Residuals Graph

The Residuals Graph is an AutoSignal Graph that graphically displays the residuals from a parametric fit.

The residuals can be displayed in five different formats:

Basic Residuals - the simple difference between the Y data value and the Y predicted from the fit

Percent Residuals - the residuals as a % of the Y data value

Standardized Residuals - the residuals as a fraction of the fit's standard error

Distribution - the residuals in a binned histogram

Delta SNP - the residuals as a delta stabilized normal probability

By default, the Basic Residuals graph also doubles as a standardized residuals graph since the Point format specifies the coloring of residuals by fit standard error.

You may also close the Residuals Graph directly. The window size and position you choose for the Residuals Graph is automatically saved across sessions. For some options, it may also be possible to toggled the graph on and off by a Residuals button.

Residuals Distribution Graph

The least-squares coefficient standard errors and confidence ranges as well as the curve's confidence and prediction intervals reported by AutoSignal contain an implicit assumption that the residuals are normally distributed. These uncertainty statistics cannot be assumed correct unless this condition of normality is verified.

The Distribution Graph option displays a histogram of the residuals. Distributions with obvious asymmetry or wide tails would readily disqualify this assumption of Gaussian errors.

AutoSignal requires at least 16 active points in order to produce a residuals distribution. Note that any histogram is of dubious merit when data table sizes are small because of the large bin spacings. The greater the number of data points, the more accurate the distribution will be.

Delta Stabilized Normal Probability Plot

AutoSignal offers this approach as the best way to assure errors are normal. A stabilized normal probability (SNP) plot uses an arctangent transformation on both X and Y to produce a normal probability plot that uses a linear scale for both the X and Y axes. On such a plot, perfectly normal errors plot as a 45 degree line. Critical limits also have a 45 degree slope, and lie equally above and below this line.

AutoSignal modifies the SNP slightly and uses a delta SNP, where the X value is subtracted from the Y. This produces a horizontal y=0 for pure normal data, and horizontal critical limit lines.

AutoSignal plots 90, 95, 99, and 99.9% critical limit lines on the SNP plot. A 99% critical limit means that in only 1 out of 100 data sets should even a single point violate this limit. You may find the 99% critical limit the most useful. If even a single data point in the SNP violates this 99% limit, it is reasonable to assume that the errors fail this normality test. By default, the 90% lines will be cyan, the 95% green, the 99% yellow, and the 99.9% red.

This is an example SNP where there is a high confidence that the errors are normally distributed:

Here is an SNP where it is safe bet that the errors are non-Gaussian:

For more information on the SNP, you may refer to:

John R. Michael, "The Stabilized Probability Plot", Biometrika, 70,1, p11-17, 1983.

Lloyd S. Nelson, "A Stabilized Normal Probability Plotting Technique", Journal of Quality Technology, 21,3, 1989.

Maximum Likelihood

When the normal assumption is invalidated due to appreciable tails in the residuals distribution, equally invalidated is the assumption that least-squares is furnishing the maximum likelihood fit. In such a case, one of AutoSignal's robust minimizations may represent a better maximum likelihood model.

If you choose to fit a robust model, please remember that AutoSignal's goodness of fit statistics are all based on a least-squares common frame of reference. As such, the goodness of fit values will fail to reflect the improvement derived from switching to a robust method. Also, you should not assume simply because a distribution of errors is Gaussian that a robust procedure is of no value. Two key reasons for using a robust minimization are to deal effectively with outliers and to manage a wide dynamic range on the y-variable.