`lsqcurvefit`

enables you to fit parameterized
nonlinear functions to data easily. You can use `lsqnonlin`

as
well; `lsqcurvefit`

is simply a convenient way
to call `lsqnonlin`

for curve fitting.

In this example, the vector `xdata`

represents
100 data points, and the vector `ydata`

represents
the associated measurements. Generate the data using the following
script:

rng(5489,'twister') % reproducible xdata = -2*log(rand(100,1)); ydata = (ones(100,1) + .1*randn(100,1)) + (3*ones(100,1)+... 0.5*randn(100,1)).*exp((-(2*ones(100,1)+... .5*randn(100,1))).*xdata);

The modeled relationship between `xdata`

and `ydata`

is

$$ydat{a}_{i}={a}_{1}+{a}_{2}\mathrm{exp}(-{a}_{3}xdat{a}_{i})+{\epsilon}_{i}.$$ | (10-25) |

The script generates `xdata`

from 100 independent
samples from an exponential distribution with mean 2. It generates `ydata`

from Equation 10-25 using `a`

= `[1;3;2]`

, perturbed by adding normal
deviates with standard deviations `[0.1;0.5;0.5]`

.

The goal is to find parameters $${\widehat{a}}_{i}$$, *i* = 1, 2, 3,
for the model that best fit the data.

In order to fit the parameters to the data using `lsqcurvefit`

,
you need to define a fitting function. Define the fitting function `predicted`

as
an anonymous function:

predicted = @(a,xdata) a(1)*ones(100,1)+a(2)*exp(-a(3)*xdata);

To fit the model to the data, `lsqcurvefit`

needs
an initial estimate `a0`

of the parameters. Enter

a0 = [2;2;2];

Run the solver `lsqcurvefit`

as follows:

[ahat,resnorm,residual,exitflag,output,lambda,jacobian] =... lsqcurvefit(predicted,a0,xdata,ydata); Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the default value of the function tolerance.

To see the resulting least-squares estimate of $$\widehat{a}$$, enter:

ahat ahat = 1.0169 3.1444 2.1596

The fitted values `ahat`

are within 8% of `a`

= `[1;3;2]`

.

If you have Statistics and Machine Learning Toolbox™ software, use the `nlparci`

function
to generate confidence intervals for the `ahat`

estimate.

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