def predict(self, x_query):
'''
Make a prediction (<f>) including uncertainty (<u>, 95% confidence interval (CI))
based on <x_query>.
'''
x_query = np.array(x_query)
X_query = PolynomialFeatures(1).fit_transform(x_query.reshape(-1, 1))
X_query2 = PolynomialFeatures(2).fit_transform(x_query.reshape(-1, 1))
f = X_query.dot(self.coefs_mean).flatten()
u = 1.96 * np.sqrt(
X_query2.dot(self.subcoefs).flatten() +
np.diag(X_query.dot(self.coefs_cov.dot(X_query.T))))
return f, u
def _fz(self, z):
powers = PolynomialFeatures(self._max_power,
include_bias=True,
interaction_only=False).fit_transform(z)
fz = powers.dot(self._coeffs).reshape((-1, 1))
fz = fz * np.exp(
-self._exp_suppression * np.sum(z**2, axis=-1))[:, np.newaxis]
return fz
def sample_data(self):
'''
:rtype (X,y): a tuple of X and y
'''
from sklearn.preprocessing import PolynomialFeatures
X = np.array(np.random.randint(low=0, high=3,
size=self.n * self.p)).reshape(
self.n, self.p).astype('float32')
X_s = PolynomialFeatures(2, interaction_only=False,
include_bias=False).fit_transform(X)
beta = np.concatenate([self.beta_a, self.beta_i])
y = X_s.dot(beta) + np.random.normal(loc=0, scale=1, size=self.n)
return (X, y)
def fit(self, ts):
'''Fits model'''
self.reset_to_default() #model clear
self.series = ts.copy()
self.true_values = ts[self.p:]
x = partition(ts[:-1], self.p, 1)
x = PolynomialFeatures(degree=self.d).fit_transform(x)
x_tr = x.transpose()
self.coef = np.linalg.inv(x_tr.dot(x)).dot(x_tr).dot(self.true_values)
self.fitted_values = x.dot(self.coef)
self.fitted_values = np.hstack((np.ones(self.p), self.fitted_values))
self.fitted_values[:self.p] = np.nan
self.residuals = self.series - self.fitted_values
return self
def sample_data(self):
"""
:rtype (X,y): a tuple of X and y
"""
if self.discretized:
X = np.array(np.random.randint(low=0, high=3,
size=self.n * self.p)).reshape(
self.n,
self.p).astype('float32')
else:
X = np.array(np.random.uniform(low=0, high=3,
size=self.n * self.p)).reshape(
self.n,
self.p).astype('float32')
X_s = PolynomialFeatures(2, interaction_only=False,
include_bias=False).fit_transform(X)
beta = np.concatenate([self.beta_a, self.beta_i])
y = X_s.dot(beta) + np.random.normal(
loc=0, scale=np.sqrt(self.noise_var), size=self.n)
return X, y
def get_ground_truth(self, X):
X_s = PolynomialFeatures(2, interaction_only=False,
include_bias=False).fit_transform(X)
beta = np.concatenate([self.beta_a, self.beta_i])
return X_s.dot(beta)
class pKa:
def __init__(self, path_to_data='./training_data.xlsx'):
'''
Load data from path (default: './training_data.xlsx') and specify variables that are being
used throughout the procedure. Display the underlying database.
'''
self.data = pd.read_excel('training_data.xlsx')
display(self.data)
self.x = np.array(
self.data['pKa_theo']).flatten() # independent variable
self.y = np.array(self.data['pKa_exp']).flatten() # dependent variable
if len(self.x) != len(self.y):
raise Exception(
"Number of instances in pKa_exp and pKa_theo is required to be identical."
)
self.N = len(self.x) # number of data points
self.X = PolynomialFeatures(1).fit_transform(self.x.reshape(
-1, 1)) # add intercept term
self.X2 = PolynomialFeatures(2).fit_transform(self.x.reshape(
-1, 1)) # ... and quadratic term
# define <x_grid> and <X_grid> (equivalent to <x> and <X>) for plotting purposes
delta = np.max(self.x) - np.min(self.x)
self.x_grid = np.linspace(
np.min(self.x) - .05 * delta,
np.max(self.x) + .05 * delta, 250)
self.X_grid = PolynomialFeatures(1).fit_transform(
self.x_grid.reshape(-1, 1))
self.rng = np.random.RandomState(
) # random number generator (local to the class)
#------------------------------------------------------------------------------------------------------#
def get_coefs(self, seed=None):
'''
Return coefficients (<model.coef_>) of a weighted linear regression model (<model>).
Weights (<weight>) are obtained on the basis of Bayesian bootstrapping.
If <x>-dependent variance (<var>) is available (from heteroscedastic regression), adjust weights.
'''
if not hasattr(self, 'var'):
self.var = 1.
if not seed is None: # sample-specific seed
self.rng.seed(seed)
weight = np.diff(
np.concatenate(
([0.], np.sort(self.rng.uniform(0., 1., self.N - 1)), [1.])))
model = LinearRegression(fit_intercept=False,
normalize=True).fit(self.X, self.y,
weight / self.var)
return model.coef_
#------------------------------------------------------------------------------------------------------#
def bootstrap(self):
'''
Draw 1000 bootstrap samples and perform weighted linear regression.
Collect regression coefficients (<coefs>) and determine the ensemble mean (<coefs_mean>)
and covariance (<coefs_cov>).
Approximate <y> on the basis of the ensemble of regression models (predictions <f>).
'''
self.coefs = []
for b in range(1000):
self.coefs.append(self.get_coefs(seed=b))
self.coefs = np.array(self.coefs)
self.coefs_mean = np.mean(self.coefs, axis=0).reshape(-1, 1)
self.coefs_cov = np.cov(self.coefs.T)
self.f = self.X.dot(self.coefs_mean).flatten()
# necessary if heteroscedastic regression has not yet been performed
if not hasattr(self, 'subcoefs'):
self.subcoefs = np.array([
self.N / (self.N - 2) * np.mean((self.y - self.f)**2), 0., 0.
])
#------------------------------------------------------------------------------------------------------#
def predict(self, x_query):
'''
Make a prediction (<f>) including uncertainty (<u>, 95% confidence interval (CI))
based on <x_query>.
'''
x_query = np.array(x_query)
X_query = PolynomialFeatures(1).fit_transform(x_query.reshape(-1, 1))
X_query2 = PolynomialFeatures(2).fit_transform(x_query.reshape(-1, 1))
f = X_query.dot(self.coefs_mean).flatten()
u = 1.96 * np.sqrt(
X_query2.dot(self.subcoefs).flatten() +
np.diag(X_query.dot(self.coefs_cov.dot(X_query.T))))
return f, u
#------------------------------------------------------------------------------------------------------#
def plot_bootstrap_results(self, show_ensemble=True):
'''
Plot the results of the bootstrapping procedure. If <show_ensemble> is True, all regression lines
will be plotted.
'''
if show_ensemble is True:
for b in range(1000):
if b == 0:
label_ = 'result for $b$th sample'
else:
label_ = None
plt.plot(self.x_grid,
self.X_grid.dot(self.coefs[b, :].reshape(-1, 1)),
color='#75bbfd',
linewidth=.5,
label=label_)
f, u = self.predict(self.x_grid)
plt.plot(self.x_grid, f, 'k-', label='regression line')
plt.plot(self.x, self.y, 'k.', label='training data')
plt.fill_between(self.x_grid, (f + u), (f - u),
facecolor='red',
alpha=0.2,
label='uncertainty (95% CI)')
plt.xlabel(r'p$K_a$ (theo)', fontsize=12)
plt.ylabel(r'p$K_a$ (exp)', fontsize=12)
plt.legend()
#------------------------------------------------------------------------------------------------------#
def query(self, x_query):
'''
Make a prediction (<f>) including uncertainty (<u>, 95% confidence interval (CI)) for a
user-specific query (<x_query>). Print statistics.
'''
x_query = np.array(x_query).flatten()
if len(x_query) != 1:
raise Exception(
"Multiple queries were made, but only one at a time is possible at the moment."
)
self.plot_bootstrap_results(show_ensemble=False)
f, u = self.predict(x_query)
plt.errorbar(x_query,
f,
u,
color='red',
mfc='black',
capsize=3,
marker='o',
label='queried prediction')
print('Prediction = ' + str(format(f.item(), '.3f')))
print('Uncertainty (95% CI) = ' + str(format(u.item(), '.3f')))
plt.legend()
#------------------------------------------------------------------------------------------------------#
def fit_variance(self):
'''
Heteroscedastic regression. Determine <var> as the <x>-dependent variance and <subcoefs> as
the coefficients of this additional regression model.
'''
model = BayesianRidge(fit_intercept=False,
normalize=True).fit(self.X2,
(self.y - self.f)**2)
self.var = model.predict(self.X2).flatten()
self.subcoefs = model.coef_
#------------------------------------------------------------------------------------------------------#
def check_query(self, query):
'''
Check whether a user-specify query is a valid number or not.
'''
try:
float(query)
return True
except ValueError:
return False
#------------------------------------------------------------------------------------------------------#
def run(self):
'''
The key method of the pKa class. Perform one run of homoscedastic regression (boostrapped)
followed by three runs of heteroscedastic regression (bootstrapped). Print statistics.
Allow users to make individual queries.
'''
if hasattr(self, 'var'):
del self.var
if hasattr(self, 'subcoefs'):
del self.subcoefs
self.bootstrap()
for i in range(3):
self.fit_variance()
self.bootstrap()
self.plot_bootstrap_results()
plt.show()
print('===============================================')
print('SUMMARY OF HETEROSCEDASTIC BOOTSTRAP REGRESSION')
print('intercept = ' + str(format(np.mean(self.coefs[:,0]), '.3f')) \
+ ' +/- ' + str(format(1.96 * np.std(self.coefs[:,0]), '.3f')) + ' (95% confidence)')
print('slope = ' + str(format(np.mean(self.coefs[:,1]), '.3f')) \
+ ' +/- ' + str(format(1.96 * np.std(self.coefs[:,1]), '.3f')) + ' (95% confidence)')
print('===============================================\n')
querying = True
while querying:
print('Enter any non-digit character to stop the procedure.')
query = input('Enter pKa value: ')
querying = self.check_query(query)
if querying:
self.query(float(query))
plt.show()
def eval_polynomial(X, w): verify_shapes(X, w=w) poly = PolynomialFeatures(poly_degree).fit_transform(X) return poly.dot(w).reshape((X.shape[0], 1))
