To add a bias term, we will add a new column to our X matrix that is full of constants.

Does it matter what constant term we choose?

The simplest way to do this is to use hstack which takes in a list of matrices and horizontally concatenates them (i.e. adds on new columns -- there exists a vstack that adds new rows). The simpleest way to construct a constant term is to use np.ones which takes in a list with the number of ones to make for each dimension.

e.g. np.ones([4,2]) will make

1 1

1 1

1 1

1 1

#TODO construct an X matrix with a bias term.

X_with_bias = np.zeros(10)

X_validation_with_bias = np.zeros(10)

#TODO replace the np.zeros() with the correct code

B_with_bias = calculate_weights_with_library(X_with_bias,Y)

Yhat_with_bias = calculate_yhat(X_with_bias, B_with_bias)

Yhat_validation_with_bias = calculate_yhat(X_validation_with_bias, B_with_bias)

residuals_with_bias = calculate_residuals(Y, Yhat_with_bias)

residuals_validation_with_bias = calculate_residuals(Y_validation, Yhat_validation_with_bias)

rmse_with_bias = calculate_rmse(residuals_with_bias)

rmse_validation_with_bias = calculate_rmse(residuals_validation_with_bias)

print('RMSE with bias term:',rmse_with_bias)

print('RMSE Validation with bias term:',rmse_validation_with_bias)

plt.plot(Y,residuals_with_bias,'x')

plt.plot(Y_validation,residuals_validation_with_bias,'ro')

plt.show()