In [2]:
import numpy as np
from matplotlib import pyplot as plt
import seaborn as sns
sns.set(color_codes=True)
%matplotlib inline
In [304]:
class GP(object):
@classmethod
def kernel_bell_shape(cls, x, y, delta=1.0):
return np.exp(-1/2.0 * np.power(x - y, 2) / delta)
@classmethod
def kernel_laplacian(cls, x, y, delta=1):
return np.exp(-1/2.0 * np.abs(x - y) / delta)
@classmethod
def generate_kernel(cls, kernel, delta=1):
def wrapper(*args, **kwargs):
kwargs.update({"delta": delta})
return kernel(*args, **kwargs)
return wrapper
def __init__(self, x, y, cov_f=None, R=0):
super().__init__()
self.x = x
self.y = y
self.N = len(self.x)
self.R = R
self.sigma = []
self.mean = []
self.cov_f = cov_f if cov_f else self.kernel_bell_shape
self.setup_sigma()
@classmethod
def calculate_sigma(cls, x, cov_f, R=0):
N = len(x)
sigma = np.ones((N, N))
for i in range(N):
for j in range(i+1, N):
cov = cov_f(x[i], x[j])
sigma[i][j] = cov
sigma[j][i] = cov
sigma = sigma + R * np.eye(N)
return sigma
def setup_sigma(self):
self.sigma = self.calculate_sigma(self.x, self.cov_f, self.R)
def predict(self, x):
cov = 1 + self.R * self.cov_f(x, x)
sigma_1_2 = np.zeros((self.N, 1))
for i in range(self.N):
sigma_1_2[i] = self.cov_f(self.x[i], x)
# SIGMA_1_2 * SIGMA_1_1.I * (Y.T -M)
# M IS ZERO
m_expt = (sigma_1_2.T * np.mat(self.sigma).I) * np.mat(self.y).T
# sigma_expt = cov - (sigma_1_2.T * np.mat(self.sigma).I) * sigma_1_2
sigma_expt = cov + self.R - (sigma_1_2.T * np.mat(self.sigma).I) * sigma_1_2
return m_expt, sigma_expt
@staticmethod
def get_probability(sigma, y, R):
multiplier = np.power(np.linalg.det(2 * np.pi * sigma), -0.5)
return multiplier * np.exp(
(-0.5) * (np.mat(y) * np.dot(np.mat(sigma).I, y).T))
def optimize(self, R_list, B_list):
def cov_f_proxy(delta, f):
def wrapper(*args, **kwargs):
kwargs.update({"delta": delta})
return f(*args, **kwargs)
return wrapper
best = (0, 0, 0)
history = []
for r in R_list:
best_beta = (0, 0)
for b in B_list:
sigma = gaus.calculate_sigma(self.x, cov_f_proxy(b, self.cov_f), r)
marginal = b* float(self.get_probability(sigma, self.y, r))
if marginal > best_beta[0]:
best_beta = (marginal, b)
history.append((best_beta[0], r, best_beta[1]))
return sorted(history)[-1], np.mat(history)
In [305]:
x = np.array([-2, -1, 0, 3.5, 4]);
y = np.array([4.1, 0.9, 2, 12.3, 15.8])
gaus = GP(x, y)
x_guess = np.linspace(-5, 16, 400)
y_pred = np.vectorize(gaus.predict)(x_guess)
plt.scatter(x, y, c="black")
plt.plot(x_guess, y_pred[0], c="b")
plt.plot(x_guess, y_pred[0] - np.sqrt(y_pred[1]) * 3, "r:")
plt.plot(x_guess, y_pred[0] + np.sqrt(y_pred[1]) * 3, "r:")
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In [306]:
def create_case(kernel, R=0):
x = np.array([-2, -1, 0, 3.5, 4]);
y = np.array([4.1, 0.9, 2, 12.3, 15.8])
gaus = GP(x, y, kernel, R=R)
x_guess = np.linspace(-4, 6, 400)
y_pred = np.vectorize(gaus.predict)(x_guess)
plt.scatter(x, y, c="black")
plt.plot(x_guess, y_pred[0], c="b")
plt.plot(x_guess, y_pred[0] - np.sqrt(y_pred[1]) * 3, "r:")
plt.plot(x_guess, y_pred[0] + np.sqrt(y_pred[1]) * 3, "r:")
In [307]:
plt.figure(figsize=(16, 16))
for i, r in enumerate([0.0001, 0.03, 0.09, 0.8, 1.5, 5.0]):
plt.subplot("32{}".format(i+1))
plt.title("kernel={}, delta={}, beta={}".format("bell shape", 1, r))
create_case(
GP.generate_kernel(GP.kernel_bell_shape, delta=1), R=r)
In [308]:
plt.figure(figsize=(16, 16))
for i, d in enumerate([0.05, 0.5, 1, 3.2, 5.0, 7.0]):
plt.subplot("32{}".format(i+1))
plt.title("kernel={}, delta={}, beta={}".format("kernel_laplacian", d, 1))
create_case(
GP.generate_kernel(GP.kernel_bell_shape, delta=d), R=0)
In [323]:
x = np.array([-2, -1, 3.5]);
y = np.array([4.1, 0.9, 12.3])
gaus = GP(x, y)
R_list = np.linspace(0.0, 1, 100)
B_list = np.linspace(0.1, 10, 20)
best_params, history = gaus.optimize(R_list, B_list)
plt.plot(history[:,1], history[:,2])
print("best parameters (probability, r, b): ", best_params)
plt.show()
create_case(
GP.generate_kernel(GP.kernel_bell_shape, delta=10), R=1.0)
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