# 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.
"""
##########################################################################
## Imports
##########################################################################
import numpy as np
import scipy as sp
from yellowbrick.base import Visualizer
from sklearn.linear_model import LinearRegression
##########################################################################
## Cook's Distance
##########################################################################
[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.draw_threshold = draw_threshold
self.linefmt = linefmt
self.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
```