Weights
is an option for various fitting and other functions which specifies weights to associate with data elements.
Details
- Weights->Automatic associates weight 1 with all data elements.
- Weights->{w1,w2,…} associates weight wi with the i
data element. - Weights->func associates weight func[xi1,xi2,…,yi] with the i
data element. - 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. »
Examples
open all close allBasic Examples (1)
Fit a model using equal weights:
LinearModelFit[Range[10] ^ 2, x, x]//NormalGive explicit weights to the data points:
LinearModelFit[Range[10] ^ 2, x, x, Weights -> 1 / Range[10]]//Normal
LinearModelFit[Range[10] ^ 2, x, x, Weights -> (Sqrt[#]&)]//NormalCompute weights from the response values:
LinearModelFit[Range[10] ^ 2, x, x, Weights -> (Sqrt[#2]&)]//NormalScope (2)
Use weights in a nonlinear model:
data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}};NonlinearModelFit[data, Log[a + b x ^ 2], {a, b}, x]//NormalGive explicit weights to the data points:
(nlm = NonlinearModelFit[data, Log[a + b x ^ 2], {a, b}, x, Weights -> {1, (1/2), (1/3), (1/4)}])//Normaldata = {{0, 1}, {1, 0}, {3, 2}, {5, 10}};GeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Poisson"]//NormalGeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Poisson", Weights -> {1, 2, 3, 4}]//NormalLogitModelFit[{.01, .3, .5, .7, .9}, x, x]//NormalLogitModelFit[{.01, .3, .5, .7, .9}, x, x, Weights -> (1 / Sqrt[#]&)]//NormalProbitModelFit[{.01, .3, .5, .7, .9}, x, x]//NormalProbitModelFit[{.01, .3, .5, .7, .9}, x, x, Weights -> (1 / Sqrt[#]&)]//NormalWeight by a function of multiple variables:
data = {{2.8, 1.8, 4.5}, {1.8, 2.2, 2.7}, {0.6, 4.6, 1.6}, {2.8, 4.1, 2.6}, {3., 0.1, 3.7}};LinearModelFit[data, {x, y}, {x, y}, Weights -> (Sqrt[#1 ^ 2 + #2 ^ 2]&)]//NormalLinearModelFit[data, {x, y}, {x, y}]//NormalCompute weights from the response values:
LinearModelFit[data, {x, y}, {x, y}, Weights -> (Log[#3]&)]//NormalApplications (1)
Fit a nonlinear model using measurement errors as weights:
data = {{2.8, 1.8, 4.5}, {1.8, 2.2, 2.7}, {0.6, 4.6, 0.6}, {2.8, 4.1, 2.6}, {3., 0.1, 3.7}};Δys = {.1, .15, .12, .2, .08};nlm = NonlinearModelFit[data, a x Exp[b Sin[y]], {a, b}, {x, y}, Weights -> 1 / Δys ^ 2, VarianceEstimatorFunction -> (1&)]Obtain standard errors for the parameters:
nlm["ParameterErrors"]Compare to estimates with weights not assumed to be from measurement errors:
nlm["ParameterErrors", VarianceEstimatorFunction -> Automatic]Properties & Relations (2)
Weights impact the relative importance of data points on the fitting:
LinearModelFit[Range[10], x ^ 2, x, Weights -> Range[10]]//NormalScaling by a constant does not change the parameter estimates:
LinearModelFit[Range[10], x ^ 2, x, Weights -> Range[10] / Total[Range[10]]]//NormalObtain parameter estimates from a weighted linear fitting:
data = RandomReal[10, {5, 2}];
wts = {1, 2, 3, 2, 1};LinearModelFit[data, {x, Sin[x]}, x, Weights -> wts]["BestFitParameters"]LeastSquares gives the equivalent result when weights are incorporated:
dm = DesignMatrix[data, {x, Sin[x]}, x];
resp = data[[All, -1]];
LeastSquares[Sqrt[wts]dm, Sqrt[wts]resp]Text
Wolfram Research (2008), Weights, Wolfram Language function, https://reference.wolfram.com/language/ref/Weights.html.
CMS
Wolfram Language. 2008. "Weights." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Weights.html.
APA
Wolfram Language. (2008). Weights. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Weights.html