10.8.3 Error Estimates for Fits

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 +/- sigma_i’, then the ‘chi^2’ statistic is now,

chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)

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 ‘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 ‘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 (‘chi^2’ is simply scaled uniformly by ‘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 modeling 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:

1. The model formula with error forms for its coefficients or parameters. This is the result that H a F would have produced.
2. A vector of “raw” parameter values for the model. These are the polynomial coefficients or other parameters as plain numbers, in the same order as the parameters appeared in the final prompt of the I a F command. For polynomials of degree ‘d’, this vector will have length ‘M = d+1’ with the constant term first.
3. The covariance matrix ‘C’ computed from the fit. This is an mxm symmetric matrix; the diagonal elements ‘C_j_j’ are the variances ‘sigma_j^2’ of the parameters. The other elements are covariances ‘sigma_i_j^2’ that describe the correlation between pairs of parameters. (A related set of numbers, the linear correlation coefficients ‘r_i_j’, are defined as ‘sigma_i_j^2 / sigma_i sigma_j’.)
4. A vector of ‘M’ “parameter filter” functions whose meanings are described below. If no filters are necessary this will instead be an empty vector; this is always the case for the polynomial and multilinear fits described so far.
5. The value of ‘chi^2’ for the fit, calculated by the formulas shown above. This gives a measure of the quality of the fit; statisticians consider ‘chi^2 = N - M’ to indicate a moderately good fit (where again ‘N’ is the number of data points and ‘M’ is the number of parameters).
6. A measure of goodness of fit expressed as a probability ‘Q’. This is computed from the utpc probability distribution function using ‘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, ‘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 ‘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.