Source code for yellowbrick.regressor.influence

# yellowbrick.regressor.influence
# Visualize the influence and leverage of individual instances on a regression model.
#
# Author:   Benjamin Bengfort
# Created:  Sun Jun 09 15:21:17 2019 -0400
#
# Copyright (C) 2019 The scikit-yb developers
# For license information, see LICENSE.txt
#
# ID: influence.py [fe14cfd] benjamin@bengfort.com $

"""
Visualize the influence and leverage of individual instances on a regression model.
"""

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## Imports
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import numpy as np
import scipy as sp

from yellowbrick.base import Visualizer
from sklearn.linear_model import LinearRegression


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## Cook's Distance
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[docs]class CooksDistance(Visualizer): """ Cook's Distance is a measure of how influential an instance is to the computation of a regression, e.g. if the instance is removed would the estimated coeficients of the underlying model be substantially changed? Because of this, Cook's Distance is generally used to detect outliers in standard, OLS regression. In fact, a general rule of thumb is that D(i) > 4/n is a good threshold for determining highly influential points as outliers and this visualizer can report the percentage of data that is above that threshold. This implementation of Cook's Distance assumes Ordinary Least Squares regression, and therefore embeds a ``sklearn.linear_model.LinearRegression`` under the hood. Distance is computed via the non-whitened leverage of the projection matrix, computed inside of ``fit()``. The results of this visualizer are therefore similar to, but not as advanced, as a similar computation using statsmodels. Computing the influence for other regression models requires leave one out validation and can be expensive to compute. .. seealso:: For a longer discussion on detecting outliers in regression and computing leverage and influence, see `linear regression in python, outliers/leverage detect <http://bit.ly/2If2fga>`_ by Huiming Song. Parameters ---------- ax : matplotlib Axes, default: None The axes to plot the figure on. If None is passed in the current axes will be used (or generated if required). draw_threshold : bool, default: True Draw a horizontal line at D(i) == 4/n to easily identify the most influential points on the final regression. This will also draw a legend that specifies the percentage of data points that are above the threshold. linefmt : str, default: 'C0-' A string defining the properties of the vertical lines of the stem plot, usually this will be a color or a color and a line style. The default is simply a solid line with the first color of the color cycle. markerfmt : str, default: ',' A string defining the properties of the markers at the stem plot heads. The default is "pixel", e.g. basically no marker head at the top of the stem plot. kwargs : dict Keyword arguments that are passed to the base class and may influence the final visualization (e.g. size or title parameters). Attributes ---------- distance_ : array, 1D The Cook's distance value for each instance specified in ``X``, e.g. an 1D array with shape ``(X.shape[0],)``. p_values_ : array, 1D The p values associated with the F-test of Cook's distance distribution. A 1D array whose shape matches ``distance_``. influence_threshold_ : float A rule of thumb influence threshold to determine outliers in the regression model, defined as It=4/n. outlier_percentage_ : float The percentage of instances whose Cook's distance is greater than the influnce threshold, the percentage is 0.0 <= p <= 100.0. Notes ----- Cook's Distance is very similar to DFFITS, another diagnostic that is meant to show how influential a point is in a statistical regression. Although the computed values of Cook's and DFFITS are different, they are conceptually identical and there even exists a closed-form formula to convert one value to another. Because of this, we have chosen to implement Cook's distance rather than or in addition to DFFITS. """ def __init__( self, ax=None, draw_threshold=True, linefmt="C0-", markerfmt=",", **kwargs ): # Initialize the visualizer super(CooksDistance, self).__init__(ax=ax, **kwargs) # Set "hyperparameters" self.set_params( draw_threshold=draw_threshold, linefmt=linefmt, markerfmt=markerfmt ) # An internal LinearRegression used to compute the residuals and MSE # This implementation doesn't support any regressor, it is OLS-specific self._model = LinearRegression()
[docs] def fit(self, X, y): """ Computes the leverage of X and uses the residuals of a ``sklearn.linear_model.LinearRegression`` to compute the Cook's Distance of each observation in X, their p-values and the number of outliers defined by the number of observations supplied. Parameters ---------- X : array-like, 2D The exogenous design matrix, e.g. training data. y : array-like, 1D The endogenous response variable, e.g. target data. Returns ------- self : CooksDistance Fit returns the visualizer instance. """ # Fit a linear model to X and y to compute MSE self._model.fit(X, y) # Leverage is computed as the diagonal of the projection matrix of X # TODO: whiten X before computing leverage leverage = (X * np.linalg.pinv(X).T).sum(1) # Compute the rank and the degrees of freedom of the OLS model rank = np.linalg.matrix_rank(X) df = X.shape[0] - rank # Compute the MSE from the residuals residuals = y - self._model.predict(X) mse = np.dot(residuals, residuals) / df # Compute Cook's distance residuals_studentized = residuals / np.sqrt(mse) / np.sqrt(1 - leverage) self.distance_ = residuals_studentized ** 2 / X.shape[1] self.distance_ *= leverage / (1 - leverage) # Compute the p-values of Cook's Distance # TODO: honestly this was done because it was only in the statsmodels # implementation... I have no idea what this is or why its important. self.p_values_ = sp.stats.f.sf(self.distance_, X.shape[1], df) # Compute the influence threshold rule of thumb self.influence_threshold_ = 4 / X.shape[0] self.outlier_percentage_ = ( sum(self.distance_ > self.influence_threshold_) / X.shape[0] ) self.outlier_percentage_ *= 100.0 self.draw() return self
[docs] def draw(self): """ Draws a stem plot where each stem is the Cook's Distance of the instance at the index specified by the x axis. Optionaly draws a threshold line. """ # Draw a stem plot with the influence for each instance _, _, baseline = self.ax.stem( self.distance_, linefmt=self.linefmt, markerfmt=self.markerfmt, use_line_collection=True ) # No padding on either side of the instance index self.ax.set_xlim(0, len(self.distance_)) # Draw the threshold for most influential points if self.draw_threshold: label = r"{:0.2f}% > $I_t$ ($I_t=\frac {{4}} {{n}}$)".format( self.outlier_percentage_ ) self.ax.axhline( self.influence_threshold_, ls="--", label=label, c=baseline.get_color(), lw=baseline.get_linewidth(), ) return self.ax
[docs] def finalize(self): """ Prepares the visualization for presentation and reporting. """ # Set the title and axis labels self.set_title("Cook's Distance Outlier Detection") self.ax.set_xlabel("instance index") self.ax.set_ylabel("influence (I)") # Only add the legend if the influence threshold has been plotted if self.draw_threshold: self.ax.legend(loc="best", frameon=True)
[docs]def cooks_distance( X, y, ax=None, draw_threshold=True, linefmt="C0-", markerfmt=",", show=True, **kwargs ): """ Cook's Distance is a measure of how influential an instance is to the computation of a regression, e.g. if the instance is removed would the estimated coeficients of the underlying model be substantially changed? Because of this, Cook's Distance is generally used to detect outliers in standard, OLS regression. In fact, a general rule of thumb is that D(i) > 4/n is a good threshold for determining highly influential points as outliers and this visualizer can report the percentage of data that is above that threshold. This implementation of Cook's Distance assumes Ordinary Least Squares regression, and therefore embeds a ``sklearn.linear_model.LinearRegression`` under the hood. Distance is computed via the non-whitened leverage of the projection matrix, computed inside of ``fit()``. The results of this visualizer are therefore similar to, but not as advanced, as a similar computation using statsmodels. Computing the influence for other regression models requires leave one out validation and can be expensive to compute. .. seealso:: For a longer discussion on detecting outliers in regression and computing leverage and influence, see `linear regression in python, outliers/leverage detect <http://bit.ly/2If2fga>`_ by Huiming Song. Parameters ---------- X : array-like, 2D The exogenous design matrix, e.g. training data. y : array-like, 1D The endogenous response variable, e.g. target data. ax : matplotlib Axes, default: None The axes to plot the figure on. If None is passed in the current axes will be used (or generated if required). draw_threshold : bool, default: True Draw a horizontal line at D(i) == 4/n to easily identify the most influential points on the final regression. This will also draw a legend that specifies the percentage of data points that are above the threshold. linefmt : str, default: 'C0-' A string defining the properties of the vertical lines of the stem plot, usually this will be a color or a color and a line style. The default is simply a solid line with the first color of the color cycle. markerfmt: str, default: ',' A string defining the properties of the markers at the stem plot heads. The default is "pixel", e.g. basically no marker head at the top of the stem plot. show: bool, default: True If True, calls ``show()``, which in turn calls ``plt.show()`` however you cannot call ``plt.savefig`` from this signature, nor ``clear_figure``. If False, simply calls ``finalize()`` kwargs : dict Keyword arguments that are passed to the base class and may influence the final visualization (e.g. size or title parameters). """ viz = CooksDistance( ax=ax, draw_threshold=draw_threshold, linefmt=linefmt, markerfmt=markerfmt, **kwargs ) viz.fit(X, y) # Draw the final visualization if show: viz.show() else: viz.finalize() # Return the visualizer return viz