
FITName:
Nonlinear fits are performed using an iterative modified LevenbergMarquardt algorithm (Dataplot implements the algorithm given in the Osborne paper listed in the References section below). This algorithm can fit linear and multilinear models as well as nonlinear models. In addition, the FIT command can perform linear and polynomoal fits using a noniterative algorithm. Since the noniterative algorithm supports a much broader range of output, this will be documented separately, Enter
for the documentation for exact linear fits. Nonlinear fits are specified by entering a function. For example,
FIT Y = A0 + A1*EXP(A2*(YEAR1950) FIT Y = (A0 + A1*X)/(1 + B1*X) The function can either be given on the FIT command or be defined with a LET FUNCTION command. For nonlinear fits, the FIT command generates the following output.
It is recommended that a FIT be followed by a residual analysis to assess the model adequacy. Specifically, the typical assumptions for the residuals are that they are independent with a common distribution having fixed location and variation. It is usually assumed that the common distribution is a normal distribution. The 4PLOT command generates 4 plots that are useful in testing these assumptions. The NIST/SEMATECH eHandbook contains a more detailed discussion of this issue at
In addition, if there is a single independent variable in the model, it can be useful to plot the data with the fitted values overlaid. For nonlinear fits, up to 15 indepedent variables can be included in the model.
where <y1> is the response (= dependent) variable; <f> is: This syntax is appropriate for all modelslinear, polynomial, multilinear (up to 15 independent variables), and nonlinear (up to 15 independent variables). It uses an iterative modified LevenbergMarquardt algorithm. Linear fits are handled as a special case (the fits are still done iteratively).
FIT Y = A*(EXP(B*TIME/10)  EXP(TIME/10)) FIT Y = B0 + B1*X**B2 FIT Y = K/(1+K*A*X**B) FIT Y = A  B*X  ATAN(C/(XD))/3.1459 FIT Y = A0*BESS0(A1*X)*BESS1(A1*X) FIT Y = (A0 + A1*X)/(1 + B1*X + B2*X**2) FIT Y = (A+B*X+C*X**D)/(SIN(EXP(ALPHA*X2+BETA*X3)))
FIT Y = F1
By default, a maximum of 50 iterations are allowed before Dataplot assumes the fit is not converging. You can change this maximum with the FIT ITERATIONS command. Dataplot checks for convergence by computing the ratio of sccuessive values of the residual standard deviation. You can specify the critierion for convergence with the FIT STANDARD DEVIATION command.
However, decent starting values can often speed up nonlinear fits. In addition, some fits may require good starting values in order to converge to accurate values. To specify starting values, simply assign values to the coefficients before doing the fit. For example:
LET A = 0.004 LET B = 0.01 FIT Y = EXP(ALPHA*X)/(A+B*X) In some cases, good starting values might be known from previous work or from theoretical considerations. However, if better starting values are needed and reasonable guesses are not available, the PREFIT command can be helpful. Sinusoidal models are one case where good starting values are needed. See the following example from the NIST/SEMATECH eHandbook for an example of fitting this kind of model
If you have a parameter in the model that you want to set to a fixed value, then enter the literal value or use the substitution character "^". For example
LET C = 1.5
To specify weights for a least squares fit, enter the command
where <var> is a variable containing the weights. Note that the RES variable contains the absolute value of the residuals after the fit. For residual plots and analysis, it may be preferrable to work with the weighted residuals. You can create this with the command
where W contains the weight variable.
Data transformations can be generated easily if needed via the LET command. The BOXCOX LINEARITY PLOT can be a useful command for determining an approriate transformation. Some analysts prefer to standardize the indpendent variables and the dependent variable by subtracting the mean and dividing by the standard deviation. This is done to provide numerical stability (note that Dataplot scales the data internally before performing the regression calculations) and also so that the data and regression coefficients are on a common scale. The original regression and standardized model are related as follows
\( y_{i}^{'} = \frac{y_{i}  \bar{y}}{s_{y}} \) with \( \bar{x} \) and \( s_x \) denoting the mean and standard deviation of the independent variable and \( \bar{y} \) and \( s_y \) denoting the mean and standard deviation of the dependent variable. The parameters are related by
\( \beta_{0}^{'} = \bar{y}  \beta_{1} \bar{x}_1  \ldots  \beta_{p} \bar{x}_p \) A variation on this is the correlation transformation (also called the standardized regression model). Specifically
\( x_{ik}^{'} = \frac{1}{\sqrt{n1}} \frac{x_{ik}  \bar{x}_{k}} {s_{x_k}} \) With this transformation, the \( X'X \) matrix reduces to a correlation matrix of the independent variables. If there are \( p \) independent variables, these transformations can be generated with the commands LET N = SIZE Y LET FACT = 1/SQRT(N1) LOOP FOR K = 1 1 P LET Z^K = STANDARDIZE X^K LET Z^K = AFACT*Z^K END OF LOOPNote:
These commands are documented separately.
where the default is 7.
Osborne (1976), "Nonlinear Least Squares  the Levenberg Algorithm Revisited", ANZIAM Journal, Vol. 19, No. 3, pp. 343357.
1987/09: Support for weighted fits 1988/03: Save LOFCDF parameter 1991/09: Expand number of allowed independent variables from 15 to 5 1992/03: Write coefficient, coefficient sd, and tvalue to dpst1f.dat 1992/03: Write coefficient, coefficient sd, and tvalue to dpst1f.dat 1997/07: Print summary information if maximum iterations reached 2001/04: Print parameter covariance matrix to dpst3f.dat 2014/06: Option to suppress output to auxillary files 2019/04: Option to suppress output to auxillary files . Step 1: Read the data . SKIP 25 READ CHWIRUT1.DAT Y X SKIP 0 . . Step 2: Perform the fit . SET WRITE DECIMALS 5 LET ALPHA = 0.15 LET A = 0.004 LET B = 0.01 FIT Y = EXP(ALPHA*X)/(A+B*X) . . Step 3: Generate diagonistic graphs . TITLE OFFSET 2 TITLE CASE ASIS LABEL CASE ASIS TITLE Predicted Values Overlaid on Raw Data (CHWIRUT1.DAT) X1LABEL Metal Distance Y1LABEL Ultrasonic Response . LINE BLANK SOLID CHARACTER X BLANK . PLOT Y PRED VS X . LABEL TITLE SET 4PLOT MULTIPLOT ON MULTIPLOT CORNER COORDINATES 0 0 100 100 TIC MARK LABEL SIZE 4 CHARACTER SIZE 4 . 4PLOT RES . JUSTIFICATION CENTER MOVE 50 97 TEXT 4Plot of Residuals (CHWIRUT1.DAT)The following output is generated. Least Squares NonLinear Fit Sample Size: 214 Model: Y =EXP(ALPHA*X)/(A+B*X) Replication Case: Replication Standard Deviation: 3.28176 Replication Degrees of Freedom: 192 Number of Distinct Subsets: 22  Residual * Iteration Convergence Standard * Parameter Number Measure Deviation * Estimates  1 0.1000000E01 0.1077871E+02 * 0.1500000E+00 0.4000000E02 0.1000000E01 2 0.5000000E02 0.3721930E+01 * 0.1807460E+00 0.5554412E02 0.1071653E01 3 0.2500000E02 0.3362018E+01 * 0.1905488E+00 0.6119125E02 0.1051960E01 4 0.1250000E02 0.3361673E+01 * 0.1904515E+00 0.6133742E02 0.1052492E01  Approximate Final Parameter Estimates Standard Deviation tValue  1 ALPHA 0.19041 0.02207 8.6266 2 A 0.00613 0.00035 17.5593 3 B 0.01053 0.00080 13.1131 Residual Standard Deviation: 3.36167 Residual Degrees of Freedom: 211 Replication Standard Deviation: 3.28176 Replication Degrees of Freedom: 192 Lack of Fit F Ratio: 1.54740 Lack of Fit F CDF (%): 92.64608 Lack of Fit Degrees of Freedom 1: 19 Lack of Fit Degrees of Freedom 2: 192 Program 2: . Step 1: Read the data . READ ROSZMAN1.DAT X T LET Q = X  SQRT(109737.3/T) . . Step 2: Perform the fit . SET WRITE DECIMALS 5 LET A = 0.2 LET B = 0.00005 LET C = 200 LET D = 123 . CAPTURE SCREEN ON CAPTURE FIT2.OUT FIT Q = A  B*T  ATAN(C/(TD))/3.14159 END OF CAPTURE . . Step 3: Generate diagonistic graphs . TITLE OFFSET 2 TITLE CASE ASIS LABEL CASE ASIS TITLE Predicted Values Overlaid on Raw Data (ROSZMAN1.DAT) X1LABEL Excited State Energy Y1LABEL Quantum Effects for Sulfur I Atom . LINE BLANK SOLID CHARACTER X BLANK . PLOT Q PRED VS T . LABEL TITLE SET 4PLOT MULTIPLOT ON MULTIPLOT CORNER COORDINATES 0 0 100 100 TIC MARK LABEL SIZE 4 CHARACTER SIZE 4 . 4PLOT RES . JUSTIFICATION CENTER MOVE 50 97 TEXT 4Plot of Residuals (ROSZMAN1.DAT)The following output is generated. Least Squares NonLinear Fit Sample Size: 25 Model: Q =A  B*T  ATAN(C/(TD))/3.14159 No Replication Case:  Residual * Iteration Convergence Standard * Parameter Number Measure Deviation * Estimates  1 0.1000000E01 0.2922875E+00 * 0.2000000E+00 0.5000000E04 0.2000000E+03 0.1230000E+03 2 0.5000000E02 0.6473232E01 * 0.2938492E+00 0.1997106E04 0.7968499E+03 0.4821014E+03 3 0.2500000E02 0.4032352E01 * 0.1419884E+00 0.4890395E06 0.1872259E+04 0.1699836E+03 4 0.1265625E01 0.3540118E01 * 0.2300597E+00 0.7388945E05 0.7415686E+03 0.2177707E+03 5 0.6328125E02 0.7476144E02 * 0.2339476E+00 0.1048361E04 0.1026235E+04 0.1199718E+03 6 0.3164063E02 0.5190951E02 * 0.2057654E+00 0.6787648E05 0.1194028E+04 0.1579526E+03 7 0.1582031E02 0.4854277E02 * 0.2019346E+00 0.6193031E05 0.1204886E+04 0.1813351E+03 8 0.7910156E03 0.4854238E02 * 0.2019425E+00 0.6191448E05 0.1204564E+04 0.1813955E+03  Approximate Final Parameter Estimates Standard Deviation tValue  1 A 0.20194 0.01927 10.4816 2 B 0.00001 0.00000 1.9250 3 C 1204.55836 74.63673 16.1389 4 D 181.39214 49.88409 3.6363 Residual Standard Deviation: 0.00485 Residual Degrees of Freedom: 21
 
Date created: 09/02/2021 Last updated: 12/04/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 