bootci

Compute the confidence interval for the capability index in statistical process control.

Generate 30 random numbers from the normal distribution with mean 1 and standard deviation 1.

rng('default') % For reproducibility
y = normrnd(1,1,30,1);

Specify the lower and upper specification limits of the process. Define the capability index.

LSL = -3;
USL = 3;
capable = @(x)(USL-LSL)./(6*std(x));

Compute the 95% confidence interval for the capability index by using 2000 bootstrap samples. By default, bootci uses the bias corrected and accelerated percentile method to construct the confidence interval.

ci = bootci(2000,capable,y)

ci = 2×1

    0.5937
    0.9900

Compute the studentized confidence interval for the capability index.

sci = bootci(2000,{capable,y},'Type','student')

sci = 2×1

    0.5193
    0.9930

Compute bootstrap confidence intervals for the coefficients of a nonlinear regression model. The technique used in this example involves bootstrapping the predictor and response values, and assumes that the predictor variable is random. For a technique that assumes the predictor variable is fixed and bootstraps the residuals, see Bootstrap Confidence Intervals for Linear Regression Model Coefficients.

Note: This example uses nlinfit, which is useful when you only need the coefficient estimates or residuals of a nonlinear regression model and you need to repeat fitting a model multiple times, as in the case of bootstrapping. If you need to investigate a fitted regression model further, create a nonlinear regression model object by using fitnlm. You can create confidence intervals for the coefficients of the resulting model by using the coefCI object function, although this function does not use bootstrapping.

Generate data from the nonlinear regression model $$y=b_1+b_2\cdot exp(-b_{3}x)+ϵ$$, where $$b_1=1$$, $$b_2=3$$, and $$b_3=2$$ are coefficients; the predictor variable x is exponentially distributed with mean 2; and the error term $$ϵ$$ is normally distributed with mean 0 and standard deviation 0.1.

modelfun = @(b,x)(b(1)+b(2)*exp(-b(3)*x));

rng('default') % For reproducibility
b = [1;3;2];
x = exprnd(2,100,1);
y = modelfun(b,x) + normrnd(0,0.1,100,1);

Create a function handle for the nonlinear regression model that uses the initial values in beta0.

beta0 = [2;2;2];
beta = @(predictor,response)nlinfit(predictor,response,modelfun,beta0)

beta = function_handle with value:
    @(predictor,response)nlinfit(predictor,response,modelfun,beta0)

Compute the 95% bootstrap confidence intervals for the coefficients of the nonlinear regression model. Create the bootstrap samples from the generated data x and y.

ci = bootci(1000,beta,x,y)

ci = 2×3

    0.9821    2.9552    2.0180
    1.0410    3.1623    2.2695

The first two confidence intervals include the true coefficient values $$b_1=1$$ and $$b_2=3$$, respectively. However, the third confidence interval does not include the true coefficient value $$b_3=2$$.

Now compute the 99% bootstrap confidence intervals for the model coefficients.

newci = bootci(1000,{beta,x,y},'Alpha',0.01)

newci = 2×3

    0.9730    2.9112    1.9562
    1.0469    3.1876    2.3133

All three confidence intervals include the true coefficient values.

Compute bootstrap confidence intervals for the coefficients of a linear regression model. The technique used in this example involves bootstrapping the residuals and assumes that the predictor variable is fixed. For a technique that assumes the predictor variable is random and bootstraps the predictor and response values, see Bootstrap Confidence Intervals for Nonlinear Regression Model Coefficients.

Note: This example uses regress, which is useful when you only need the coefficient estimates or residuals of a regression model and you need to repeat fitting a model multiple times, as in the case of bootstrapping. If you need to investigate a fitted regression model further, create a linear regression model object by using fitlm. You can create confidence intervals for the coefficients of the resulting model by using the coefCI object function, although this function does not use bootstrapping.

Load the sample data.

load hald

Perform a linear regression and compute the residuals.

x = [ones(size(heat)),ingredients];
y = heat;
b = regress(y,x);
yfit = x*b;
resid = y - yfit;

Compute the 95% bootstrap confidence intervals for the coefficients of the linear regression model. Create the bootstrap samples from the residuals. Use normal approximated intervals with bootstrapped bias and standard error by specifying 'Type','normal'. You cannot use the default confidence interval type in this case.

ci = bootci(1000,{@(bootr)regress(yfit+bootr,x),resid}, ...
    'Type','normal')

ci = 2×5

  -47.7130    0.3916   -0.6298   -1.0697   -1.2604
  172.4899    2.7202    1.6495    1.2778    0.9704

Plot the estimated coefficients b, omitting the intercept term, and display error bars showing the coefficient confidence intervals.

slopes = b(2:end)';
lowerBarLengths = slopes-ci(1,2:end);
upperBarLengths = ci(2,2:end)-slopes;
errorbar(1:4,slopes,lowerBarLengths,upperBarLengths)
xlim([0 5])
title('Coefficient Confidence Intervals')

Only the first nonintercept coefficient is significantly different from 0.

Compute the mean and standard deviation of 100 bootstrap samples. Find the 95% confidence interval for each statistic.

Generate 100 random numbers from the exponential distribution with mean 5.

rng('default') % For reproducibility
y = exprnd(5,100,1);

Draw 100 bootstrap samples from the vector y. For each bootstrap sample, compute the mean and standard deviation. Find the 95% bootstrap confidence interval for the mean and standard deviation.

[ci,bootstat] = bootci(100,@(x)[mean(x) std(x)],y);

ci(:,1) contains the lower and upper bounds of the mean confidence interval, and c(:,2) contains the lower and upper bounds of the standard deviation confidence interval. Each row of bootstat contains the mean and standard deviation of a bootstrap sample.

Plot the mean and standard deviation of each bootstrap sample as a point. Plot the lower and upper bounds of the mean confidence interval as dotted vertical lines, and plot the lower and upper bounds of the standard deviation confidence interval as dotted horizontal lines.

plot(bootstat(:,1),bootstat(:,2),'o')
xline(ci(1,1),':')
xline(ci(2,1),':')
yline(ci(1,2),':')
yline(ci(2,2),':')
xlabel('Mean')
ylabel('Standard Deviation')