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NonlinearModelFit
✖
NonlinearModelFit

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✖
NonlinearModelFit[{{x1,y1},{x2,y2},…},form,{β1,…},x]

constructs a nonlinear model with formula form that fits the yi for each xi using the free parameters βi.

✖
NonlinearModelFit[data,form,params,{x1,…}]

constructs a nonlinear model where form depends on the variables xk.

✖
NonlinearModelFit[data,{form,cons},params,{x1,…}]

constructs a nonlinear model subject to the parameter constraints cons.

Details and Options

  • NonlinearModelFit attempts to model the input data using a general mathematical formula with free parameters.
  • NonlinearModelFit produces a nonlinear model of the form under the assumption that the original are independent normally distributed with mean and common standard deviation.
  • NonlinearModelFit returns a symbolic FittedModel object to represent the nonlinear model it constructs. The properties and diagnostics of the model can be obtained from model["property"].
  • The value of the best-fit function from NonlinearModelFit at a particular point x1, … can be found from model[x1,…].
  • The best-fit function from NonlinearModelFit[data,form,pars,vars] is the same as the result from FindFit[data,form,pars,vars].
  • NonlinearModelFit[data,form,{{β1,val1},…},vars] starts the search for a fit with {β1->val1,…}.
  • Possible forms of data are:
  • {y1,y2,…}equivalent to the form {{1,y1},{2,y2},…}
    {{x11,x12,…,y1},…}a list of independent values xij and the responses yi
    {{x11,x12,…}y1,…}a list of rules between input values and response
    {{x11,x12,…},…}{y1,y2,…}a rule between a list of input values and responses
    {{x11,…,y1,…},…}nfit the n^(th) column of a matrix
    Tabular[…]namefit the column name in a tabular object
  • With multivariate data such as {{x_(11),x_(12),... ,y_(1)},{x_(21),x_(22),... ,y_(2)},...}, the number of coordinates xi1, xi2, … should equal the number of variables xi.
  • The data points can be approximate real numbers. Uncertainty can be specified using Around.
    • Options
  • NonlinearModelFit takes the following options:
  • AccuracyGoalAutomaticthe number of digits of accuracy sought
    ConfidenceLevel 95/100confidence level for parameters and predictions
    EvaluationMonitorNoneexpression to evaluate whenever form is evaluated
    GradientAutomaticthe list of gradient components for form
    MaxIterationsAutomaticmaximum number of iterations to use
    Method Automaticmethod to use
    PrecisionGoalAutomaticthe precision sought
    StepMonitorNonethe expression to evaluate whenever a step is taken
    VarianceEstimatorFunction Automaticfunction for estimating the error variance
    Weights Automaticweights for data elements
    WorkingPrecision Automaticthe precision used in internal computations
  • With ConfidenceLevel->p, probability-p confidence intervals are computed for parameter and prediction intervals.
  • With the setting Weights->{w1,w2,…}, the error variance for yi is assumed to be proportional to .
  • With the setting Weights->Automatic, the weights will be set to 1 if the data contains exact values. If the data contains Around values, the weights will be set to , with the total response variance.
  • The total response variance is a function of the initial response variance Δyi2 and the independent values variance .
  • The uncertainties are propagated through the model using AroundReplace, and the resulting variance is added to response variance Δyi2. The function FindRoot is used internally to find a self-consistent solution according to the Fasano and Vio method.
  • With the setting VarianceEstimatorFunction->f, the common variance is estimated by f[res,w], where res={y1-,y2-,…} is the list of residuals and w is the list of weights.
  • Using VarianceEstimatorFunction->(1&) and Weights->{1/Δy12,1/Δy22,…}, Δyi is treated as the known uncertainty of measurement yi, and parameter standard errors are effectively computed only from the weights.
  • Possible settings for Method include:
  • "ConjugateGradient"nonlinear conjugate gradient
    "Gradient"gradient descent
    "LevenbergMarquardt"Gauss–Newton method for least squares
    "Newton"Newton method
    "QuasiNewton"quasi-Newton BFGS
    "InteriorPoint"interior point method
    "NMinimize"use NMinimize for optimization
    Automaticautomatic default method
  • Additional method suboptions can be given in the form Method->{…,opts}.
  • The Method option can take any local optimization method as specified in the tutorial Unconstrained Optimization: Methods for Local Minimization. »
  • Any global optimization method can be specified as a submethod to the "NMinimize" method. They can be found in the Numerical Algorithms for Constrained Global Optimization tutorial. »
    • Properties
  • For constrained models, properties based on approximate normality assumptions may not be valid. When such values are computed, the values are generated along with a warning message.
  • Properties related to data and the fitted function using model["property"] include:
  • "BestFit"fitted function
    "BestFitParameters"parameter estimates
    "Data"the input data or design matrix and response vector
    "Function"best-fit pure function
    "Response"response values in the input data
    "Weights"weights used to fit the data
  • Types of residuals include:
  • "FitResiduals"difference between actual and predicted responses
    "StandardizedResiduals"fit residuals divided by the standard error for each residual
    "StudentizedResiduals"fit residuals divided by single deletion error estimates
  • Properties related to the sum of squared errors include:
  • "ANOVA"analysis of variance data
    "EstimatedVariance"estimate of the error variance
  • Properties and diagnostics for parameter estimates include:
  • "CorrelationMatrix"asymptotic parameter correlation matrix
    "CovarianceMatrix"asymptotic parameter covariance matrix
    "ParameterEstimates"table of fitted parameter information
    "ParameterBias"estimated bias in the parameter estimates
    "ParameterConfidenceRegion"ellipsoidal parameter confidence region
  • Properties for curvature diagnostics include:
  • "CurvatureConfidenceRegion"confidence region for curvature diagnostics
    "FitCurvature"table of fit curvature information
    "MaxIntrinsicCurvature"measure of maximum intrinsic curvature
    "MaxParameterEffectsCurvature"measure of maximum parameter effects curvature
  • Properties related to influence measures include:
  • "HatDiagonal"diagonal elements of the hat matrix
    "SingleDeletionVariances"list of variance estimates with the ^(th) data point omitted
  • Properties of predicted values include:
  • "MeanPredictionBands"confidence bands for mean predictions
    "MeanPredictions"confidence intervals for the mean predictions
    "PredictedResponse"fitted values for the data
    "SinglePredictionBands"confidence bands based on single observations
    "SinglePredictions"confidence intervals for the predicted response of single observations
  • Properties that measure goodness of fit include:
  • "AdjustedRSquared" adjusted for the number of model parameters
    "AIC"Akaike Information Criterion
    "AICc"finite sample corrected AIC
    "BIC"Bayesian Information Criterion
    "RSquared"coefficient of determination

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Fit a nonlinear model to some data:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fr8gq2
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ito2gu
Out[2]=2

Obtain the functional form:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-i19ql9
Out[3]=3

Evaluate the model at a point:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-cy4qr5
Out[4]=4

Visualize the fitted function with the data:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ky8uld
Out[5]=5

Extract and plot the residuals:

In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-o0cv4
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jxmybb
Out[7]=7

Scope  (15)Survey of the scope of standard use cases

Data  (7)

Fit a model of one variable, assuming increasing integer-independent values:

In[17]:=17
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hf4cmk
Out[17]=17

Fit a model of more than one variable:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bvlb1y
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-dco39z
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-d3u3zn
Out[3]=3

Fit a list of rules:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-n6f6mz
Out[1]=1

Fit a rule of input values and responses:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jqt9yr
Out[1]=1

Specify a column as the response:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ccidut
Out[1]=1

Give starting values when parameters are far from the default value 1:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c1p3pj
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-o9latu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-idjus
Out[3]=3

With the default starting values, the model is effectively 0:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ybfg4
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hj2clp
Out[5]=5

Obtain a list of available properties for a nonlinear model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-haavoo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hmjjxg
Out[2]=2

Properties  (8)

Data & Fitted Functions  (1)

Fit a nonlinear model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hu5oz
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fj8t12
Out[2]=2

Extract the original data:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-na9fdf
Out[3]=3

Obtain and plot the best fit:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-btkjeg
Out[4]=4

Obtain the fitted function as a pure function:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-2573y
Out[5]=5

Residuals  (1)

Examine residuals for a fit:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c96hoj
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-e4h1qz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c6mebv

Visualize the raw residuals:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-iyrir
Out[4]=4

Visualize scaled residuals in stem plots:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fc0pug
Out[5]=5

Plot the absolute differences between the standardized and Studentized residuals:

In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-zwlex
Out[6]=6

Sums of Squares  (1)

Fit a nonlinear model to some data:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bvfu5x
Out[1]=1

Extract the estimated error variance:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c0a7u8
Out[2]=2

Obtain the analysis of variance table:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-xre76
Out[3]=3

Get the sums of squares column from the table:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-dv5b70
Out[4]=4

Parameter Estimation Diagnostics  (1)

Obtain a formatted table of parameter information:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-yeijh
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-g1mvmy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-vvtd9
Out[3]=3

Extract the column of -statistic values:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-m5b3hb
Out[4]=4

Curvature Diagnostics  (1)

Fit a nonlinear model to some data:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-dea3ag
Out[1]=1

Obtain a table of curvature measures for the fitted model:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bh3yuu
Out[2]=2

Extract the list of numeric values from the table:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-66fxz
Out[3]=3

Extract the max parameter effects curvature value:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-2ux26
Out[4]=4

Influence Measures  (1)

Fit some data containing extreme values to a nonlinear model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-dxjd2
Out[1]=1

Use single deletion variances to check the impact on the error variance of removing each point:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bw2le3
Out[2]=2

Check the diagonal elements of the hat matrix to assess influence of points on the fitting:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fbslf9
Out[3]=3

Prediction Values  (1)

Fit a nonlinear model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hzioq
Out[1]=1

Plot the predicted values against the observed values:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c2gdcw
Out[2]=2

Obtain tabular results for mean- and single-prediction confidence intervals:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ho9ak9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ietrop
Out[4]=4

Get the single-prediction intervals from the table:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fhixbf
Out[5]=5

Extract 99% mean prediction bands:

In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-vx0x8
Out[6]=6

Goodness-of-Fit Measures  (1)

Obtain a table of goodness-of-fit measures for a nonlinear model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-f1m5pa
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-oijifv
Out[2]=2
In[8]:=8
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-cys2tt
Out[8]=8

Generalizations & Extensions  (2)Generalized and extended use cases

Fit data to a model defined by a numerical operation:

In[22]:=22
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-d7mjw9
In[28]:=28
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jlw8
In[30]:=30
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jq2e9w
Out[30]=30
In[31]:=31
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b7fqic
Out[31]=31

Use ParametricNDSolveValue to make the computation much faster by caching solutions of the differential equation:

In[11]:=11
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ghv7u3
Out[12]=12
In[21]:=21
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-f36tv4
Out[21]=21

Perform other mathematical operations on the functional form of the model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-i4rgx5
Out[1]=1

Integrate symbolically and numerically:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-brsld4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ly2tj7
Out[3]=3

Find a predictor value that gives a particular value for the model:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-pmp2d
Out[4]=4

Options  (10)Common values & functionality for each option

ConfidenceLevel  (1)

The default gives 95% confidence intervals:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ecoitr
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-j4wn3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b58ly3
Out[3]=3

Use 99% intervals instead:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-l2hd6e
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-qrf46
Out[5]=5

Set the level to 90% within FittedModel:

In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ihsmr4
Out[6]=6

Method  (3)

Use the default method for minimizing the least-squares objective function:

In[14]:=14
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-y6xuco
Out[14]=14

Use Newton's method for optimization:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-tgll73
Out[1]=1

Configure the step control method for Newton's algorithm:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-i339qo
Out[2]=2

Use the interior point method with constraints:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-64ysmv
Out[3]=3

Perform a more exhaustive search with the global optimization methods from NMinimize:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-tvuiy4
Out[1]=1

Use the submethod "RandomSearch":

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fmqwot
Out[2]=2

Specify the number of initial search points for the "RandomSearch" algorithm:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b38cit
Out[3]=3

VarianceEstimatorFunction  (1)

Use the default unbiased estimate of error variance:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-d4of89
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-gya2cd
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jcrgh
Out[3]=3

Assume a known error variance:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-em28ru
Out[4]=4

Estimate the variance by the mean squared error:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-eykbej
Out[5]=5

Weights  (4)

Fit a model using equal weights:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-d74zow
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-io2mln
Out[2]=2

Give explicit weights for the data points:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-gr7fv
Out[3]=3

Use Around values to give different weights to data points:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b88ln0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ex1yjq
Out[2]=2

Find the weights that were used to account for the uncertainty in the data:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ee2z8f
Out[3]=3

Use Around values in both the independent values and responses:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b8p7lg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ehp5a
Out[2]=2

In some cases, the root finding algorithm that handles uncertainty in the independent variates does not converge using the standard option settings:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-i712o3
Out[1]=1

Use the FixedPoint algorithm with a low DampingFactor and high MaxIterations to reach convergence:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-oifq1n
Out[2]=2

WorkingPrecision  (1)

Use WorkingPrecision to get higher precision in parameter estimates:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-olb75
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-cpb4wl
Out[2]=2

Obtain the fitted function:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-uefph
Out[3]=3

Reduce the precision in property computations after the fitting:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-cjjn8p
Out[4]=4

Applications  (1)Sample problems that can be solved with this function

Simulate some data:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-iiinqn

Fit a nonlinear model to the data:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-fyrh8
Out[2]=2

Obtain and visualize 90% confidence bands for the fit:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-lp2zjz
Out[4]=4

Obtain 95%, 99%, and 99.9% confidence bands:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hcwlgq

Visualize the confidence bands for the various levels:

In[9]:=9
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-l7uyg3
Out[9]=9

Properties & Relations  (5)Properties of the function, and connections to other functions

NonlinearModelFit fits linear and nonlinear models assuming normally distributed errors:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-23q98
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-xyy7q
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-jw4mzi
Out[3]=3

LinearModelFit fits linear models assuming normally distributed errors:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hq5ql8
Out[4]=4

FindFit and NonlinearModelFit fit equivalent models:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-f4ja0x
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-oov7mh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-k8uh75
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-imbp8
Out[4]=4

NonlinearModelFit allows for extraction of additional information about the fitting:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bdsgz8
Out[5]=5

NonlinearModelFit assumes normally distributed responses:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-mojlid
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-g4yusx
Out[2]=2

LogitModelFit assumes binomially distributed responses:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-ifep8u
Out[3]=3

The fits are not identical:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-1zgny
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-d7xe8
Out[5]=5

The same is true for ProbitModelFit:

In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-e9wuy
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-dqhfj3
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b06stb
Out[8]=8

NonlinearModelFit will use the time stamps of a TimeSeries as variables:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-55rjns
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-8vunzb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c50i1w
Out[3]=3

Rescale the time stamps and fit again:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hiel46
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-7d00cg
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-emgg00
Out[6]=6

Find fit for the values:

In[7]:=7
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-siexdh
Out[7]=7

NonlinearModelFit acts pathwise on a multipath TemporalData:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-1mqe20
Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

Distributional assumptions are based upon an unconstrained model:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-frkfex
Out[2]=2

Here the confidence interval for contains points that violate the constraint:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-b4mfkg
Out[3]=3

The coefficient of determination in NonlinearModelFit is calculated with uncorrected data:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-vxuola
In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-wrf255
Out[2]=2

The coefficient of determination:

In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-462hwt
Out[3]=3

Direct calculation using residuals and data:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-mv0odk
Out[4]=4

Sometimes is defined using centralized data:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-f1la5p
Out[5]=5

Fit data that spans over multiple orders of magnitude:

In[1]:=1
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-bjl91a
Out[1]=1

An exponential fit might appear correct at first glance:

In[2]:=2
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-c41j92
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-hdg3de
Out[3]=3

But shows significant deviations on a log scale:

In[4]:=4
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-1t40hh
Out[4]=4

This can be addressed by using weights inversely proportional to the variance:

In[5]:=5
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-83s7cy
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bzrhw01segbnw4ng-pvn6q1
Out[6]=6
Wolfram Research (2008), NonlinearModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/NonlinearModelFit.html (updated 2025).
✖
Wolfram Research (2008), NonlinearModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/NonlinearModelFit.html (updated 2025).

Text

Wolfram Research (2008), NonlinearModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/NonlinearModelFit.html (updated 2025).

✖
Wolfram Research (2008), NonlinearModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/NonlinearModelFit.html (updated 2025).

CMS

Wolfram Language. 2008. "NonlinearModelFit." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/NonlinearModelFit.html.

✖
Wolfram Language. 2008. "NonlinearModelFit." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/NonlinearModelFit.html.

APA

Wolfram Language. (2008). NonlinearModelFit. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NonlinearModelFit.html

✖
Wolfram Language. (2008). NonlinearModelFit. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NonlinearModelFit.html

BibTeX

@misc{reference.wolfram_2025_nonlinearmodelfit, author="Wolfram Research", title="{NonlinearModelFit}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/NonlinearModelFit.html}", note=[Accessed: 23-April-2025 ]}

✖
@misc{reference.wolfram_2025_nonlinearmodelfit, author="Wolfram Research", title="{NonlinearModelFit}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/NonlinearModelFit.html}", note=[Accessed: 23-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_nonlinearmodelfit, organization={Wolfram Research}, title={NonlinearModelFit}, year={2025}, url={https://reference.wolfram.com/language/ref/NonlinearModelFit.html}, note=[Accessed: 23-April-2025 ]}

✖
@online{reference.wolfram_2025_nonlinearmodelfit, organization={Wolfram Research}, title={NonlinearModelFit}, year={2025}, url={https://reference.wolfram.com/language/ref/NonlinearModelFit.html}, note=[Accessed: 23-April-2025 ]}