Curve fitting

The Adjustment Theory (including compensation calculation, adjustment, parameter estimation, adaptation, regression or fit ( ting ) called ) is a mathematical optimization method to determine the unknown parameters of their geometric and physical model or the parameters of a given function for a number of measurement data ( to estimate ). In general, be solved with her about certain problems.

The aim of the adjustment is that the final model or function adjusts the data and their inevitable contradictions little as possible. In general, the calculation is carried out with the least squares method. This method means that the parameters are small improvements attached so that the sum of the squares of all the individual deviations between measured and model data to be minimal. For randomly distributed model or measurement errors, this leads to the most probable value for the unknowns to be calculated. The remaining small residues are called residuals and statements about the accuracy and reliability of the measurement and data model to.

• 3.1 regression
• 3.2 Fit
• 3.3 Summary

Equalization and approximation

Since small contradictions in all redundant, tested for reliability data occur (see also over-determination ), the handling of these usually randomly distributed residual deviations from the important task in various science and technology has become. In addition to the smoothing effect on scattering data, the curve fitting is also used to mitigate the effects of discrepancies about in the social sciences.

This search for the natural, probable values ​​of systems or measurement series is in the language of approximation theory, the estimation of unknown parameters of a mathematical model. The method chosen most commonly used least squares ( least mean squares english or shortly least squares ) corresponds to the Gauss-Markov model. In the simplest case, an adjustment to the target, to describe a large number of empirical measurement or collection of data by a curve, and to minimize the residual deviation ( residual category ). Such a curve fitting can also be amazingly performed exactly freiäugig - graphically by looking at the data series, which highlights the natural characteristics of the square deviation minimization.

The curve fitting was developed around 1800 by Carl Friedrich Gauss for a surveying network of Geodesy and for the orbit determination of asteroids. Since then compensations are performed in all natural sciences and engineering, and even at times the economic and social sciences. The adjustment by the Gauss-Markov model gives the best result, if the residuals are random and follow a normal distribution. Varying degrees of accuracy values ​​are adjusted by weighting.

Include the measurements or data but also systematic influences or gross error, then the balanced result is distorted and the residuals show a trend in terms of interference. In such cases, further analysis is required, such as analysis of variance or the choice of a robust estimation method.

Introduction

In the simplest case is the equalization of the measurement errors (improvement, residual ) by the least squares method. Here the unknowns (parameters ) of the model is determined such that the sum of squares of measured deviations of all observations will be minimal. The estimated parameters are correct then is unbiased in agreement with the theoretical model. Alternatively, the adjustment can be carried out also in accordance with another Residuenbewertungsfunktion, for example by minimizing the sum or the maximum of the magnitudes of the measurement errors.

This is an optimization problem. The calculation steps of an adjustment simplify significantly when the observations can be considered normally distributed and uncorrelated. If unequal accuracies of the measured variables are present, this can be taken into account by weighting. Further stochastic properties of the observations can be explored in the regression analysis.

Functional and stochastic model

Each adjustment is preceded by a modeling. A distinction is generally between functional model and stochastic model.

• A functional model here describes the mathematical relations between the known (constant), the unknown and the observed parameters. The observations represent this stochastic variables ( random variable ) is, eg, with random disturbances superimposed measurements. As a simple example, a triangle is called, in the excess measurements to geometrical contradictions lead (eg, sum of angles equal to 180 °). The functional model to the formulas of trigonometry, the disturbances may be small, for example, slippage at each angular measurement.

The objective of the adjustment is optimal derivative of the unknown values ​​( parameters, such as the coordinates of the measuring points) and the dimensions for their accuracy and reliability in the sense of an objective function. For the latter one chooses usually the minimum sum of squares, but can use it for special cases for example, minimum absolute values ​​or other objective functions to be.

Solution method

Depending on the functional and stochastic model different Ausgleichungsmodelle be used.

The main distinguishing feature of this is

• Whether all the observations can be represented as functions of unknowns and constants
• Whether the observations are mutually stochastically independent or correlated,
• Whether the relations have only observations and constants, however, do not contain any unknown,
• Whether there are also those among the crowd of relations that exclusively describe relationships among constants and unknowns and thus describe restrictions between the unknown.
• In mixed occurrence of very different metrics - such as geometrical and physical measurements - the methods of compensation calculation of some mathematicians and surveyors around 1970 the so-called collocation have been extended. It is used among other things for the geoid determination, see H. Moritz, H. Sunkel and CC Tscherning.

The Ausgleichungsmodelle read:

• Adjustment by mediating observations: The individual observations are functions of the unknown parameters.
• Adjustment by switching observations with conditions between the unknowns: There are additional conditions between the unknown parameters.
• Adjustment by related observations ( Constrained Adjustment ): There are equations of condition for the observations set up, in which do not occur, the unknown parameters. The unknown parameters can then be calculated from the adjusted observations.
• General case of the Stabilization: There are functional relationships between observations and parameters set up in which the observations are not explicitly occur as a function of the parameters.

Graphical method

While the mathematical solution method, a model must be based on the graphical method is possible without such an assumption. Here is a continuously curved balancing is approximated the measured points. Depending on the background knowledge ( expectations of the course ) or personal evaluation ( individual measurement points as " outliers" ) can vary the line, however, quite.

Differences between regression and Fit

Regression and fit and least squares method are not synonymous and differ in the problems and the data to be evaluated.

Regression

A regression examined a possible correlation between data points with an assumed intrinsic connection (in this case two-dimensional). The data points have no uncertainties or measurement error; they are assumed to be constant and firm. With an assumed continuous function examines the regression how much can be described by the assumed function of the data points. The resulting regression parameters (with a linear relationship, for example ) of the regression function are stochastic variables.

Fit

Under a fit is understood in consideration of measurement errors or uncertainties in the measurement points is a function adjustment. The resulting function parameters, such as the measured values ​​with an uncertainty are then prone. The most common method of fits is the method of least squares using a Gaussian distributed measurement uncertainty is assumed. The result of fits is always a family of curves in which the "true" functional relationship lies with a certain probability:

Specifically: the result to a fit function is a set of most probable function parameters and the covariance matrix of the errors and correlations of parameters:

Is comparable to a single measured value with an error, here is the "true" value with a certain probability within the error bars; the central value here is only the most probable value.

Summary

Both of these methods can provide the same function in special cases parameter, for example in linear relationships with the same degree of uncertainty of the measurement values ​​in Fit.

In Fits the method of least squares is only suitable when data points have errors that are not distributed Gaussian. Even with a transformation of data points must be noted that the uncertainties of the data points must be transformed as well (example: an exponential distribution with uncertainties to be linearized ).