def directional_variance_gradient(matrix: Matrix, w: Vector) -> float:
w_dir = direction(w)
return [
sum(2 * dot_product(v, w_dir) * v[i] for v in matrix)
for i in range(len(w))
]
def directional_variance(matrix: Matrix, v: Vector) -> float:
v_dir = direction(v)
return sum(dot_product(v, v_dir)**2 for v in matrix)
def ridge_penalty(beta: Vector, alpha: float) -> float:
"""The larger alpha is, the larger the penalty is"""
return alpha * dot_product(beta[1:], beta[1:])
def transform_vector(v: Vector, components: Matrix) -> Vector:
return [dot_product(v, w) for w in components]
def predict(xs: Vector, beta: Vector) -> float:
assert xs[0] == 1, "The first element of xs must be 1"
return dot_product(xs, beta)