer graphics such as computer-aided design and computer-aided manufacturing, surfaces are one way of representing objects. The other ways are wireframe and solids...

article.

## Software

Many statistical packages such as R
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

and numerical software such as the GNU Scientific Library
GNU Scientific Library
In computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...

, SciPy
SciPy
SciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...

and OpenOpt
OpenOpt
OpenOpt is an open-source framework for numerical optimization, nonlinear equations and systems of them. It is licensed under the BSD license, making it available to be used in both open- and closed-code software. The package already has some essential ....

include commands for doing curve fitting in a variety of scenarios. There are also programs specifically written to do curve fitting; they can be found in the lists of statistical and numerical analysis programs as well as in :Category:Regression and curve fitting software.

• Levenbergâ€“Marquardt algorithm
• Nonlinear regression
Nonlinear regression
In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables...

• Smoothing
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. Many different algorithms are used in smoothing...

• Total least squares
• Overfitting
Overfitting
In statistics, overfitting occurs when a statistical model describes random error or noise instead of the underlying relationship. Overfitting generally occurs when a model is excessively complex, such as having too many parameters relative to the number of observations...