### Train your model

$*$ The following was mentioned above, however, it is important and is worth repeating

$*$ The goal of curve fitting is to systematically search through the parameter space p$=[$p$0,$p$1,$p$2...$ p$6]$ to find a special set of $7$ numbers that minimizes the MSE

$*$ The parameterization will be our "best fit model"

$*$ the numerical search process, i$.$e$.$ numerical optimization, is called "training"

$*$ The minimization search which finds the "optimal" parameter vector p$=[$p$0,$p$1,$p$2...$ p$6]$ can be done in two ways

$***$Analytical optimization$**$: i$.$e$.$ manually do the optimization problem with pen and paper and a lot of calculus

$***$Numerical optimization:$**$ Use a computer and a numerical optimization algorithm $($e$.$g$.$ the scipy miminizer$).$ This is the option we will use

$**$ASSIGNMENT$-7**$

$*$ Choose an initial guess for the parameter vector p

$*$ When you are fitting the normalized data, most of the x and y data values should be roughly in the range $[-3$ to $3].$

$*$ As discussed in the lecture, the parameters are associated with the stretches and shifts along either the x or y directions.

$*$ Therefore the parameters values when fitting the normalized data should also be roughly on the order of $[-3$ to $3].$

$***$YOU WILL LIKELY HAVE TO RUN THE CODE SEVERAL TIMES WITH DIFFERENT RANDOM INITIAL GUESSES BEFORE THE OPTIMIZER FINDS THE OPTIMAL FIT $($i$.$e GLOBAL MINIMA$)**$

$*$ Just keep doing "Run All" until you get a fit where both peaks are fit by the model

$*$ Use the Scipy minimizer to minimize the loss function, by varying the components of the parameter vector p$,$ to find the best fit parameters for the model

$```$

res $=$ minimize$($loss$,$ p$0,$ method$=\text{'}$BFGS$\text{'},$ tol$=1$e$-5)$

popt$=$res.x

$```$

$*$ when the minimizer is done, it outputs the $7$ parameters $($i$.$e$.$ numbers$)$ that correspond to the "best fit" parameters. these are stored in "popt$=$res.x$".$

$*$ To get predictions for this particular parameterization, you need to evaluate the model one last time ypred$\_$opt$=$m$($x$\_$train, popt$)$

$*$ Make predictions with the fitted model for the test and validation data

$*$ Print the MSE and median percent error for the training AND validation sets $($the $0\mathrm{.}00001)$ is included to avoid division by zero

$```$

print$("$MSE TRAIN:",np$.$mean$(($yp$\_$train$-$y$\_$train$)**2\mathrm{.}0))$

print$("$MSE VALIDATION:",np$.$mean$(($yp$\_$val$-$y$\_$val$)**2\mathrm{.}0))$

print$("$MEDIAN PERCENT ERROR TRAIN:",$100*$np$.$median$($np$.$absolute$(($yp$\_$train$-$y$\_$train$)/($y$\_$train$+0\mathrm{.}00001))))$