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description= NonlinearModelFit attempts to model the input data using a general mathematical formula with free parameters.;
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wolfram language educational programs for adults summer school winter school educational programs for youth middle school camp high school research program computational adventures read stephen wolfram s writings wolfram blog wolfram tech books wolfram media complex systems educational resources wolfram mathworld wolfram in stem wolfram challenges wolfram problem generator wolfram initiatives wolfram science wolfram foundation history of mathematics project events stephen wolfram livestreams online in person events contact us connect follow for ais wolfram cloud your account user portal enable javascript to interact with content and submit forms on wolfram websites learn how wolfram language system documentation center close nonlinearmodelfit see also linearmodelfit generalizedlinearmodelfit fittedmodel findfit fit timeseriesmodelfit findformula findminimum nminimize related guides statistical model analysis statistical data analysis numerical data matrix based minimization scientific models scientific data analysis time series processing supervised machine learning tabular modeling image computation for microscopy tech notes statistical model analysis curve fitting constrained optimization unconstrained optimization see also linearmodelfit generalizedlinearmodelfit fittedmodel findfit fit timeseriesmodelfit findformula findminimum nminimize related guides statistical model analysis statistical data analysis numerical data matrix based minimization scientific models scientific data analysis time series processing supervised machine learning tabular modeling image computation for microscopy tech notes statistical model analysis curve fitting constrained optimization unconstrained optimization nonlinearmodelfit x 1 y 1 x 2 y 2 form β 1 x constructs a nonlinear model with formula form that fits the y i for each x i using the free parameters β i nonlinearmodelfit data form params x 1 constructs a nonlinear model where form depends on the variables x k nonlinearmodelfit data form cons params x 1 constructs a nonlinear model subject to the parameter constraints cons details and options examples basic examples scope data constraints properties data fitted functions residuals sums of squares parameter estimation diagnostics curvature diagnostics influence measures prediction values goodness of fit measures generalizations extensions options confidencelevel evaluationmonitor gradient maxiterations method stepmonitor varianceestimatorfunction weights workingprecision applications properties relations possible issues see also tech notes related guides history cite this page built in symbol see also linearmodelfit generalizedlinearmodelfit fittedmodel findfit fit timeseriesmodelfit findformula findminimum nminimize related guides statistical model analysis statistical data analysis numerical data matrix based minimization scientific models scientific data analysis time series processing supervised machine learning tabular modeling image computation for microscopy tech notes statistical model analysis curve fitting constrained optimization unconstrained optimization see also linearmodelfit generalizedlinearmodelfit fittedmodel findfit fit timeseriesmodelfit findformula findminimum nminimize related guides statistical model analysis statistical data analysis numerical data matrix based minimization scientific models scientific data analysis time series processing supervised machine learning tabular modeling image computation for microscopy tech notes statistical model analysis curve fitting constrained optimization unconstrained optimization nonlinearmodelfit nonlinearmodelfit x 1 y 1 x 2 y 2 form β 1 x constructs a nonlinear model with formula form that fits the y i for each x i using the free parameters β i nonlinearmodelfit data form params x 1 constructs a nonlinear model where form depends on the variables x k nonlinearmodelfit data form cons params x 1 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 x 1 can be found from model x 1 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 β 1 val 1 vars starts the search for a fit with β 1 val 1 data possible forms of data are y 1 y 2 equivalent to the form 1 y 1 2 y 2 x 11 x 12 y 1 a list of independent values x ij and the responses y i x 11 x 12 y 1 a list of rules between input values and response x 11 x 12 y 1 y 2 a rule between a list of input values and responses x 11 y 1 n fit the n column of a matrix tabular name fit the column name in a tabular object with multivariate data such as the number of coordinates x i 1 x i 2 should equal the number of variables x i the data points can be approximate real numbers uncertainty can be specified using around options nonlinearmodelfit takes the following options accuracygoal automatic the number of digits of accuracy sought confidencelevel 95 100 confidence level for parameters and predictions evaluationmonitor none expression to evaluate whenever form is evaluated gradient automatic the list of gradient components for form maxiterations automatic maximum number of iterations to use method automatic method to use precisiongoal automatic the precision sought stepmonitor none the expression to evaluate whenever a step is taken varianceestimatorfunction automatic function for estimating the error variance weights automatic weights for data elements workingprecision automatic the precision used in internal computations with confidencelevel p probability p confidence intervals are computed for parameter and prediction intervals with the setting weights w 1 w 2 the error variance for y i 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 δ y i 2 and the independent values variance the uncertainties are propagated through the model using aroundreplace and the resulting variance is added to response variance δ y i 2 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 y 1 y 2 is the list of residuals and w is the list of weights using varianceestimatorfunction 1 and weights 1 δ y 1 2 1 δ y 2 2 δ y i is treated as the known uncertainty of measurement y i 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 automatic automatic 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 bestfitaround fitted function and mean uncertainty bestfitdataaround fitted function and data uncertainty 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 data point omitted properties of predicted values include meanpredictionbands confidence bands for mean predictions meanpredictions data about mean predictions predictedresponse fitted values for the data singlepredictionbands confidence bands based on single observations singlepredictions data about 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 the properties bestfit bestfitaround bestfitdataaround singlepredictionbands and meanpredictionbands can also be called as prop x or prop x 1 x 2 to evaluate these properties at specific independent values examples open all close all basic examples 2 fit a nonlinear model to some data wolfram language code nlm nonlinearmodelfit log a b x 2 a b x evaluate the model at a point wolfram language code nlm 2 3 visualize the fitted function with the data wolfram language code show listplot nlm data plot nlm x x 0 7 frame true fit a model of two variables wolfram language code nlm nonlinearmodelfit a x b y c a b c x y obtain the functional form as an expression wolfram language code normal nlm extract information about the fitting wolfram language code nlm fitresiduals scope 17 data 7 fit a model of one variable assuming increasing integer independent values wolfram language code nonlinearmodelfit log 3 log 5 log 7 log 9 log 11 log a x b a b x fit a model of more than one variable wolfram language code seedrandom 0 data flatten table x y exp 3x 7y randomreal 1 1 x 5 y 5 1 wolfram language code nonlinearmodelfit data exp a x b y a b x y wolfram language code normal fit a list of rules wolfram language code nonlinearmodelfit 0 0 1 1 2 2 3 3 4 3 5 4 log a b x 2 a b x fit a rule of input values and responses wolfram language code nonlinearmodelfit 0 1 2 3 4 5 0 1 2 3 3 4 log a b x 2 a b x specify a column as the response wolfram language code nonlinearmodelfit 1 0010978759544942 1 1 0 8949819317774829 1 2 0 6851122183910994 1 3 0 7639434166108392 3 1 1 3567934196980815 3 2 0 18537374908770277 3 3 1 exp a x b y a b x y give starting values when parameters are far from the default value 1 wolfram language code data 25 0 001 25 5 0 002 26 0 011 26 5 0 045 27 0 112 27 5 0 215 28 0 259 28 5 0 206 29 0 112 29 5 0 044 30 0 011 wolfram language code nlm nonlinearmodelfit data a exp x b 2 a 5 b 25 x wolfram language code plot nlm x x 25 30 epilog point data plotstyle orange thick with the default starting values the model is effectively 0 wolfram language code nlmd nonlinearmodelfit data a exp x b 2 a b x wolfram language code plot chop nlmd x x 25 30 epilog point data plotstyle orange thick obtain a list of available properties for a nonlinear model wolfram language code nlm nonlinearmodelfit range 10 exp a x a x wolfram language code nlm properties constraints 2 specify a constraint on a model parameter to ensure that the model expression is well defined wolfram language code nonlinearmodelfit a log b x b 0 01 a b x use a constraint to link model parameters together wolfram language code nonlinearmodelfit a x b y c a b 0 a b c x y properti...
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