Lecture 27 – Data 100, Fall 2023¶

Data 100, Fall 2023

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
1 4.9 3.0 1.4 0.2
32 5.2 4.1 1.5 0.1
42 4.4 3.2 1.3 0.2
37 4.9 3.6 1.4 0.1
127 6.1 3.0 4.9 1.8
106 4.9 2.5 4.5 1.7
39 5.1 3.4 1.5 0.2
52 6.9 3.1 4.9 1.5
143 6.8 3.2 5.9 2.3
7 5.0 3.4 1.5 0.2

K-Means Clustering¶

In this section, we will use the data from the iris dataset 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 center for each color to 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');
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))
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))
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))
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))
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))
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))
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))
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))
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))
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))
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))
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))

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)
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)
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)
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)
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)
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))
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))
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))
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");
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)
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)

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)
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');
In [70]:
fig = plot_clusters(iris)
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_)

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');

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 : 0.7669465622770762
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 : 0.6573949269287823
In [78]:
min(sample_silhouette_values)
Out[78]:
-0.14062817006789843
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');