Curve fitting

Curve fitting^[1][2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,^[3] possibly subject to constraints.^[4][5] Curve fitting can involve either interpolation,^[6][7] where an exact fit to the data is required, or smoothing,^[8][9] in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,^[10][11] which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fitted to data observed with random errors. Fitted curves can be used as an aid for data visualization,^[12][13] to infer values of a function where no data are available,^[14] and to summarize the relationships among two or more variables.^[15] Extrapolation refers to the use of a fitted curve beyond the range of the observed data,^[16] and is subject to a degree of uncertainty^[17] since it may reflect the method used to construct the curve as much as it reflects the observed data.

For linear-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). 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.^[18][19][20]