This mini-project is based on this blog post by yhat. Please feel free to refer to the post for additional information, and solutions.
In [1]:
%matplotlib inline
import pandas as pd
import sklearn
import matplotlib.pyplot as plt
import seaborn as sns
# Setup Seaborn
sns.set_style("whitegrid")
sns.set_context("poster")
The dataset contains information on marketing newsletters/e-mail campaigns (e-mail offers sent to customers) and transaction level data from customers. The transactional data shows which offer customers responded to, and what the customer ended up buying. The data is presented as an Excel workbook containing two worksheets. Each worksheet contains a different dataset.
In [2]:
df_offers = pd.read_excel("./WineKMC.xlsx", sheetname=0)
df_offers.columns = ["offer_id", "campaign", "varietal", "min_qty", "discount", "origin", "past_peak"]
df_offers.head()
Out[2]:
We see that the first dataset contains information about each offer such as the month it is in effect and several attributes about the wine that the offer refers to: the variety, minimum quantity, discount, country of origin and whether or not it is past peak. The second dataset in the second worksheet contains transactional data -- which offer each customer responded to.
In [3]:
df_transactions = pd.read_excel("./WineKMC.xlsx", sheetname=1)
df_transactions.columns = ["customer_name", "offer_id"]
df_transactions['n'] = 1
df_transactions.head()
Out[3]:
We're trying to learn more about how our customers behave, so we can use their behavior (whether or not they purchased something based on an offer) as a way to group similar minded customers together. We can then study those groups to look for patterns and trends which can help us formulate future offers.
The first thing we need is a way to compare customers. To do this, we're going to create a matrix that contains each customer and a 0/1 indicator for whether or not they responded to a given offer.
Exercise: Create a data frame where each row has the following columns (Use the pandas [`merge`](http://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.merge.html) and [`pivot_table`](http://pandas.pydata.org/pandas-docs/stable/generated/pandas.pivot_table.html) functions for this purpose):
Make sure you also deal with any weird values such as `NaN`. Read the documentation to develop your solution.
In [4]:
# your turn
# Create a data frame where each row has a customer_name column and one column
# for each offer, with a 1 if the customer responded to the offer.
# Use pandas merge and pivot table functions for this purpose.
df_responses = pd.pivot_table(data=df_offers.merge(df_transactions, how='left', on='offer_id'),
values='n', index='customer_name', columns='offer_id', fill_value=0, dropna=False)
Recall that in K-Means Clustering we want to maximize the distance between centroids and minimize the distance between data points and the respective centroid for the cluster they are in. True evaluation for unsupervised learning would require labeled data; however, we can use a variety of intuitive metrics to try to pick the number of clusters K. We will introduce two methods: the Elbow method, the Silhouette method and the gap statistic.
Exercise:
In [5]:
# your turn
# Create numpy matrix with only the columns representing the offers.
from sklearn.cluster import KMeans
import numpy as np
x_cols = np.matrix(df_responses)
# Set-up a placeholder dataframe for SS scores.
# The scores will be populated by NaN for now.
SS_scores = pd.DataFrame(columns=['K', 'score'])
SS_scores['K'] = range(2, 11)
# Apply KMeans clustering method to above matrix.
# The exercise suggests testing for 2 <= K <= 10.
for K in range(2, 11):
cluster = KMeans(n_clusters=K, random_state=0).fit(x_cols)
SS_scores['score'][SS_scores.K == K] = cluster.inertia_
# Plot SS for each K.
plt.scatter(data=SS_scores, x='K', y='score')
# Based on this SSE plot, there is no clear elbow wherein the
# the SSE decreases rapidly. The score is still dropping
# from K =8 to K =10. So I guess at this point, I'm choosing
# K = 10 as the best K.
Out[5]:
In [6]:
# Make a bar chart showing the number of points in each cluster for k-means
# under the best K, which I chose to be 10.
cluster = KMeans(n_clusters=10, random_state=0).fit(x_cols)
plt.hist(x=cluster.labels_, bins=10)
# What challenges did you experience using the Elbow method to pick K
# Answer: The elbow method didn't really work since I couldn't find
# that elbow inflection point.
Out[6]:
There exists another method that measures how well each datapoint $x_i$ "fits" its assigned cluster and also how poorly it fits into other clusters. This is a different way of looking at the same objective. Denote $a_{x_i}$ as the average distance from $x_i$ to all other points within its own cluster $k$. The lower the value, the better. On the other hand $b_{x_i}$ is the minimum average distance from $x_i$ to points in a different cluster, minimized over clusters. That is, compute separately for each cluster the average distance from $x_i$ to the points within that cluster, and then take the minimum. The silhouette $s(x_i)$ is defined as
$$s(x_i) = \frac{b_{x_i} - a_{x_i}}{\max{\left( a_{x_i}, b_{x_i}\right)}}$$The silhouette score is computed on every datapoint in every cluster. The silhouette score ranges from -1 (a poor clustering) to +1 (a very dense clustering) with 0 denoting the situation where clusters overlap. Some criteria for the silhouette coefficient is provided in the table below.
Fortunately, scikit-learn provides a function to compute this for us (phew!) called sklearn.metrics.silhouette_score. Take a look at this article on picking $K$ in scikit-learn, as it will help you in the next exercise set.
Exercise: Using the documentation for the `silhouette_score` function above, construct a series of silhouette plots like the ones in the article linked above.
Exercise: Compute the average silhouette score for each $K$ and plot it. What $K$ does the plot suggest we should choose? Does it differ from what we found using the Elbow method?
In [7]:
# Your turn.
# Using the documentation for the silhoutte_score function above,
# construct a series of silhouetter plots like the ones in the
# article linked above. By doing silhouette analysis, we can
# study the separation distance between the resulting clusters.
from __future__ import print_function
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_samples, silhouette_score
import matplotlib.cm as cm
import numpy as np
print(__doc__)
# Generating the sample data from make_blobs
# This particular setting has one distinct cluster and 3 clusters placed close
# together.
X, y = make_blobs(n_samples=500,
n_features=2,
centers=4,
cluster_std=1,
center_box=(-10.0, 10.0),
shuffle=True,
random_state=1) # For reproducibility
range_n_clusters = range(2, 10)
for n_clusters in range_n_clusters:
# Create a subplot with 1 row and 2 columns.
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.set_size_inches(18, 7)
# The 1st subplot is the silhouette plot.
# The silhouette coefficient can range from -1, 1 but in this example all
# lie within [-0.2, 0.5]
ax1.set_xlim([-0.1, 1.0])
# 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])
# Initialize the clusterer with n_clusters value and a random generator
# seed of 10 for reproducibility.
clusterer = KMeans(n_clusters=n_clusters, random_state=10)
cluster_labels = clusterer.fit_predict(x_cols)
# 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_cols, 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_cols, 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.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="--")
ax1.set_yticks([]) # Clear the yaxis labels / ticks
ax1.set_xticks([])
# 2nd Plot show the actual clusters formed
colors = cm.spectral(cluster_labels.astype(float) / n_clusters)
ax2.scatter(X[:, 0], X[:, 1], marker='.', s=30, lw=0, alpha=0.7,
c=colors)
# Labeling the clusters
centers = clusterer.cluster_centers_
# Draw white circles at cluster centers
ax2.scatter(centers[:, 0], centers[:, 1],
marker='o', c="white", alpha=1, s=200)
for i, c in enumerate(centers):
ax2.scatter(c[0], c[1], marker='$%d$' % i, alpha=1, s=50)
ax2.set_title("The visualization of the clustered data.")
ax2.set_xlabel("Feature space for the 1st feature")
ax2.set_ylabel("Feature space for the 2nd feature")
plt.suptitle(("Silhouette analysis for KMeans clustering on sample data "
"with n_clusters = %d" % n_clusters),
fontsize=14, fontweight='bold')
plt.show()