FYS-STK4155/doc/LectureNotes/_build/jupyter_execute/Clustering.ipynb

Clustering Analysis

In this chapter we will concern ourselves with the study of cluster analysis. In general terms cluster analysis, or clustering, is the task of grouping a data-set into different distinct categories based on some measure of equality of the data. This measure is often referred to as a metric or similarity measure in the literature (note: sometimes we deal with a dissimilarity measure instead). Usually, these metrics are formulated as some kind of distance function between points in a high-dimensional space.

There exists a lot of such distance measures. The simplest, and also the most common is the Euclidean distance (i.e. Pythagoras). A good source for those of you wanting a thorough overview is the article (DOI:10.5120/ijca2016907841 Irani, Pise, Phatak). A few other metrics mentioned there are: cosine similarity, Manhattan distance, Chebychev distance and the Minkowski distance. The Minkowski distance is a general formulation which encapsulates a range of metrics. All of these, and many more, can be used in clustering. There exists different categories of clustering algorithms. A few of the most common are: centroid-, distribution-, density- and hierarchical- clustering. We will concern ourselves primarily with the first one.

Basic Idea of the K-means Clustering Algorithm

The simplest of all clustering algorithms is the aptly named k-means algorithm , sometimes also referred to as Lloyds algorithm. It is the simplest and also the most common. From its simplicity it obtains both strengths and weaknesses. These will be discussed in more detail later. The k-means algorithm is a centroid based clustering algorithm.

Assume, we are given n data points and we wish to split the data into K < n different categories, or clusters. We label each cluster by an integer $k\in{ 1, \cdots, K }$. In the basic k-means algorithm each point is assigned to only one cluster k, and these assignments are non-injective i.e. many-to-one. We can think of these mappings as an encoder k = C(i), which assigns the $i$-th data-point \bf x_i to the $k$-th cluster. Before we jump into the mathematics let us describe the k-means algorithm in words:

1. We start with guesses / random initializations of our k cluster centers / centroids

2. For each centroid the points that are most similar are identified

3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.

4. Iterate this points 2, 3) until the centroids no longer move (to some tolerance)

Now we consider the method formally. Again, we assume we have n data-points (vectors)

\begin{equation}\label{eq:kmeanspoints} \tag{1}
  \boldsymbol{x_i}  = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p.
\end{equation}

which we wish to group into K < n clusters. For our dissimilarity measure we will use the squared Euclidean distance

\begin{equation}\label{eq:squaredeuclidean} \tag{2}
  d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2
                         = ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2
\end{equation}

Next we define the so called within-cluster point scatter which gives us a measure of how close each data point assigned to the same cluster tends to be to the all the others.

\begin{equation}\label{eq:withincluster} \tag{3}
  W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
          \sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) =
          \sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2
\end{equation}

where \boldsymbol{\overline{x_k}} is the mean vector associated with the $k$-th cluster, and N_k = \sum_{i=1}^nI(C(i) = k), where the I() notation is similar to the Kronecker delta (Commonly used in statistics, it just means that when i = k we have the encoder $C(i)$). In other words, the within-cluster scatter measures the compactness of each cluster with respect to the data points assigned to each cluster. This is the quantity that the $k$-means algorithm aims to minimize. We refer to this quantity W(C) as the within cluster scatter because of its relation to the total scatter.

\begin{equation}\label{eq:totalscatter} \tag{4}
  T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n
                    \sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
                  = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
                    \Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
                  + \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big)
\end{equation}

Which is a quantity that is conserved throughout the $k$-means algorithm. It can be thought of as the total amount of information in the data, and it is composed of the aforementioned within-cluster scatter and the between-cluster scatter B(C). In methods such as principle component analysis the total scatter is not conserved.

Given a cluster mean \boldsymbol{m_k} we define the total cluster variance

\begin{equation}\label{eq:totalclustervariance} \tag{5}
  \min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2
\end{equation}

Now we have all the pieces necessary to formally revisit the k-means algorithm. If you at this point feel like some of the above definitions came a bit out of no-where, don't fret, the method does get a whole lot simpler once we start programming.

The K-means Clustering Algorithm

The k-means clustering algorithm goes as follows (note in my opinion this description is a bit complicated and is lifted directly out of ESL HASTIE for deeper understanding purposes)

1. For a given cluster assignment C, and k cluster means \{m_1, \cdots, m_k\}. We minimize the total cluster variance with respect to the cluster means \{m_k\} yielding the means of the currently assigned clusters.

2. Given a current set of k means \{m_k\} the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2

3. Steps 1 and 2 are repeated until the assignments do not change.

As previously stated the above formulation can be a bit difficult to understand, at least the first time, due to the dense notation used. But all in all the concept is fairly simple when explained in words. The math needs to be understood but to help you along the way we summarize the algorithm as follows (try to look at the terms above to match with the summary).

1. Before we start we specify a number k which is the number of clusters we want to try to separate our data into.

2. We initially choose k random data points in our data as our initial centroids, or means (this is where the name comes from).

3. Assign each data point to their closest centroid, based on the squared Euclidean distance.

4. For each of the k cluster we update the centroid by calculating new mean values for all the data points in the cluster.

5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.

That's it, nothing magical happening.

Writing Our Own Code

In the following section we will work to develop a deeper understanding of the previously discussed mathematics through developing codes to do k-means cluster analysis.

Basic Python

Let us now program the most basic version of the algorithm using nothing but Python with numpy arrays. This code is kept intentionally simple to gradually progress our understanding. There is no vectorization of any kind, and even most helper functions are not utilized. Throughout our implementation process it will be helpful to keep in mind both the mathematical description of the algorithm and our summary from above. In addition, try to think of ways to optimize this while reading the next section. We will get to it, take it as a challenge to see if your optimizations are better.

First of all we need a dataset to do our cluster analysis on, for clarity (and lack of googling beforehand) we generate it ourselves using Gaussians. First we import

In [1]:

%matplotlib inline

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import time
from IPython.display import display

np.random.seed(2021)

Next we define functions, for ease of use later, to generate Gaussians and to set up our toy data set.

In [2]:

def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),
                    sample_variance=1):
    """
    Very simple custom function to generate gaussian distributed point clusters
    with variable dimension, number of points, means in each direction
    (must match dim) and sample variance.

    Inputs:
        dim (int)
        n_points (int)
        mean_vector (np.array) (where index 0 is x, index 1 is y etc.)
        sample_variance (float)

    Returns:
        data (np.array): with dimensions (dim x n_points)
    """

    mean_matrix = np.zeros(dim) + mean_vector
    covariance_matrix = np.eye(dim) * sample_variance
    data = np.random.multivariate_normal(mean_matrix, covariance_matrix,
                                    n_points)
    return data



def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,
                                    return_data=True):
    """
    Toy model to illustrate k-means clustering
    """

    data1 = gaussian_points(mean_vector=np.array([5, 5]))
    data2 = gaussian_points()
    data3 = gaussian_points(mean_vector=np.array([1, 4.5]))
    data4 = gaussian_points(mean_vector=np.array([5, 1]))
    data = np.concatenate((data1, data2, data3, data4), axis=0)

    if plotting:
        fig, ax = plt.subplots()
        ax.scatter(data[:, 0], data[:, 1], alpha=0.2)
        ax.set_title('Toy Model Dataset')
        plt.show()


    if return_data:
        return data


data = generate_simple_clustering_dataset()

Now that we are our, albeit very simple, dataset we are ready to start implementing the k-means algorithm.

In [3]:

n_samples, dimensions = data.shape
n_clusters = 4

# we randomly initialize our centroids
np.random.seed(2021)
centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]
distances = np.zeros((n_samples, n_clusters))

# first we need to calculate the distance to each centroid from our data
for k in range(n_clusters):
    for n in range(n_samples):
        dist = 0
        for d in range(dimensions):
            dist += np.abs(data[n, d] - centroids[k, d])**2
            distances[n, k] = dist

# we initialize an array to keep track of to which cluster each point belongs
# the way we set it up here the index tracks which point and the value which
# cluster the point belongs to
cluster_labels = np.zeros(n_samples, dtype='int')

# next we loop through our samples and for every point assign it to the cluster
# to which it has the smallest distance to
for n in range(n_samples):
    # tracking variables (all of this is basically just an argmin)
    smallest = 1e10
    smallest_row_index = 1e10
    for k in range(n_clusters):
        if distances[n, k] < smallest:
            smallest = distances[n, k]
            smallest_row_index = k

    cluster_labels[n] = smallest_row_index

Let's plot and see

In [4]:

fig = plt.figure()
ax = fig.add_subplot()
unique_cluster_labels = np.unique(cluster_labels)
for i in unique_cluster_labels:
    ax.scatter(data[cluster_labels == i, 0],
               data[cluster_labels == i, 1],
               label = i,
               alpha = 0.2)
    ax.scatter(centroids[:, 0], centroids[:, 1], c='black')

ax.set_title("First Grouping of Points to Centroids")

plt.show()

So what do we have so far? We have 'picked' k centroids at random from our data points. There are other ways of more intelligently choosing their initializations, however for our purposes randomly is fine. Then we have initialized an array 'distances' which holds the information of the distance, or dissimilarity, of every point to of our centroids. Finally, we have initialized an array 'cluster_labels' which according to our distances array holds the information of to which centroid every point is assigned. This was the first pass of our algorithm. Essentially, all we need to do now is repeat the distance and assignment steps above until we have reached a desired convergence or a maximum amount of iterations.

In [5]:

max_iterations = 100
tolerance = 1e-8
start_time = time.time()

for iteration in range(max_iterations):
    prev_centroids = centroids.copy()
    for k in range(n_clusters):
        # this array will be used to update our centroid positions
        vector_mean = np.zeros(dimensions)
        mean_divisor = 0
        for n in range(n_samples):
            if cluster_labels[n] == k:
                vector_mean += data[n, :]
                mean_divisor += 1

        # update according to the k means
        centroids[k, :] = vector_mean / mean_divisor

    # we find the dissimilarity
    for k in range(n_clusters):
        for n in range(n_samples):
            dist = 0
            for d in range(dimensions):
                dist += np.abs(data[n, d] - centroids[k, d])**2
                distances[n, k] = dist

    # assign each point
    for n in range(n_samples):
        smallest = 1e10
        smallest_row_index = 1e10
        for k in range(n_clusters):
            if distances[n, k] < smallest:
                smallest = distances[n, k]
                smallest_row_index = k

        cluster_labels[n] = smallest_row_index

    # convergence criteria
    centroid_difference = np.sum(np.abs(centroids - prev_centroids))
    if centroid_difference < tolerance:
        print(f'Converged at iteration {iteration}')
        print(f'Runtime: {time.time() - start_time} seconds')
        break

    elif iteration == max_iterations:
        print(f'Did not converge in {max_iterations} iterations')
        print(f'Runtime: {time.time() - start_time} seconds')

Converged at iteration 5
Runtime: 0.23237395286560059 seconds

And thats it! We now have an extremely barebones, un-optimized k-means clustering implementation. Lets plot the final result

In [6]:

fig = plt.figure()
ax = fig.add_subplot()
unique_cluster_labels = np.unique(cluster_labels)
for i in unique_cluster_labels:
    ax.scatter(data[cluster_labels == i, 0],
               data[cluster_labels == i, 1],
               label = i,
               alpha = 0.2)
    ax.scatter(centroids[:, 0], centroids[:, 1], c='black')

ax.set_title("Final Result of K-means Clustering")

plt.show()

Now there are a few glaring improvements to be done here. First of all is organizing things into functions for better readability. Second is getting rid of the small inefficiencies like manually calculating distances and argmin. And finally, we need to optimize for better run-time. It's like we always say: the best way of looping in Python is to not loop in Python. Let us tackle the first two improvements.

Towards a More Numpythonic Code

In [7]:

def get_distances_to_clusters(data, centroids):
    """
    Function that for each cluster finds the squared Euclidean distance
    from every data point to the cluster centroid and returns a numpy array
    containing the distances such that distance[i, j] means the distance between
    the i-th point and the j-th centroid.
    Inputs:
        data (np.array): with dimensions (n_samples x dim)
        centroids (np.array): with dimensions (n_clusters x dim)

    Returns:
        distances (np.array): with dimensions (n_samples x n_clusters)
    """

    n_samples, dimensions = data.shape
    n_clusters = centroids.shape[0]
    distances = np.zeros((n_samples, n_clusters))
    for k in range(n_clusters):
        for i in range(n_samples):
            dist = 0
            for j in range(dimensions):
                dist += np.abs(data[i, j] - centroids[k, j])**2
                distances[i, k] = dist

    return distances



def assign_points_to_clusters(distances):
    """
    Function to assign each data point to the cluster to which it is the closest
    based on the squared Euclidean distance from the get_distances_to_clusters
    method.
    Inputs:
        distances (np.array): with dimensions (n_samples x n_clusters)

    Returns:
        cluster_labels (np.array): with dimensions (n_samples)
    """
    cluster_labels = np.argmin(distances, axis=1)

    return cluster_labels



def k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8):
    """
    Naive implementation of the k-means clustering algorithm. A short summary of
    the algorithm is as follows: we randomly initialize k centroids / means.
    Then we assign, using the squared Euclidean distance, every data-point to a
    cluster. We then update the position of the k centroids / means, and repeat
    until convergence or we reach our desired maximum iterations. The method
    returns the cluster assignments of our data-points and a sequence of
    centroids.
    Inputs:
        data (np.array): with dimesions (n_samples x dim)
        n_clusters (int): hyperparameter which depends on dataset
        max_iterations (int): hyperparameter which depends on dataset
        tolerance (float): convergence measure

    Returns:
        cluster_labels (np.array): with dimension (n_samples)
        centroid_list (list): list of centroids (np.array)
                              with dimensions (n_clusters x dim)
    """

    samples, dimensions = data.shape
    np.random.seed(2021)
    centroids = data[np.random.choice(len(data), n_clusters, replace=False), :]
    distances = get_distances_to_clusters(data, centroids)
    cluster_labels = assign_points_to_clusters(distances)

    start_time = time.time()

    for iteration in range(max_iterations):
        prev_centroids = centroids.copy()
        for k in range(n_clusters):
            vector_mean = np.zeros(dimensions)
            mean_divisor = 0
            for n in range(n_samples):
                if cluster_labels[n] == k:
                    vector_mean += data[n, :]
                    mean_divisor += 1
            # And update according to the new means
            centroids[k, :] = vector_mean / mean_divisor

        distances = get_distances_to_clusters(data, centroids)
        cluster_labels = assign_points_to_clusters(distances)

        centroid_difference = np.sum(np.abs(centroids - prev_centroids))
        if centroid_difference < tolerance:
            print(f'Converged at iteration: {iteration}')
            print(f'Runtime: {time.time() - start_time} seconds')

            return cluster_labels, centroids

    print(f'Did not converge in {max_iterations} iterations')
    print(f'Runtime: {time.time() - start_time} seconds')

    return cluster_labels, centroids


# quirk of numpy / Jupyter need to set seed again
cluster_labels, centroids = k_means(data)

Converged at iteration: 5
Runtime: 0.19273090362548828 seconds

Note: the start of the timing is after the random initialization, and first 'cycle' of our algorithm. This is technically not the correct way to time it but due to this being in a Jupyter notebook and the way it is structured this way of comparing our algorithms will produce a more equal result. When timing code we should always encapsulate our whole computation block.

So we see an improvement from just switching to numpy's argmin function. There is a very nice tool (or category of tools) called profilers. These can be utilized to make clearer which improvements to our code we should care most about here is an excellent source on the topic. Even before optimizing we can understand which parts of our code will be taking the most of the run-time. It will be the longest Python loop, i.e. the loop over all the samples. Nonetheless, let us do some profiling!

In [8]:

test_data = generate_simple_clustering_dataset(n_points=10000, plotting=False)
%prun -l 10 cluster_labels, centroids = k_means(test_data)

Converged at iteration: 11
Runtime: 0.386091947555542 seconds
 

Here we can see the reason for profiling. We now know for certain a lot can be gained just by vectorizing our distance function. Ideally we wish to perform most of our loops in numpy, i.e. C. To do this we need our array shapes to match and clever reshaping will let us do so.

In [9]:

def np_get_distances_to_clusters(data, centroids):
    """
    Squared Euclidean distance between all data-points and every centroid. For
    the function to work properly it needs data and centroids to be numpy
    broadcastable. We sum along the dimension axis.
    Inputs:
        data (np.array): with dimensions (samples x 1 x dim)
        centroids (np.array): with dimensions (1 x n_clusters x dim)

    Returns:
        distances (np.array): with dimensions (samples x n_clusters)
    """

    distances = np.sum(np.abs((data - centroids))**2, axis=2)
    return distances

def np_assign_points_to_clusters(distances):
    """
    Assigning each data-point to a cluster given an array distances containing
    the squared Euclidean distance from every point to each centroid. We do
    np.argmin along the cluster axis to find the closest cluster. Returns a
    numpy array with corresponding labels.
    Inputs:
        distances (np.array): with dimensions (samples x n_clusters)

    Returns:
        cluster_labels (np.array): with dimensions (samples x None)
    """
    cluster_labels = np.argmin(distances, axis=1)
    return cluster_labels


def np_k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8):
    """
    Numpythonic implementation of the k-means clusting algorithm.
    Inputs:
        data (np.array): with dimesions (samples x dim)
        n_clusters (int): hyperparameter which depends on dataset
        max_iterations (int): hyperparameter which depends on dataset
        tolerance (float): convergence measure
        progression_plot (bool): activation flag for plotting
    Returns:
        cluster_labels (np.array): with dimension (samples)
        centroid_list (list): list of centroids (np.array)
                              with dimensions (n_clusters x dim)
    """
    n_samples, dimensions = data.shape
    np.random.seed(2021)
    centroids = data[np.random.choice(len(data), n_clusters, replace=False), :]

    distances = np_get_distances_to_clusters(np.reshape(data,
                                            (n_samples, 1, dimensions)),
                                          np.reshape(centroids,
                                            (1, n_clusters, dimensions)))
    cluster_labels = np_assign_points_to_clusters(distances)

    start_time = time.time()

    for iteration in range(max_iterations):
        prev_centroids = centroids.copy()
        for k in range(n_clusters):
            points_in_cluster = data[cluster_labels == k]
            mean_vector = np.mean(points_in_cluster, axis=0)
            centroids[k] = mean_vector

        distances = np_get_distances_to_clusters(np.reshape(data,
                                                (n_samples, 1, dimensions)),
                                              np.reshape(centroids,
                                                (1, n_clusters, dimensions)))
        cluster_labels = np_assign_points_to_clusters(distances)

        centroid_difference = np.sum(np.abs(centroids - prev_centroids))
        if centroid_difference < tolerance:
            print(f'Converged at iteration: {iteration}')
            print(f'Runtime: {time.time() - start_time} seconds')

            return cluster_labels, centroids

    print(f'Did not converge in {max_iterations} iterations')
    print(f'Runtime: {time.time() - start_time} seconds')

    return cluster_labels, centroids

When working towards becoming a data scientist using Python this last step is arguably one of the most important. Thinking of ways to avoid explicitly looping by adding dimensions to our arrays in such a way that they become broadcastable using numpy (also tensorflow and many others). Let us take a look at our the fruits of our labor.

In [10]:

cluster_labels, centroids = np_k_means(data)

Converged at iteration: 5
Runtime: 0.002377033233642578 seconds

In [ ]: