Gower clustering

Exploring the gower distance metric

A distance metric used to find clusters in ordinal data

\(G S_{i j}=\frac{1}{m} \sum_{f=1}^{m} p s_{i j}^{(f)}\)

Similarity between observations i and j. Having each observation m different features, either numerical, categorical or mixed.

\(p s_{i j}^{(f)}=1-\frac{\left|x_{i f}-x_{j f}\right|}{R_{f}}\)

Similarity between observation i and j in feature f when f is numerical. For a categorical feature, the partial similarity between two individuals is one only when both observations have exactly the same value for this feature. Zero otherwise.

\(R_{f}=\max f-\min f\)

Range of a feature f.

[1]:

import pandas as pd
from pandas.api.types import CategoricalDtype
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.cluster import DBSCAN
from sklearn.metrics import roc_auc_score, roc_curve, confusion_matrix
from sklearn.datasets import make_classification
import gower
from scipy.cluster.hierarchy import linkage, fcluster, dendrogram

[2]:

from extras import plot_confusion_matrix, tSNE

Multiclass example 1.

[3]:

# Creating a dictionary with the data
df = pd.DataFrame({"age": [22, 25, 30, 38, 42, 47, 55, 62, 61, 90],
              "gender": ["M", "M", "F", "F", "F", "M", "M", "M", "M", "M"],
              "civil_status": ["SINGLE", "SINGLE", "SINGLE", "MARRIED", "MARRIED", "SINGLE", "MARRIED", "DIVORCED", "MARRIED", "DIVORCED"],
              "salary": [18000, 23000, 27000, 32000, 34000, 20000, 40000, 42000, 25000, 70000],
              "has_children": [False, False, False, True, True, False, False, False, False, True],
              "purchaser_type": ["LOW_PURCHASER", "LOW_PURCHASER", "LOW_PURCHASER", "HEAVY_PURCHASER", "HEAVY_PURCHASER", "LOW_PURCHASER", "MEDIUM_PURCHASER", "MEDIUM_PURCHASER", "MEDIUM_PURCHASER", "LOW_PURCHASER"]})

[4]:

df

[4]:
age gender civil_status salary has_children purchaser_type
0 22 M SINGLE 18000 False LOW_PURCHASER
1 25 M SINGLE 23000 False LOW_PURCHASER
2 30 F SINGLE 27000 False LOW_PURCHASER
3 38 F MARRIED 32000 True HEAVY_PURCHASER
4 42 F MARRIED 34000 True HEAVY_PURCHASER
5 47 M SINGLE 20000 False LOW_PURCHASER
6 55 M MARRIED 40000 False MEDIUM_PURCHASER
7 62 M DIVORCED 42000 False MEDIUM_PURCHASER
8 61 M MARRIED 25000 False MEDIUM_PURCHASER
9 90 M DIVORCED 70000 True LOW_PURCHASER 
[5]:

d_matrix = gower.gower_matrix(df.drop("purchaser_type", axis=1))

[6]:

customer_names = [f"c_{x}" for x in range(d_matrix.shape[0])]

[7]:

_ = plt.figure(figsize=(7,4))
ax = sns.heatmap(data=pd.DataFrame(d_matrix, index=customer_names, columns=customer_names),
               annot=True,
                fmt='.2g')
ax.xaxis.tick_top()
ax.xaxis.set_label_position('top')
_images/gower_clustering_15_0.png 
[8]:

# Configuring the parameters of the clustering algorithm
dbscan_cluster = DBSCAN(eps=0.3,
                        min_samples=2,
                        metric="precomputed")

[9]:

# Fitting the clustering algorithm
dbscan_cluster.fit(d_matrix)

[9]:

DBSCAN(eps=0.3, metric='precomputed', min_samples=2)

[10]:

# Adding the results to a new column in the dataframe
df["DBSCAN_cluster"] = dbscan_cluster.labels_

[11]:

l_matrix = linkage(d_matrix)

<ipython-input-11-df338ac2b59e>:1: ClusterWarning: scipy.cluster: The symmetric non-negative hollow observation matrix looks suspiciously like an uncondensed distance matrix
  l_matrix = linkage(d_matrix)

[12]:

cld = fcluster(l_matrix, 3, criterion='maxclust')
df["linkage_cluster"] = cld

[13]:

_ = plt.figure(figsize=(20, 4))
dn = dendrogram(l_matrix)
_images/gower_clustering_21_0.png 
[14]:

df

[14]:
age gender civil_status salary has_children purchaser_type DBSCAN_cluster linkage_cluster
0 22 M SINGLE 18000 False LOW_PURCHASER 0 2
1 25 M SINGLE 23000 False LOW_PURCHASER 0 2
2 30 F SINGLE 27000 False LOW_PURCHASER 0 2
3 38 F MARRIED 32000 True HEAVY_PURCHASER 1 1
4 42 F MARRIED 34000 True HEAVY_PURCHASER 1 1
5 47 M SINGLE 20000 False LOW_PURCHASER 0 2
6 55 M MARRIED 40000 False MEDIUM_PURCHASER 0 2
7 62 M DIVORCED 42000 False MEDIUM_PURCHASER 0 2
8 61 M MARRIED 25000 False MEDIUM_PURCHASER 0 2
9 90 M DIVORCED 70000 True LOW_PURCHASER -1 3 
[15]:

ordered_cat = CategoricalDtype(['HEAVY_PURCHASER', 'LOW_PURCHASER', 'MEDIUM_PURCHASER'], ordered=True)

[16]:

y_test = df["purchaser_type"].astype(ordered_cat).cat.codes + 1
y_pred = df["linkage_cluster"]

[17]:

cf_matrix = confusion_matrix(y_test, y_pred)

fig, ax = plot_confusion_matrix(cf=cf_matrix, title="Confusion Matrix clustering buying groups")
_images/gower_clustering_25_0.png

Multiclass example 2.

[18]:

# Creating a dictionary with the data
df = sns.load_dataset("iris")

[19]:

df.head()

[19]:
sepal_length sepal_width petal_length petal_width species
0 5.1 3.5 1.4 0.2 setosa
1 4.9 3.0 1.4 0.2 setosa
2 4.7 3.2 1.3 0.2 setosa
3 4.6 3.1 1.5 0.2 setosa
4 5.0 3.6 1.4 0.2 setosa 
[20]:

d_matrix = gower.gower_matrix(df.drop("species", axis=1))

[21]:

customer_names = [f"c_{x}" for x in range(d_matrix.shape[0])]

[22]:

_ = plt.figure(figsize=(7,5))
ax = sns.heatmap(data=pd.DataFrame(d_matrix),
               annot=False,
                fmt='.2g')
ax.xaxis.tick_top()
ax.xaxis.set_label_position('top')
_images/gower_clustering_31_0.png 
[23]:

# Configuring the parameters of the clustering algorithm
dbscan_cluster = DBSCAN(eps=0.3,
                        min_samples=10,
                        metric="precomputed")

[24]:

# Fitting the clustering algorithm
dbscan_cluster.fit(d_matrix)

[24]:

DBSCAN(eps=0.3, metric='precomputed', min_samples=10)

[25]:

# Adding the results to a new column in the dataframe
df["DBSCAN_cluster"] = dbscan_cluster.labels_

[26]:

l_matrix = linkage(d_matrix)

<ipython-input-26-df338ac2b59e>:1: ClusterWarning: scipy.cluster: The symmetric non-negative hollow observation matrix looks suspiciously like an uncondensed distance matrix
  l_matrix = linkage(d_matrix)

[27]:

cld = fcluster(l_matrix, 3, criterion='maxclust')
df["linkage_cluster"] = cld

[28]:

_ = plt.figure(figsize=(20, 4))
dn = dendrogram(l_matrix)
_images/gower_clustering_37_0.png 
[29]:

df.head()

[29]:
sepal_length sepal_width petal_length petal_width species DBSCAN_cluster linkage_cluster
0 5.1 3.5 1.4 0.2 setosa 0 1
1 4.9 3.0 1.4 0.2 setosa 0 1
2 4.7 3.2 1.3 0.2 setosa 0 1
3 4.6 3.1 1.5 0.2 setosa 0 1
4 5.0 3.6 1.4 0.2 setosa 0 1 
[30]:

df["linkage_cluster"].unique()

[30]:

array([1, 3, 2], dtype=int32)

[31]:

ordered_cat = CategoricalDtype(['setosa', 'virginica', 'versicolor',], ordered=True)

[32]:

y_test = df["species"].astype(ordered_cat).cat.codes + 1
y_pred = df["linkage_cluster"]

[33]:

y_test.unique()

[33]:

array([1, 3, 2], dtype=int8)

[34]:

y_pred.unique()

[34]:

array([1, 3, 2], dtype=int32)

[35]:

cf_matrix = confusion_matrix(y_test, y_pred)

fig, ax = plot_confusion_matrix(cf=cf_matrix, title="Confusion Matrix clustering flower types")
_images/gower_clustering_44_0.png 
[36]:

df["linkage_cluster"] = df["linkage_cluster"].astype(str)

[37]:

fig, ax = tSNE(data=df, n_components=2, hue='species', tag='linkage_cluster', figsize=(15, 5))
_images/gower_clustering_46_0.png 
[38]:

fig, ax = tSNE(data=df, n_components=2, hue='species', figsize=(7, 5))
_images/gower_clustering_47_0.png 
[39]:

fig, ax = tSNE(data=df
               .assign(species_int = lambda d: d["species"].astype("category").cat.codes)
               .drop("species", axis=1),
               n_components=3, hue='species_int')

/Users/jamestwose/Coding/Data-Science/Machine_learning/extras.py:302: MatplotlibDeprecationWarning: Adding an axes using the same arguments as a previous axes currently reuses the earlier instance.  In a future version, a new instance will always be created and returned.  Meanwhile, this warning can be suppressed, and the future behavior ensured, by passing a unique label to each axes instance.
  ax = fig.add_subplot(111, projection="3d")
_images/gower_clustering_48_1.png

Binary example

[40]:

X, y = make_classification(n_samples=100,
                             n_features=5,
                             n_informative=5,
                           n_redundant=0,
                            n_repeated=0,
                            n_classes=2,
                            n_clusters_per_class=2,
                             shuffle=True,
                             random_state=42)

[41]:

df = (pd.DataFrame(X, columns=[f"feat_{x}" for x in range(0, X.shape[1])]).round(0).astype(str)
.merge(pd.DataFrame(y, columns=["target"]),
      left_index=True,
      right_index=True))

[42]:

df.head()

[42]:
feat_0 feat_1 feat_2 feat_3 feat_4 target
0 -2.0 1.0 1.0 1.0 1.0 0
1 -1.0 -1.0 -1.0 1.0 2.0 1
2 -3.0 2.0 2.0 -1.0 1.0 0
3 -2.0 2.0 -3.0 -0.0 -1.0 1
4 -2.0 1.0 2.0 -1.0 1.0 0 
[43]:

d_matrix = gower.gower_matrix(df.drop("target", axis=1))

[44]:

_ = plt.figure(figsize=(7,4))
ax = sns.heatmap(data=pd.DataFrame(d_matrix),
               annot=False,
                fmt='.2g')
ax.xaxis.tick_top()
ax.xaxis.set_label_position('top')
_images/gower_clustering_54_0.png 
[45]:

# Configuring the parameters of the clustering algorithm
dbscan_cluster = DBSCAN(eps=0.3,
                        min_samples=2,
                        metric="precomputed")

[46]:

# Fitting the clustering algorithm
dbscan_cluster.fit(d_matrix)

[46]:

DBSCAN(eps=0.3, metric='precomputed', min_samples=2)

[47]:

# Adding the results to a new column in the dataframe
df["DBSCAN_cluster"] = dbscan_cluster.labels_

[48]:

l_matrix = linkage(d_matrix)

<ipython-input-48-df338ac2b59e>:1: ClusterWarning: scipy.cluster: The symmetric non-negative hollow observation matrix looks suspiciously like an uncondensed distance matrix
  l_matrix = linkage(d_matrix)

[49]:

cld = fcluster(l_matrix, 3, criterion='maxclust')
df["linkage_cluster"] = cld

[50]:

# _ = plt.figure(figsize=(20, 4))
# dn = dendrogram(l_matrix)

[51]:

df.head()

[51]:
feat_0 feat_1 feat_2 feat_3 feat_4 target DBSCAN_cluster linkage_cluster
0 -2.0 1.0 1.0 1.0 1.0 0 0 1
1 -1.0 -1.0 -1.0 1.0 2.0 1 -1 1
2 -3.0 2.0 2.0 -1.0 1.0 0 1 1
3 -2.0 2.0 -3.0 -0.0 -1.0 1 -1 1
4 -2.0 1.0 2.0 -1.0 1.0 0 1 1 
[52]:

y_test = df["target"]
y_pred = df["DBSCAN_cluster"] * -1

[53]:

# Compute False postive rate, and True positive rate
fpr, tpr, thresholds = roc_curve(y_test, y_pred)
# Calculate Area under the curve to display on the plot
auc_score = roc_auc_score(y_test, y_pred, average="macro")
# Now, plot the computed values
_ = plt.plot(fpr,
             tpr,
             label="ROC curve (area = %0.2f)" % auc_score,)
# Custom settings for the plot
_ = plt.plot([0, 1], [0, 1], c="grey", ls="--")
_ = plt.xlim([0.0, 1.0])
_ = plt.ylim([0.0, 1.05])
_ = plt.xlabel("1-Specificity (False Positive Rate)")
_ = plt.ylabel("Sensitivity (True Positive Rate)")
_ = plt.title("Receiver Operating Characteristics")
_ = plt.legend(loc="lower right")
_images/gower_clustering_63_0.png