Question $1$: Curve Fitting

You can interactively fit curves and surfaces to data using MATLAB's Curve Fitting Toolbox $($CFT$)$ without writing much code. The step$-$by$-$step guide on how to use the toolbox for curve fitting is as follows:

a$)2$D Data:

$x=[1,2,3,4,5,6,7,8,9,10]$

$y=[2\mathrm{.}2,3\mathrm{.}1,4\mathrm{.}4,5\mathrm{.}4,6\mathrm{.}1,7\mathrm{.}2,8\mathrm{.}0,9\mathrm{.}6,10\mathrm{.}3,12\mathrm{.}3]$

Step:

Load the data: create the data in the MATLAB Command Windows

Type cftool in Command Windows to open the toolbox.

Fit a Polynomial Curve:

a$.$ Select Polynomial from the model options.

b$.$ Fit a $2$nd$-$degree polynomial $($quadratic curve$).$

c$.$ Observe the plot, residuals, and goodness$-$of$-$fit metrics.

Fit an Exponential Curve:

a$.$ Select Exponential from the model options.

b$.$ Fit the data using the model $y=a{e}^{bx}$

c$.$ Compare the fit quality with the polynomial fit.

Export Results:

a$.$ Export both fitted models to the workspace for analysis:

$\%$ Example command for exported fit

$y_{{?}_p\text{o}\text{l}\text{y}}=$ polyFit $(x)$; $\%$ Polynomial fit

Exponential fit

b$)3$D Data:

Generate a $3$D dataset using the following command:

$[x,y]=$ meshgrid $(1$:$10,1$:$10)$;

$z=3**x^2+2**y+$randn$(size(x))$

$\%$ Add some noise

$2.$ Load the Data: Save $x,y,$ and $z$ into the MATLAB workspace.

$3.$ Type cftool in Command Windows to open the toolbox.

a$.$ Import $x,y,$ and $z.$

b$.$ Select a Polynomial Surface and specify the degree $($e$.$g$.,2).$

$4.$ Fit a Custom Surface:

a$.$ Define a custom equation like $z=a{x}^2+by+c$

b$.$ Adjust the fit options as needed.

$5.$ Visualize the Fit:

a$.$ Overlay the fitted surface on the original data.

b$.$ Examine residuals and diagnostic plots.

$6.$ Export Results: Export the models to the workspace for further analysis.