nlparci

Nonlinear regression parameter confidence intervals

Syntax

ci = nlparci(beta,r,"Covar",CovB)

ci = nlparci(beta,r,"Jacobian",J)

ci = nlparci(___,"Alpha",alpha)

Description

ci = nlparci(beta,r,"Covar",CovB) returns the 95% confidence intervals ci for the nonlinear least-squares parameter estimates beta. Before calling nlparci, get the estimated coefficients beta, residuals r, and estimated covariance matrix CovB by using the nlinfit function to fit a nonlinear regression model.

If you use a robust option with nlinfit, you must use this syntax for nlparci. The covariance matrix CovB is required with robust fitting.

ci = nlparci(beta,r,"Jacobian",J) returns the 95% confidence intervals ci for the nonlinear least-squares parameter estimates beta. Before calling nlparci, get the estimated coefficients beta, residuals r, and Jacobian J by using the nlinfit function to fit a nonlinear regression model.

ci = nlparci(___,"Alpha",alpha) returns the 100(1 — alpha)% confidence intervals, using any of the input argument combinations in the previous syntaxes.

example

Examples

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Fit Exponential Decay Model

Open Live Script

Fit an exponential decay model of the form

$y_{i} = β_{1} + β_{2} \exp (- β_{3} x_{i}) + ϵ_{i}$ ,

where $β_{j}$ are the parameters to estimate, $x_{i}$ are the data points, $y_{i}$ are the responses, and $ε_{i}$ is the noise term.

Write a function handle that represents the model.

mdl = @(beta,x)(beta(1) + beta(2)*exp(-beta(3)*x));

Generate synthetic data with true parameter values beta = [1;3;2], x data points distributed exponentially with parameter 2, and noise distributed normally with mean 0 and standard deviation 0.1.

rng(9845,'twister') % For reproducibility
beta = [1;3;2];
x = exprnd(2,100,1);
epsn = normrnd(0,0.1,100,1);
y = mdl(beta,x) + epsn;

Fit the model to data starting from the arbitrary value beta0 = [2;2;2].

beta0 = [2;2;2];
[betahat,r,J,CovB,mse] = nlinfit(x,y,mdl,beta0);
betahat

betahat = 3×1

    1.0153
    3.0229
    2.1070

Check the 95% confidence intervals using the covariance matrix. Note that the true parameter values [1;3;2] are within all three intervals.

ci = nlparci(betahat,r,"Covar",CovB)

ci = 3×2

    0.9869    1.0438
    2.9401    3.1058
    1.9963    2.2177

You can obtain the same result using the Jacobian.

ci = nlparci(betahat,r,"Jacobian",J)

ci = 3×2

    0.9869    1.0438
    2.9401    3.1058
    1.9963    2.2177

Determine if the true parameter values [1;3;2] are in the narrower 90% confidence intervals. Note that $β_{1}$ and $β_{2}$ are within these intervals, but $β_{3}$ is not.

ci = nlparci(betahat,r,"Covar",CovB,"Alpha",0.1)

ci = 3×2

    0.9915    1.0392
    2.9536    3.0923
    2.0144    2.1996

Plot the estimated regression coefficients with the 90% confidence intervals.

lowerBars = betahat - ci(:,1);
upperBars = ci(:,2) - betahat;
errorbar(1:3,betahat,lowerBars,upperBars,'*'), grid
title('Estimated Regression Coefficients with 90% Confidence Intervals')
ylabel('Coefficient Value')
xlabel('Estimated Regression Coefficient \beta_j, j = 1,2,3')
xticks([1 2 3])
xlim([.8 3.2])

Figure contains an axes object. The axes object with title Estimated Regression Coefficients with 90% Confidence Intervals, xlabel Estimated Regression Coefficient beta indexOf j baseline , blank j blank = 1,2,3, ylabel Coefficient Value contains an object of type errorbar.

Input Arguments

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`beta` — Estimated regression coefficients
numeric vector

Estimated regression coefficients, specified as a numeric vector returned by the nlinfit function.