Lecture 26 – Data 100, Spring 2024¶

Data 100, Spring 2024

Acknowledgments Page

In [1]:
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import random
from sklearn import datasets

import warnings
warnings.filterwarnings('ignore')
In [2]:
iris, _ = datasets.load_iris(return_X_y=True, as_frame=True)
iris.rename(columns={"sepal length (cm)": "sepal_length", "sepal width (cm)": "sepal_width", 
                     "petal length (cm)": "petal_length", "petal width (cm)": "petal_width"}, inplace=True)
iris.sample(10)
Out[2]:
sepal_length sepal_width petal_length petal_width
60 5.0 2.0 3.5 1.0
128 6.4 2.8 5.6 2.1
44 5.1 3.8 1.9 0.4
19 5.1 3.8 1.5 0.3
137 6.4 3.1 5.5 1.8
125 7.2 3.2 6.0 1.8
117 7.7 3.8 6.7 2.2
96 5.7 2.9 4.2 1.3
6 4.6 3.4 1.4 0.3
58 6.6 2.9 4.6 1.3

K-Means Clustering¶

In today's lecture, we will use the data from the iris dataset from sklearn to perform clustering using two features, petal_length and petal_width. Summary of the algorithm:

  • Repeat until convergence:
    • Color points according to the closest center.
    • Move the center for each color to the center of points with that color.
In [3]:
sns.scatterplot(data=iris, x="petal_length", y="petal_width", color="black")
plt.xlabel('x')
plt.ylabel('y');
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In [4]:
class Center():
    def __init__(self, data):
        """generates a random center inside the region bounded by the data"""        
        num_dimensions = data.shape[1]
        self.coordinates = np.array([0.0] * num_dimensions)
        for i in range(num_dimensions):
            min_value = np.min(data[:, i])
            max_value = np.max(data[:, i])            
            random_value = random.uniform(min_value, max_value)            
            self.coordinates[i] = random_value
    
    def __str__(self):
        return str(self.coordinates)

    def __repr__(self):
        return repr(self.coordinates)

    def dist(self, data_point):
        return np.sqrt(np.sum((self.coordinates - data_point)**2, axis = 1))
    
    def dist_sq(self, data_point):
        return np.sum((self.coordinates - data_point)**2, axis = 1)
In [5]:
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
In [6]:
# Force coordinates from the lecture demo
c1.coordinates = np.array([2.52364007, 2.31040024])
c2.coordinates = np.array([6.53276402, 1.211463])
In [7]:
def plot_centers_and_black_data(iris, centers):
    for center in centers:
        plt.plot(center.coordinates[0], center.coordinates[1], '*', markersize=10)    
    sns.scatterplot(data=iris, x="petal_length", y="petal_width", color="black")
    plt.xlabel('petal_length')
    plt.ylabel('petal_width')
    legend_text = ['c' + str(i) for i in range(1, len(centers) + 1)]
    legend_text.append('data')
    plt.legend(legend_text)
In [8]:
plot_centers_and_black_data(iris, (c1, c2))
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In [9]:
def get_cluster_number(dists):
    return np.where(dists == np.min(dists))[0][0]
In [10]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
iris.head(10)
Out[10]:
sepal_length sepal_width petal_length petal_width dist1 dist2 cluster
0 5.1 3.5 1.4 0.2 2.390890 5.231474 0
1 4.9 3.0 1.4 0.2 2.390890 5.231474 0
2 4.7 3.2 1.3 0.2 2.439484 5.329623 0
3 4.6 3.1 1.5 0.2 2.345555 5.133398 0
4 5.0 3.6 1.4 0.2 2.390890 5.231474 0
5 5.4 3.9 1.7 0.4 2.080387 4.900416 0
6 4.6 3.4 1.4 0.3 2.303101 5.213064 0
7 5.0 3.4 1.5 0.2 2.345555 5.133398 0
8 4.4 2.9 1.4 0.2 2.390890 5.231474 0
9 4.9 3.1 1.5 0.1 2.435920 5.154034 0
In [11]:
iris["cluster"].value_counts()
Out[11]:
cluster
0    79
1    71
Name: count, dtype: int64
In [12]:
def plot_centers_and_colorized_data(iris, centers):
    plt.figure()
    for center in centers:
        plt.plot(center.coordinates[0], center.coordinates[1], 
                 marker='*', markersize=10, linestyle="None")  
    current_palette = sns.color_palette()[0:len(centers)]
    sns.scatterplot(data=iris, x="petal_length", y="petal_width", hue="cluster", palette=current_palette)  
    plt.xlabel('petal_length')
    plt.ylabel('petal_width')
    legend_text = ['c' + str(i) for i in range(1, len(centers) + 1)]
    legend_text.append('data')
    plt.legend(legend_text)
In [13]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [14]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [15]:
plot_centers_and_black_data(iris, (c1, c2))
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In [16]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [17]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [18]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [19]:
plot_centers_and_black_data(iris, (c1, c2))
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In [20]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [21]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [22]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [23]:
plot_centers_and_black_data(iris, (c1, c2))
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In [24]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [25]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [26]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [27]:
plot_centers_and_black_data(iris, (c1, c2))
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In [28]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [29]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [30]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [31]:
plot_centers_and_black_data(iris, (c1, c2))
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In [32]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [33]:
plot_centers_and_colorized_data(iris, (c1, c2))
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Example for K > 2¶

In [34]:
import copy
def compute_centers_after_N_iterations(data, column_names, centers, N):
    centers = copy.deepcopy(centers)
    
    for i in range(N):
        # Recompute clusters        
        dist_names = []
        for center_num in range(len(centers)):        
            data["dist" + str(center_num)] = centers[center_num].dist(data[column_names])
            dist_names.append("dist" + str(center_num))
        
        data["cluster"] = data[dist_names].apply(get_cluster_number, axis = 1)    
        
        # Update centers
        for center_num in range(len(centers)):
            for col_num in range(len(column_names)):
                col_name = column_names[col_num]
    
                centers[center_num].coordinates[col_num] = np.mean(data[data["cluster"] == center_num][col_name])

    return centers
In [35]:
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c1.coordinates = np.array([2.52364007, 2.31040024])
c2.coordinates = np.array([6.53276402, 1.211463])
In [36]:
iris
Out[36]:
sepal_length sepal_width petal_length petal_width dist1 dist2 cluster
0 5.1 3.5 1.4 0.2 0.111489 3.824028 0
1 4.9 3.0 1.4 0.2 0.111489 3.824028 0
2 4.7 3.2 1.3 0.2 0.202142 3.916407 0
3 4.6 3.1 1.5 0.2 0.063233 3.732042 0
4 5.0 3.6 1.4 0.2 0.111489 3.824028 0
... ... ... ... ... ... ... ...
145 6.7 3.0 5.2 2.3 4.230663 0.676487 1
146 6.3 2.5 5.0 1.9 3.871120 0.230631 1
147 6.5 3.0 5.2 2.0 4.094650 0.420388 1
148 6.2 3.4 5.4 2.3 4.407000 0.779445 1
149 5.9 3.0 5.1 1.8 3.921694 0.210959 1

150 rows × 7 columns

In [37]:
def inertia(data, centers):
    total_inertia = 0
    for center_num in range(len(centers)):
        data_in_this_cluster = data[data["cluster"] == center_num]        
        total_inertia += np.sum(centers[center_num].dist(data_in_this_cluster[["petal_length", "petal_width"]]))
    return total_inertia
In [38]:
def distortion(data, centers):
    total_distortion = 0
    for center_num in range(len(centers)):
        data_in_this_cluster = data[data["cluster"] == center_num]        
        total_distortion += np.sum(centers[center_num].dist(data_in_this_cluster[["petal_length", "petal_width"]]))/len(data_in_this_cluster)
    return total_distortion
In [39]:
random.seed(25)
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c3 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c4 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
new_centers = compute_centers_after_N_iterations(iris, ['petal_length', 'petal_width'], [c1, c2, c3, c4], 12)
print(f"inertia: {inertia(iris, new_centers)}, distortion: {distortion(iris, new_centers)})")
plot_centers_and_colorized_data(iris, new_centers)

inertia: 44.88363871576328, distortion: 1.2530251953116613)
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In [40]:
random.seed(29)
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c3 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c4 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
new_centers = compute_centers_after_N_iterations(iris, ['petal_length', 'petal_width'], [c1, c2, c3, c4], 12)
print(f"inertia: {inertia(iris, new_centers)}, distortion: {distortion(iris, new_centers)})")
plot_centers_and_colorized_data(iris, new_centers)

inertia: 45.87509130916156, distortion: 1.3068391699161572)
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In [41]:
random.seed(40)
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c3 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c4 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
new_centers = compute_centers_after_N_iterations(iris, ['petal_length', 'petal_width'], [c1, c2, c3, c4], 12)
print(f"inertia: {inertia(iris, new_centers)}, distortion: {distortion(iris, new_centers)})")
plot_centers_and_colorized_data(iris, new_centers)

inertia: 54.272527867765156, distortion: 1.4992328098338596)
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In [42]:
random.seed(75)
c1 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c3 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
c4 = Center(iris.loc[:, ['petal_length', 'petal_width']].values)
new_centers = compute_centers_after_N_iterations(iris, ['petal_length', 'petal_width'], [c1, c2, c3, c4], 12)
print(f"inertia: {inertia(iris, new_centers)}, distortion: {distortion(iris, new_centers)})")
plot_centers_and_colorized_data(iris, new_centers)

inertia: 44.88363871576328, distortion: 1.2530251953116613)
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In [43]:
random.seed(20)
np.random.seed(20)
iris_small = iris.sample(7)
c1 = Center(iris_small.loc[:, ['petal_length', 'petal_width']].values)
c2 = Center(iris_small.loc[:, ['petal_length', 'petal_width']].values)
new_centers2 = compute_centers_after_N_iterations(iris_small, ['petal_length', 'petal_width'], [c1, c2], 12)
plot_centers_and_colorized_data(iris_small, new_centers2)
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In [44]:
def print_distances_squared(data, centers):
    for center_num in range(len(centers)):
        data_in_this_cluster = data[data["cluster"] == center_num]        
        print(centers[center_num].dist(data_in_this_cluster[["petal_length", "petal_width"]])**2)

print_distances_squared(iris_small, new_centers2)

73     0.837778
129    0.121111
143    0.547778
dtype: float64
47    4.765
74    0.845
67    0.425
89    0.425
dtype: float64
In [45]:
inertia(iris_small, new_centers2)
Out[45]:
6.409399633729917
In [46]:
distortion(iris_small, new_centers2)
Out[46]:
1.7693026035868602
In [47]:
iris_small
Out[47]:
sepal_length sepal_width petal_length petal_width dist1 dist2 cluster dist0 dist3
47 4.6 3.2 1.4 0.2 2.182888 4.783274 1 4.334487 3.637045
73 6.1 2.8 4.7 1.2 1.274755 1.415877 0 0.915302 0.401564
74 6.4 2.9 4.3 1.3 0.919239 1.691724 1 1.233333 0.545981
129 7.2 3.0 5.8 1.6 2.438237 0.508523 0 0.348010 1.042109
67 5.8 2.7 4.1 1.0 0.651920 2.017764 1 1.535506 0.888636
89 5.5 2.5 4.0 1.3 0.651920 1.960509 1 1.520234 0.814145
143 6.8 3.2 5.9 2.3 2.797320 0.222950 0 0.740120 1.340931

Example of Inertia Failing to Match Intuition¶

In [48]:
c1.coordinates = [1.2, 0.15]
c2.coordinates = [4.906000000000001, 1.6760000000000006]
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [49]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [50]:
print(f"inertia: {inertia(iris, [c1, c2])}, distortion: {distortion(iris, [c1, c2])})")

inertia: 94.3164648130483, distortion: 1.0959547804008838)
In [51]:
average_c1_length = np.mean(iris[iris["cluster"] == 0]["petal_length"])
average_c1_width = np.mean(iris[iris["cluster"] == 0]["petal_width"])
c1.coordinates = (average_c1_length, average_c1_width)

average_c2_length = np.mean(iris[iris["cluster"] == 1]["petal_length"])
average_c2_width = np.mean(iris[iris["cluster"] == 1]["petal_width"])
c2.coordinates = (average_c2_length, average_c2_width)
In [52]:
plot_centers_and_black_data(iris, (c1, c2))
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In [53]:
iris["dist1"] = c1.dist(iris[["petal_length", "petal_width"]])
iris["dist2"] = c2.dist(iris[["petal_length", "petal_width"]])
iris["cluster"] = iris[["dist1", "dist2"]].apply(get_cluster_number, axis = 1)
In [54]:
plot_centers_and_colorized_data(iris, (c1, c2))
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In [55]:
print(f"inertia: {inertia(iris, [c1, c2])}, distortion: {distortion(iris, [c1, c2])})")

inertia: 87.2103463131798, distortion: 0.9775403068856574)

Hierarchical Agglomerative Clustering¶

In [56]:
np.random.seed(42)
iris_small = iris.sample(13).loc[:, 'sepal_length':'petal_width'].reset_index(drop=True)
iris_small = iris_small.drop(8).reset_index(drop=True)
In [57]:
sns.scatterplot(data=iris_small, x="petal_length", y="petal_width", color="black");
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In [58]:
iris_small["cluster"] = np.array(range(0, len(iris_small)))
In [59]:
iris_small
Out[59]:
sepal_length sepal_width petal_length petal_width cluster
0 6.1 2.8 4.7 1.2 0
1 5.7 3.8 1.7 0.3 1
2 7.7 2.6 6.9 2.3 2
3 6.0 2.9 4.5 1.5 3
4 6.8 2.8 4.8 1.4 4
5 5.4 3.4 1.5 0.4 5
6 5.6 2.9 3.6 1.3 6
7 6.9 3.1 5.1 2.3 7
8 5.8 2.7 3.9 1.2 8
9 6.5 3.2 5.1 2.0 9
10 4.8 3.0 1.4 0.1 10
11 5.5 3.5 1.3 0.2 11
In [60]:
def plot_clusters(data):
    fig = plt.figure()
    p1 = sns.scatterplot(data=data, x="petal_length", y="petal_width")
    plt.axis('equal')
    for line in range(0,data.shape[0]):
         p1.text(data["petal_length"][line]+0.05, data["petal_width"][line] - 0.03,   
         data["cluster"][line], horizontalalignment='left', 
         size='medium', color='black', weight='semibold')
    return fig
            
fig = plot_clusters(iris_small)
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In [61]:
from scipy.spatial import distance

def dist_between_clusters(data, cnum1, cnum2):
    cluster1 = data[data["cluster"] == cnum1]
    cluster2 = data[data["cluster"] == cnum2]    
    return distance.cdist(cluster1[["petal_length", "petal_width"]], cluster2[["petal_length", "petal_width"]]).max()
In [62]:
def closest_clusters(data):
    cluster_values = data["cluster"].unique()
    smallest_distance = float("inf")
    best_pair = [-1, -1]
    for cnum1 in cluster_values:
        for cnum2 in cluster_values:
            if cnum1 == cnum2:
                continue
            cur_dist = dist_between_clusters(data, cnum1, cnum2)
            if cur_dist < smallest_distance:
                best_pair = [cnum1, cnum2]
                smallest_distance = cur_dist
                
    return best_pair
In [63]:
def merge_clusters(data, cnum1, cnum2):
    data.loc[data["cluster"] == cnum2, "cluster"] = cnum1
In [64]:
i = 0
while len(iris_small["cluster"].unique()) != 2:
    i += 1
    cnum1, cnum2 = closest_clusters(iris_small)
    merge_clusters(iris_small, cnum1, cnum2)
    plot_clusters(iris_small)
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Run on full dataset¶

In [65]:
iris_small = iris.copy()
iris_small["cluster"] = np.array(range(0, len(iris_small)))
fig = plot_clusters(iris_small)
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In [66]:
from sklearn.cluster import AgglomerativeClustering
clustering = AgglomerativeClustering().fit(iris[["petal_length", "petal_width"]])
In [67]:
clustering.labels_
Out[67]:
array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
In [68]:
iris["cluster"] = clustering.labels_
In [69]:
sns.scatterplot(data=iris, x="petal_length", y="petal_width", 
                hue ="cluster", legend=None);
plt.axis('equal');
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In [70]:
fig = plot_clusters(iris)
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In [71]:
# From https://github.com/scikit-learn/scikit-learn/blob/70cf4a676caa2d2dad2e3f6e4478d64bcb0506f7/examples/cluster/plot_hierarchical_clustering_dendrogram.py
from scipy.cluster.hierarchy import dendrogram

def plot_dendrogram(model, **kwargs):
    # Children of hierarchical clustering
    children = model.children_

    # Distances between each pair of children
    # Since we don't have this information, we can use a uniform one for plotting
    distance = np.arange(children.shape[0])

    # The number of observations contained in each cluster level
    no_of_observations = np.arange(2, children.shape[0]+2)

    # Create linkage matrix and then plot the dendrogram
    linkage_matrix = np.column_stack([children, distance, no_of_observations]).astype(float)

    # Plot the corresponding dendrogram
    dendrogram(linkage_matrix, **kwargs)



plt.title('Hierarchical Clustering Dendrogram')
plot_dendrogram(clustering, labels=clustering.labels_)
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Elbow Method¶

In [72]:
from sklearn.cluster import KMeans 
distortions = [] 
inertias = [] 
mapping1 = {} 
mapping2 = {} 
K = range(1,10) 
X = iris[["petal_length", "petal_width"]]
    
for k in K: 
    # Building and fitting the model 
    kmeanModel = KMeans(n_clusters=k).fit(X)     
      
    distortions.append(sum(np.min(distance.cdist(X, kmeanModel.cluster_centers_, 
                      'euclidean'),axis=1)) / X.shape[0]) 
    inertias.append(kmeanModel.inertia_) 
  
    mapping1[k] = sum(np.min(distance.cdist(X, kmeanModel.cluster_centers_, 
                 'euclidean'),axis=1)) / X.shape[0] 
    mapping2[k] = kmeanModel.inertia_
In [73]:
plt.plot(K, inertias, 'bx-') 
plt.xlabel('Values of K') 
plt.ylabel('Inertia') 
plt.title('The Elbow Method using Inertia');
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Silhouette Scores¶

In [74]:
X.loc[(X["petal_length"] < 3.2) & (X["petal_length"] > 2)]
Out[74]:
petal_length petal_width
98 3.0 1.1
In [75]:
from sklearn.metrics import silhouette_samples, silhouette_score
import matplotlib.cm as cm

fig, ax1 = plt.subplots(1, 1)

n_clusters = 2

# The 1st subplot is the silhouette plot.
# The silhouette coefficient can range from -1, 1 but in this example all
# lie within [-0.1, 1]
ax1.set_xlim([-0.1, 1])
# The (n_clusters+1)*10 is for inserting blank space between silhouette
# plots of individual clusters, to demarcate them clearly.
ax1.set_ylim([0, len(X) + (n_clusters + 1) * 10])

cluster_labels = clustering.fit_predict(X)


# The silhouette_score gives the average value for all the samples.
# This gives a perspective into the density and separation of the formed clusters
silhouette_avg = silhouette_score(X, cluster_labels)
print("For n_clusters =", n_clusters,
      "The average silhouette_score is :", silhouette_avg)

# Compute the silhouette scores for each sample
sample_silhouette_values = silhouette_samples(X, cluster_labels)

y_lower = 10
for i in range(n_clusters):
    # Aggregate the silhouette scores for samples belonging to
    # cluster i, and sort them
    ith_cluster_silhouette_values = \
        sample_silhouette_values[cluster_labels == i]

    ith_cluster_silhouette_values.sort()

    size_cluster_i = ith_cluster_silhouette_values.shape[0]
    y_upper = y_lower + size_cluster_i

    color = cm.nipy_spectral(float(i) / n_clusters)
    ax1.fill_betweenx(np.arange(y_lower, y_upper),
                      0, ith_cluster_silhouette_values,
                      facecolor=color, edgecolor=color, alpha=0.7)

    # Label the silhouette plots with their cluster numbers at the middle
    ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))

    # Compute the new y_lower for next plot
    y_lower = y_upper + 10  # 10 for the 0 samples

ax1.set_title("The silhouette plot for the various clusters.")
ax1.set_xlabel("The silhouette coefficient values")
ax1.set_ylabel("Cluster label")

# The vertical line for average silhouette score of all the values
ax1.axvline(x=silhouette_avg, color="red", linestyle="--");

For n_clusters = 2 The average silhouette_score is : 0.7669465622893307
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In [76]:
min(sample_silhouette_values)
Out[76]:
-0.13132990547983361
In [77]:
from sklearn.cluster import AgglomerativeClustering
clustering = AgglomerativeClustering(n_clusters = 3).fit(iris[["petal_length", "petal_width"]])


fig, ax1 = plt.subplots(1, 1)

n_clusters = 3

# The 1st subplot is the silhouette plot
# The silhouette coefficient can range from -1, 1 but in this example all
# lie within [-0.1, 1]
ax1.set_xlim([-0.1, 1])
# The (n_clusters+1)*10 is for inserting blank space between silhouette
# plots of individual clusters, to demarcate them clearly.
ax1.set_ylim([0, len(X) + (n_clusters + 1) * 10])

cluster_labels = clustering.fit_predict(X)

# The silhouette_score gives the average value for all the samples.
# This gives a perspective into the density and separation of the formed clusters
silhouette_avg = silhouette_score(X, cluster_labels)
print("For n_clusters =", n_clusters,
      "The average silhouette_score is :", silhouette_avg)

# Compute the silhouette scores for each sample
sample_silhouette_values = silhouette_samples(X, cluster_labels)

y_lower = 10
for i in range(n_clusters):
    # Aggregate the silhouette scores for samples belonging to
    # cluster i, and sort them
    ith_cluster_silhouette_values = \
        sample_silhouette_values[cluster_labels == i]

    ith_cluster_silhouette_values.sort()

    size_cluster_i = ith_cluster_silhouette_values.shape[0]
    y_upper = y_lower + size_cluster_i

    color = cm.nipy_spectral(float(i) / n_clusters)
    ax1.fill_betweenx(np.arange(y_lower, y_upper),
                      0, ith_cluster_silhouette_values,
                      facecolor=color, edgecolor=color, alpha=0.7)

    # Label the silhouette plots with their cluster numbers at the middle
    ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))

    # Compute the new y_lower for next plot
    y_lower = y_upper + 10  # 10 for the 0 samples

ax1.set_title("The silhouette plot for the various clusters.")
ax1.set_xlabel("The silhouette coefficient values")
ax1.set_ylabel("Cluster label")

# The vertical line for the average silhouette score of all the values
ax1.axvline(x=silhouette_avg, color="red", linestyle="--");

For n_clusters = 3 The average silhouette_score is : 0.6573949270307473
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In [78]:
min(sample_silhouette_values)
Out[78]:
-0.14062817006789816
In [79]:
iris["cluster"] = cluster_labels
current_palette = sns.color_palette()[0:3]
sns.scatterplot(data = iris, x = "petal_length", y= "petal_width", hue="cluster", palette = current_palette);
plt.axis('equal');
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