Least Squares - Problem Statement

Problem Statement

The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of n points (data pairs), i = 1, ..., n, where is an independent variable and is a dependent variable whose value is found by observation. The model function has the form, where the m adjustable parameters are held in the vector . The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, S, of squared residuals

is a minimum. A residual is defined as the difference between the actual value of the dependent variable and the value predicted by the model.

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An example of a model is that of the straight line. Denoting the intercept as and the slope as, the model function is given by . See linear least squares for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, x and z, say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point.