def plot_with_cost(x, y, theta):
for i in range(len(x)):
plt.plot(x[i], y[i], 'bo')
plt.plot(x, predict_(x, theta), 'r')
plt.vlines(x, y, predict_(x, theta), 'r', "dashed")
plt.title("Cost: {}".format(cost_(y, predict_(x, theta))))
plt.show()
def main():
x1 = np.array([[0.], [1.], [2.], [3.], [4.]])
theta1 = np.array([[2.], [4.]])
y_hat1 = predict_(x1, theta1)
y1 = np.array([[2.], [7.], [12.], [17.], [22.]])
print(cost_elem_(y1, y_hat1))
print(cost_(y1, y_hat1))
plot(x1, y1, theta1)
x2 = np.array([[0.2, 2., 20.], [0.4, 4., 40.], [0.6, 6., 60.],
[0.8, 8., 80.]])
theta2 = np.array([[0.05], [1.], [1.], [1.]])
y_hat2 = predict_(x2, theta2)
y2 = np.array([[19.], [42.], [67.], [93.]])
print(cost_elem_(y2, y_hat2))
print(cost_(y2, y_hat2))
plot(x2, y2, theta2)
x3 = np.array([0, 15, -9, 7, 12, 3, -21])
theta3 = np.array([[0.], [1.]])
y_hat3 = predict_(x3, theta3)
y3 = np.array([2, 14, -13, 5, 12, 4, -19])
print(cost_(y3, y_hat3))
print(cost_(y3, y3))
plot(x3, y3, theta3)
def reg_linear_grad(y, x, theta, lambda_):
"""Computes the regularized linear gradient of three non-empty numpy.ndarray, with two
,→ for-loop. The three arrays must have compatible dimensions.
Args:
y: has to be a numpy.ndarray, a vector of dimension m * 1.
x: has to be a numpy.ndarray, a matrix of dimesion m * n.
theta: has to be a numpy.ndarray, a vector of dimension n * 1.
lambda_: has to be a float.
Returns:
A numpy.ndarray, a vector of dimension n * 1, containing the results of the formula for all
,→ j.
None if y, x, or theta are empty numpy.ndarray.
None if y, x or theta does not share compatibles dimensions.
Raises:
This function should not raise any Exception.
"""
if x.size == 0 or y.size == 0 or theta.size == 0 or x.shape[0] != y.shape[
0] or x is None or y is None:
return None
gr_vec = np.zeros((theta.shape[0], 1))
y_hat = predict_(x, theta)
gr_vec[0] = np.sum((y_hat - y)) / float(y.shape[0])
for j in range(1, theta.shape[0]):
gr_vec[j] = (np.sum((y_hat - y) * x[:, j - 1].reshape(-1, 1)) +
(lambda_ * theta[j])) / y.shape[0]
return gr_vec
def vec_reg_linear_grad(y, x, theta, lambda_):
"""Computes the regularized linear gradient of three non-empty numpy.ndarray, without any
,→ for-loop. The three arrays must have compatible dimensions.
Args:
y: has to be a numpy.ndarray, a vector of dimension m * 1.
x: has to be a numpy.ndarray, a matrix of dimesion m * n.
theta: has to be a numpy.ndarray, a vector of dimension n * 1.
lambda_: has to be a float.
Returns:
A numpy.ndarray, a vector of dimension n * 1, containing the results of the formula for all
,→ j.
None if y, x, or theta are empty numpy.ndarray.
None if y, x or theta does not share compatibles dimensions.
Raises:
This function should not raise any Exception.
"""
if x.size == 0 or y.size == 0 or theta.size == 0 or x.shape[0] != y.shape[
0] or x is None or y is None:
return None
y_hat = predict_(x, theta)
theta2 = np.copy(theta)
theta2[0] = 0
gr_vec = (np.matmul(np.transpose(add_intercept(x)),
(y_hat - y)) + (lambda_ * theta2)) / y.shape[0]
return gr_vec
def plot_with_cost(x, y, theta):
y_hat = predict_(x, theta)
# const_elm = cost_elem_(y, y_hat)
plt.title(f'Cost: {cost_(y, y_hat)}')
plt.scatter(x, y)
plt.plot(x, y_hat, c='r')
plt.show()
def simple_gradient(x, y, theta):
"""Computes a gradient vector from three non-empty numpy.ndarray,
without any for-loop. The three arrays must have compatible dimensions.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a 2 * 1 vector.
Returns:
The gradient as a numpy.ndarray, a vector of dimension 2 * 1.
None if x, y, or theta are empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
if (not isinstance(x, np.ndarray) or not isinstance(y, np.ndarray)
or not isinstance(theta, np.ndarray) or theta.size != 2
or x.size != y.size or x.ndim != 1 or y.ndim != 1
or theta.ndim != 1):
return None
else:
h = predict_(x, theta)
j0 = (1 / y.size) * (h - y)
j0 = abs(j0.sum())
j1 = (1 / y.size) * ((h - y) * x)
j1 = abs(j1.sum())
return np.array((j0, j1))
def simple_gradient(x: np.ndarray, y: np.ndarray,
theta: np.ndarray) -> np.ndarray:
"""Computes a gradient vector from three non-empty numpy.ndarray.
The three arrays must have compatible dimensions.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a 2 * 1 vector.
Returns:
The gradient as a numpy.ndarray, a vector of dimension 2 * 1.
None if x, y, or theta are empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
if (0 in [len(x), len(y), len(theta)] or x.shape != y.shape
or (x.shape[1] + 1) != theta.shape[0]):
return None
res = np.zeros(shape=(theta.shape))
m = x.shape[0]
y_hat = predict_(x, theta)
for i in range(m):
res[0][0] += (y_hat[i][0] - y[i][0])
res[1][0] += (y_hat[i][0] - y[i][0]) * (x[i][0])
res = res / m
return res
def plot_with_cost(x, y, theta):
plt.title("Cost : " + str(cost_(y, predict_(x, theta))))
plt.plot(x, y, 'ro', color='blue')
pred = predict_(x, theta)
plt.plot(x, pred, color='orange')
zeroes = np.zeros_like(x)
difs = np.concatenate((zeroes, x))
for valx, valpred, valy in zip(x, pred, y):
if valpred > valy:
mini = valy
maxi = valpred
else:
mini = valpred
maxi = valy
plt.vlines(x=valx, ymin=mini, ymax=maxi, linestyle='--', color='red')
plt.show()
def plot_with_cost(x, y, theta):
plt.plot(x, y, 'o')
plt.plot(x, (theta[1] * x + theta[0]), '-')
y_hat = predict_(x, theta)
cost = half_mse_(y, y_hat)
plt.plot((x, x), (y, y_hat), '--r')
plt.title(("Cost: " + str(round(cost, 6))))
plt.show()
def plot_with_cost(x, y, theta):
y_hat = predict_(x, theta)
cost = cost_(y, y_hat)
plt.plot(x, y, 'bo')
plt.plot(x, y_hat, 'r', c='orangered')
plt.title(str(cost))
plt.plot((x, x), (y, y_hat), '--r')
plt.show()
def simple_gradient(x, y, theta):
h = predict_(x, theta)
if h is None:
return None
check, y, h = check_size_and_shape(y, h)
if not check:
return None
#x[:, :1] is the first column of x values (as x here does not have column of 1s)
return np.array([np.sum(h - y) / y.size, np.sum((h - y) * x[:, :1]) / y.size])
def test_correc():
x = np.arange(1, 6)
# Example 1:
theta1 = np.array([5, 0])
assert np.equal(predict_(x, theta1), np.array([5., 5., 5., 5., 5.])).all()
# Do you understand why y_hat contains only 5's here?
# Example 2:
theta2 = np.array([0, 1])
assert (predict_(x, theta2) == np.array([1., 2., 3., 4., 5.])).all()
# Do you understand why y_hat == x here?
# Example 3:
theta3 = np.array([5, 3])
assert (predict_(x, theta3) == np.array([8., 11., 14., 17., 20.])).all()
# Example 4:
theta4 = np.array([-3, 1])
assert (predict_(x, theta4) == np.array([-2., -1., 0., 1., 2.])).all()
def simple_gradient(x, y, theta):
"""Computes a gradient vector from three non-empty numpy.ndarray, without any for-loop. The
,→ three arrays must have compatible dimensions.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
11
theta: has to be an numpy.ndarray, a 2 * 1 vector.
Returns:
The gradient as a numpy.ndarray, a vector of dimension 2 * 1.
None if x, y, or theta are empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
# try:
return np.array([
1.0 / len(x) * np.sum(predict_(x, theta) - y), 1.0 / len(x) * np.sum(
(predict_(x, theta) - y) * x)
])
def simple_gradient(x, y, theta1):
ret = []
s0 = 0
s1 = 0
y_hat = predict_(x, theta1)
for i in range(len(y)):
s0 += y_hat[i] - y[i]
s1 += (y_hat[i] - y[i]) * x[i]
ret.append(s0 / len(y))
ret.append(s1 / len(y))
return (np.array(ret))
def simple_gradient(x, y, theta): """Computes a gradient vector from three non-empty numpy.ndarray, without any for-loop. ,→ The three arrays must have compatible dimensions. Args: x: has to be an numpy.ndarray, a vector of dimension m * 1. y: has to be an numpy.ndarray, a vector of dimension m * 1. theta: has to be an numpy.ndarray, a 2 * 1 vector. Returns: The gradient as a numpy.ndarray, a vector of dimension 2 * 1. None if x, y, or theta are empty numpy.ndarray. None if x, y and theta do not have compatible dimensions. Raises: This function should not raise any Exception. """ m = x.shape[0] # print(predict_(x, theta)) # print(y) # print(predict_(x, theta) - y) j0 = (predict_(x, theta) - y).sum() / m j1 = ((predict_(x, theta) - y).dot(x)) / m # print(j1) return np.array([j0, j1])
def plot(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exceptions.
"""
plt.plot(x, y, 'bo')
plt.plot(x, predict_(x, theta), 'r-')
plt.show()
def test():
# Example 0:
theta1 = fit_(x, y, alpha=5e-8, max_iter=1500000)
print("theta1:", theta1)
# Output:
print("should be:", np.array([[1.40709365], [1.1150909]]))
# Example 1:
print("y_pred:", predict_(x, theta1))
# Output:
print(
"should be:",
np.array([[15.3408728], [25.38243697], [36.59126492], [55.95130097],
[65.53471499]]))
assert True == False
def main():
x1 = np.array([0., 1., 2., 3., 4.])
theta1 = np.array([2., 4.])
y_hat1 = predict_(x1, theta1)
y1 = np.array([2., 7., 12., 17., 22.])
print(cost_elem_(y1, y_hat1))
print(cost_(y1, y_hat1))
x2 = np.array([[0.2, 2., 20.], [0.4, 4., 40.], [0.6, 6., 60.],
[0.8, 8., 80.]])
theta2 = np.array([[0.05], [1.], [1.], [1.]])
y_hat2 = predict_(x2, theta2)
y2 = np.array([[19.], [42.], [67.], [93.]])
print(cost_elem_(y2, y_hat2))
print(cost_(y2, y_hat2))
x3 = np.array([[0], [15], [-9], [7], [12], [3], [-21]])
theta3 = np.array([[0.], [1.]])
y_hat3 = predict_(x3, theta3)
y3 = np.array([[2], [14], [-13], [5], [12], [4], [-19]])
print(cost_elem_(y3, y_hat3))
print(cost_(y3, y_hat3))
print(cost_(y3, y3))
def test2():
theta2 = np.array([1, -0.4])
res = simple_gradient(x, y, theta2)
loss = mse_(x, predict_(x, theta2))
print(res, loss)
theta2 -= 0.001 * res
res = simple_gradient(x, y, theta2)
loss = mse_(x, predict_(x, theta2))
print(res, loss)
theta2 -= 0.001 * res
res = simple_gradient(x, y, theta2)
loss = mse_(x, predict_(x, theta2))
print(res, loss)
theta2 -= 0.0001 * res
res = simple_gradient(x, y, theta2)
loss = mse_(x, predict_(x, theta2))
print(res, loss)
theta2 -= 0.0001 * res
res = simple_gradient(x, y, theta2)
loss = mse_(x, predict_(x, theta2))
print(res, loss)
assert (res == np.array([58.86823748, 2229.72297889])).all()
def plot(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray. Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exceptions.
"""
for x_, y_ in zip(x, y):
plt.scatter(x_, y_, color='b', s=30)
predX = predict_(x, theta)
plt.plot(x, predX, color='r')
plt.show()
def mse_(x, theta, y):
"""
Description:
Calculate the MSE between the predicted output and the real output.
Args:
y: has to be a numpy.ndarray, a vector of dimension m * 1.
y_hat: has to be a numpy.ndarray, a vector of dimension m * 1.
Returns:
mse: has to be a float.
None if there is a matching dimension problem.
Raises:
This function should not raise any Exceptions.
"""
y_hat = predict_(x, theta)
if len(y) < 1 or len(y_hat) < 1:
return None
return np.sum((y_hat - y)**2) / float(y.shape[0])
def plot(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exceptions.
"""
plt.plot(x, y, 'ro')
linex = np.linspace(np.min(x), np.max(x), 2)
plt.plot(linex, predict_(linex, theta), color="blue", linestyle="dotted")
plt.show()
def plot_with_cost(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exception.
"""
yb = predict_(x, theta)
plt.plot(x, yb, 'y')
plt.plot(x, y, 'bo')
for i in range(0, y.size):
x1, y1 = [x[i], x[i]], [y[i], yb[i]]
plt.plot(x1, y1, "r--")
plt.title(f"Cost : {cost_(y,yb)}")
plt.show()
def plot_with_cost(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exception.
"""
y_hat = predict_(x, theta)
plt.title(f'Cost : {cost_(y, y_hat)}')
for i in range(x.shape[0]):
plt.plot([x[i], x[i]], [y[i], y_hat[i]], 'r--', color='r')
plt.plot(x, y, '.')
plt.plot(x, y_hat)
plt.show()
def plot_with_cost(x, y, theta):
"""
Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exception.
"""
y_hat = predict_(x, theta)
plt.title(f'Cost: {cost_(y, y_hat)}')
plt.plot(x, y_hat, color='r')
for x_, y_, y_hat in zip(x, y, y_hat):
plt.scatter(x_, y_, color='b', s=30)
plt.plot([x_, x_], [y_hat, y_], 'r--', color='g')
plt.show()
def gradient(x, y, theta):
"""
Computes a gradient vector from three non-empty numpy.ndarray,
without any for-loop. The three arrays must have the compatible dimensions.
Args:
x: has to be an numpy.ndarray, a matrix of dimension m * n.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector (n +1) * 1.
Returns:
The gradient as a numpy.ndarray, a vector of dimensions n * 1,
containg the result of the formula for all j.
None if x, y, or theta are empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
if x.shape[0] != y.shape[0] or x.shape[1] + 1 != theta.shape[0]:
return None
xi = add_ones(x)
return (1 / y.shape[0]) * np.sum(xi.T * (predict_(x, theta) - y), axis=1)
def vec_gradient(x, y, theta): """Computes a gradient vector from three non-empty numpy.ndarray, without any for-loop. The → three arrays must have the compatible dimensions. Args: x: has to be an numpy.ndarray, a matrix of dimension m * n. y: has to be an numpy.ndarray, a vector of dimension m * 1. theta: has to be an numpy.ndarray, a vector (n +1) * 1. Returns: The gradient as a numpy.ndarray, a vector of dimensions n * 1, containg the result of the → formula for all j. None if x, y, or theta are empty numpy.ndarray. None if x, y and theta do not have compatible dimensions. Raises: This function should not raise any Exception. """ if len(x) < 1 or len(y) < 1 or len(theta) < 1 or x is None or y is None or theta is None or x.shape[0] != y.shape[0]: return None y_hat = predict_(x, theta) gr_vec = (np.matmul(add_intercept(x).transpose(), (y_hat - y))) / y.shape[0] return gr_vec
def plot_with_cost(x, y, theta):
"""Plot the data and prediction line from three non-empty numpy.ndarray.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1.
y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a vector of dimension 2 * 1.
Returns:
Nothing.
Raises:
This function should not raise any Exception.
"""
plt.plot(x, y, 'o')
plt.plot(x, theta[1] * x + theta[0])
y_hat = predict_(x, theta)
for x_i, y_hat_i, y_i in zip(x, y_hat, y):
plt.plot([x_i, x_i], [y_i, y_hat_i], 'r--')
plt.show()
def gradient(x, y, theta):
"""Computes a gradient vector from three non-empty numpy.ndarray, without any for loop. The
,→ three arrays must have compatible dimensions.
Args:
x: has to be a numpy.ndarray, a matrix of dimension m * 1.
y: has to be a numpy.ndarray, a vector of dimension m * 1.
theta: has to be a numpy.ndarray, a 2 * 1 vector.
Returns:
The gradient as a numpy.ndarray, a vector of dimension 2 * 1.
None if x, y, or theta is an empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
if len(x) < 1 or len(y) < 1 or len(
theta) < 1 or x.shape != y.shape or theta.shape[
0] < 1 or x is None or y is None:
return None
y_hat = predict_(x, theta)
gr_vec = (np.matmul(np.transpose(add_intercept(x)),
(y_hat - y))) / y.shape[0]
return gr_vec
def simple_gradient(x, y, theta):
"""Computes a gradient vector from three non-empty numpy.ndarray, without any for-loop. The
→ three arrays must have compatible dimensions.
Args:
x: has to be an numpy.ndarray, a vector of dimension m * 1. y: has to be an numpy.ndarray, a vector of dimension m * 1.
theta: has to be an numpy.ndarray, a 2 * 1 vector.
Returns:
The gradient as a numpy.ndarray, a vector of dimension 2 * 1.
None if x, y, or theta are empty numpy.ndarray.
None if x, y and theta do not have compatible dimensions.
Raises:
This function should not raise any Exception.
"""
if len(x) < 1 or len(y) < 1 or len(
theta) < 1 or x.shape != y.shape or theta.shape[
0] < 1 or x is None or y is None:
return None
gr_vec = np.zeros((2, ))
y_hat = predict_(x, theta)
gr_vec[0] = np.sum((y_hat - y)) / float(y.shape[0])
gr_vec[1] = np.sum((y_hat - y) * x) / float(y.shape[0])
return gr_vec
