Customer Segmentation using Clustering

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This mini-project is based on this blog post by yhat. Please feel free to refer to the post for additional information, and solutions.

Data

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.

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.

Data wrangling

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.

K-Means Clustering

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.

Choosing K: The Elbow Sum-of-Squares Method

The first method looks at the sum-of-squares error in each cluster against $K$. We compute the distance from each data point to the center of the cluster (centroid) to which the data point was assigned.

$$SS = \sum_k \sum_{x_i \in C_k} \sum_{x_j \in C_k} \left( x_i - x_j \right)^2 = \sum_k \sum_{x_i \in C_k} \left( x_i - \mu_k \right)^2$$

where $x_i$ is a point, $C_k$ represents cluster $k$ and $\mu_k$ is the centroid for cluster $k$. We can plot SS vs. $K$ and choose the elbow point in the plot as the best value for $K$. The elbow point is the point at which the plot starts descending much more slowly.

Choosing K: The Silhouette Method

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.

Range
Interpretation




0.71 - 1.0
A strong structure has been found.


0.51 - 0.7
A reasonable structure has been found.


0.26 - 0.5
The structure is weak and could be artificial.


< 0.25
No substantial structure has been found.



</pre>
Source: http://www.stat.berkeley.edu/~spector/s133/Clus.html

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.

Choosing $K$: The Gap Statistic

There is one last method worth covering for picking $K$, the so-called Gap statistic. The computation for the gap statistic builds on the sum-of-squares established in the Elbow method discussion, and compares it to the sum-of-squares of a "null distribution," that is, a random set of points with no clustering. The estimate for the optimal number of clusters $K$ is the value for which $\log{SS}$ falls the farthest below that of the reference distribution:

$$G_k = E_n^*\{\log SS_k\} - \log SS_k$$

In other words a good clustering yields a much larger difference between the reference distribution and the clustered data. The reference distribution is a Monte Carlo (randomization) procedure that constructs $B$ random distributions of points within the bounding box (limits) of the original data and then applies K-means to this synthetic distribution of data points.. $E_n^*\{\log SS_k\}$ is just the average $SS_k$ over all $B$ replicates. We then compute the standard deviation $\sigma_{SS}$ of the values of $SS_k$ computed from the $B$ replicates of the reference distribution and compute

$$s_k = \sqrt{1+1/B}\sigma_{SS}$$

Finally, we choose $K=k$ such that $G_k \geq G_{k+1} - s_{k+1}$.

Aside: Choosing $K$ when we Have Labels

Unsupervised learning expects that we do not have the labels. In some situations, we may wish to cluster data that is labeled. Computing the optimal number of clusters is much easier if we have access to labels. There are several methods available. We will not go into the math or details since it is rare to have access to the labels, but we provide the names and references of these measures.

• Adjusted Rand Index
• Mutual Information
• V-Measure
• Fowlkes–Mallows index

See this article for more information about these metrics.

Visualizing Clusters using PCA

How do we visualize clusters? If we only had two features, we could likely plot the data as is. But we have 100 data points each containing 32 features (dimensions). Principal Component Analysis (PCA) will help us reduce the dimensionality of our data from 32 to something lower. For a visualization on the coordinate plane, we will use 2 dimensions. In this exercise, we're going to use it to transform our multi-dimensional dataset into a 2 dimensional dataset.

This is only one use of PCA for dimension reduction. We can also use PCA when we want to perform regression but we have a set of highly correlated variables. PCA untangles these correlations into a smaller number of features/predictors all of which are orthogonal (not correlated). PCA is also used to reduce a large set of variables into a much smaller one.

What we've done is we've taken those columns of 0/1 indicator variables, and we've transformed them into a 2-D dataset. We took one column and arbitrarily called it x and then called the other y. Now we can throw each point into a scatterplot. We color coded each point based on it's cluster so it's easier to see them.

Other Clustering Algorithms

k-means is only one of a ton of clustering algorithms. Below is a brief description of several clustering algorithms, and the table provides references to the other clustering algorithms in scikit-learn.

• Affinity Propagation does not require the number of clusters $K$ to be known in advance! AP uses a "message passing" paradigm to cluster points based on their similarity.

• Spectral Clustering uses the eigenvalues of a similarity matrix to reduce the dimensionality of the data before clustering in a lower dimensional space. This is tangentially similar to what we did to visualize k-means clusters using PCA. The number of clusters must be known a priori.

• Ward's Method applies to hierarchical clustering. Hierarchical clustering algorithms take a set of data and successively divide the observations into more and more clusters at each layer of the hierarchy. Ward's method is used to determine when two clusters in the hierarchy should be combined into one. It is basically an extension of hierarchical clustering. Hierarchical clustering is divisive, that is, all observations are part of the same cluster at first, and at each successive iteration, the clusters are made smaller and smaller. With hierarchical clustering, a hierarchy is constructed, and there is not really the concept of "number of clusters." The number of clusters simply determines how low or how high in the hierarchy we reference and can be determined empirically or by looking at the dendogram.

• Agglomerative Clustering is similar to hierarchical clustering but but is not divisive, it is agglomerative. That is, every observation is placed into its own cluster and at each iteration or level or the hierarchy, observations are merged into fewer and fewer clusters until convergence. Similar to hierarchical clustering, the constructed hierarchy contains all possible numbers of clusters and it is up to the analyst to pick the number by reviewing statistics or the dendogram.

• DBSCAN is based on point density rather than distance. It groups together points with many nearby neighbors. DBSCAN is one of the most cited algorithms in the literature. It does not require knowing the number of clusters a priori, but does require specifying the neighborhood size.

Clustering Algorithms in Scikit-learn

</colgroup>

Method name

Parameters

Scalability

Use Case

Geometry (metric used) </tr> </thead>

K-Means

number of clusters

Very largen_samples, medium n_clusters with MiniBatch code

General-purpose, even cluster size, flat geometry, not too many clusters

Distances between points </tr>

Affinity propagation

damping, sample preference

Not scalable with n_samples

Many clusters, uneven cluster size, non-flat geometry

Graph distance (e.g. nearest-neighbor graph) </tr>

Mean-shift

bandwidth

Not scalable with n_samples

Many clusters, uneven cluster size, non-flat geometry

Distances between points </tr>

Spectral clustering

number of clusters

Medium n_samples, small n_clusters

Few clusters, even cluster size, non-flat geometry

Graph distance (e.g. nearest-neighbor graph) </tr>

Ward hierarchical clustering

number of clusters

Large n_samples and n_clusters

Many clusters, possibly connectivity constraints

Distances between points </tr>

Agglomerative clustering

number of clusters, linkage type, distance

Large n_samples and n_clusters

Many clusters, possibly connectivity constraints, non Euclidean distances

Any pairwise distance </tr>

DBSCAN

neighborhood size

Very large n_samples, medium n_clusters

Non-flat geometry, uneven cluster sizes

Distances between nearest points </tr>

Gaussian mixtures

many

Not scalable

Flat geometry, good for density estimation

Mahalanobis distances to centers </tr>

Birch

branching factor, threshold, optional global clusterer.

Large n_clusters and n_samples

Large dataset, outlier removal, data reduction.

Euclidean distance between points </tr> </tbody> </table> Source: http://scikit-learn.org/stable/modules/clustering.html