|
Fit
Fit[
data
,
funs
,
vars
] finds a least-squares fit to a list of data as a linear combination of the functions funs of variables vars.
The data can have the form 
,
, ... ,
,
,
, ... ,
, ...  , where the number of coordinates x, y, ... 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[
,
, ... 
,
1,
x,
x^2
,
x] gives a quadratic fit to a sequence of values  . The result is of the form 
+
x
+
x^2, where the  are real numbers. The successive values of x needed to obtain the  are assumed to be 1, 2, ... .
Fit[
,
,
,
, ... 
,
1,
x,
x^2
,
x] does a quadratic fit, assuming a sequence of x values  .
Fit[
,
,
, ... 
,
1,
x,
y
,
x,
y
] finds a fit of the form 
+
x
+
y.
Fit always finds the linear combination of the functions in the list forms 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.
See the Mathematica book: Section 1.6.6, Section 3.8.1.
See also Implementation NotesA.9.44.25MainBookLinkOldButtonDataA.9.44.25.
See also: Interpolation, InterpolatingPolynomial, Solve, PseudoInverse, QRDecomposition, FindMinimum.
Related packages: Statistics`NonlinearFit`, Statistics`LinearRegression`.
Further Examples
You will have to evaluate all the cells in this example to regenerate the data used for the plots.
Here is a table of the first 20 primes.
In[1]:= 
Here is a plot of this data.
Evaluate the cell to see the graphic.
In[2]:= 
This gives a quadratic fit to the data.
In[3]:= 
Here is a plot of the quadratic fit.
Evaluate the cell to see the graphic.
In[4]:= 
This shows the quadratic fit superimposed on the original data. The quadratic fit is better than the linear one.
Evaluate the cell to see the graphic.
In[5]:= 
This gives a table of the values of an exponential function for x from 1 to 10 in steps of 1.
In[6]:= 
Out[6]= 
This fit recovers the original functional form.
In[7]:= 
Out[7]= 
If you include other functions in the list, Fit determines that they occur with small coefficients.
In[8]:= 
Out[8]= 
You can use Chop to get rid of the terms with small coefficients.
In[9]:= 
Out[9]= 
In[10]:= 
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