With the Hyperbolic flag, H a F [efit
] performs the same
fitting operation as a F, but reports the coefficients as error
forms instead of plain numbers. Fitting our two data matrices (first
with 13, then with 14) to a line with H a F gives the results,
3. + 2. x 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
In the first case the estimated errors are zero because the linear fit is perfect. In the second case, the errors are nonzero but moderately small, because the data are still very close to linear.
It is also possible for the input to a fitting operation to contain error forms. The data values must either all include errors or all be plain numbers. Error forms can go anywhere but generally go on the numbers in the last row of the data matrix. If the last row contains error forms `y_i +/- @c{$\sigma_i$} sigma_i', then the @c{$\chi^2$} chi^2 statistic is now,
so that data points with larger error estimates contribute less to the fitting operation.
If there are error forms on other rows of the data matrix, all the errors for a given data point are combined; the square root of the sum of the squares of the errors forms the @c{$\sigma_i$} sigma_i used for the data point.
Both a F and H a F can accept error forms in the input matrix, although if you are concerned about error analysis you will probably use H a F so that the output also contains error estimates.
If the input contains error forms but all the @c{$\sigma_i$} sigma_i values are the same, it is easy to see that the resulting fitted model will be the same as if the input did not have error forms at all (@c{$\chi^2$} chi^2 is simply scaled uniformly by @c{$1 / \sigma^2$} 1 / sigma^2, which doesn't affect where it has a minimum). But there will be a difference in the estimated errors of the coefficients reported by H a F.
Consult any text on statistical modelling of data for a discussion of where these error estimates come from and how they should be interpreted.
With the Inverse flag, I a F [xfit
] produces even more
information. The result is a vector of six items:
utpc
probability distribution
function using @c{$\chi^2$}
chi^2 with N - M degrees of freedom. A
value of 0.5 implies a good fit; some texts recommend that often
Q = 0.1 or even 0.001 can signify an acceptable fit. In
particular, @c{$\chi^2$}
chi^2 statistics assume the errors in your inputs
follow a normal (Gaussian) distribution; if they don't, you may
have to accept smaller values of Q.
The Q value is computed only if the input included error
estimates. Otherwise, Calc will report the symbol nan
for Q. The reason is that in this case the @c{$\chi^2$}
chi^2
value has effectively been used to estimate the original errors
in the input, and thus there is no redundant information left
over to use for a confidence test.