def backpropagate(network, input_vector, target):
hidden_outputs, outputs = feed_forward(network, input_vector)
# the output * (1 - output) is from the derivative of sigmoid
output_deltas = [
output * (1 - output) * (output - target[i])
for i, output in enumerate(outputs)
]
# adjust weights for output layer (network[-1])
for i, output_neuron in enumerate(network[-1]):
for j, hidden_output in enumerate(hidden_outputs + [1]):
output_neuron[j] -= output_deltas[i] * hidden_output
# back-propagate errors to hidden layer
hidden_deltas = [
hidden_output * (1 - hidden_output) *
dot(output_deltas, [n[i] for n in network[-1]])
for i, hidden_output in enumerate(hidden_outputs)
]
# adjust weights for hidden layer (network[0])
for i, hidden_neuron in enumerate(network[0]):
for j, input in enumerate(input_vector + [1]):
hidden_neuron[j] -= hidden_deltas[i] * input
def covariance(x, y):
n = len(x)
return dot(de_mean(x), de_mean(y)) / (n - 1)
def predict(x_i, beta):
return dot(x_i, beta)
def ridge_penalty(beta, alpha):
return alpha * dot(beta[1:], beta[1:])
def transform_vector(v, components):
return [dot(v, w) for w in components]
def project(v, w):
"""return the projection of v onto w"""
coefficient = dot(v, w)
return scalar_multiply(coefficient, w)
def directional_variance_gradient_i(x_i, w):
"""the contribution of row x_i to the gradient of
the direction-w variance"""
projection_length = dot(x_i, direction(w))
return [2 * projection_length * x_ij for x_ij in x_i]
def directional_variance_i(x_i, w):
"""the variance of the row x_i in the direction w"""
return dot(x_i, direction(w))**2
print()
random.seed(0) # so that you get the same results as me
bootstrap_betas = bootstrap_statistic(list(zip(x, daily_minutes_good)),
estimate_sample_beta, 100)
bootstrap_standard_errors = [
standard_deviation([beta[i] for beta in bootstrap_betas])
for i in range(4)
]
print("bootstrap standard errors", bootstrap_standard_errors)
print()
print("p_value(30.63, 1.174)", p_value(30.63, 1.174))
print("p_value(0.972, 0.079)", p_value(0.972, 0.079))
print("p_value(-1.868, 0.131)", p_value(-1.868, 0.131))
print("p_value(0.911, 0.990)", p_value(0.911, 0.990))
print()
print("regularization")
random.seed(0)
for alpha in [0.0, 0.01, 0.1, 1, 10]:
beta = estimate_beta_ridge(x, daily_minutes_good, alpha=alpha)
print("alpha", alpha)
print("beta", beta)
print("dot(beta[1:],beta[1:])", dot(beta[1:], beta[1:]))
print("r-squared", multiple_r_squared(x, daily_minutes_good, beta))
print()
def cosine_similarity(v, w):
return dot(v, w) / math.sqrt(dot(v, v) * dot(w, w))
def neuron_output(weights, inputs):
return sigmoid(dot(weights, inputs))
def perceptron_output(weights, bias, x):
"""returns 1 if the perceptron 'fires', 0 if not"""
return step_function(dot(weights, x) + bias)
# and maximize using gradient descent
beta_hat = maximize_batch(fn, gradient_fn, beta_0)
print("beta_batch", beta_hat)
beta_0 = [1, 1, 1]
beta_hat = maximize_stochastic(logistic_log_likelihood_i,
logistic_log_gradient_i, x_train, y_train,
beta_0)
print("beta stochastic", beta_hat)
true_positives = false_positives = true_negatives = false_negatives = 0
for x_i, y_i in zip(x_test, y_test):
predict = logistic(dot(beta_hat, x_i))
if y_i == 1 and predict >= 0.5: # TP: paid and we predict paid
true_positives += 1
elif y_i == 1: # FN: paid and we predict unpaid
false_negatives += 1
elif predict >= 0.5: # FP: unpaid and we predict paid
false_positives += 1
else: # TN: unpaid and we predict unpaid
true_negatives += 1
precision = true_positives / (true_positives + false_positives)
recall = true_positives / (true_positives + false_negatives)
print("precision", precision)
print("recall", recall)
def matrix_product_entry(A, B, i, j):
return dot(get_row(A, i), get_column(B, j))
def logistic_log_partial_ij(x_i, y_i, beta, j):
"""here i is the index of the data point,
j the index of the derivative"""
return (y_i - logistic(dot(x_i, beta))) * x_i[j]
def logistic_log_likelihood_i(x_i, y_i, beta):
if y_i == 1:
return math.log(logistic(dot(x_i, beta)))
else:
return math.log(1 - logistic(dot(x_i, beta)))