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391 lines
13 KiB
391 lines
13 KiB
# Authors: Fabian Pedregosa <fabian@fseoane.net>
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# Alexandre Gramfort <alexandre.gramfort@inria.fr>
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# Nelle Varoquaux <nelle.varoquaux@gmail.com>
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# License: BSD 3 clause
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import numpy as np
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from scipy import interpolate
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from scipy.stats import spearmanr
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from .base import BaseEstimator, TransformerMixin, RegressorMixin
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from .utils import as_float_array, check_array, check_consistent_length
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from ._isotonic import _inplace_contiguous_isotonic_regression, _make_unique
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import warnings
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import math
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__all__ = ['check_increasing', 'isotonic_regression',
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'IsotonicRegression']
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def check_increasing(x, y):
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"""Determine whether y is monotonically correlated with x.
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y is found increasing or decreasing with respect to x based on a Spearman
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correlation test.
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Parameters
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----------
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x : array-like, shape=(n_samples,)
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Training data.
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y : array-like, shape=(n_samples,)
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Training target.
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Returns
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-------
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increasing_bool : boolean
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Whether the relationship is increasing or decreasing.
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Notes
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-----
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The Spearman correlation coefficient is estimated from the data, and the
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sign of the resulting estimate is used as the result.
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In the event that the 95% confidence interval based on Fisher transform
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spans zero, a warning is raised.
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References
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----------
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Fisher transformation. Wikipedia.
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https://en.wikipedia.org/wiki/Fisher_transformation
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"""
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# Calculate Spearman rho estimate and set return accordingly.
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rho, _ = spearmanr(x, y)
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increasing_bool = rho >= 0
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# Run Fisher transform to get the rho CI, but handle rho=+/-1
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if rho not in [-1.0, 1.0] and len(x) > 3:
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F = 0.5 * math.log((1. + rho) / (1. - rho))
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F_se = 1 / math.sqrt(len(x) - 3)
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# Use a 95% CI, i.e., +/-1.96 S.E.
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# https://en.wikipedia.org/wiki/Fisher_transformation
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rho_0 = math.tanh(F - 1.96 * F_se)
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rho_1 = math.tanh(F + 1.96 * F_se)
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# Warn if the CI spans zero.
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if np.sign(rho_0) != np.sign(rho_1):
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warnings.warn("Confidence interval of the Spearman "
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"correlation coefficient spans zero. "
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"Determination of ``increasing`` may be "
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"suspect.")
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return increasing_bool
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def isotonic_regression(y, sample_weight=None, y_min=None, y_max=None,
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increasing=True):
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"""Solve the isotonic regression model::
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min sum w[i] (y[i] - y_[i]) ** 2
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subject to y_min = y_[1] <= y_[2] ... <= y_[n] = y_max
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where:
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- y[i] are inputs (real numbers)
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- y_[i] are fitted
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- w[i] are optional strictly positive weights (default to 1.0)
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Read more in the :ref:`User Guide <isotonic>`.
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Parameters
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----------
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y : iterable of floats
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The data.
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sample_weight : iterable of floats, optional, default: None
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Weights on each point of the regression.
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If None, weight is set to 1 (equal weights).
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y_min : optional, default: None
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If not None, set the lowest value of the fit to y_min.
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y_max : optional, default: None
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If not None, set the highest value of the fit to y_max.
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increasing : boolean, optional, default: True
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Whether to compute ``y_`` is increasing (if set to True) or decreasing
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(if set to False)
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Returns
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-------
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y_ : list of floats
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Isotonic fit of y.
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References
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----------
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"Active set algorithms for isotonic regression; A unifying framework"
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by Michael J. Best and Nilotpal Chakravarti, section 3.
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"""
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order = np.s_[:] if increasing else np.s_[::-1]
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y = np.array(y[order], dtype=np.float64)
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if sample_weight is None:
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sample_weight = np.ones(len(y), dtype=np.float64)
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else:
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sample_weight = np.array(sample_weight[order], dtype=np.float64)
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_inplace_contiguous_isotonic_regression(y, sample_weight)
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if y_min is not None or y_max is not None:
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# Older versions of np.clip don't accept None as a bound, so use np.inf
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if y_min is None:
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y_min = -np.inf
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if y_max is None:
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y_max = np.inf
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np.clip(y, y_min, y_max, y)
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return y[order]
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class IsotonicRegression(BaseEstimator, TransformerMixin, RegressorMixin):
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"""Isotonic regression model.
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The isotonic regression optimization problem is defined by::
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min sum w_i (y[i] - y_[i]) ** 2
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subject to y_[i] <= y_[j] whenever X[i] <= X[j]
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and min(y_) = y_min, max(y_) = y_max
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where:
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- ``y[i]`` are inputs (real numbers)
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- ``y_[i]`` are fitted
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- ``X`` specifies the order.
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If ``X`` is non-decreasing then ``y_`` is non-decreasing.
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- ``w[i]`` are optional strictly positive weights (default to 1.0)
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Read more in the :ref:`User Guide <isotonic>`.
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Parameters
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----------
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y_min : optional, default: None
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If not None, set the lowest value of the fit to y_min.
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y_max : optional, default: None
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If not None, set the highest value of the fit to y_max.
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increasing : boolean or string, optional, default: True
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If boolean, whether or not to fit the isotonic regression with y
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increasing or decreasing.
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The string value "auto" determines whether y should
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increase or decrease based on the Spearman correlation estimate's
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sign.
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out_of_bounds : string, optional, default: "nan"
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The ``out_of_bounds`` parameter handles how x-values outside of the
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training domain are handled. When set to "nan", predicted y-values
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will be NaN. When set to "clip", predicted y-values will be
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set to the value corresponding to the nearest train interval endpoint.
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When set to "raise", allow ``interp1d`` to throw ValueError.
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Attributes
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----------
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X_min_ : float
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Minimum value of input array `X_` for left bound.
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X_max_ : float
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Maximum value of input array `X_` for right bound.
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f_ : function
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The stepwise interpolating function that covers the input domain ``X``.
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Notes
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-----
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Ties are broken using the secondary method from Leeuw, 1977.
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References
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----------
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Isotonic Median Regression: A Linear Programming Approach
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Nilotpal Chakravarti
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Mathematics of Operations Research
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Vol. 14, No. 2 (May, 1989), pp. 303-308
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Isotone Optimization in R : Pool-Adjacent-Violators
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Algorithm (PAVA) and Active Set Methods
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Leeuw, Hornik, Mair
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Journal of Statistical Software 2009
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Correctness of Kruskal's algorithms for monotone regression with ties
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Leeuw, Psychometrica, 1977
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"""
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def __init__(self, y_min=None, y_max=None, increasing=True,
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out_of_bounds='nan'):
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self.y_min = y_min
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self.y_max = y_max
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self.increasing = increasing
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self.out_of_bounds = out_of_bounds
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def _check_fit_data(self, X, y, sample_weight=None):
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if len(X.shape) != 1:
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raise ValueError("X should be a 1d array")
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def _build_f(self, X, y):
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"""Build the f_ interp1d function."""
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# Handle the out_of_bounds argument by setting bounds_error
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if self.out_of_bounds not in ["raise", "nan", "clip"]:
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raise ValueError("The argument ``out_of_bounds`` must be in "
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"'nan', 'clip', 'raise'; got {0}"
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.format(self.out_of_bounds))
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bounds_error = self.out_of_bounds == "raise"
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if len(y) == 1:
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# single y, constant prediction
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self.f_ = lambda x: y.repeat(x.shape)
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else:
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self.f_ = interpolate.interp1d(X, y, kind='linear',
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bounds_error=bounds_error)
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def _build_y(self, X, y, sample_weight, trim_duplicates=True):
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"""Build the y_ IsotonicRegression."""
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check_consistent_length(X, y, sample_weight)
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X, y = [check_array(x, ensure_2d=False) for x in [X, y]]
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y = as_float_array(y)
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self._check_fit_data(X, y, sample_weight)
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# Determine increasing if auto-determination requested
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if self.increasing == 'auto':
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self.increasing_ = check_increasing(X, y)
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else:
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self.increasing_ = self.increasing
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# If sample_weights is passed, removed zero-weight values and clean
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# order
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if sample_weight is not None:
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sample_weight = check_array(sample_weight, ensure_2d=False)
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mask = sample_weight > 0
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X, y, sample_weight = X[mask], y[mask], sample_weight[mask]
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else:
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sample_weight = np.ones(len(y))
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order = np.lexsort((y, X))
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X, y, sample_weight = [array[order].astype(np.float64, copy=False)
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for array in [X, y, sample_weight]]
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unique_X, unique_y, unique_sample_weight = _make_unique(
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X, y, sample_weight)
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# Store _X_ and _y_ to maintain backward compat during the deprecation
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# period of X_ and y_
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self._X_ = X = unique_X
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self._y_ = y = isotonic_regression(unique_y, unique_sample_weight,
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self.y_min, self.y_max,
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increasing=self.increasing_)
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# Handle the left and right bounds on X
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self.X_min_, self.X_max_ = np.min(X), np.max(X)
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if trim_duplicates:
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# Remove unnecessary points for faster prediction
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keep_data = np.ones((len(y),), dtype=bool)
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# Aside from the 1st and last point, remove points whose y values
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# are equal to both the point before and the point after it.
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keep_data[1:-1] = np.logical_or(
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np.not_equal(y[1:-1], y[:-2]),
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np.not_equal(y[1:-1], y[2:])
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)
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return X[keep_data], y[keep_data]
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else:
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# The ability to turn off trim_duplicates is only used to it make
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# easier to unit test that removing duplicates in y does not have
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# any impact the resulting interpolation function (besides
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# prediction speed).
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return X, y
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def fit(self, X, y, sample_weight=None):
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"""Fit the model using X, y as training data.
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Parameters
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----------
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X : array-like, shape=(n_samples,)
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Training data.
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y : array-like, shape=(n_samples,)
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Training target.
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sample_weight : array-like, shape=(n_samples,), optional, default: None
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Weights. If set to None, all weights will be set to 1 (equal
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weights).
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Returns
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-------
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self : object
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Returns an instance of self.
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Notes
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-----
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X is stored for future use, as `transform` needs X to interpolate
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new input data.
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"""
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# Transform y by running the isotonic regression algorithm and
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# transform X accordingly.
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X, y = self._build_y(X, y, sample_weight)
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# It is necessary to store the non-redundant part of the training set
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# on the model to make it possible to support model persistence via
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# the pickle module as the object built by scipy.interp1d is not
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# picklable directly.
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self._necessary_X_, self._necessary_y_ = X, y
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# Build the interpolation function
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self._build_f(X, y)
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return self
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def transform(self, T):
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"""Transform new data by linear interpolation
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Parameters
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----------
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T : array-like, shape=(n_samples,)
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Data to transform.
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Returns
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-------
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T_ : array, shape=(n_samples,)
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The transformed data
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"""
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T = as_float_array(T)
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if len(T.shape) != 1:
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raise ValueError("Isotonic regression input should be a 1d array")
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# Handle the out_of_bounds argument by clipping if needed
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if self.out_of_bounds not in ["raise", "nan", "clip"]:
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raise ValueError("The argument ``out_of_bounds`` must be in "
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"'nan', 'clip', 'raise'; got {0}"
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.format(self.out_of_bounds))
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if self.out_of_bounds == "clip":
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T = np.clip(T, self.X_min_, self.X_max_)
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return self.f_(T)
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def predict(self, T):
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"""Predict new data by linear interpolation.
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Parameters
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----------
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T : array-like, shape=(n_samples,)
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Data to transform.
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Returns
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-------
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T_ : array, shape=(n_samples,)
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Transformed data.
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"""
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return self.transform(T)
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def __getstate__(self):
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"""Pickle-protocol - return state of the estimator. """
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state = super(IsotonicRegression, self).__getstate__()
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# remove interpolation method
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state.pop('f_', None)
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return state
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def __setstate__(self, state):
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"""Pickle-protocol - set state of the estimator.
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We need to rebuild the interpolation function.
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"""
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super(IsotonicRegression, self).__setstate__(state)
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if hasattr(self, '_necessary_X_') and hasattr(self, '_necessary_y_'):
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self._build_f(self._necessary_X_, self._necessary_y_)
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