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import numpy as np
import scipy.linalg
from scipy.optimize import minimize
from math import sqrt, pi
import time
import matplotlib.pyplot as plt
import seaborn
seaborn.set_style("white")
data = np.loadtxt('CASP.csv', delimiter=',', skiprows=1)
y = data[:, 0]
X = data[:, 1:]
def split_train_test(X, y, fraction_train=9.0 / 10.0):
end_train = round(X.shape[0] * fraction_train)
X_train = X[0:end_train, ]
y_train = y[0:end_train]
X_test = X[end_train:, ]
y_test = y[end_train:]
return X_train, y_train, X_test, y_test
def normalize_features(X_train, X_test):
mean_X_train = np.mean(X_train, 0)
std_X_train = np.std(X_train, 0)
std_X_train[std_X_train == 0] = 1
X_train_normalized = (X_train - mean_X_train) / std_X_train
X_test_normalized = (X_test - mean_X_train) / std_X_train
return X_train_normalized, X_test_normalized
X_train, y_train, X_test, y_test = split_train_test(X, y)
X_train, X_test = normalize_features(X_train, X_test)
X_train = np.concatenate((X_train, np.ones((X_train.shape[0], 1))), 1)
X_test = np.concatenate((X_test, np.ones((X_test.shape[0], 1))), 1)
def map_estimate(X, y, sigma=1.0, tau=10.):
lamb = sqrt(tau) * np.identity(X.shape[1])
X_tilde = np.concatenate((X/sqrt(sigma), lamb))
Y_tilde = np.concatenate((y/sqrt(sigma), np.zeros(X.shape[1])))
q, r = np.linalg.qr(X_tilde)
return scipy.linalg.solve_triangular(r, np.dot(q.T, Y_tilde),
check_finite=False)
def train_qr(X, y):
return map_estimate(X, y)
def log_posterior(w, X, y, sigma=1.0, tau=10):
return (1 / (2 * sigma) * np.linalg.norm(y - np.dot(X, w)) ** 2
+ 1 / (2 * tau) * np.linalg.norm(w) ** 2)
def gradient(w, X, y, sigma=1.0, tau=10):
return (- 1 / sigma * np.dot(X.T, y - np.dot(X, w)) + 1 / tau * w)
def verify_gradient(eps=1e-10):
w = np.ones(X_train.shape[1])
for i in range(X_train.shape[1]):
epsilon = np.zeros(X_train.shape[1])
epsilon[i] = eps
print (log_posterior(w + epsilon, X_train, y_train)
- log_posterior(w - epsilon, X_train, y_train)) / (2 * eps)
print gradient(w, X_train, y_train)[i]
def rmse(X, y, w):
predictions = np.dot(X, w)
return np.linalg.norm(predictions - y) / sqrt(len(y))
def train_lbfgs(X, y):
x0 = np.zeros(X.shape[1])
def f(w):
return log_posterior(w, X, y)
def g(w):
return gradient(w, X, y)
return minimize(f, x0, jac=g, method='L-BFGS-B',
options={'maxiter': 100}).x
def map_features(X, d):
A = np.random.normal(0, 1, size=(d, X.shape[1]))
b = np.random.uniform(0, 2 * pi, d)
return np.cos(np.dot(A, X.T).T + b)
if __name__ == "__main__":
# problem 4
wqr = train_qr(X_train, y_train)
print wqr
print rmse(X_test, y_test, wqr)
# problem 5
wlbfgs = train_lbfgs(X_train, y_train)
print wlbfgs
print rmse(X_test, y_test, wlbfgs)
# problem 6
lqr = []
llbfgs = []
for d in [100, 200, 400, 600]:
X = map_features(X_train, d)
X_t = map_features(X_test, d)
t_start = time.time()
wqr = train_qr(X, y_train)
rqr = time.time() - t_start
t_start = time.time()
wlbfgs = train_lbfgs(X, y_train)
rlbfgs = time.time() - t_start
lqr.append((rqr, rmse(X_t, y_test, wqr)))
llbfgs.append((rlbfgs, rmse(X_t, y_test, wlbfgs)))
plt.figure(figsize=(8, 8))
x, y = zip(*lqr)
plt.plot(x, y, "-ro", label="QR")
x, y = zip(*llbfgs)
plt.plot(x, y, "-bo", label="L-BFGS")
plt.xlabel("Time (s)")
plt.ylabel("RMSE")
plt.legend()
plt.savefig("plot.pdf", bbox_inches="tight")
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