Key Word(s): Clustering, KMeans, K-Means, Hierarchical, Agglomerative, Silhouette, Unsupervised

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CS109B Data Science 2: Advanced Topics in Data Science

Lab 3 - Clustering

Harvard University
Spring 2020
Instructors: Mark Glickman, Pavlos Protopapas, and Chris Tanner
Lab Instructors: Chris Tanner and Eleni Kaxiras
Content: Chris Tanner and Will Claybaugh

In [1]:
## RUN THIS CELL TO PROPERLY HIGHLIGHT THE EXERCISES
import requests
from IPython.core.display import HTML
styles = requests.get("https://raw.githubusercontent.com/Harvard-IACS/2019-CS109B/master/content/styles/cs109.css").text
HTML(styles)
Out[1]:
In [2]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn import preprocessing

%matplotlib inline

Learning Objectives

By the end of this lab, you should be able to:

  • Explain what PCA is and know the differences between it and clustering
  • Understand the common distance metrics (e.g., Euclidean, Manhattan, Hamming)
  • Understand how different clustering algorithms work (e.g., k-means, Hierarchical, DBScan)
  • Explain the trade-offs between the clustering approaches
  • Quantitatively describe the quality clusters' fit, according to different metrics
  • Comfortably cluster any new data that you are presented with

This lab corresponds to Lectures #4 and #5 and maps to Homework #2.

Table of Contents

  1. PCA Refresher
  2. Distant Metrics
  3. Clustering Algorithms and Measuring Quality of Clusters

1. PCA Refresher


Discussion #1 What is PCA? How can it be useful?

How to use it ( via sklearn):

# assume we a DataFrame df

a. Instantiate a new PCA object:

pca_transformer = PCA()

b. Fit some data (learns the transformation based on this data):

fitted_pca = pca_transformer.fit(df)

c. Transform the data to the reduced dimensions:

pca_df = fitted_pca.transform(df)

Using two distinct steps (i.e., (b) and (c)) to fit and transform our data allows one the flexibility to transform any dataset according to our learned fit(). Alternatively, if you know you only want to transform a single dataset, you can combine (b) and (c) into one step:

Fit and transform:

pca_df = pca_transformer.fit_transform(df)

Example:

In [3]:
ms_df = pd.read_csv("../data/multishapes.csv")[['x','y']] # loads x,y columns of a dataset
pca_transformer = PCA()
fitted_pca = pca_transformer.fit(ms_df)
pca_df = fitted_pca.transform(ms_df)

NOTE: The above PCA transformation is a bit silly because we started with 2 dimensions and are transforming it to 2 dimensions -- no reduction. The data is still transforming the original data by applying a linear transformation so as to capture the most variance, but PCA is even more useful when the original data is high-dimensional. This example was just to remind you of the syntax.


Discussion #2: We didn't scale our data before applying PCA. Should we usually do so? Why or why not?



2. Distance Metrics

In the picture below, we are concerned with measuring the distance between two points, p and q.

(edited from Wikipedia.org)

Euclidean Distance:

The Euclidean distance measures the shortest path between the two points, navigating through all dimensions:

Manhattan Distance:

The Manhattan distance measures the cumulative difference between the two points, across all dimensions.

Discussion #3: Where have we seen something like this before in CS109A? What are the effects of using one versus another?


Hamming Distance (extra credit):

If our two elements of comparison can be represented a sequence of discrete items, it can be useful to measure how many of their elements differ.

For example:

  • Mahmoud and Mahmood differ by just 1 character and thus have a hamming distance of 1.
  • 10101 and 01101 have a hamming distance of 2.
  • Mary and Barry have a hamming distance of 3 (m->b, y->r, null->y).

Note: the last example may seem sub-optimal, as we could transform Mary to Barry by just 2 operations (substituting the M with a B, then adding an 'r'). The very related Levenshtein distance can handle this, and thus tends to be more appropriate for Strings.



3. Clustering Algorithms


Question: Why do we care about clustering? How/why is it useful?




We will now walk through three clustering algorithms, first discussing them at a high-level, then showing how to implement them with Python libraries. Let's first load and scale our data, so that particular dimensions don't naturally dominate in their contributions in the distant calculations:

In [4]:
# loads and displays our summary statistics of our data
multishapes = pd.read_csv("../data/multishapes.csv")
ms_df = multishapes[['x','y']]
ms_df.describe()
Out[4]:
x y
count 1100.000000 1100.000000
mean -0.081222 -0.625431
std 0.644967 1.176170
min -1.489180 -3.353462
25% -0.478839 -1.126752
50% -0.132920 -0.297040
75% 0.366072 0.250817
max 1.492208 1.253874
In [5]:
# scales our data
scaled_df = pd.DataFrame(preprocessing.scale(ms_df), index=multishapes['shape'], columns = ms_df.columns)
scaled_df.describe()
Out[5]:
x y
count 1.100000e+03 1100.000000
mean 6.459479e-18 0.000000
std 1.000455e+00 1.000455
min -2.183985e+00 -2.320473
25% -6.167723e-01 -0.426425
50% -8.019252e-02 0.279331
75% 6.938298e-01 0.745340
max 2.440659e+00 1.598544
In [6]:
# plots our data
msplot = scaled_df.plot.scatter(x='x',y='y',c='Black',title="Multishapes data",figsize=(11,8.5))
msplot.set_xlabel("X")
msplot.set_ylabel("Y")
plt.show()

3a. k-Means clustering:


Table Exercise #1: With your table, collectively discuss how k-means works. Use a whiteboard, draw a bunch of dots, and walk through each step of how the algorithm works. When you're confident of your answer, speak with a TF to verify its correctness.

Code (via sklearn):

In [7]:
from sklearn.cluster import KMeans
ms_kmeans = KMeans(n_clusters=3, init='random', n_init=3, random_state=109).fit(scaled_df)

That's it! Just 1 line of code!

Now that we've run k-Means, we can look at various attributes of our clusters. Full documenation is here.

In [8]:
display(ms_kmeans.cluster_centers_)
display(ms_kmeans.labels_[0:10])

array([[ 1.06623356,  0.17403387],
       [-0.46343097,  0.54170897],
       [-0.98352608, -1.57663916]])
array([1, 0, 0, 1, 0, 0, 1, 1, 1, 1], dtype=int32)

Plotting

Take note of matplotlib's c= argument to color items in the plot, along with our stacking two different plotting functions in the same plot.

In [9]:
plt.figure(figsize=(10,10))
plt.scatter(scaled_df['x'],scaled_df['y'], c=ms_kmeans.labels_);
plt.scatter(ms_kmeans.cluster_centers_[:,0],ms_kmeans.cluster_centers_[:,1], c='r', marker='h', s=100);
Question: Is this expected or did something go wrong? Should we always scale our data before clustering?

Lessons:

  • Initializations matter; run multiple times
  • Total Squared distance should never get worse during an update
  • k-Means can struggle with clusters that are close together; they can get lumped into one
  • There's no notion of 'not part of any cluster' or 'part of two clusters'
  • Visualization here

Quality of Clusters: Inertia

Inertia measures the total squared distance from points to their cluster's centroid. We obviously want this distance to be relatively small. If we increase the number of clusters, it will naturally make the average distance smaller. If every point has its own cluster, then our distance would be 0. That's obviously not an ideal way to cluster. One way to determine a reasonable number of clusters to simply try many different clusterings as we vary k, and each time, measure the overall inertia.

In [10]:
wss = []
for i in range(1,11):
    fitx = KMeans(n_clusters=i, init='random', n_init=5, random_state=109).fit(scaled_df)
    wss.append(fitx.inertia_)

plt.figure(figsize=(11,8.5))
plt.plot(range(1,11), wss, 'bx-')
plt.xlabel('Number of clusters $k$')
plt.ylabel('Inertia')
plt.title('The Elbow Method showing the optimal $k$')
plt.show()

Look for the place(s) where distance stops decreasing as much (i.e., the 'elbow' of the curve). It seems that 4 would be a good number of clusters, as a higher k yields diminishing returns.

Quality of Clusters: Silhouette

Let's say we have a data point $i$, and the cluster it belongs to is referred to as $C(i)$. One way to measure the quality of a cluster $C(i)$ is to measure how close its data points are to each other (within-cluster) compared to nearby, other clusters $C(j)$. This is what Silhouette Scores provide for us. The range is [-1,1]; 0 indicates a point on the decision boundary (equal average closeness to points intra-cluster and out-of-cluster), and negative values mean that datum might be better in a different cluster.

Specifically, let $a(i)$ denote the average distance data point $i$ is to the other points in the same cluster:

Similarly, we can also compute the average distance that data point $i$ is to all other clusters. The cluster that yields the minimum distance is denoted by $b(i)$:

Hopefully our data point $i$ is much closer, on average, to points within its own cluster (i.e., $a(i)$ than it is to its closest neighboring cluster $b(i)$). The silhouette score quantifies this as $s(i)$:

NOTE: If data point $i$ belongs to its own cluster (no other points), then the silhouette score is set to 0 (otherwise, $a(i)$ would be undefined).

The silhouette score plotted below is the overall average across all points in our dataset.

The silhouette_score() function is available in sklearn. We can manually loop over values of K (for applying k-Means algorithm), then plot its silhouette score. This should allow us to make a reasonable choice for selecting the 'optimal' number of clusters.

In [11]:
from sklearn.metrics import silhouette_score

scores = [0]
for i in range(2,11):
    fitx = KMeans(n_clusters=i, init='random', n_init=5, random_state=109).fit(scaled_df)
    score = silhouette_score(scaled_df, fitx.labels_)
    scores.append(score)
    
plt.figure(figsize=(11,8.5))
plt.plot(range(1,11), np.array(scores), 'bx-')
plt.xlabel('Number of clusters $k$')
plt.ylabel('Average Silhouette')
plt.title('Silhouette Scores for varying $k$ clusters')
plt.show()

Visualizing all Silhoutte scores for a particular clustering

Below, we borrow from an sklearn example. The second plot may be overkill.

  • The second plot is just the scaled data. It is not a PCA plot
  • If you only need the raw silhouette scores, use the silhouette_samples() function
In [12]:
from sklearn.metrics import silhouette_samples, silhouette_score
import matplotlib.cm as cm
#modified code from http://scikit-learn.org/stable/auto_examples/cluster/plot_kmeans_silhouette_analysis.html

def silplot(X, clusterer, pointlabels=None):
    cluster_labels = clusterer.labels_
    n_clusters = clusterer.n_clusters
    
    # Create a subplot with 1 row and 2 columns
    fig, (ax1, ax2) = plt.subplots(1, 2)
    fig.set_size_inches(11,8.5)

    # 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])

    # 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,".",sep="")

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

    y_lower = 10
    for i in range(0,n_clusters+1):
        # 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="--")

    ax1.set_yticks([])  # Clear the yaxis labels / ticks
    ax1.set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])

    # 2nd Plot showing the actual clusters formed
    colors = cm.nipy_spectral(cluster_labels.astype(float) / n_clusters)
    ax2.scatter(X[:, 0], X[:, 1], marker='.', s=200, lw=0, alpha=0.7,
                c=colors, edgecolor='k')
    xs = X[:, 0]
    ys = X[:, 1]
    
    if pointlabels is not None:
        for i in range(len(xs)):
            plt.text(xs[i],ys[i],pointlabels[i])

    # 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, edgecolor='k')

    for i, c in enumerate(centers):
        ax2.scatter(c[0], c[1], marker='$%d$' % int(i), alpha=1,
                    s=50, edgecolor='k')

    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')
In [13]:
# run k-means with 3 clusters
ms_kmeans = KMeans(n_clusters=3, init='random', n_init=3, random_state=109).fit(scaled_df)

# plot a fancy silhouette plot
silplot(scaled_df.values, ms_kmeans)

For n_clusters = 3, the average silhouette_score is 0.4269854455072775.
Exercise #1:

Using the silhouette scores' optimal number of clusters (per the elbow plot above):

  • Fit a new k-Means model with that many clusters
  • Plot the clusters like we originally did with k-means
  • Plot the silhouette scores just like the above cells
  • Which seems like a better clustering (i.e., 3 clusters or the number returned by the elbow plot above)?
In [14]:
# your code here
# %load solutions/exercise1-solution.py

Quality of Clusters: Gap Statistic

The gap statistic compares within-cluster distances (like in silhouette), but instead of comparing against the second-best existing cluster for that point, it compares our clustering's overall average to the average we'd see if the data were generated at random (we'd expect randomly generated data to not necessarily have any inherit patterns that can be easily clustered). For full details, you can read the original research paper.

In essence, the within-cluster distances (in the elbow plot) will go down just becuse we have more clusters. We additionally calculate how much they'd go down on non-clustered data with the same spread as our data and subtract that trend out to produce the plot below.

In [15]:
from gap_statistic import OptimalK
from sklearn.datasets.samples_generator import make_blobs

gs_obj = OptimalK()

n_clusters = gs_obj(scaled_df.values, n_refs=50, cluster_array=np.arange(1, 15))
print('Optimal clusters: ', n_clusters)

Optimal clusters:  14
In [16]:
gs_obj.gap_df
Out[16]:
n_clusters gap_value gap* ref_dispersion_std diff diff*
0 1.0 -2.485489 -2016.745254 3.197501 -0.048743 327.709212
1 2.0 -2.414842 -1166.422655 2.485720 -0.047171 201.216617
2 3.0 -2.343297 -680.433189 1.740576 0.124498 278.323626
3 4.0 -2.428315 -477.003229 1.929003 0.125511 419.030416
4 5.0 -2.527884 -445.798296 1.003146 -0.098308 221.349425
5 6.0 -2.406904 -331.922050 0.739480 -0.003661 215.946444
6 7.0 -2.376581 -272.577609 0.732733 -0.050279 155.749504
7 8.0 -2.289622 -213.102403 0.879548 0.148301 215.475149
8 9.0 -2.392151 -213.226757 0.991139 -0.066456 126.100923
9 10.0 -2.287996 -168.823140 0.714412 0.113461 167.342742
10 11.0 -2.366609 -167.250275 0.605772 -0.023348 122.170160
11 12.0 -2.306497 -143.993202 0.591874 0.067395 128.290998
12 13.0 -2.332732 -135.467448 0.598679 -0.071119 89.515897
13 14.0 -2.224408 -111.934186 0.509789 NaN NaN
In [17]:
gs_obj.plot_results() # makes nice plots

If we wish to add error bars to help us decide how many clusters to use, the following code displays such:

In [18]:
def display_gapstat_with_errbars(gap_df):
    gaps = gap_df["gap_value"].values
    diffs = gap_df["diff"]
    
    err_bars = np.zeros(len(gap_df))
    err_bars[1:] = diffs[:-1] - gaps[:-1] + gaps[1:]

    plt.scatter(gap_df["n_clusters"], gap_df["gap_value"])
    plt.errorbar(gap_df["n_clusters"], gap_df["gap_value"], yerr=err_bars, capsize=6)
    plt.xlabel("Number of Clusters")
    plt.ylabel("Gap Statistic")
    plt.show()
    
display_gapstat_with_errbars(gs_obj.gap_df)

For more information about the gap_stat package, please see the full documentation here.

3b. Agglomerative Clustering


Table Exercise #2: With your table, collectively discuss how agglomerative clustering works. Use a whiteboard, draw a bunch of dots, and walk through each step of how the algorithm works. When you're confident of your answer, speak with a TF to verify its correctness.

Code (via scipy):

There are many different cluster-merging criteria, one of which is Ward's criteria. Ward's optimizes having the lowest total within-cluster distances, so it merges the two clusters that will harm this objective least. scipy's agglomerative clustering function implements Ward's method.

In [19]:
import scipy.cluster.hierarchy as hac
from scipy.spatial.distance import pdist

plt.figure(figsize=(11,8.5))
dist_mat = pdist(scaled_df, metric="euclidean")
ward_data = hac.ward(dist_mat)

hac.dendrogram(ward_data);
Discussion #4: How do you read a plot like the above? What are valid options for number of clusters, and how can you tell? Are some more valid than others? Does it make sense to compute silhouette scores for an agglomerative clustering? If we wanted to compute silhouette scores, what would we need for this to be possible?

Lessons:

  • It's expensive: O(n^3) time complexity and O(n^2) space complexity.
  • Many choices for linkage criteria
  • Every node gets clustered (no child left behind)
In [20]:
# %load solutions/discussion4-solution.py

3c. DBscan Clustering

DBscan uses an intuitive notion of denseness to define clusters, rather than defining clusters by a central point as in k-means.

Code (via sklearn):

DBscan is implemented in good 'ol sklearn, but there aren't great automated tools for searching for the optimal epsilon parameter. For full documentation, please visit this page

In [21]:
from sklearn.cluster import DBSCAN
plt.figure(figsize=(11,8.5))
fitted_dbscan = DBSCAN(eps=0.2).fit(scaled_df)
plt.scatter(scaled_df['x'],scaled_df['y'], c=fitted_dbscan.labels_);

Note: the dark purple dots are not clustered with anything else. They are lone singletons. You can validate such by setting epsilon to a very small value, and increase the min_samples to a high value. Under these conditions, nothing would cluster, and yet all dots become dark purple.


Exercise #2: Experiment with the above code by changing its epsilon value and the min_samples (what is the default value for it, since the above code doesn't specify a value?)



Instead of just empirically observing how the epsilon value affects the clustering (which would be very costly for large, high-dimensional data), we can also inspect how far each data point is to its $N^{th}$ closest neighbor:

In [22]:
from sklearn.neighbors import NearestNeighbors

# x-axis is each individual data point, numbered by an artificial index
# y-axis is the distance to its 2nd closest neighbor
def plot_epsilon(df, min_samples):
    fitted_neigbors = NearestNeighbors(n_neighbors=min_samples).fit(df)
    distances, indices = fitted_neigbors.kneighbors(df)
    dist_to_nth_nearest_neighbor = distances[:,-1]
    plt.plot(np.sort(dist_to_nth_nearest_neighbor))
    plt.xlabel("Index\n(sorted by increasing distances)")
    plt.ylabel("{}-NN Distance (epsilon)".format(min_samples-1))
    plt.tick_params(right=True, labelright=True)
In [23]:
plot_epsilon(scaled_df, 3)

Lessons:

  • Can cluster non-linear relationships very well; potential for more natural, arbritrarily shaped groupings
  • Does not require specifying the # of clusters (i.e., k); the algorithm determines such
  • Robust to outliers
  • Very sensitive to the parameters (requires strong knowledge of the data)
  • Doesn't guarantee that every (or ANY) item will be clustered


Discussion #5:
When should we prefer one type of clustering over another? Should we always just try all of them? Imagine you work at Spotify and you want to create personalized playlists for each person. One could imagine a dataset exists whereby each row is a particular song, and the columns are features (e.g., tempo (BPM), average vocal frequency, amount of bass, sentiment of lyrics, duration in seconds, etc). Let's use clustering to group one's catalog of favorite music, which will serve as disjoint starting points for suggesting future songs. Specifically, imagine that you've 'liked' 500 songs on Spotify so far, and your recommendation algorithm needs to cluster those 500 songs. Would you first experiment with k-Means, Agglomerative, or DBScan? Why?