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:

Find the best fit parameters given a design matrix and response vector:

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:

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:

Compute a high-degree polynomial fit to data using a Chebyshev basis:

Use regularization to stabilize the numerical solution to where is perturbed:

The solution found by LinearSolve has very large entries:

Regularized solutions stay much closer to the solution of the unperturbed problem:

Use variation regularization to find a smooth approximation for an input signal:

The output target is a step function:

The input is determined from the output via convolution with :

Without regularization, the predicated response matches the signal very closely, but the computed input has a lot of oscillation:

With regularization of the variation, a much smoother approximation can be found:

Regularization of the input size may also be included:

Smooth a corrupted signal using variation regularization:

Plot the tradeoff between the norm of the residuals and the norm of the variation:

Choose a value of near where the curve bends sharply:

Use total variation regularization to smooth a corrupted signal with jumps:

Regularize with the parameter :

A smaller value of gives less smoothing but the residual norm is smaller:

Use LASSO (L1) regularization to find a sparse fit (basis pursuit):

Here is a signal:

The goal is to approximate the signal with just of few of the thousands of Gabor basis functions:

With , a fit is found using only 41 of the basis elements:

The error is quite small:

Once the important elements of the basis have been found, error can be reduced by finding the least squares fit to these elements:

With a smaller value , a fit is found using more of the basis elements:

The error is even smaller:

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 a+b x:

Find the minimum symbolically:

These are the coefficients given by Fit:

The exact coefficients may be found by using WorkingPrecision->Infinity:

This is the sum of squares error for a quadratic a+b x+c x^2:

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:

Fit will use the time stamps of a TimeSeries as variables:

Rescale the time stamps and fit again:

Find fit for the values:

Fit acts pathwise on a multipath TemporalData: