Numeric Summary¶

The Numeric Summary is used to display all numerical information associated with the curve-fit for the current equation. This summary includes goodness of fit criteria, coefficient standard errors and confidence limits for the fitted parameters, measured values such as area and extrema of the function and its derivatives, the fitting method used, an analysis of variance, and data table statistics.

The Numeric Summary is toggled on and off by the Numeric button in the Curve-Fit graph window. The window size and position you choose for the Numeric Summary is automatically saved across sessions. The Numeric Summary uses the TableCurve Text Viewer to display, modify, and print the information.

The Numeric Summary offers four Options menu items for controlling the amount of information presented:

Probabilities - adds probability values to coefficient summary and analysis of variance table
Area and Extrema - adds function area and measured function and derivative extrema to summary
Analysis of Variance - adds ANOVA table to summary
Data Table Statistics - adds basic data table statistics and titles to summary

The initial confidence level will be the value set in the Confidence Intervals option of the Curve-Fit graph. To change to a different confidence level, select from one of the five confidence level items in the Options menu:

90%
95%
99%
99.9%
99.99%

Goodness of Fit Statistics and Coefficient Summary¶

The standard numeric summary information includes the four goodness of fit statistics used through TableCurve 2D and a fitted parameter table that includes standard errors, t-values (parameter divided by std. error), and confidence limits.

Rank 14 Eqn 8051 Gamma(a,b,c,d,e)

r² Coef Det DF Adj r² Fit Std Err F-value

0.9930667028 0.9926939449 2.6723443324 3365.9407391

Parm Value Std Error t-value 95% Confidence Limits P>|t|

a 10.21013877 0.401680699 25.41854460 9.412592322 11.00768521 0.00000

b 98.51427854 0.910244052 108.2284232 96.70696764 100.3215894 0.00000

c 3.682986825 0.012626218 291.6935934 3.657917174 3.708056477 0.00000

d 0.801518981 0.027243214 29.42086668 0.747426940 0.855611022 0.00000

e 1.806320747 0.052683021 34.28658230 1.701717373 1.910924121 0.00000

Area and Function and Derivative Extrema¶

The area is computed using a successive Gaussian Quadrature integration from the lowest active X value in the data set to the highest active X active. The precision attained in the area computation is also shown. One dimensional minimization is used to determine the minimum and maximum function, first derivative, and second derivative values and the X-values where these extrema occur. The derivative values will be based upon analytic derivatives if these are set in the Reference menu of the Curve-Fit graph.

Area Xmin-Xmax Area Precision

287.24061570 2.995478e-18

Function min X-Value Function max X-Value

13.221606759 9.9090910000 105.51788491 4.0632864709

1st Deriv min X-Value 1st Deriv max X-Value

-76.43073838 4.7957195384 76.430738375 3.3308490285

2nd Deriv min X-Value 2nd Deriv max X-Value

-172.0467226 4.0632847674 76.777625516 5.3318999291

Rational Function Singularities¶

The poles of rational functions, which represent the roots of the denominator, are also reported in the numeric summary.

When the rational's denominator is a polynomial in x, all real roots are found and reported using an eigenvalue procedure. These are broken down by those within the X range of the data and by those anywhere outside of such. For example:

Singularities [Data Range]

3.0433089735 6.7586369700 18.155326487 26.402880316

Singularities [All Other]

30.691233986 36.520959952

When the rational denominator is not a polynomial in x, such as is the case for ln(x), half order, even order, Chebyshev, and Fourier series rationals, a brute force partitioning is used to isolate pertinent real roots. 100 linear partitions are used within the data range, 50 more in a half-length range below the minimum X, and 50 more within a half-length range above the maximum X. For example:

Singularities [Data Range]

6.6040821944 20.327000216 27.751943411

Singularities [±½ Data Range]

-6.604081091 30.074267286

As with all root-finding algorithms, this approach is not absolutely foolproof. The reported roots will most definitely exist, but there is a finite possibility that a root may go undetected. You can successively zoom in regions of concern to check for undetected poles. You can also use the Review's Evaluation procedure to generate up to 16384 function evaluations between any two X-values of concern. You should generate directly to file and then Import the file to graphically check for a pole that may have eluded detection.

ANOVA¶

The analysis of variance list includes the procedure, matrix method, minimization, iterations, and SVD condition number, depending on type of fit. The goodness of fit statistics not included in the ANOVA are then listed along with the maximum error (the largest residual) in the fit. The ANOVA table follows.

Procedure Minimization Iterations

LevMarqdt Least Squares 12

r² Coef Det DF Adj r² Fit Std Err Max Abs Err

0.8890484160 0.8843270720 10.633870945 32.682979967

Source Sum of Squares DF Mean Square F Statistic P>F

Regr 86079.212 3 28693.071 253.743 0.00000

Error 10742.525 95 113.07921

Total 96821.737 98

Pure Error and Lack of Fit¶

When there are multiple Y observations at a specific X value, the variance in these Y values can be said to consist of pure or random error. When such replicates are present in a data set, it is possible to separate the portion of the variance attributable to pure error from that portion associated with the fitted model. This expanded analysis of variance will be automatically included in the numeric summary when such replicates are present and all weights remain set at unity.

Procedure Minimization Iterations

LevMarqdt Least Squares 7

r² Coef Det DF Adj r² Fit Std Err Max Abs Err

0.9946085954 0.9944681942 2.0290264352 7.0584260241

r² Attainable

0.9973156573

Source Sum of Squares DF Mean Square F Statistic P>F

Regr 146582.8 4 36645.699 8901.18 0.00000

Error 794.57102 193 4.1169483

Total 147377.37 197

Lack Fit 398.95966 94 4.2442517 1.06211 0.38327

Pure Err 395.61136 99 3.9960743

The r² Attainable is the maximum r² that can be achieved with any model. The Pure Err sum of squares will reflect that portion of the sum of squared residuals attributable to pure error in these repeat observations. The Lack Fit sum of squares is simply the difference between the overall sum of squared residuals and this pure error sum of squares. When this lack of fit F-statistic is significant (>>1), the model may be inadequate. In such cases, the residuals should be checked closely for systematic trends which would confirm the insufficiency of the model.