10.8.2 Polynomial and Multilinear Fits

To fit the data to higher-order polynomials, just type one of the digits 2 through 9 when prompted for a model. For example, we could fit the original data matrix from the previous section (with 13, not 14) to a parabola instead of a line by typing a F 2 RET.

2.00000000001 x - 1.5e-12 x^2 + 2.99999999999

Note that since the constant and linear terms are enough to fit the data exactly, it’s no surprise that Calc chose a tiny contribution for ‘x^2’. (The fact that it’s not exactly zero is due only to roundoff error. Since our data are exact integers, we could get an exact answer by typing m f first to get Fraction mode. Then the ‘x^2’ term would vanish altogether. Usually, though, the data being fitted will be approximate floats so Fraction mode won’t help.)

Doing the a F 2 fit on the data set with 14 instead of 13 gives a much larger ‘x^2’ contribution, as Calc bends the line slightly to improve the fit.

0.142857142855 x^2 + 1.34285714287 x + 3.59999999998

An important result from the theory of polynomial fitting is that it is always possible to fit n data points exactly using a polynomial of degree n-1, sometimes called an interpolating polynomial. Using the modified (14) data matrix, a model number of 4 gives a polynomial that exactly matches all five data points:

0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.

The actual coefficients we get with a precision of 12, like ‘0.0416666663588’, clearly suffer from loss of precision. It is a good idea to increase the working precision to several digits beyond what you need when you do a fitting operation. Or, if your data are exact, use Fraction mode to get exact results.

You can type i instead of a digit at the model prompt to fit the data exactly to a polynomial. This just counts the number of columns of the data matrix to choose the degree of the polynomial automatically.

Fitting data “exactly” to high-degree polynomials is not always a good idea, though. High-degree polynomials have a tendency to wiggle uncontrollably in between the fitting data points. Also, if the exact-fit polynomial is going to be used to interpolate or extrapolate the data, it is numerically better to use the a p command described below. See Polynomial Interpolation.

Another generalization of the linear model is to assume the ‘y’ values are a sum of linear contributions from several ‘x’ values. This is a multilinear fit, and it is also selected by the 1 digit key. (Calc decides whether the fit is linear or multilinear by counting the rows in the data matrix.)

Given the data matrix,

[ [  1,   2,   3,    4,   5  ]
  [  7,   2,   3,    5,   2  ]
  [ 14.5, 15, 18.5, 22.5, 24 ] ]

the command a F 1 RET will call the first row ‘x’ and the second row ‘y’, and will fit the values in the third row to the model ‘a + b x + c y’.

8. + 3. x + 0.5 y

Calc can do multilinear fits with any number of independent variables (i.e., with any number of data rows).

Yet another variation is homogeneous linear models, in which the constant term is known to be zero. In the linear case, this means the model formula is simply ‘a x’; in the multilinear case, the model might be ‘a x + b y + c z’; and in the polynomial case, the model could be ‘a x + b x^2 + c x^3’. You can get a homogeneous linear or multilinear model by pressing the letter h followed by a regular model key, like 1 or 2. This will be indicated by an “h” in the minibuffer after the help message.

It is certainly possible to have other constrained linear models, like ‘2.3 + a x’ or ‘a - 4 x’. While there is no single key to select models like these, a later section shows how to enter any desired model by hand. In the first case, for example, you would enter a F ' 2.3 + a x.

Another class of models that will work but must be entered by hand are multinomial fits, e.g., ‘a + b x + c y + d x^2 + e y^2 + f x y’.