Scientist: Examples: Data Analysis/Fitting

This example illustrates a typical working session with Scientist, including the generation of statistical output. The equation to be fit is:

Y = M*X + B

After loading Scientist, we choose the Model option from the New submenu of the which enables us to enter a model at the keyboard. We indicate that X is to be the independent variable, Y is to be the dependent variable, and M and B are parameters. The model file should appear as follows:

We then choose the Spreadsheet option from the New submenu on the File Menu to input data. We enter X and Y as the names of the first two columns of the Spreadsheet window, respectively. After data entry is complete, we elect to save the data to disk for possible future use. The data entered is the following:

Having defined the model and entered the data, the only step left before the model can be fitted is to specify initial estimates of the parameters. We choose New / Parameter Set from the model interface menu and enter the default estimates of 0.0 for M and B. We could just as easily have specified the parameter values in the model itself, but for demonstration purposes, we have chosen not to do so.

We then choose the Least Squares Fit option from the Calculate Menu. Since the model is linear with respect to both parameters, iteration is not required and Scientist immediately reports that the best fit values are 9.2857 and 9.0952 for M and B, respectively. A simulation is then performed by Scientist using the calculated parameter values and the results are shown in the Spreadsheet window that we created.

After least squares fitting, we can choose the Statistics option from the Calculate Menu to generate a statistics report. We select descriptive statistics in addition to the default options as shown and run the statistics report by selecting the Run function from the function toolbar.

Scientist calculates descriptive statistics for the data set as shown.

The Goodness-of-Fit statistics are displayed next in the Statistics Report.

The parameter estimates, the standard errors of the estimates, and 95% confidence limits for the parameters, based on the assumption that the model is linear near the solution, are displayed next in the Statistics Report.

The univariate and support plane limits are describd in the Scientist documentation. In this case, they indicate that the slope (M) is reasonably well determined, but that the intercept (B) is only poorly determined. This is to be expected since the data is so far removed from the origin.

The parameter variance-covariance matrix and the parameter correlation matrix are displayed next in the Statistics Report.

The near unity value of the off-diagonal term in the correlation matrix indicates that the parameter estimates are inversely correlated, meaning that there exists a near linear dependence between a and 1/b. That is, a small increase in a in one could be readily compensated for by an appropriately small decrease in b.

Next, we chart the result of the calculation by selecting the Quick Chart function from the function toolbar of the variable and parameter interface.

Scientist quickly plots the fitted data and a smooth curve computed using the calculated values of the parameters.