Function Repository Resource:

MultiNonlinearModelFit

Fit multiple datasets with multiple expressions that share parameters

Contributed by: Sjoerd Smit

ResourceFunction["MultiNonlinearModelFit"][{dat₁,dat₂,…},{form₁,form₂,…},{β₁,…},{x₁,…}]

is a generalized form of NonlinearModelFit where each dataset dat_i is fitted with expression form_i. The expressions are allowed to share parameter values β_i with each other.

ResourceFunction["MultiNonlinearModelFit"][{dat₁,dat₂, …}, <|"Expressions"→{form₁,form₂,…}, "Constraints"→cons |>, {β₁,…},{x₁,…}]

fits the model subject to parameter constraints cons.

Details and Options

ResourceFunction["MultiNonlinearModelFit"] works by recasting the data and fitting forms into a form suitable for NonlinearModelFit.

The datasets dat_i must all be numeric matrices.

The number of datasets dat_i must match the number of expressions form_i. It is also possible to specify a function that evaluates to a list when provided with numerical arguments.

The parameters β_i can be specified in the same way as in NonlinearModelFit. ResourceFunction["MultiNonlinearModelFit"] also inherits all options from NonlinearModelFit.

Internally, the datasets dat_i are joined together into a single matrix, so if a Weights vector is specified as an option, it should have the same length as Join[dat₁,dat₂,…]. It is also possible to specify a Weights vector that has a length matching the number of datasets. There is a special option Weights → "InverseLengthWeights" to specify that each dataset should weigh equally in the fit, regardless of the number of points in each. Finally, it is also possible to specify the Weights as a list of vectors matching the input data.

The returned model uses the symbol

(\[FormalN]) to switch between the different expressions form_i. This symbol should therefore be avoided by the user when using ResourceFunction["MultiNonlinearModelFit"]. This symbol can be changed with the "DatasetIndexSymbol" option.

Examples

Basic Examples (2)

Fit some random data to simple linear models with a shared slope parameter:

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data = RandomVariate[BinormalDistribution[0.7], {2, 100}];
data[[1, All, 2]] += 1.;
data[[2, All, 2]] += -1.;

fit = ResourceFunction["MultiNonlinearModelFit"][
data,
{a x + b1, a x + b2}, {a, b1, b2}, {x}
]

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Show[
ListPlot[data],
Plot[{fit[1, x], fit[2, x]}, {x, -5, 5}]
]

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Fit the same model with parameter constraints to allow for a small difference between the slopes of the lines (this works best by using a quadratic form for the difference between the slope parameters, since (a₁-a₂)²is differentiable whereas Abs[a₁–a₂] is not):

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$fit = ResourceFunction["MultiNonlinearModelFit"][ data, <| "Expressions" -> {a1 x + b1, a2 x + b2}, "Constraints" -> (a1 - a2)^2 < 0.1^2 && b1 > 0 && b2 < 0 |>, {a1, a2, b1, b2}, {x} ]$

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Scope (2)

It is possible to define a fitting function that evaluates to a list of values equal to the number of datasets when provided with numerical parameters:

The fit function does not evaluate symbolically, only numerically:

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Use this function to fit with:

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fit = ResourceFunction["MultiNonlinearModelFit"][
data,
fitfun[{a, b1, b2}, x],
{a, b1, b2},
{x}
]

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Fit two Gaussian peaks with a shared location parameter:

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$xvals = Range[-5, 5, 0.1]; gauss[x_] := Evaluate@PDF[NormalDistribution[], x]; With[{amp1 = 1.2, amp2 = 0.5, width1 = 1, width2 = 2, sharedOffset = 0.5, eps = 0.05}, dat1 = Table[{x, amp1 gauss[(x - sharedOffset)/width1] + eps RandomVariate[NormalDistribution[]]}, {x, xvals}]; dat2 = Table[{x, amp2 gauss[(x - sharedOffset)/width2] + eps RandomVariate[NormalDistribution[]]}, {x, xvals}] ]; plot = ListPlot[{dat1, dat2}]$

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Fit with the models that were used to generate the data:

In[14]:=

fit = ResourceFunction["MultiNonlinearModelFit"][
{dat1, dat2},
{
amp1 gauss[(x - sharedOffset)/width1],
amp2 gauss[(x - sharedOffset)/width2]
},
{amp1, amp2, width1, width2, sharedOffset},
{x}
]

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Extract the fits as a list of expressions:

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Compare the fits to the data:

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Options (6)

Weights (5)

The Weights option can be specified in a number of ways. First generate datasets with an unequal number of points and offset them slightly:

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$gauss[x_] := Evaluate@PDF[NormalDistribution[], x]; With[{amp1 = 1.2, amp2 = 0.5, width1 = 1, width2 = 2, sharedOffset = 0.5, eps = 0.025}, dat1 = Table[ {x, amp1 gauss[(x - sharedOffset)/width1] + eps RandomVariate[NormalDistribution[]]}, {x, -5, 5, 0.1} ]; dat2 = Table[ {x, amp2 gauss[(x - sharedOffset + 1)/width2] + eps RandomVariate[NormalDistribution[]]}, {x, -5, 5, 0.2} ] ]; fitfuns = { amp1 gauss[(x - sharedOffset)/width1], amp2 gauss[(x - sharedOffset)/width2] }; fitParams = {amp1, amp2, width1, width2, sharedOffset}; variables = {x}; plot = ListPlot[{dat1, dat2}]$

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Fit the data normally. In this case, each individual data point has equal weights in the fit, so the first dataset gets more weight overall since it has more points:

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