281 lines
7.9 KiB
Python
281 lines
7.9 KiB
Python
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"""
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Module contains classes for invertible (and differentiable) link functions.
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"""
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# Author: Christian Lorentzen <lorentzen.ch@gmail.com>
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from abc import ABC, abstractmethod
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from dataclasses import dataclass
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import numpy as np
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from scipy.special import expit, logit
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from scipy.stats import gmean
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from ..utils.extmath import softmax
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@dataclass
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class Interval:
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low: float
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high: float
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low_inclusive: bool
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high_inclusive: bool
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def __post_init__(self):
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"""Check that low <= high"""
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if self.low > self.high:
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raise ValueError(
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f"One must have low <= high; got low={self.low}, high={self.high}."
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)
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def includes(self, x):
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"""Test whether all values of x are in interval range.
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Parameters
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----------
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x : ndarray
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Array whose elements are tested to be in interval range.
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Returns
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-------
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result : bool
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"""
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if self.low_inclusive:
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low = np.greater_equal(x, self.low)
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else:
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low = np.greater(x, self.low)
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if not np.all(low):
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return False
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if self.high_inclusive:
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high = np.less_equal(x, self.high)
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else:
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high = np.less(x, self.high)
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# Note: np.all returns numpy.bool_
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return bool(np.all(high))
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def _inclusive_low_high(interval, dtype=np.float64):
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"""Generate values low and high to be within the interval range.
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This is used in tests only.
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Returns
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-------
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low, high : tuple
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The returned values low and high lie within the interval.
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"""
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eps = 10 * np.finfo(dtype).eps
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if interval.low == -np.inf:
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low = -1e10
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elif interval.low < 0:
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low = interval.low * (1 - eps) + eps
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else:
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low = interval.low * (1 + eps) + eps
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if interval.high == np.inf:
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high = 1e10
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elif interval.high < 0:
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high = interval.high * (1 + eps) - eps
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else:
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high = interval.high * (1 - eps) - eps
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return low, high
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class BaseLink(ABC):
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"""Abstract base class for differentiable, invertible link functions.
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Convention:
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- link function g: raw_prediction = g(y_pred)
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- inverse link h: y_pred = h(raw_prediction)
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For (generalized) linear models, `raw_prediction = X @ coef` is the so
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called linear predictor, and `y_pred = h(raw_prediction)` is the predicted
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conditional (on X) expected value of the target `y_true`.
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The methods are not implemented as staticmethods in case a link function needs
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parameters.
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"""
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is_multiclass = False # used for testing only
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# Usually, raw_prediction may be any real number and y_pred is an open
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# interval.
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# interval_raw_prediction = Interval(-np.inf, np.inf, False, False)
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interval_y_pred = Interval(-np.inf, np.inf, False, False)
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@abstractmethod
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def link(self, y_pred, out=None):
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"""Compute the link function g(y_pred).
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The link function maps (predicted) target values to raw predictions,
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i.e. `g(y_pred) = raw_prediction`.
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Parameters
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----------
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y_pred : array
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Predicted target values.
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out : array
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A location into which the result is stored. If provided, it must
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have a shape that the inputs broadcast to. If not provided or None,
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a freshly-allocated array is returned.
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Returns
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-------
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out : array
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Output array, element-wise link function.
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"""
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@abstractmethod
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def inverse(self, raw_prediction, out=None):
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"""Compute the inverse link function h(raw_prediction).
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The inverse link function maps raw predictions to predicted target
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values, i.e. `h(raw_prediction) = y_pred`.
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Parameters
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----------
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raw_prediction : array
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Raw prediction values (in link space).
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out : array
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A location into which the result is stored. If provided, it must
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have a shape that the inputs broadcast to. If not provided or None,
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a freshly-allocated array is returned.
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Returns
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-------
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out : array
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Output array, element-wise inverse link function.
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"""
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class IdentityLink(BaseLink):
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"""The identity link function g(x)=x."""
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def link(self, y_pred, out=None):
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if out is not None:
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np.copyto(out, y_pred)
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return out
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else:
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return y_pred
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inverse = link
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class LogLink(BaseLink):
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"""The log link function g(x)=log(x)."""
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interval_y_pred = Interval(0, np.inf, False, False)
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def link(self, y_pred, out=None):
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return np.log(y_pred, out=out)
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def inverse(self, raw_prediction, out=None):
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return np.exp(raw_prediction, out=out)
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class LogitLink(BaseLink):
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"""The logit link function g(x)=logit(x)."""
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interval_y_pred = Interval(0, 1, False, False)
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def link(self, y_pred, out=None):
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return logit(y_pred, out=out)
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def inverse(self, raw_prediction, out=None):
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return expit(raw_prediction, out=out)
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class HalfLogitLink(BaseLink):
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"""Half the logit link function g(x)=1/2 * logit(x).
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Used for the exponential loss.
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"""
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interval_y_pred = Interval(0, 1, False, False)
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def link(self, y_pred, out=None):
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out = logit(y_pred, out=out)
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out *= 0.5
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return out
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def inverse(self, raw_prediction, out=None):
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return expit(2 * raw_prediction, out)
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class MultinomialLogit(BaseLink):
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"""The symmetric multinomial logit function.
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Convention:
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- y_pred.shape = raw_prediction.shape = (n_samples, n_classes)
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Notes:
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- The inverse link h is the softmax function.
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- The sum is over the second axis, i.e. axis=1 (n_classes).
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We have to choose additional constraints in order to make
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y_pred[k] = exp(raw_pred[k]) / sum(exp(raw_pred[k]), k=0..n_classes-1)
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for n_classes classes identifiable and invertible.
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We choose the symmetric side constraint where the geometric mean response
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is set as reference category, see [2]:
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The symmetric multinomial logit link function for a single data point is
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then defined as
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raw_prediction[k] = g(y_pred[k]) = log(y_pred[k]/gmean(y_pred))
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= log(y_pred[k]) - mean(log(y_pred)).
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Note that this is equivalent to the definition in [1] and implies mean
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centered raw predictions:
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sum(raw_prediction[k], k=0..n_classes-1) = 0.
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For linear models with raw_prediction = X @ coef, this corresponds to
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sum(coef[k], k=0..n_classes-1) = 0, i.e. the sum over classes for every
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feature is zero.
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Reference
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---------
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.. [1] Friedman, Jerome; Hastie, Trevor; Tibshirani, Robert. "Additive
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logistic regression: a statistical view of boosting" Ann. Statist.
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28 (2000), no. 2, 337--407. doi:10.1214/aos/1016218223.
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https://projecteuclid.org/euclid.aos/1016218223
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.. [2] Zahid, Faisal Maqbool and Gerhard Tutz. "Ridge estimation for
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multinomial logit models with symmetric side constraints."
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Computational Statistics 28 (2013): 1017-1034.
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http://epub.ub.uni-muenchen.de/11001/1/tr067.pdf
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"""
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is_multiclass = True
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interval_y_pred = Interval(0, 1, False, False)
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def symmetrize_raw_prediction(self, raw_prediction):
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return raw_prediction - np.mean(raw_prediction, axis=1)[:, np.newaxis]
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def link(self, y_pred, out=None):
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# geometric mean as reference category
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gm = gmean(y_pred, axis=1)
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return np.log(y_pred / gm[:, np.newaxis], out=out)
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def inverse(self, raw_prediction, out=None):
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if out is None:
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return softmax(raw_prediction, copy=True)
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else:
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np.copyto(out, raw_prediction)
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softmax(out, copy=False)
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return out
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_LINKS = {
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"identity": IdentityLink,
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"log": LogLink,
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"logit": LogitLink,
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"half_logit": HalfLogitLink,
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"multinomial_logit": MultinomialLogit,
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}
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