def test_regression_mixed(self):
multivariate = np.array([[0, 0], [2, 7], [1, 7], [3, 9], [4, 16],
[2, 14], [3, 5]])
X = [
multivariate,
FDataBasis(Monomial(n_basis=3),
[[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 0, 0],
[0, 1, 0], [0, 0, 1]])
]
# y = 2 + sum([3, 1] * array) + int(3 * function)
intercept = 2
coefs_multivariate = np.array([3, 1])
coefs_functions = FDataBasis(Monomial(n_basis=3), [[3, 0, 0]])
y_integral = np.array([3, 3 / 2, 1, 4, 3, 3 / 2, 1])
y_sum = multivariate @ coefs_multivariate
y = 2 + y_sum + y_integral
scalar = LinearRegression()
scalar.fit(X, y)
np.testing.assert_allclose(scalar.intercept_, intercept, atol=0.01)
np.testing.assert_allclose(scalar.coef_[0],
coefs_multivariate,
atol=0.01)
np.testing.assert_allclose(scalar.coef_[1].coefficients,
coefs_functions.coefficients,
atol=0.01)
y_pred = scalar.predict(X)
np.testing.assert_allclose(y_pred, y, atol=0.01)
def test_regression_single_explanatory(self):
x_basis = Monomial(n_basis=7)
x_fd = FDataBasis(x_basis, np.identity(7))
beta_basis = Fourier(n_basis=5)
beta_fd = FDataBasis(beta_basis, [1, 1, 1, 1, 1])
y = [
0.9999999999999993, 0.162381381441085, 0.08527083481359901,
0.08519946930844623, 0.09532291032042489, 0.10550022969639987,
0.11382675064746171
]
scalar = LinearRegression(coef_basis=[beta_basis])
scalar.fit(x_fd, y)
np.testing.assert_allclose(scalar.coef_[0].coefficients,
beta_fd.coefficients)
np.testing.assert_allclose(scalar.intercept_, 0.0, atol=1e-6)
y_pred = scalar.predict(x_fd)
np.testing.assert_allclose(y_pred, y)
scalar = LinearRegression(coef_basis=[beta_basis], fit_intercept=False)
scalar.fit(x_fd, y)
np.testing.assert_allclose(scalar.coef_[0].coefficients,
beta_fd.coefficients)
np.testing.assert_equal(scalar.intercept_, 0.0)
y_pred = scalar.predict(x_fd)
np.testing.assert_allclose(y_pred, y)
def test_error_beta_not_basis(self):
""" Test that all beta are Basis objects. """
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = [1 for _ in range(7)]
beta = FDataBasis(Monomial(n_basis=7), np.identity(7))
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(TypeError):
scalar.fit([x_fd], y)
def test_error_y_is_FData(self):
"""Tests that none of the explained variables is an FData object
"""
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = list(FDataBasis(Monomial(n_basis=7), np.identity(7)))
scalar = LinearRegression(coef_basis=[Fourier(n_basis=5)])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y)
def test_error_weights_negative(self):
""" Test that none of the weights are negative. """
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = [1 for _ in range(7)]
weights = [-1 for _ in range(7)]
beta = Monomial(n_basis=7)
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y, weights)
def test_error_X_not_FData(self):
"""Tests that at least one of the explanatory variables
is an FData object. """
x_fd = np.identity(7)
y = np.zeros(7)
scalar = LinearRegression(coef_basis=[Fourier(n_basis=5)])
with np.testing.assert_warns(UserWarning):
scalar.fit([x_fd], y)
def test_error_weights_lenght(self):
""" Test that the number of weights is equal to the
number of samples """
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = [1 for _ in range(7)]
weights = [1 for _ in range(8)]
beta = Monomial(n_basis=7)
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y, weights)
def test_error_X_beta_len_distinct(self):
""" Test that the number of beta bases and explanatory variables
are not different """
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = [1 for _ in range(7)]
beta = Fourier(n_basis=5)
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd, x_fd], y)
scalar = LinearRegression(coef_basis=[beta, beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y)
def test_error_y_X_samples_different(self):
""" Test that the number of response samples and explanatory samples
are not different """
x_fd = FDataBasis(Monomial(n_basis=7), np.identity(7))
y = [1 for _ in range(8)]
beta = Fourier(n_basis=5)
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y)
x_fd = FDataBasis(Monomial(n_basis=8), np.identity(8))
y = [1 for _ in range(7)]
beta = Fourier(n_basis=5)
scalar = LinearRegression(coef_basis=[beta])
with np.testing.assert_raises(ValueError):
scalar.fit([x_fd], y)
def test_multivariate(self):
def ignore_scalar_warning():
warnings.filterwarnings(
"ignore", category=UserWarning,
message="All the covariates are scalar.")
X, y = make_regression(n_samples=20, n_features=10,
random_state=1, bias=3.5)
X_train, X_test, y_train, _ = train_test_split(
X, y, random_state=2)
for regularization_parameter in [0, 1, 10, 100]:
with self.subTest(
regularization_parameter=regularization_parameter):
sklearn_l2 = Ridge(alpha=regularization_parameter)
skfda_l2 = LinearRegression(
regularization=L2Regularization(
regularization_parameter=regularization_parameter),
)
sklearn_l2.fit(X_train, y_train)
with warnings.catch_warnings():
ignore_scalar_warning()
skfda_l2.fit(X_train, y_train)
sklearn_y_pred = sklearn_l2.predict(X_test)
with warnings.catch_warnings():
ignore_scalar_warning()
skfda_y_pred = skfda_l2.predict(X_test)
np.testing.assert_allclose(
sklearn_l2.coef_, skfda_l2.coef_[0])
np.testing.assert_allclose(
sklearn_l2.intercept_, skfda_l2.intercept_)
np.testing.assert_allclose(
sklearn_y_pred, skfda_y_pred)
def test_regression_mixed_regularization(self):
multivariate = np.array([[0, 0], [2, 7], [1, 7], [3, 9], [4, 16],
[2, 14], [3, 5]])
X = [
multivariate,
FDataBasis(Monomial(n_basis=3),
[[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 0, 0],
[0, 1, 0], [0, 0, 1]])
]
# y = 2 + sum([3, 1] * array) + int(3 * function)
intercept = 2
coefs_multivariate = np.array([3, 1])
y_integral = np.array([3, 3 / 2, 1, 4, 3, 3 / 2, 1])
y_sum = multivariate @ coefs_multivariate
y = 2 + y_sum + y_integral
scalar = LinearRegression(regularization=[
TikhonovRegularization(lambda x: x),
TikhonovRegularization(LinearDifferentialOperator(2))
])
scalar.fit(X, y)
np.testing.assert_allclose(scalar.intercept_, intercept, atol=0.01)
np.testing.assert_allclose(scalar.coef_[0], [2.536739, 1.072186],
atol=0.01)
np.testing.assert_allclose(scalar.coef_[1].coefficients,
[[2.125676, 2.450782, 5.808745e-4]],
atol=0.01)
y_pred = scalar.predict(X)
np.testing.assert_allclose(y_pred, [
5.349035, 16.456464, 13.361185, 23.930295, 32.650965, 23.961766,
16.29029
],
atol=0.01)
def test_regression_multiple_explanatory(self):
y = [1, 2, 3, 4, 5, 6, 7]
X = FDataBasis(Monomial(n_basis=7), np.identity(7))
beta1 = BSpline(domain_range=(0, 1), n_basis=5)
scalar = LinearRegression(coef_basis=[beta1])
scalar.fit(X, y)
np.testing.assert_allclose(scalar.intercept_.round(4),
np.array([32.65]),
rtol=1e-3)
np.testing.assert_allclose(
scalar.coef_[0].coefficients.round(4),
np.array([[-28.6443, 80.3996, -188.587, 236.5832, -481.3449]]),
rtol=1e-3)
y_pred = scalar.predict(X)
np.testing.assert_allclose(y_pred, y, atol=0.01)
class BasisRegression(object):
"""Class implementing functional linear models with basis functions for vector-valued covariates.
Parameters
----------
nbasis: int
Number of basis functions.
basis_type: str, default='bspline'
Type of basis used, possible values are 'bspline', 'fourier' and 'fPCA'
Attributes
----------
reg: object
Instance of skfda.ml.regression.LinearRegression
coef: array, default=None
Regression coefficients
fpca_basis: object
If basis_type='fPCA', instance of skfda.preprocessing.dim_reduction.projection.FPCA().
"""
def __init__(self, nbasis, basis_type='bspline'):
self.nbasis = nbasis
self.reg = LinearRegression()
self.basis_type = basis_type
self.coef = None
if self.basis_type == 'fPCA':
self.fpca_basis = FPCA(self.nbasis)
def data_to_basis(self, X, fit_fPCA=True):
"""Project the data to basis functions.
Parameters
----------
X: array, shape (n,n_points,d)
Array of paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise linear paths,
each composed of n_points.
fit_fPCA: boolean, default=True
If n_basis='fPCA' and fit_fPCA=True, the basis functions are fitted to be the functional principal
components of X.
Returns
-------
fd_basis: object
Instance of skfda.representation.basis.FDataBasis, the basis representation of X, where the type of basis is
determined by self.n_basis.
"""
grid_points = np.linspace(0, 1, X.shape[1])
fd = FDataGrid(X, grid_points)
basis_vec = []
for i in range(X.shape[2]):
if self.basis_type == 'bspline':
basis_vec.append(BSpline(n_basis=self.nbasis))
elif self.basis_type == 'fourier':
basis_vec.append(Fourier(n_basis=self.nbasis))
elif self.basis_type == 'fPCA':
basis_vec.append(BSpline(n_basis=7))
basis = VectorValued(basis_vec)
fd_basis = fd.to_basis(basis)
if self.basis_type == 'fPCA':
if fit_fPCA:
self.fpca_basis = self.fpca_basis.fit(fd_basis)
fd_basis = self.fpca_basis.transform(fd_basis)
return fd_basis
def fit(self, X, Y):
"""Fit the functional linear model to X and Y
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Y: array, shape (n)
Array of target values.
Returns
-------
reg: object
Instance of skfda.ml.regression.LinearRegression
"""
fd_basis = self.data_to_basis(X)
self.reg.fit(fd_basis, Y)
self.coef = self.reg.coef_
return self.reg
def predict(self, X):
"""Predict the output of self.reg for X.
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Returns
-------
Ypred: array, shape (n)
Array of predicted values.
"""
fd_basis = self.data_to_basis(X, fit_fPCA=False)
return self.reg.predict(fd_basis)
def get_loss(self, X, Y, plot=False):
"""Computes the empirical squared loss obtained with the functional linear model
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Y: array, shape (n)
Array of target values.
plot: boolean, default=False
If True, plots the regression coefficients and a scatter plot of the target values Y against its predicted
values Ypred to assess the quality of the fit.
Returns
-------
hatL: float
The squared loss, that is the sum of the squares of Y-Ypred, where Ypred are the fitted values of the Ridge
regression of Y against signatures of X truncated at k.
"""
Ypred = self.predict(X)
if plot:
plt.scatter(Y, Ypred)
plt.plot([0.9 * np.min(Y), 1.1 * np.max(Y)],
[0.9 * np.min(Y), 1.1 * np.max(Y)],
'--',
color='black')
plt.title("Ypred against Y")
plt.show()
return np.mean((Y - Ypred)**2)
def test_regression_regularization(self):
x_basis = Monomial(n_basis=7)
x_fd = FDataBasis(x_basis, np.identity(7))
beta_basis = Fourier(n_basis=5)
beta_fd = FDataBasis(beta_basis, [1.0403, 0, 0, 0, 0])
y = [
1.0000684777229512, 0.1623672257830915, 0.08521053851548224,
0.08514200869281137, 0.09529138749665378, 0.10549625973303875,
0.11384314859153018
]
y_pred_compare = [
0.890341, 0.370162, 0.196773, 0.110079, 0.058063, 0.023385,
-0.001384
]
scalar = LinearRegression(coef_basis=[beta_basis],
regularization=TikhonovRegularization(
LinearDifferentialOperator(2)))
scalar.fit(x_fd, y)
np.testing.assert_allclose(scalar.coef_[0].coefficients,
beta_fd.coefficients,
atol=1e-3)
np.testing.assert_allclose(scalar.intercept_, -0.15, atol=1e-4)
y_pred = scalar.predict(x_fd)
np.testing.assert_allclose(y_pred, y_pred_compare, atol=1e-4)
x_basis = Monomial(n_basis=3)
x_fd = FDataBasis(x_basis,
[[1, 0, 0], [0, 1, 0], [0, 0, 1], [2, 0, 1]])
beta_fd = FDataBasis(x_basis, [3, 2, 1])
y = [1 + 13 / 3, 1 + 29 / 12, 1 + 17 / 10, 1 + 311 / 30]
# Non regularized
scalar = LinearRegression()
scalar.fit(x_fd, y)
np.testing.assert_allclose(scalar.coef_[0].coefficients,
beta_fd.coefficients)
np.testing.assert_allclose(scalar.intercept_, 1)
y_pred = scalar.predict(x_fd)
np.testing.assert_allclose(y_pred, y)
# Regularized
beta_fd_reg = FDataBasis(x_basis, [2.812, 3.043, 0])
y_reg = [5.333, 3.419, 2.697, 11.366]
scalar_reg = LinearRegression(regularization=TikhonovRegularization(
LinearDifferentialOperator(2)))
scalar_reg.fit(x_fd, y)
np.testing.assert_allclose(scalar_reg.coef_[0].coefficients,
beta_fd_reg.coefficients,
atol=0.001)
np.testing.assert_allclose(scalar_reg.intercept_, 0.998, atol=0.001)
y_pred = scalar_reg.predict(x_fd)
np.testing.assert_allclose(y_pred, y_reg, atol=0.001)