1.6.6 Manipulating Numerical DataWhen you have numerical data, it is often convenient to find a simple formula that approximates it. For example, you can try to "fit" a line or curve through the points in your data.
Fit[{ ,  , ... }, { ,  , ... }, x]
| | fit the values  to a linear combination of functions  |
Fit[{{ ,  }, { ,  }, ... }, { ,  , ... }, x]
| | fit the points  to a linear combination of the  |
Fitting curves to linear combinations of functions. | This generates a table of the numerical values of the exponential function. Table will be discussed in Section 1.8.2. | |
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data = Table[ Exp[x/5.] , {x, 7}]
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This finds a fit of the form  . | |
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Fit[data, {1, x, x^3, x^5}, x]
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This gives a table of  ,  pairs. | |
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data = Table[ {x, Exp[Sin[x]]} , {x, 0., 1., 0.2}]
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This finds a fit to the new data, of the form  . | |
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Fit[%, {1, Sin[x], Sin[2x]}, x]
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FindFit[data, form, { ,  , ... }, x]
| | find a fit to form with parameters  |
Fitting data to general forms. | This finds the best parameters for a linear fit. | |
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FindFit[data, a + b x + c x^2, {a, b, c}, x]
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| This does a nonlinear fit. | |
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FindFit[data, a + b^(c + d x), {a, b, c, d}, x]
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One common way of picking out "signals" in numerical data is to find the Fourier transform, or frequency spectrum, of the data.
| Fourier[data] | numerical Fourier transform | | InverseFourier[data] | inverse Fourier transform |
Fourier transforms. | Here is a simple square pulse. | |
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data = {1, 1, 1, 1, -1, -1, -1, -1}
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| This takes the Fourier transform of the pulse. | |
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Note that the Fourier function in Mathematica is defined with the sign convention typically used in the physical sciences--opposite to the one often used in electrical engineering. Section 3.8.4 gives more details.
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