def ridge_penalty(beta, alpha):
return alpha * dot(beta[1:], beta[1:])
def predict(x_i, beta):
return dot(x_i, beta)
def predict(x_i, beta):
"""assumes that the firts element of each x_i is 1"""
return dot(x_i, beta)
def transform_vector(v, components):
return [dot(v, w) for w in components]
print
random.seed(0) # so that you get the same results as me
bootstrap_betas = bootstrap_statistic(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 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 project(v, w):
"""return the projection of v onto w"""
coefficient = dot(v, w)
return scalar_multiply(coefficient, w)
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
def covariance(x, y):
n = len(x)
return dot(de_mean(x), de_mean(y)) / (n - 1)
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(log(x_i, beta)))
# 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