def test_fit(data, optimizer):
x, x_dot = data
if len(x.shape) == 1:
x = x.reshape(-1, 1)
opt = SINDyOptimizer(optimizer, unbias=False)
opt.fit(x, x_dot)
check_is_fitted(opt)
assert opt.complexity >= 0
if len(x_dot.shape) > 1:
assert opt.coef_.shape == (x.shape[1], x_dot.shape[1])
else:
assert opt.coef_.shape == (1, x.shape[1])
def test_complexity_parameter(opt_cls, reg_name, n_samples, n_features,
n_informative, random_state):
"""Behaviour test for complexity 2.
We assume that a model with a bigger regularization parameter is less complex.
"""
assume(n_informative <= n_features)
x, y = make_regression(n_samples,
n_features,
n_informative,
1,
0,
noise=0.1,
random_state=random_state)
y = y.reshape(-1, 1)
optimizers = [
SINDyOptimizer(opt_cls(**{reg_name: reg_value}), unbias=True)
for reg_value in [3, 1, 0.3, 0.1, 0.01]
]
for opt in optimizers:
opt.fit(x, y)
for less_complex, more_complex in zip(optimizers, optimizers[1:]):
assert less_complex.complexity <= more_complex.complexity
def test_bad_optimizers(data_derivative_1d):
x, x_dot = data_derivative_1d
x = x.reshape(-1, 1)
with pytest.raises(AttributeError):
opt = SINDyOptimizer(DummyEmptyModel())
with pytest.raises(AttributeError):
opt = SINDyOptimizer(DummyModelNoCoef())
opt.fit(x, x_dot)
def test_unbias_external(data_derivative_1d):
x, x_dot = data_derivative_1d
x = x.reshape(-1, 1)
optimizer_biased = SINDyOptimizer(Lasso(alpha=0.1,
fit_intercept=False,
max_iter=1),
unbias=False)
optimizer_biased.fit(x, x_dot)
optimizer_unbiased = SINDyOptimizer(Lasso(alpha=0.1,
fit_intercept=False,
max_iter=1),
unbias=True)
optimizer_unbiased.fit(x, x_dot)
assert (norm(optimizer_biased.coef_ - optimizer_unbiased.coef_) /
(norm(optimizer_unbiased.coef_) + 1e-5) > 1e-9)
def test_unbias(data_derivative_1d):
x, x_dot = data_derivative_1d
x = x.reshape(-1, 1)
optimizer_biased = SINDyOptimizer(STLSQ(threshold=0.01,
alpha=0.1,
max_iter=1),
unbias=False)
optimizer_biased.fit(x, x_dot)
optimizer_unbiased = SINDyOptimizer(STLSQ(threshold=0.01,
alpha=0.1,
max_iter=1),
unbias=True)
optimizer_unbiased.fit(x, x_dot)
assert (norm(optimizer_biased.coef_ - optimizer_unbiased.coef_) /
norm(optimizer_unbiased.coef_) > 1e-9)
def test_complexity(n_samples, n_features, n_informative, random_state):
"""Behaviour test for complexity.
We assume that more regularized optimizers are less complex on the same dataset.
"""
assume(n_informative < n_features)
# Average complexity over multiple datasets
n_datasets = 5
complexities = [0] * 7
seed(random_state)
for rs in randint(low=0, high=2**32 - 1, size=n_datasets):
x, y = make_regression(
n_samples=n_samples,
n_features=n_features,
n_informative=n_informative,
n_targets=1,
bias=0,
noise=0.1,
random_state=rs,
)
y = y.reshape(-1, 1)
opt_kwargs = dict(fit_intercept=True, normalize=False)
optimizers = [
SR3(thresholder="l0", threshold=0.1, **opt_kwargs),
SR3(thresholder="l1", threshold=0.1, **opt_kwargs),
Lasso(**opt_kwargs),
STLSQ(**opt_kwargs),
ElasticNet(**opt_kwargs),
Ridge(**opt_kwargs),
LinearRegression(**opt_kwargs),
]
optimizers = [SINDyOptimizer(o, unbias=True) for o in optimizers]
for k, opt in enumerate(optimizers):
opt.fit(x, y)
complexities[k] += opt.complexity
for less_complex, more_complex in zip(complexities, complexities[1:]):
# relax the condition to account for
# noise and non-normalized threshold parameters
assert less_complex <= more_complex + 5
def test_sr3_trimming(optimizer, data_linear_oscillator_corrupted):
X, X_dot, trimming_array = data_linear_oscillator_corrupted
optimizer_without_trimming = SINDyOptimizer(optimizer(), unbias=False)
optimizer_without_trimming.fit(X, X_dot)
optimizer_trimming = SINDyOptimizer(optimizer(trimming_fraction=0.15),
unbias=False)
optimizer_trimming.fit(X, X_dot)
# Check that trimming found the right samples to remove
np.testing.assert_array_equal(optimizer_trimming.optimizer.trimming_array,
trimming_array)
# Check that the coefficients found by the optimizer with trimming
# are closer to the true coefficients than the coefficients found by the
# optimizer without trimming
true_coef = np.array([[-2.0, 0.0], [0.0, 1.0]])
assert norm(true_coef - optimizer_trimming.coef_) < norm(
true_coef - optimizer_without_trimming.coef_)
def test_sr3_trimming(data_linear_oscillator_corrupted):
X, X_dot, trimming_array = data_linear_oscillator_corrupted
optimizer_without_trimming = SINDyOptimizer(SR3(), unbias=False)
optimizer_without_trimming.fit(X, X_dot)
optimizer_trimming = SINDyOptimizer(SR3(trimming_fraction=0.15),
unbias=False)
optimizer_trimming.fit(X, X_dot)
# Check that trimming found the right samples to remove
assert (np.sum(
np.abs(optimizer_trimming.optimizer.trimming_array -
trimming_array)) == 0.0)
# Check that the coefficients found by the optimizer with trimming are closer to
# the true coefficients than the coefficients found by the optimizer without
# trimming
true_coef = np.array([[-2.0, 0.0], [0.0, 1.0]])
assert norm(true_coef - optimizer_trimming.coef_) < norm(
true_coef - optimizer_without_trimming.coef_)
def test_complexity(n_samples, n_features, n_informative, random_state):
"""Behaviour test for complexity.
We assume that more regularized optimizers are less complex on the same dataset.
"""
assume(n_informative < n_features)
x, y = make_regression(n_samples,
n_features,
n_informative,
1,
0,
noise=0.1,
random_state=random_state)
y = y.reshape(-1, 1)
opt_kwargs = dict(fit_intercept=True, normalize=False)
optimizers = [
SR3(thresholder="l0", threshold=0.1, **opt_kwargs),
SR3(thresholder="l1", threshold=0.1, **opt_kwargs),
Lasso(**opt_kwargs),
STLSQ(**opt_kwargs),
ElasticNet(**opt_kwargs),
Ridge(**opt_kwargs),
LinearRegression(**opt_kwargs),
]
optimizers = [SINDyOptimizer(o, unbias=True) for o in optimizers]
for opt in optimizers:
opt.fit(x, y)
for less_complex, more_complex in zip(optimizers, optimizers[1:]):
# relax the condition to account for
# noise and non-normalized threshold parameters
assert less_complex.complexity <= more_complex.complexity + 1
def fit(
self,
x,
t=None,
x_dot=None,
u=None,
multiple_trajectories=False,
unbias=True,
quiet=False,
):
"""
Fit the SINDy model.
Parameters
----------
x: array-like or list of array-like, shape (n_samples, n_input_features)
Training data. If training data contains multiple trajectories,
x should be a list containing data for each trajectory. Individual
trajectories may contain different numbers of samples.
t: float, numpy array of shape [n_samples], or list of numpy arrays, optional \
(default None)
If t is a float, it specifies the timestep between each sample.
If array-like, it specifies the time at which each sample was
collected.
In this case the values in t must be strictly increasing.
In the case of multi-trajectory training data, t may also be a list
of arrays containing the collection times for each individual
trajectory.
If None, the default time step ``t_default`` will be used.
x_dot: array-like or list of array-like, shape (n_samples, n_input_features), \
optional (default None)
Optional pre-computed derivatives of the training data. If not
provided, the time derivatives of the training data will be
computed using the specified differentiation method. If x_dot is
provided, it must match the shape of the training data and these
values will be used as the time derivatives.
u: array-like or list of array-like, shape (n_samples, n_control_features), \
optional (default None)
Control variables/inputs. Include this variable to use sparse
identification for nonlinear dynamical systems for control (SINDYc).
If training data contains multiple trajectories (i.e. if x is a list of
array-like), then u should be a list containing control variable data
for each trajectory. Individual trajectories may contain different
numbers of samples.
multiple_trajectories: boolean, optional, (default False)
Whether or not the training data includes multiple trajectories. If
True, the training data must be a list of arrays containing data
for each trajectory. If False, the training data must be a single
array.
unbias: boolean, optional (default True)
Whether to perform an extra step of unregularized linear regression to
unbias the coefficients for the identified support.
If the optimizer (``SINDy.optimizer``) applies any type of regularization,
that regularization may bias coefficients toward particular values,
improving the conditioning of the problem but harming the quality of the
fit. Setting ``unbias=True`` enables an extra step wherein unregularized
linear regression is applied, but only for the coefficients in the support
identified by the optimizer. This helps to remove the bias introduced by
regularization.
quiet: boolean, optional (default False)
Whether or not to suppress warnings during model fitting.
Returns
-------
self: returns an instance of self
"""
if t is None:
t = self.t_default
if u is None:
self.n_control_features_ = 0
else:
trim_last_point = self.discrete_time and (x_dot is None)
u = validate_control_variables(
x,
u,
multiple_trajectories=multiple_trajectories,
trim_last_point=trim_last_point,
)
self.n_control_features_ = u.shape[1]
if multiple_trajectories:
x, x_dot = self._process_multiple_trajectories(x, t, x_dot)
else:
x = validate_input(x, t)
if self.discrete_time:
if x_dot is None:
x_dot = x[1:]
x = x[:-1]
else:
x_dot = validate_input(x_dot)
else:
if x_dot is None:
x_dot = self.differentiation_method(x, t)
else:
x_dot = validate_input(x_dot, t)
# Append control variables
if self.n_control_features_ > 0:
x = concatenate((x, u), axis=1)
# Drop rows where derivative isn't known
x, x_dot = drop_nan_rows(x, x_dot)
optimizer = SINDyOptimizer(self.optimizer, unbias=unbias)
steps = [("features", self.feature_library), ("model", optimizer)]
self.model = Pipeline(steps)
action = "ignore" if quiet else "default"
with warnings.catch_warnings():
warnings.filterwarnings(action, category=ConvergenceWarning)
warnings.filterwarnings(action, category=LinAlgWarning)
warnings.filterwarnings(action, category=UserWarning)
self.model.fit(x, x_dot)
self.n_input_features_ = self.model.steps[0][1].n_input_features_
self.n_output_features_ = self.model.steps[0][1].n_output_features_
if self.feature_names is None:
feature_names = []
for i in range(self.n_input_features_ - self.n_control_features_):
feature_names.append("x" + str(i))
for i in range(self.n_control_features_):
feature_names.append("u" + str(i))
self.feature_names = feature_names
return self
