20.07.2020       Выпуск 344 (20.07.2020 - 26.07.2020)       Статьи

K-Means Clustering in Python: A Practical Guide


Экспериментальная функция:

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To follow along with the examples below, you can download the source code by clicking on the following link:

Building a K-Means Clustering Pipeline

Assuming you want to start with a fresh namespace, import all the modules needed to build and evaluate the pipeline, including pandas and seaborn for more advanced visualizations:

In [1]: import tarfile
   ...: import urllib
   ...: import numpy as np
   ...: import matplotlib.pyplot as plt
   ...: import pandas as pd
   ...: import seaborn as sns
   ...: from sklearn.cluster import KMeans
   ...: from sklearn.decomposition import PCA
   ...: from sklearn.metrics import silhouette_score, adjusted_rand_score
   ...: from sklearn.pipeline import Pipeline
   ...: from sklearn.preprocessing import LabelEncoder, MinMaxScaler

Download and extract the TCGA dataset from UCI:

In [2]: uci_tcga_url = "https://archive.ics.uci.edu/ml/machine-learning-databases/00401/"
   ...: archive_name = "TCGA-PANCAN-HiSeq-801x20531.tar.gz"

   ...: # Build the url
   ...: full_download_url = urllib.parse.urljoin(uci_tcga_url, archive_name)
   ...: # Download the file
   ...: r = urllib.request.urlretrieve (full_download_url, archive_name)

   ...: # Extract the data from the archive
   ...: tar = tarfile.open(archive_name, "r:gz")
   ...: tar.extractall()
   ...: tar.close()

After the download and extraction is completed, you should have a directory that looks like this:

├── data.csv
└── labels.csv

The KMeans class in scikit-learn requires a NumPy array as an argument. The NumPy package has a helper function to load the data from the text file into memory as NumPy arrays:

In [3]: datafile = "TCGA-PANCAN-HiSeq-801x20531/data.csv"
   ...: labels_file = "TCGA-PANCAN-HiSeq-801x20531/labels.csv"
   ...: data = np.genfromtxt(
   ...:     datafile,
   ...:     delimiter=",",
   ...:     usecols=range(1, 20532),
   ...:     skip_header=1
   ...: )
   ...: true_label_names = np.genfromtxt(
   ...:     labels_file,
   ...:     delimiter=",",
   ...:     usecols=(1,),
   ...:     skip_header=1,
   ...:     dtype="str"
   ...: )

Check out the first three columns of data for the first five samples as well as the labels for the first five samples:

In [4]: data[:5, :3]
array([[0.        , 2.01720929, 3.26552691],
       [0.        , 0.59273209, 1.58842082],
       [0.        , 3.51175898, 4.32719872],
       [0.        , 3.66361787, 4.50764878],
       [0.        , 2.65574107, 2.82154696]])

In [5]: true_label_names[:5]
Out[5]: array(['PRAD', 'LUAD', 'PRAD', 'PRAD', 'BRCA'], dtype='<U4')

The data variable contains all the gene expression values from 20,531 genes. The true_label_names are the cancer types for each of the 881 samples. The first record in data corresponds with the first label in true_labels.

The labels are strings containing abbreviations of cancer types:

  • BRCA: Breast invasive carcinoma
  • COAD: Colon adenocarcinoma
  • KIRC: Kidney renal clear cell carcinoma
  • LUAD: Lung adenocarcinoma
  • PRAD: Prostate adenocarcinoma

To use these labels in the evaluation methods, you first need to convert the abbreviations to integers with LabelEncoder:

In [6]: label_encoder = LabelEncoder()

In [7]: true_labels = label_encoder.fit_transform(true_label_names)

In [8]: true_labels[:5]
Out[8]: array([4, 3, 4, 4, 0])

Since the label_encoder has been fitted to the data, you can see the unique classes represented using .classes_. Store the length of the array to the variable n_clusters for later use:

In [9]: label_encoder.classes_
Out[9]: array(['BRCA', 'COAD', 'KIRC', 'LUAD', 'PRAD'], dtype='<U4')

In [10]: n_clusters = len(label_encoder.classes_)

In practical machine learning pipelines, it’s common for the data to undergo multiple sequences of transformations before it feeds into a clustering algorithm. You learned about the importance of one of these transformation steps, feature scaling, earlier in this tutorial. An equally important data transformation technique is dimensionality reduction, which reduces the number of features in the dataset by either removing or combining them.

Dimensionality reduction techniques help to address a problem with machine learning algorithms known as the curse of dimensionality. In short, as the number of features increases, the feature space becomes sparse. This sparsity makes it difficult for algorithms to find data objects near one another in higher-dimensional space. Since the gene expression dataset has over 20,000 features, it qualifies as a great candidate for dimensionality reduction.

Principal Component Analysis (PCA) is one of many dimensionality reduction techniques. PCA transforms the input data by projecting it into a lower number of dimensions called components. The components capture the variability of the input data through a linear combination of the input data’s features.

Note: A full description of PCA is out of scope for this tutorial, but you can learn more about it in the scikit-learn user guide.

The next code block introduces you to the concept of scikit-learn pipelines. The scikit-learn Pipeline class is a concrete implementation of the abstract idea of a machine learning pipeline.

Your gene expression data aren’t in the optimal format for the KMeans class, so you’ll need to build a preprocessing pipeline. The pipeline will implement an alternative to the StandardScaler class called MinMaxScaler for feature scaling. You use MinMaxScaler when you do not assume that the shape of all your features follows a normal distribution.

The next step in your preprocessing pipeline will implement the PCA class to perform dimensionality reduction:

In [11]: preprocessor = Pipeline(
   ...:     [
   ...:         ("scaler", MinMaxScaler()),
   ...:         ("pca", PCA(n_components=2, random_state=42)),
   ...:     ]
   ...: )

Now that you’ve built a pipeline to process the data, you’ll build a separate pipeline to perform k-means clustering. You’ll override the following default arguments of the KMeans class:

  • init: You’ll use "k-means++" instead of "random" to ensure centroids are initialized with some distance between them. In most cases, this will be an improvement over "random".

  • n_init: You’ll increase the number of initializations to ensure you find a stable solution.

  • max_iter: You’ll increase the number of iterations per initialization to ensure that k-means will converge.

Build the k-means clustering pipeline with user-defined arguments in the KMeans constructor:

In [12]: clusterer = Pipeline(
   ...:    [
   ...:        (
   ...:            "kmeans",
   ...:            KMeans(
   ...:                n_clusters=n_clusters,
   ...:                init="k-means++",
   ...:                n_init=50,
   ...:                max_iter=500,
   ...:                random_state=42,
   ...:            ),
   ...:        ),
   ...:    ]
   ...: )

The Pipeline class can be chained to form a larger pipeline. Build an end-to-end k-means clustering pipeline by passing the "preprocessor" and "clusterer" pipelines to Pipeline:

In [13]: pipe = Pipeline(
   ...:     [
   ...:         ("preprocessor", preprocessor),
   ...:         ("clusterer", clusterer)
   ...:     ]
   ...: )

Calling .fit() with data as the argument performs all the pipeline steps on the data:

In [14]: pipe.fit(data)
                 Pipeline(steps=[('scaler', MinMaxScaler()),
                                  PCA(n_components=2, random_state=42))])),
                                  KMeans(max_iter=500, n_clusters=5, n_init=50,

The pipeline performs all the necessary steps to execute k-means clustering on the gene expression data! Depending on your Python REPL, .fit() may print a summary of the pipeline. Objects defined inside pipelines are accessible using their step name.

Evaluate the performance by calculating the silhouette coefficient:

In [15]: preprocessed_data = pipe["preprocessor"].transform(data)

In [16]: predicted_labels = pipe["clusterer"]["kmeans"].labels_

In [17]: silhouette_score(preprocessed_data, predicted_labels)
Out[17]: 0.5118775528450304

Calculate ARI, too, since the ground truth cluster labels are available:

In [18]: adjusted_rand_score(true_labels, predicted_labels)
Out[18]: 0.722276752060253

As mentioned earlier, the scale for each of these clustering performance metrics ranges from -1 to 1. A silhouette coefficient of 0 indicates that clusters are significantly overlapping one another, and a silhouette coefficient of 1 indicates clusters are well-separated. An ARI score of 0 indicates that cluster labels are randomly assigned, and an ARI score of 1 means that the true labels and predicted labels form identical clusters.

Since you specified n_components=2 in the PCA step of the k-means clustering pipeline, you can also visualize the data in the context of the true labels and predicted labels. Plot the results using a pandas DataFrame and the seaborn plotting library:

In [19]: pcadf = pd.DataFrame(
   ...:     pipe["preprocessor"].transform(data),
   ...:     columns=["component_1", "component_2"],
   ...: )
   ...: pcadf["predicted_cluster"] = pipe["clusterer"]["kmeans"].labels_
   ...: pcadf["true_label"] = label_encoder.inverse_transform(true_labels)

In [20]: plt.style.use("fivethirtyeight")
   ...: plt.figure(figsize=(8, 8))
   ...: scat = sns.scatterplot(
   ...:     "component_1",
   ...:     "component_2",
   ...:     s=50,
   ...:     data=pcadf,
   ...:     hue="predicted_cluster",
   ...:     style="true_label",
   ...:     palette="Set2",
   ...: )
   ...: scat.set_title(
   ...:     "Clustering results from TCGA Pan-Cancer\nGene Expression Data"
   ...: )
   ...: plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.0)
   ...: plt.show()

Here’s what the plot looks like:

k means clustering pca

The visual representation of the clusters confirms the results of the two clustering evaluation metrics. The performance of your pipeline was pretty good. The clusters only slightly overlapped, and cluster assignments were much better than random.

Tuning a K-Means Clustering Pipeline

Your first k-means clustering pipeline performed well, but there’s still room to improve. That’s why you went through the trouble of building the pipeline: you can tune the parameters to get the most desirable clustering results.

The process of parameter tuning consists of sequentially altering one of the input values of the algorithm’s parameters and recording the results. At the end of the parameter tuning process, you’ll have a set of performance scores, one for each new value of a given parameter. Parameter tuning is a powerful method to maximize performance from your clustering pipeline.

By setting the PCA parameter n_components=2, you squished all the features into two components, or dimensions. This value was convenient for visualization on a two-dimensional plot. But using only two components means that the PCA step won’t capture all of the explained variance of the input data.

Explained variance measures the discrepancy between the PCA-transformed data and the actual input data. The relationship between n_components and explained variance can be visualized in a plot to show you how many components you need in your PCA to capture a certain percentage of the variance in the input data. You can also use clustering performance metrics to evaluate how many components are necessary to achieve satisfactory clustering results.

In this example, you’ll use clustering performance metrics to identify the appropriate number of components in the PCA step. The Pipeline class is powerful in this situation. It allows you to perform basic parameter tuning using a for loop.

Iterate over a range of n_components and record evaluation metrics for each iteration:

In [21]: # Empty lists to hold evaluation metrics
   ...: silhouette_scores = []
   ...: ari_scores = []
   ...: for n in range(2, 11):
   ...:     # This set the number of components for pca,
   ...:     # but leaves other steps unchanged
   ...:     pipe["preprocessor"]["pca"].n_components = n
   ...:     pipe.fit(data)
   ...:     silhouette_coef = silhouette_score(
   ...:         pipe["preprocessor"].transform(data),
   ...:         pipe["clusterer"]["kmeans"].labels_,
   ...:     )
   ...:     ari = adjusted_rand_score(
   ...:         true_labels,
   ...:         pipe["clusterer"]["kmeans"].labels_,
   ...:     )
   ...:     # Add metrics to their lists
   ...:     silhouette_scores.append(silhouette_coef)
   ...:     ari_scores.append(ari)

Plot the evaluation metrics as a function of n_components to visualize the relationship between adding components and the performance of the k-means clustering results:

In [22]: plt.style.use("fivethirtyeight")
   ...: plt.figure(figsize=(6, 6))
   ...: plt.plot(
   ...:     range(2, 11),
   ...:     silhouette_scores,
   ...:     c="#008fd5",
   ...:     label="Silhouette Coefficient",
   ...: )
   ...: plt.plot(range(2, 11), ari_scores, c="#fc4f30", label="ARI")
   ...: plt.xlabel("n_components")
   ...: plt.legend()
   ...: plt.title("Clustering Performance as a Function of n_components")
   ...: plt.tight_layout()
   ...: plt.show()

The above code generates the a plot showing performance metrics as a function of n_components:

k means performance evaluation

There are two takeaways from this figure:

  1. The silhouette coefficient decreases linearly. The silhouette coefficient depends on the distance between points, so as the number of dimensions increases, the sparsity increases.

  2. The ARI improves significantly as you add components. It appears to start tapering off after n_components=7, so that would be the value to use for presenting the best clustering results from this pipeline.

Like most machine learning decisions, you must balance optimizing clustering evaluation metrics with the goal of the clustering task. In situations when cluster labels are available, as is the case with the cancer dataset used in this tutorial, ARI is a reasonable choice. ARI quantifies how accurately your pipeline was able to reassign the cluster labels.

The silhouette coefficient, on the other hand, is a good choice for exploratory clustering because it helps to identify subclusters. These subclusters warrant additional investigation, which can lead to new and important insights.

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