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# __new__ methods have different arguments
# pylint: disable=arguments-differ
from warnings import warn
from collections import namedtuple
from itertools import chain
from time import time
import numpy as np
import sklearn.model_selection as skl
from Orange.data import Domain, ContinuousVariable, DiscreteVariable
from Orange.data.util import get_unique_names
__all__ = ["Results", "CrossValidation", "LeaveOneOut", "TestOnTrainingData",
"ShuffleSplit", "TestOnTestData", "sample", "CrossValidationFeature"]
_MpResults = namedtuple('_MpResults', ('fold_i', 'learner_i', 'model',
'failed', 'n_values', 'values',
'probs', 'train_time', 'test_time'))
def _identity(x):
return x
def _mp_worker(fold_i, train_data, test_data, learner_i, learner,
store_models, suppresses_exceptions=True):
predicted, probs, model, failed = None, None, None, False
train_time, test_time = None, None
try:
if not train_data or not test_data:
raise RuntimeError('Test fold is empty')
# training
t0 = time()
model = learner(train_data)
train_time = time() - t0
t0 = time()
# testing
class_var = train_data.domain.class_var
if class_var and class_var.is_discrete:
predicted, probs = model(test_data, model.ValueProbs)
else:
predicted = model(test_data, model.Value)
test_time = time() - t0
# Different models can fail at any time raising any exception
except Exception as ex: # pylint: disable=broad-except
if not suppresses_exceptions:
raise ex
failed = ex
return _MpResults(fold_i, learner_i, store_models and model,
failed, len(test_data), predicted, probs,
train_time, test_time)
class Results:
"""
Class for storing predictions in model testing.
Attributes:
data (Optional[Table]): Data used for testing.
models (Optional[List[Model]]): A list of induced models.
row_indices (np.ndarray): Indices of rows in `data` that were used in
testing, stored as a numpy vector of length `nrows`.
Values of `actual[i]`, `predicted[i]` and `probabilities[i]` refer
to the target value of instance, that is, the i-th test instance
is `data[row_indices[i]]`, its actual class is `actual[i]`, and
the prediction by m-th method is `predicted[m, i]`.
nrows (int): The number of test instances (including duplicates);
`nrows` equals the length of `row_indices` and `actual`, and the
second dimension of `predicted` and `probabilities`.
actual (np.ndarray): true values of target variable in a vector of
length `nrows`.
predicted (np.ndarray): predicted values of target variable in an array
of shape (number-of-methods, `nrows`)
probabilities (Optional[np.ndarray]): predicted probabilities
(for discrete target variables) in an array of shape
(number-of-methods, `nrows`, number-of-classes)
folds (List[Slice or List[int]]): a list of indices (or slice objects)
corresponding to testing data subsets, that is,
`row_indices[folds[i]]` contains row indices used in fold i, so
`data[row_indices[folds[i]]]` is the corresponding testing data
train_time (np.ndarray): training times of batches
test_time (np.ndarray): testing times of batches
"""
def __init__(self, data=None, *,
nmethods=None, nrows=None, nclasses=None,
domain=None,
row_indices=None, folds=None, score_by_folds=True,
learners=None, models=None, failed=None,
actual=None, predicted=None, probabilities=None,
# pylint: disable=unused-argument
store_data=None, store_models=None,
train_time=None, test_time=None):
"""
Construct an instance.
The constructor stores the given data, and creates empty arrays
`actual`, `predicted` and `probabilities` if ther are not given but
sufficient data is provided to deduct their shapes.
The function
- set any attributes specified directly through arguments.
- infers the number of methods, rows and classes from other data
and/or check their overall consistency.
- Prepare empty arrays `actual`, `predicted`, `probabilities` and
`failed` if the are not given. If not enough data is available,
the corresponding arrays are `None`.
Args:
data (Orange.data.Table): stored data from which test was sampled
nmethods (int): number of methods; can be inferred (or must match)
the size of `learners`, `models`, `failed`, `predicted` and
`probabilities`
nrows (int): number of data instances; can be inferred (or must
match) `data`, `row_indices`, `actual`, `predicted` and
`probabilities`
nclasses (int): number of class values (`None` if continuous); can
be inferred (or must match) from `domain.class_var` or
`probabilities`
domain (Orange.data.Domain): data domain; can be inferred (or must)
match `data.domain`
row_indices (np.ndarray): see class documentation
folds (np.ndarray): see class documentation
score_by_folds (np.ndarray): see class documentation
learners (np.ndarray): see class documentation
models (np.ndarray): see class documentation
failed (list of str): see class documentation
actual (np.ndarray): see class documentation
predicted (np.ndarray): see class documentation
probabilities (np.ndarray): see class documentation
store_data (bool): ignored; kept for backward compatibility
store_models (bool): ignored; kept for backward compatibility
"""
# Set given data directly from arguments
self.data = data
self.domain = domain
self.row_indices = row_indices
self.folds = folds
self.score_by_folds = score_by_folds
self.learners = learners
self.models = models
self.actual = actual
self.predicted = predicted
self.probabilities = probabilities
self.failed = failed
self.train_time = train_time
self.test_time = test_time
# Guess the rest -- or check for ambguities
def set_or_raise(value, exp_values, msg):
for exp_value in exp_values:
if exp_value is False:
continue
if value is None:
value = exp_value
elif value != exp_value:
raise ValueError(msg)
return value
domain = self.domain = set_or_raise(
domain, [data is not None and data.domain],
"mismatching domain")
self.nrows = nrows = set_or_raise(
nrows, [actual is not None and len(actual),
row_indices is not None and len(row_indices),
predicted is not None and predicted.shape[1],
probabilities is not None and probabilities.shape[1]],
"mismatching number of rows")
if domain is not None and domain.has_continuous_class:
if nclasses is not None:
raise ValueError(
"regression results cannot have non-None 'nclasses'")
if probabilities is not None:
raise ValueError(
"regression results cannot have 'probabilities'")
nclasses = set_or_raise(
nclasses, [domain is not None and domain.has_discrete_class and
len(domain.class_var.values),
probabilities is not None and probabilities.shape[2]],
"mismatching number of class values")
nmethods = set_or_raise(
nmethods, [learners is not None and len(learners),
models is not None and models.shape[1],
failed is not None and len(failed),
predicted is not None and predicted.shape[0],
probabilities is not None and probabilities.shape[0]],
"mismatching number of methods")
# Prepare empty arrays
if actual is None \
and nrows is not None:
self.actual = np.empty(nrows)
if predicted is None \
and nmethods is not None and nrows is not None:
self.predicted = np.empty((nmethods, nrows))
if probabilities is None \
and nmethods is not None and nrows is not None \
and nclasses is not None:
self.probabilities = \
np.empty((nmethods, nrows, nclasses))
if failed is None \
and nmethods is not None:
self.failed = [False] * nmethods
def get_fold(self, fold):
results = Results()
results.data = self.data
if self.folds is None:
raise ValueError("This 'Results' instance does not have folds.")
if self.models is not None:
results.models = self.models[fold]
results.row_indices = self.row_indices[self.folds[fold]]
results.actual = self.actual[self.folds[fold]]
results.predicted = self.predicted[:, self.folds[fold]]
results.domain = self.domain
if self.probabilities is not None:
results.probabilities = self.probabilities[:, self.folds[fold]]
return results
def get_augmented_data(self, model_names,
include_attrs=True, include_predictions=True,
include_probabilities=True):
"""
Return the test data table augmented with meta attributes containing
predictions, probabilities (if the task is classification) and fold
indices.
Args:
model_names (list of str): names of models
include_attrs (bool):
if set to `False`, original attributes are removed
include_predictions (bool):
if set to `False`, predictions are not added
include_probabilities (bool):
if set to `False`, probabilities are not added
Returns:
augmented_data (Orange.data.Table):
data augmented with predictions, probabilities and fold indices
"""
assert self.predicted.shape[0] == len(model_names)
data = self.data[self.row_indices]
domain = data.domain
class_var = domain.class_var
classification = class_var and class_var.is_discrete
new_meta_attr = []
new_meta_vals = np.empty((len(data), 0))
names = [var.name for var in chain(domain.attributes,
domain.metas,
domain.class_vars)]
if classification:
# predictions
if include_predictions:
uniq_new, names = self.create_unique_vars(names, model_names, class_var.values)
new_meta_attr += uniq_new
new_meta_vals = np.hstack((new_meta_vals, self.predicted.T))
# probabilities
if include_probabilities:
proposed = [f"{name} ({value})" for name in model_names for value in class_var.values]
uniq_new, names = self.create_unique_vars(names, proposed)
new_meta_attr += uniq_new
for i in self.probabilities:
new_meta_vals = np.hstack((new_meta_vals, i))
elif include_predictions:
# regression
uniq_new, names = self.create_unique_vars(names, model_names)
new_meta_attr += uniq_new
new_meta_vals = np.hstack((new_meta_vals, self.predicted.T))
# add fold info
if self.folds is not None:
values = [str(i + 1) for i in range(len(self.folds))]
uniq_new, names = self.create_unique_vars(names, ["Fold"], values)
new_meta_attr += uniq_new
fold = np.empty((len(data), 1))
for i, s in enumerate(self.folds):
fold[s, 0] = i
new_meta_vals = np.hstack((new_meta_vals, fold))
# append new columns to meta attributes
new_meta_attr = list(data.domain.metas) + new_meta_attr
new_meta_vals = np.hstack((data.metas, new_meta_vals))
attrs = data.domain.attributes if include_attrs else []
domain = Domain(attrs, data.domain.class_vars, metas=new_meta_attr)
predictions = data.transform(domain)
with predictions.unlocked(predictions.metas):
predictions.metas = new_meta_vals
predictions.name = data.name
return predictions
def create_unique_vars(self, names, proposed_names, values=()):
unique_vars = []
for proposed in proposed_names:
uniq = get_unique_names(names, proposed)
if values:
unique_vars.append(DiscreteVariable(uniq, values))
else:
unique_vars.append(ContinuousVariable(uniq))
names.append(uniq)
return unique_vars, names
def split_by_model(self):
"""
Split evaluation results by models.
The method generates instances of `Results` containing data for single
models
"""
data = self.data
nmethods = len(self.predicted)
for i in range(nmethods):
res = Results()
res.data = data
res.domain = self.domain
res.learners = [self.learners[i]]
res.row_indices = self.row_indices
res.actual = self.actual
res.folds = self.folds
res.score_by_folds = self.score_by_folds
res.test_time = self.test_time[i]
res.train_time = self.train_time[i]
res.predicted = self.predicted[(i,), :]
if getattr(self, "probabilities", None) is not None:
res.probabilities = self.probabilities[(i,), :, :]
if self.models is not None:
res.models = self.models[:, i:i + 1]
res.failed = [self.failed[i]]
yield res
class Validation:
"""
Base class for different testing schemata such as cross validation and
testing on separate data set.
If `data` is some data table and `learners` is a list of learning
algorithms. This will run 5-fold cross validation and store the results
in `res`.
cv = CrossValidation(k=5)
res = cv(data, learners)
If constructor was given data and learning algorithms (as in
`res = CrossValidation(data, learners, k=5)`, it used to automagically
call the instance after constructing it and return `Results` instead
of an instance of `Validation`. This functionality
is deprecated and will be removed in the future.
Attributes:
store_data (bool): a flag defining whether the data is stored
store_models (bool): a flag defining whether the models are stored
"""
score_by_folds = False
def __new__(cls,
data=None, learners=None, preprocessor=None, test_data=None,
*, callback=None, store_data=False, store_models=False,
n_jobs=None, **kwargs):
self = super().__new__(cls)
if (learners is None) != (data is None):
raise ValueError(
"learners and train_data must both be present or not")
if learners is None:
if preprocessor is not None:
raise ValueError("preprocessor cannot be given if learners "
"and train_data are omitted")
if callback is not None:
raise ValueError("callback cannot be given if learners "
"and train_data are omitted")
return self
warn("calling Validation's constructor with data and learners "
"is deprecated;\nconstruct an instance and call it",
DeprecationWarning, stacklevel=2)
# Explicitly call __init__ because Python won't
self.__init__(store_data=store_data, store_models=store_models,
**kwargs)
if test_data is not None:
test_data_kwargs = {"test_data": test_data}
else:
test_data_kwargs = {}
return self(data, learners=learners, preprocessor=preprocessor,
callback=callback, **test_data_kwargs)
# Note: this will be called only if __new__ doesn't have data and learners
def __init__(self, *, store_data=False, store_models=False):
self.store_data = store_data
self.store_models = store_models
def fit(self, *args, **kwargs):
warn("Validation.fit is deprecated; use the call operator",
DeprecationWarning)
return self(*args, **kwargs)
def __call__(self, data, learners, preprocessor=None, *, callback=None,
suppresses_exceptions=True):
"""
Args:
data (Orange.data.Table): data to be used (usually split) into
training and testing
learners (list of Orange.Learner): a list of learning algorithms
preprocessor (Orange.preprocess.Preprocess): preprocessor applied
on training data
callback (Callable): a function called to notify about the progress
suppresses_exceptions (bool): suppress the exceptions if True
Returns:
results (Result): results of testing
"""
if preprocessor is None:
preprocessor = _identity
if callback is None:
callback = _identity
indices = self.get_indices(data)
folds, row_indices, actual = self.prepare_arrays(data, indices)
data_splits = (
(fold_i, preprocessor(data[train_i]), data[test_i])
for fold_i, (train_i, test_i) in enumerate(indices))
args_iter = (
(fold_i, data, test_data, learner_i, learner, self.store_models)
for (fold_i, data, test_data) in data_splits
for (learner_i, learner) in enumerate(learners))
part_results = []
parts = np.linspace(.0, .99, len(learners) * len(indices) + 1)[1:]
for progress, part in zip(parts, args_iter):
part_results.append(
_mp_worker(*(part + ()),
suppresses_exceptions=suppresses_exceptions)
)
callback(progress)
callback(1)
results = Results(
data=data if self.store_data else None,
domain=data.domain,
nrows=len(row_indices), learners=learners,
row_indices=row_indices, folds=folds, actual=actual,
score_by_folds=self.score_by_folds,
train_time=np.zeros((len(learners),)),
test_time=np.zeros((len(learners),)))
if self.store_models:
results.models = np.tile(None, (len(indices), len(learners)))
self._collect_part_results(results, part_results)
return results
@classmethod
def prepare_arrays(cls, data, indices):
"""Prepare `folds`, `row_indices` and `actual`.
The method is used by `__call__`. While functional, it may be
overriden in subclasses for speed-ups.
Args:
data (Orange.data.Table): data use for testing
indices (list of vectors):
indices of data instances in each test sample
Returns:
folds: (np.ndarray): see class documentation
row_indices: (np.ndarray): see class documentation
actual: (np.ndarray): see class documentation
"""
folds = []
row_indices = []
ptr = 0
for _, test in indices:
folds.append(slice(ptr, ptr + len(test)))
row_indices.append(test)
ptr += len(test)
row_indices = np.concatenate(row_indices, axis=0)
return folds, row_indices, data[row_indices].Y
@staticmethod
def get_indices(data):
"""
Return a list of arrays of indices of test data instance
For example, in k-fold CV, the result is a list with `k` elements,
each containing approximately `len(data) / k` nonoverlapping indices
into `data`.
This method is abstract and must be implemented in derived classes
unless they provide their own implementation of the `__call__`
method.
Args:
data (Orange.data.Table): test data
Returns:
indices (list of np.ndarray):
a list of arrays of indices into `data`
"""
raise NotImplementedError()
def _collect_part_results(self, results, part_results):
part_results = sorted(part_results)
ptr, prev_fold_i, prev_n_values = 0, 0, 0
for res in part_results:
if res.fold_i != prev_fold_i:
ptr += prev_n_values
prev_fold_i = res.fold_i
result_slice = slice(ptr, ptr + res.n_values)
prev_n_values = res.n_values
if res.failed:
results.failed[res.learner_i] = res.failed
continue
if len(res.values.shape) > 1 and res.values.shape[1] > 1:
msg = "Multiple targets are not supported."
results.failed[res.learner_i] = ValueError(msg)
continue
if self.store_models:
results.models[res.fold_i][res.learner_i] = res.model
results.predicted[res.learner_i][result_slice] = res.values
results.train_time[res.learner_i] += res.train_time
results.test_time[res.learner_i] += res.test_time
if res.probs is not None:
results.probabilities[res.learner_i][result_slice, :] = \
res.probs
class CrossValidation(Validation):
"""
K-fold cross validation
Attributes:
k (int): number of folds (default: 10)
random_state (int):
seed for random number generator (default: 0). If set to `None`,
a different seed is used each time
stratified (bool):
flag deciding whether to perform stratified cross-validation.
If `True` but the class sizes don't allow it, it uses non-stratified
validataion and adds a list `warning` with a warning message(s) to
the `Result`.
"""
# TODO: list `warning` contains just repetitions of the same message
# replace with a flag in `Results`?
def __init__(self, k=10, stratified=True, random_state=0,
store_data=False, store_models=False, warnings=None):
super().__init__(store_data=store_data, store_models=store_models)
self.k = k
self.stratified = stratified
self.random_state = random_state
self.warnings = [] if warnings is None else warnings
def get_indices(self, data):
if self.stratified and data.domain.has_discrete_class:
try:
splitter = skl.StratifiedKFold(
self.k, shuffle=True, random_state=self.random_state
)
splitter.get_n_splits(data.X, data.Y)
return list(splitter.split(data.X, data.Y))
except ValueError:
self.warnings.append("Using non-stratified sampling.")
splitter = skl.KFold(
self.k, shuffle=True, random_state=self.random_state)
splitter.get_n_splits(data)
return list(splitter.split(data))
class CrossValidationFeature(Validation):
"""
Cross validation with folds according to values of a feature.
Attributes:
feature (Orange.data.Variable): the feature defining the folds
"""
def __init__(self, feature=None,
store_data=False, store_models=False, warnings=None):
super().__init__(store_data=store_data, store_models=store_models)
self.feature = feature
def get_indices(self, data):
data = data.transform(Domain([self.feature], None))
values = data[:, self.feature].X
indices = []
for v in range(len(self.feature.values)):
test_index = np.where(values == v)[0]
train_index = np.where((values != v) & (~np.isnan(values)))[0]
if test_index.size and train_index.size:
indices.append((train_index, test_index))
if not indices:
raise ValueError(
f"'{self.feature.name}' does not have at least two distinct "
"values on the data")
return indices
class LeaveOneOut(Validation):
"""Leave-one-out testing"""
score_by_folds = False
def get_indices(self, data):
splitter = skl.LeaveOneOut()
splitter.get_n_splits(data)
return list(splitter.split(data))
@staticmethod
def prepare_arrays(data, indices):
# sped up version of super().prepare_arrays(data)
row_indices = np.arange(len(data))
return row_indices, row_indices, data.Y.flatten()
class ShuffleSplit(Validation):
"""
Test by repeated random sampling
Attributes:
n_resamples (int): number of repetitions
test_size (float, int, None):
If float, should be between 0.0 and 1.0 and represent the proportion
of the dataset to include in the test split. If int, represents the
absolute number of test samples. If None, the value is set to the
complement of the train size. By default, the value is set to 0.1.
The default will change in version 0.21. It will remain 0.1 only
if ``train_size`` is unspecified, otherwise it will complement
the specified ``train_size``.
(from documentation of scipy.sklearn.StratifiedShuffleSplit)
train_size : float, int, or None, default is None
If float, should be between 0.0 and 1.0 and represent the
proportion of the dataset to include in the train split. If
int, represents the absolute number of train samples. If None,
the value is automatically set to the complement of the test size.
(from documentation of scipy.sklearn.StratifiedShuffleSplit)
stratified (bool):
flag deciding whether to perform stratified cross-validation.
random_state (int):
seed for random number generator (default: 0). If set to `None`,
a different seed is used each time
"""
def __init__(self, n_resamples=10, train_size=None, test_size=0.1,
stratified=True, random_state=0,
store_data=False, store_models=False):
super().__init__(store_data=store_data, store_models=store_models)
self.n_resamples = n_resamples
self.train_size = train_size
self.test_size = test_size
self.stratified = stratified
self.random_state = random_state
def get_indices(self, data):
if self.stratified and data.domain.has_discrete_class:
splitter = skl.StratifiedShuffleSplit(
n_splits=self.n_resamples, train_size=self.train_size,
test_size=self.test_size, random_state=self.random_state
)
splitter.get_n_splits(data.X, data.Y)
return list(splitter.split(data.X, data.Y))
splitter = skl.ShuffleSplit(
n_splits=self.n_resamples, train_size=self.train_size,
test_size=self.test_size, random_state=self.random_state
)
splitter.get_n_splits(data)
return list(splitter.split(data))
class TestOnTestData(Validation):
"""
Test on separately provided test data
Note that the class has a different signature for `__call__`.
"""
# get_indices is not needed in this class, pylint: disable=abstract-method
def __new__(cls, data=None, test_data=None, learners=None,
preprocessor=None, **kwargs):
if "train_data" in kwargs:
if data is None:
data = kwargs.pop("train_data")
else:
raise ValueError(
"argument 'data' is given twice (once as 'train_data')")
return super().__new__(
cls,
data=data, learners=learners, preprocessor=preprocessor,
test_data=test_data, **kwargs)
def __call__(self, data, test_data, learners, preprocessor=None,
*, callback=None, suppresses_exceptions=True):
"""
Args:
data (Orange.data.Table): training data
test_data (Orange.data.Table): test_data
learners (list of Orange.Learner): a list of learning algorithms
preprocessor (Orange.preprocess.Preprocess): preprocessor applied
on training data
callback (Callable): a function called to notify about the progress
suppresses_exceptions (bool): suppress the exceptions if True
Returns:
results (Result): results of testing
"""
if preprocessor is None:
preprocessor = _identity
if callback is None:
callback = _identity
train_data = preprocessor(data)
part_results = []
for (learner_i, learner) in enumerate(learners):
part_results.append(
_mp_worker(0, train_data, test_data, learner_i, learner,
self.store_models, suppresses_exceptions))
callback((learner_i + 1) / len(learners))
callback(1)
results = Results(
data=test_data if self.store_data else None,
domain=test_data.domain,
nrows=len(test_data), learners=learners,
row_indices=np.arange(len(test_data)),
folds=(Ellipsis, ),
actual=test_data.Y,
score_by_folds=self.score_by_folds,
train_time=np.zeros((len(learners),)),
test_time=np.zeros((len(learners),)))
if self.store_models:
results.models = np.tile(None, (1, len(learners)))
self._collect_part_results(results, part_results)
return results
class TestOnTrainingData(TestOnTestData):
"""Test on training data"""
# get_indices is not needed in this class, pylint: disable=abstract-method
# signature is such as on the base class, pylint: disable=signature-differs
def __new__(cls, data=None, learners=None, preprocessor=None, **kwargs):
return super().__new__(
cls,
data, test_data=data, learners=learners, preprocessor=preprocessor,
**kwargs)
def __call__(self, data, learners, preprocessor=None, *, callback=None,
suppresses_exceptions=True, **kwargs):
kwargs.setdefault("test_data", data)
# if kwargs contains anything besides test_data, this will be detected
# (and complained about) by super().__call__
return super().__call__(
data=data, learners=learners, preprocessor=preprocessor,
callback=callback, suppresses_exceptions=suppresses_exceptions,
**kwargs)
def sample(table, n=0.7, stratified=False, replace=False,
random_state=None):
"""
Samples data instances from a data table. Returns the sample and
a dataset from input data table that are not in the sample. Also
uses several sampling functions from
`scikit-learn <http://scikit-learn.org>`_.
table : data table
A data table from which to sample.
n : float, int (default = 0.7)
If float, should be between 0.0 and 1.0 and represents
the proportion of data instances in the resulting sample. If
int, n is the number of data instances in the resulting sample.
stratified : bool, optional (default = False)
If true, sampling will try to consider class values and
match distribution of class values
in train and test subsets.
replace : bool, optional (default = False)
sample with replacement
random_state : int or RandomState
Pseudo-random number generator state used for random sampling.
"""
if isinstance(n, float):
n = int(n * len(table))
if replace:
if random_state is None:
rgen = np.random
else:
rgen = np.random.mtrand.RandomState(random_state)
a_sample = rgen.randint(0, len(table), n)
o = np.ones(len(table))
o[a_sample] = 0
others = np.nonzero(o)[0]
return table[a_sample], table[others]
n = len(table) - n
if stratified and table.domain.has_discrete_class:
test_size = max(len(table.domain.class_var.values), n)
splitter = skl.StratifiedShuffleSplit(
n_splits=1, test_size=test_size, train_size=len(table) - test_size,
random_state=random_state)
splitter.get_n_splits(table.X, table.Y)
ind = splitter.split(table.X, table.Y)
else:
splitter = skl.ShuffleSplit(
n_splits=1, test_size=n, random_state=random_state)
splitter.get_n_splits(table)
ind = splitter.split(table)
ind = next(ind)
return table[ind[0]], table[ind[1]]
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