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%matplotlib inline
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import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs
from sklearn import datasets
from yellowbrick.cluster import KElbowVisualizer, SilhouetteVisualizer
mpl.rcParams["figure.figsize"] = (9,6)
The Yellowbrick library is a diagnostic visualization platform for machine learning that allows data scientists to steer the model selection process. It extends the scikit-learn API with a new core object: the Visualizer. Visualizers allow models to be fit and transformed as part of the scikit-learn pipeline process, providing visual diagnostics throughout the transformation of high-dimensional data.
In machine learning, clustering models are unsupervised methods that attempt to detect patterns in unlabeled data. There are two primary classes of clustering algorithms: agglomerative clustering which links similar data points together, and centroidal clustering which attempts to find centers or partitions in the data.
Currently, Yellowbrick provides two visualizers to evaluate centroidal mechanisms, particularly K-Means clustering, that help users discover an optimal $K$ parameter in the clustering metric:
KElbowVisualizer visualizes the clusters according to a scoring function, looking for an "elbow" in the curve. SilhouetteVisualizer visualizes the silhouette scores of each cluster in a single model.For the following examples, we'll use the widely famous Iris dataset. The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. You can learn more about it here: Iris Data Set
The dataset is loaded using scikit-learn's datasets.load_iris() function to create a sample two-dimensional dataset with 8 random clusters of points.
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# Load iris flower dataset
iris = datasets.load_iris()
X = iris.data #clustering is unsupervised learning hence we load only X(i.e.iris.data) and not Y(i.e. iris.target)
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# Converting the data into dataframe
feature_names = iris.feature_names
iris_dataframe = pd.DataFrame(X, columns=feature_names)
iris_dataframe.head(10)
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# Fitting the model with a dummy model, with 3 clusters (we already know there are 3 classes in the Iris dataset)
k_means = KMeans(n_clusters=3)
k_means.fit(X)
# Plotting a 3d plot using matplotlib to visualize the data points
fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(111, projection='3d')
# Setting the colors to match cluster results
colors = ['red' if label == 0 else 'purple' if label==1 else 'green' for label in k_means.labels_]
ax.scatter(X[:,3], X[:,0], X[:,2], c=colors)
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In the above example plot, one of the clusters is linearly seperable and at a good seperation from other two clusters. Two of the clusters are close by and not linearly seperable.
Also the dataset is 4-dimensional i.e. it has 4 features, but for the sake of visualization using matplotlib, one of dimensions has been ignored. Therefore, it can be said that just visualization of data-points is not always enough for knowing optimal number of clusters $K$.
Yellowbrick's KElbowVisualizer implements the “elbow” method of selecting the optimal number of clusters by fitting the K-Means model with a range of values for $K$. If the line chart looks like an arm, then the “elbow” (the point of inflection on the curve) is a good indication that the underlying model fits best at that point.
In the following example, the KElbowVisualizer fits the model for a range of $K$ values from 2 to 10, which is set by the parameter k=(2,11). When the model is fit with 3 clusters we can see an "elbow" in the graph, which in this case we know to be the optimal number since our dataset has 3 clusters of points.
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# Instantiate the clustering model and visualizer
model = KMeans()
visualizer = KElbowVisualizer(model, k=(2,11))
visualizer.fit(X) # Fit the data to the visualizer
visualizer.show() # Draw/show/show the data
By default, the scoring parameter metric is set to distortion, which computes the sum of squared distances from each point to its assigned center. However, two other metrics can also be used with the KElbowVisualizer—silhouette and calinski_harabaz. The silhouette score is the mean silhouette coefficient for all samples, while the calinski_harabaz score computes the ratio of dispersion between and within clusters.
The KElbowVisualizer also displays the amount of time to fit the model per $K$, which can be hidden by setting timings=False. In the following example, we'll use the calinski_harabaz score and hide the time to fit the model.
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# Instantiate the clustering model and visualizer
model = KMeans()
visualizer = KElbowVisualizer(model, k=(2,11), metric='calinski_harabaz', timings=False)
visualizer.fit(X) # Fit the data to the visualizer
visualizer.show() # Draw/show/show the data
It is important to remember that the Elbow method does not work well if the data is not very clustered. In such cases, you might see a smooth curve and the optimal value of $K$ will be unclear.
You can learn more about the Elbow method at Robert Grove's Blocks.
Silhouette analysis can be used to evaluate the density and separation between clusters. The score is calculated by averaging the silhouette coefficient for each sample, which is computed as the difference between the average intra-cluster distance and the mean nearest-cluster distance for each sample, normalized by the maximum value. This produces a score between -1 and +1, where scores near +1 indicate high separation and scores near -1 indicate that the samples may have been assigned to the wrong cluster.
The SilhouetteVisualizer displays the silhouette coefficient for each sample on a per-cluster basis, allowing users to visualize the density and separation of the clusters. This is particularly useful for determining cluster imbalance or for selecting a value for $K$ by comparing multiple visualizers.
Since we created the sample dataset for these examples, we already know that the data points are grouped into 8 clusters. So for the first SilhouetteVisualizer example, we'll set $K$ to 3 in order to show how the plot looks when using the optimal value of $K$.
Notice that graph contains homogeneous and long silhouettes. In addition, the vertical red-dotted line on the plot indicates the average silhouette score for all observations.
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# Instantiate the clustering model and visualizer
model = KMeans(3)
visualizer = SilhouetteVisualizer(model)
visualizer.fit(X) # Fit the data to the visualizer
visualizer.show() # Draw/show/show the data
For the next example, let's see what happens when using a non-optimal value for $K$, in this case, 6.
Now we see that the width of clusters 1 to 6 have become narrow, of unequal width and their silhouette coefficient scores have dropped. This occurs because the width of each silhouette is proportional to the number of samples assigned to the cluster. The model is trying to fit our data into a larger than optimal number of clusters, making some of the clusters narrower but much less cohesive as seen from the drop in average-silhouette score.
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# Instantiate the clustering model and visualizer
model = KMeans(6)
visualizer = SilhouetteVisualizer(model)
visualizer.fit(X) # Fit the data to the visualizer
visualizer.show() # Draw/show/show the data
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import numpy as np
import matplotlib.pyplot as plt
from yellowbrick.style import color_palette
from yellowbrick.cluster.base import ClusteringScoreVisualizer
from sklearn.metrics import silhouette_score, silhouette_samples
## Packages for export
__all__ = [
"SilhouetteVisualizer"
]
##########################################################################
## Silhouette Method for K Selection
##########################################################################
class SilhouetteVisualizer(ClusteringScoreVisualizer):
"""
The Silhouette Visualizer displays the silhouette coefficient for each
sample on a per-cluster basis, visually evaluating the density and
separation between clusters. The score is calculated by averaging the
silhouette coefficient for each sample, computed as the difference
between the average intra-cluster distance and the mean nearest-cluster
distance for each sample, normalized by the maximum value. This produces a
score between -1 and +1, where scores near +1 indicate high separation
and scores near -1 indicate that the samples may have been assigned to
the wrong cluster.
In SilhouetteVisualizer plots, clusters with higher scores have wider
silhouettes, but clusters that are less cohesive will fall short of the
average score across all clusters, which is plotted as a vertical dotted
red line.
This is particularly useful for determining cluster imbalance, or for
selecting a value for K by comparing multiple visualizers.
Parameters
----------
model : a Scikit-Learn clusterer
Should be an instance of a centroidal clustering algorithm (``KMeans``
or ``MiniBatchKMeans``).
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).
kwargs : dict
Keyword arguments that are passed to the base class and may influence
the visualization as defined in other Visualizers.
Attributes
----------
silhouette_score_ : float
Mean Silhouette Coefficient for all samples. Computed via scikit-learn
`sklearn.metrics.silhouette_score`.
silhouette_samples_ : array, shape = [n_samples]
Silhouette Coefficient for each samples. Computed via scikit-learn
`sklearn.metrics.silhouette_samples`.
n_samples_ : integer
Number of total samples in the dataset (X.shape[0])
n_clusters_ : integer
Number of clusters (e.g. n_clusters or k value) passed to internal
scikit-learn model.
Examples
--------
>>> from yellowbrick.cluster import SilhouetteVisualizer
>>> from sklearn.cluster import KMeans
>>> model = SilhouetteVisualizer(KMeans(10))
>>> model.fit(X)
>>> model.show()
"""
def __init__(self, model, ax=None, **kwargs):
super(SilhouetteVisualizer, self).__init__(model, ax=ax, **kwargs)
# Visual Properties
# TODO: Fix the color handling
self.colormap = kwargs.get('colormap', 'set1')
self.color = kwargs.get('color', None)
def fit(self, X, y=None, **kwargs):
"""
Fits the model and generates the silhouette visualization.
"""
# TODO: decide to use this method or the score method to draw.
# NOTE: Probably this would be better in score, but the standard score
# is a little different and I'm not sure how it's used.
# Fit the wrapped estimator
self.estimator.fit(X, y, **kwargs)
# Get the properties of the dataset
self.n_samples_ = X.shape[0]
self.n_clusters_ = self.estimator.n_clusters
# Compute the scores of the cluster
labels = self.estimator.predict(X)
self.silhouette_score_ = silhouette_score(X, labels)
self.silhouette_samples_ = silhouette_samples(X, labels)
# Draw the silhouette figure
self.draw(labels)
# Return the estimator
return self
def draw(self, labels):
"""
Draw the silhouettes for each sample and the average score.
Parameters
----------
labels : array-like
An array with the cluster label for each silhouette sample,
usually computed with ``predict()``. Labels are not stored on the
visualizer so that the figure can be redrawn with new data.
"""
# Track the positions of the lines being drawn
y_lower = 10 # The bottom of the silhouette
# Get the colors from the various properties
# TODO: Use resolve_colors instead of this
colors = color_palette(self.colormap, self.n_clusters_)
# For each cluster, plot the silhouette scores
for idx in range(self.n_clusters_):
# Collect silhouette scores for samples in the current cluster .
values = self.silhouette_samples_[labels == idx]
values.sort()
# Compute the size of the cluster and find upper limit
size = values.shape[0]
y_upper = y_lower + size
color = colors[idx]
self.ax.fill_betweenx(
np.arange(y_lower, y_upper), 0, values,
facecolor=color, edgecolor=color, alpha=0.5
)
# Label the silhouette plots with their cluster numbers
self.ax.text(-0.05, y_lower + 0.5 * size, str(idx))
# Compute the new y_lower for next plot
y_lower = y_upper + 10
# The vertical line for average silhouette score of all the values
self.ax.axvline(
x=self.silhouette_score_, color="red", linestyle="--"
)
return self.ax
def finalize(self):
"""
Prepare the figure for rendering by setting the title and adjusting
the limits on the axes, adding labels and a legend.
"""
# Set the title
self.set_title((
"Silhouette Plot of {} Clustering for {} Samples in {} Centers"
).format(
self.name, self.n_samples_, self.n_clusters_
))
# Set the X and Y limits
# The silhouette coefficient can range from -1, 1;
# but here we scale the plot according to our visualizations
# l_xlim and u_xlim are lower and upper limits of the x-axis,
# set according to our calculated maximum and minimum silhouette score along with necessary padding
l_xlim = max(-1, min(-0.1, round(min(self.silhouette_samples_) - 0.1, 1)))
u_xlim = min(1, round(max(self.silhouette_samples_) + 0.1, 1))
self.ax.set_xlim([l_xlim, u_xlim])
# The (n_clusters_+1)*10 is for inserting blank space between
# silhouette plots of individual clusters, to demarcate them clearly.
self.ax.set_ylim([0, self.n_samples_ + (self.n_clusters_ + 1) * 10])
# Set the x and y labels
self.ax.set_xlabel("silhouette coefficient values")
self.ax.set_ylabel("cluster label")
# Set the ticks on the axis object.
self.ax.set_yticks([]) # Clear the yaxis labels / ticks
self.ax.xaxis.set_major_locator(plt.MultipleLocator(0.1)) # Set the ticks at multiples of 0.1
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# Instantiate the clustering model and visualizer
model = KMeans(6)
visualizer = SilhouetteVisualizer(model)
visualizer.fit(X) # Fit the data to the visualizer
visualizer.show() # Draw/show/show the data
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