BUILT-IN MATHEMATICA SYMBOL

Fit

finds a least-squares fit to a list of data as a linear combination of the functions funs of variables vars.

MORE INFORMATION

The data can have the form , where the number of coordinates , , ... is equal to the number of variables in the list vars.

The data can also be of the form , with a single coordinate assumed to take values 1, 2, ....

The argument funs can be any list of functions that depend only on the objects vars.

Fit gives a quadratic fit to a sequence of values . The result is of the form , where the are real numbers. The successive values of needed to obtain the are assumed to be 1, 2, ... . »

Fit does a quadratic fit, assuming a sequence of values . »

Fit finds a fit of the form . »

Fit always finds the linear combination of the functions in the list funs that minimizes the sum of the squares of deviations from the values . »

Exact numbers given as input to Fit are converted to approximate numbers with machine precision. »

EXAMPLES

CLOSE ALL

Basic Examples (1)

Here is some data:

Find the line that best fits the data:

Find the quadratic that best fits the data:

Show the data with the two curves:

Here is some data:

In[1]:=

Find the line that best fits the data:

In[2]:=

Out[2]=

Find the quadratic that best fits the data:

In[3]:=

Out[3]=

Show the data with the two curves:

In[4]:=

Out[4]=

Scope (2)

Here is some data defined with exact values:

Fit the data to a linear combination of sine functions using machine arithmetic:

Fit the data using 24-digit precision arithmetic:

Show the data with the curve:

Here is some data in three dimensions:

Find the plane that best fits the data:

Show the plane with the data points:

Find the quadratic that best fits the data:

The quadratic actually interpolates the data:

Generalizations & Extensions (1)

Here is a list of values:

Fit to a quadratic. When coordinates are not given, the values are assumed to be paired up with 1, 2, ...:

Fit to a quartic:

Show the data with the curve:

Properties & Relations (3)

Fit gives the best fit function:

LinearModelFit allows for extraction of additional information about the fitting:

Extract the fitted function:

Extract additional results and diagnostics:

Here is some data:

This is the sum of squares error for a line

:

Find the minimum symbolically:

These are the coefficients given by Fit:

This is the sum of squares error for a quadratic

:

Find the minimum symbolically:

These are the coefficients given by Fit:

When a polynomial fit is done to a high enough degree, Fit returns the interpolating polynomial:

The result is consistent with that given by InterpolatingPolynomial:

Possible Issues (1)

Here is some data from a random perturbation of a Gaussian:

This is a function that gives the standard basis for polynomials:

Show the fits computed for successively higher-degree polynomials:

The problem is that the coefficients get very small for higher powers:

Giving the basis in terms of scaled and shifted values helps with this problem: