how to find the best order for a polynomial fitting in matlab

main.m
% Load dataset
load('mydata.mat')

% Split data into training and test sets
N = length(y);
idx = randperm(N);
idx_train = idx(1:floor(0.8*N));
idx_test = idx(floor(0.8*N)+1:end);

x_train = x(idx_train);
y_train = y(idx_train);
x_test = x(idx_test);
y_test = y(idx_test);

% Maximum order to consider
max_order = 10;

% Initialize arrays to store error metrics
train_err = zeros(max_order, 1);
test_err = zeros(max_order, 1);

% Loop over orders and fit polynomial
for i=1:max_order
   p = polyfit(x_train, y_train, i);
   y_train_fit = polyval(p, x_train);
   y_test_fit = polyval(p, x_test);
   
   % Compute error metrics for training and test sets
   train_err(i) = mean((y_train - y_train_fit).^2);
   test_err(i) = mean((y_test - y_test_fit).^2);
end

% Plot error metrics
figure
plot(1:max_order, train_err, 'bo-', 'LineWidth', 2)
hold on
plot(1:max_order, test_err, 'ro-', 'LineWidth', 2)
xlabel('Polynomial order')
ylabel('Error')
legend('Training set', 'Test set')
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