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Before you begin…

Download and install Titus and Numpy. This article was tested with Titus 0.5.14 and Numpy 1.8.2; newer versions should work with no modification. Python >= 2.6 and < 3.0 is required.

Launch a Python prompt and import titus.producer.kmeans, titus.producer.tools, and numpy:

Python 2.7.6
Type "help", "copyright", "credits" or "license" for more information.
>>> import titus.producer.kmeans as k
>>> import titus.producer.tools as t
>>> import numpy as np

Download the Iris dataset, which we use for examples. The first line contains field names (so strip it off with 1:), the last column contains flower classifications (so strip it off with :-1), and the remaining numbers should be converted to np.double). The dataset has 150 rows and 4 columns.

>>> import csv
>>> iris = list(csv.reader(open("iris.csv")))
>>> np_iris = np.array(np.array(iris)[1:,:-1], dtype=np.double)
>>> print np_iris.shape
(150, 4)

Learn about k-means clustering (Lloyd’s algorithm) if you’re not already familiar with it.

Simple k-means

Let’s do the most straightforward example first.

A k-means fit is performed using a KMeans object. It is constructed from a number of clusters (k = 3 for our example) and a Numpy dataset whose first index runs over points (150 in our example) and whose second index runs over dimensions (4 in our example).

>>> kmeans = k.KMeans(3, np_iris)

The cluster centers are initially set to randomly chosen points from the dataset. Your initial clusters will almost certainly be different from the ones below.

>>> print kmeans.centers()
[[6.5, 2.7999999999999998, 4.5999999999999996, 1.5], [7.4000000000000004, 2.7999999999999998, 6.0999999999999996, 1.8999999999999999], [7.7000000000000002, 3.0, 6.0999999999999996, 2.2999999999999998]]

Run the k-means fit by calling optimize, giving it your choice of stopping condition. The simplest condition is to stop when the clusters stop moving (data points stop crossing boundaries between clusters).

>>> kmeans.optimize(k.moving())

For such a small dataset, this should be instantaneous. The resulting clusters may be close to, but not necessarily exactly equal to, the following.

>>> print kmeans.centers()
[[5.0060000000000002, 3.4180000000000001, 1.464, 0.24400000000000002], [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355], [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473]]

To make a model that can be used in a PFA scoring engine, call pfaDocument with an arbitrary cluster type name and three cluster instance names.

>>> clusterTypeName = "Cluster"
>>> clusterNames = ["cluster_" + str(i) for i in range(3)]
>>> pfa = kmeans.pfaDocument(clusterTypeName, clusterNames)

Optionally look at it with t.look.

>>> t.look(pfa, indexWidth=20)
index                data
----------------------------------------
                     {
input                  "input": {"items": "double", "type": "array"},
output                 "output": "string",
cells                  "cells": {
cells,clusters           "clusters": {
cells,clusters,init        "init": [
cells,clusters,in...         {"id": "cluster_0", "center": [5.0060000000000002, 3.4180000000000001, 1.464, 0.24400000000000002]},
cells,clusters,in...         {"id": "cluster_1", "center": [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355]},
cells,clusters,in...         {"id": "cluster_2", "center": [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473]}
                           ]
...

And write it to a file with json.dump.

>>> import json
>>> json.dump(pfa, open("myClusterModel.pfa", "w"))

That’s it! Now try applying it to your favorite dataset.

Testing it

To test the clustering classifier, build a PFAEngine from it and run the original data through it. (In actual model-building, you would probably reserve part of the dataset for training and another part for cross-validation.)

>>> import titus.genpy
>>> engine, = titus.genpy.PFAEngine.fromJson(pfa)
>>> 
>>> for row in iris[1:]:
...     sepalL, sepalW, petalL, petalW = map(float, row[:-1])
...     flowerType = row[-1]
...     result = engine.action([sepalL, sepalW, petalL, petalW])
...     print result, flowerType
...

Stopping conditions

A stopping condition is a function that tells KMeans when it should stop iterating. With large datasets, you may want to tune this to avoid wasting time on unimportant fine modifications of the final result.

The function takes four arguments and returns a boolean: True means keep iterating, False means stop. The arguments are

You could write your own or construct one from the following functors:

In addition, two fake stopping conditions are provided for diagnostics.

Similarity and metrics

PFA can score clusters with user-specified metrics, so Titus can also produce clusters with user-specified metrics. The KMeans constructor takes a metric object with keyword metric; the default is Euclidean:

>>> kmeans = k.KMeans(3, np_iris, metric=k.Euclidean(k.AbsDiff()))

We use “similarity” to mean a function that takes a one-dimensional cluster center component and a data component and returns their distance. The most common of these is the absolute difference,

>>> absdiff = k.AbsDiff()
>>> print absdiff.calculate(-3, 5)
8

And we use “metric” to mean a function that takes a d-dimensional cluster and a d-dimensional data element and returns thier distance. The most common of these is the Euclidean metric, which we build up by composing k.AbsDiff with k.Euclidean:

>>> euclidean = k.Euclidean(k.AbsDiff())
>>> print euclidean.calculate(np.array([[101, 102, 103, 104]]), np.array([[100, 100, 100, 100]]))
>>> 5.4772255750516612

which is the square root of 1 squared plus 2 squared plus 3 squared plus 4 squared.

Metrics are built this way to provide more opportunities for code reuse: similarity objects are one-dimensional differences and metric objects are dimensional combiners. The built-in suite of similarity and metric classes have a pfa method so that the combined function can be transcribed into PFA.

>>> print euclidean.pfa("x", "y")
{'metric.euclidean': [{'fcn': 'metric.absDiff'}, 'x', 'y']}

The built-in similarity classes are:

The built-in metric classes are:

Better starting conditions

Many variations on k-means clustering differ only in how they choose seed clusters. By default, the KMeans object randomly selects numberOfClusters points from the dataset, but you can change this by explicitly setting kmeans.clusters before optimizing. It is a Python list of one-dimensional Numpy arrays.

>>> kmeans = k.KMeans(3, np_iris)
>>> kmeans.clusters = [np.array([0.0, 0.0, 0.0, 0.0]) for i in range(3)]
>>> kmeans.optimize(k.whileall(k.moving(), k.maxIterations(1000)))

If the clustering procedure encounters any nan values while optimizing, it will call kmeans.newCluster(), which calls k.kmeans.randomPoint(), so if you want to change the seeding value for these special cases as well, override one of these methods.

One seeding method that I have found useful for large datasets is to run the clustering procedure on randomly selected subsets of the data, keeping the partially optimized clusters while increasing the size of the subset. This takes advantage of the following facts:

Thus, we iterate on a small subset to get most of the way to the solution and then don’t need as many iterations on the whole dataset to refine the result.

The KMeans object has a randomSubset(subsetSize) method for selecting random subsets (without replacement) and a stepup(condition, base) method to apply the whole procedure. The condition argument is a stopping condition for each subproblem and base is the multiplicative size of each subproblem (base = 2 doubles the size of the dataset with each subproblem).

So for a large dataset, you may want to do something like the following:

>>> kmeans = k.KMeans(3, np_iris)
>>> stoppingCondition = k.whileall(k.moving(), k.maxIterations(1000))
>>> kmeans.stepup(stoppingCondition)
>>> kmeans.optimize(stoppingCondition)

Normalization and other preprocessing

Because k-means clustering assumes clusters to be approximately spherical, it is often necessary to normalize the components of the raw data such that they all have the same range. The metric is a good place to do this because it guarantees that the same normalization will be used for training and scoring.

For instance, suppose we want to shift and scale the Iris dataset so that it is contained within the unit interval. The following expressions would need to be applied to each component:

>>> lows = np_iris.min(axis=0)
>>> highs = np_iris.max(axis=0)
>>> normExprs = ["(x - {lo})/({hi} - {lo})".format(**locals()) for lo, hi in zip(lows, highs)]
>>> print "\n".join(normExprs)
(x - 4.3)/(7.9 - 4.3)
(x - 2.0)/(4.4 - 2.0)
(x - 1.0)/(6.9 - 1.0)
(x - 0.1)/(2.5 - 0.1)

We can provide these expressions directly to the constructor of a KMeans object through the preprocess keyword.

>>> kmeans = k.KMeans(3, np_iris, preprocess=normExprs)
>>> kmeans.optimize(k.moving())

The KMeans object uses these strings to transform its private copy of the dataset and it adds the transformation to the PFA document, thus ensuring that the same transformation is applied to the training procedure and the scoring procedure.

>>> pfa = kmeans.pfaDocument(clusterTypeName, clusterNames)
>>> t.look(t.find({"metric.euclidean": t.Any()}, pfa), indexWidth=10, inlineDepth=6)
index      data
--------------------
           {
metric....   "metric.euclidean": [
metric....     {"fcn": "metric.absDiff"},
metric....     {
metric....       "new": [
metric....         {"/": [{"-": [{"path": [0], "attr": "datum"}, 4.3]}, {"-": [7.9, 4.3]}]},
metric....         {"/": [{"-": [{"path": [1], "attr": "datum"}, 2.0]}, {"-": [4.4, 2.0]}]},
metric....         {"/": [{"-": [{"path": [2], "attr": "datum"}, 1.0]}, {"-": [6.9, 1.0]}]},
metric....         {"/": [{"-": [{"path": [3], "attr": "datum"}, 0.1]}, {"-": [2.5, 0.1]}]}
                 ],
metric....       "type": {"items": "double", "type": "array"}
               },
metric....     "clusterCenter"
             ]
           }

Any PFA functions that have a Numpy equivalent are available for preprocessing because the expression strings are converted into Numpy operations using Python’s eval and PFA operations using titus.producer.expression.pfa.

We could, for instance, apply a logarithm to each component, which is m.ln in PFA and np.log in Python:

>>> normExprs = ["m.ln(x)"] * np_iris.shape[1]
>>> print "\n".join(normExprs)
m.ln(x)
m.ln(x)
m.ln(x)
m.ln(x)
>>> kmeans = k.KMeans(3, np_iris, preprocess=normExprs)
>>> kmeans.optimize(k.moving())
>>> pfa = kmeans.pfaDocument(clusterTypeName, clusterNames)
>>> t.look(t.find({"metric.euclidean": t.Any()}, pfa), indexWidth=10, inlineDepth=3)
index      data
--------------------
           {
metric....   "metric.euclidean": [
metric....     {"fcn": "metric.absDiff"},
metric....     {
metric....       "new": [
metric....         {"m.ln": {"path": [0], "attr": "datum"}},
metric....         {"m.ln": {"path": [1], "attr": "datum"}},
metric....         {"m.ln": {"path": [2], "attr": "datum"}},
metric....         {"m.ln": {"path": [3], "attr": "datum"}}
                 ],
metric....       "type": {"items": "double", "type": "array"}
               },
metric....     "clusterCenter"
             ]
           }

Modifying the scoring procedure

We have seen the kmeans.pfaDocument(clusterTypeName, clusterNames) method, which turns the final result of clustering into the following scoring procedure:

  1. Given a d-dimensional array, find the closest cluster.
  2. Return the clusterName of the closest cluster (a string).

You may want to do something different, like return the distance to the closest cluster, or compare the input data to two different cluster sets and report which of the two cluster sets yields a smaller distance (et cetera, ad infinitum).

That is a general PFA-building problem, which is the topic of most articles on this website. To approach this problem from a clustering point of view, let’s start by looking at the PFA generated by the default method.

>>> clusterTypeName = "Cluster"
>>> clusterNames = ["cluster_" + str(i) for i in range(3)]
>>> pfa = kmeans.pfaDocument(clusterTypeName, clusterNames)
>>> t.look(pfa["action"], indexWidth=10)
index      data
--------------------
           {
attr         "attr": {
attr,mo...     "model.cluster.closest": [
attr,mo...       "input",
attr,mo...       {"cell": "clusters"},
attr,mo...       {
attr,mo...         "params": [
attr,mo...           {"datum": {"items": "double", "type": "array"}},
attr,mo...           {"clusterCenter": {"items": "double", "type": "array"}}
                   ],
attr,mo...         "ret": "double",
attr,mo...         "do": {
attr,mo...           "metric.euclidean": [{"fcn": "metric.absDiff"}, "datum", "clusterCenter"]
                   }
                 }
               ]
             },
path         "path": [{"string": "id"}]
           }

The primary function here is model.cluster.closest, which takes a data array (input), a set of clusters ({"cell": "clusters"}), and a metric function (parameters datum and clusterCenter, which are passed to metric.euclidean with metric.absDiff similarity, as described above).

The model.cluster.closest function returns a record representing the cluster, so the attr-path extracts its id. The record’s type is defined in the declaration of the cell named clusters.

>>> t.look(pfa["cells"]["clusters"], indexWidth=10)
index      data
--------------------
           {
type         "type": {
type,items     "items": {
type,it...       "fields": [
type,it...         {"type": {"items": "double", "type": "array"}, "name": "center"},
type,it...         {"type": "string", "name": "id"}
                 ],
type,it...       "type": "record",
type,it...       "name": "Cluster"
               },
type,type      "type": "array"
             },
init         "init": [
init,0         {"center": [5.0060000000000002, 3.4180000000000001, 1.464, 0.24399999999999999], "id": "cluster_0"},
init,1         {"center": [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355], "id": "cluster_1"},
init,2         {"center": [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473], "id": "cluster_2"}
             ]
           }

A Cluster is a record that contains an array named center and a string named id. All the k-means procedure did was set the initial values of the centers and supply the appropriate metric.

To make custom scoring procedures, we need a function that only outputs the relevant data. The method kmeans.pfaValue(clusterNames) returns just the snippet in init above:

>>> justCenters = kmeans.pfaValue(clusterNames)
>>> t.look(justCenters, indexWidth=10)
index      data
--------------------
           [
0            {"center": [5.0060000000000002, 3.4180000000000001, 1.464, 0.24399999999999999], "id": "cluster_0"},
1            {"center": [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355], "id": "cluster_1"},
2            {"center": [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473], "id": "cluster_2"}
           ]

kmeans.centers() returns the centers with no PFA formatting at all:

>>> print kmeans.centers()
[[5.0060000000000002, 3.4180000000000001, 1.464, 0.24399999999999999], [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355], [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473]]

and kmeans.metric.pfa(datumRef, clusterRef) returns the metric as PFA code, including any normalization or preprocessing that was passed through the preprocess option (described above):

>>> kmeans.metric.pfa("datum", "clusterCenter")
{'metric.euclidean': [{'fcn': 'metric.absDiff'}, 'datum', 'clusterCenter']}

Building a scoring engine using PrettyPFA

PrettyPFA is a method of building PFA from more human-friendly source code. The default scoring procedure can be built as follows:

>>> import titus.prettypfa
>>> 
>>> pfa = titus.prettypfa.jsonNode("""
... types:
...   record(Cluster,
...          id: string,
...          center: array(double))
... input: array(double)
... output: string
... cells:
...  clusters(array(Cluster)) = CLUSTERS_HERE
... action:
...   model.cluster.closest(
...     input,
...     clusters,
...     fcn(datum: array(double), clusterCenter: array(double) -> double)
...       METRIC_HERE).id
... """, check=False)
>>> 
>>> t.assignAt("CLUSTERS_HERE", pfa, kmeans.pfaValue(clusterNames))
>>> t.assignAt("METRIC_HERE", pfa, kmeans.metric.pfa("datum", "clusterCenter"))

Variations through PrettyPFA

Suppose you want to report only the distance to the closest cluster. The output of the model should then be double and you would use the metric to compute the distance again.

>>> pfa = titus.prettypfa.jsonNode("""
... types:
...   record(Cluster,
...          id: string,
...          center: array(double))
... input: array(double)
... output: double        // NOTE: double!
... cells:
...   clusters(array(Cluster)) = CLUSTERS_HERE
... action:
...   var bestClusterCenter =
...     model.cluster.closest(
...       input,
...       clusters,
...       fcn(datum: array(double), clusterCenter: array(double) -> double)
...         METRIC_HERE)["center"];
...   METRIC_HERE_2;
... """, check=False)
>>> 
>>> t.assignAt("CLUSTERS_HERE", pfa, kmeans.pfaValue(clusterNames))
>>> t.assignAt("METRIC_HERE", pfa, kmeans.metric.pfa("datum", "clusterCenter"))
>>> t.assignAt("METRIC_HERE_2", pfa, kmeans.metric.pfa("input", "bestClusterCenter"))

Suppose you have two cluster sets, named good and bad (KMeans objects trained from suggestively labeled datasets), and you want to find out which of the two is closer for each input.

>>> pfa = titus.prettypfa.jsonNode("""
... types:
...   record(Cluster,
...          id: string,
...          center: array(double))
... input: array(double)
... output: string
... cells:
...   good(array(Cluster)) = GOOD_HERE;
...   bad(array(Cluster)) = BAD_HERE;
... action:
...   var closestGoodCenter =
...     model.cluster.closest(
...       input,
...       good,
...       fcn(datum: array(double), clusterCenter: array(double) -> double)
...         METRIC_HERE)["center"];
...   var closestBadCenter =
...     model.cluster.closest(
...       input,
...       good,
...       fcn(datum: array(double), clusterCenter: array(double) -> double)
...         METRIC_HERE)["center"];
...   if (METRIC_HERE_2 < METRIC_HERE_3)
...     "good"
...   else
...     "bad";
>>> """, check=False)
>>> 
>>> t.assignAt("GOOD_HERE", pfa, good.pfaValue(clusterNames))
>>> t.assignAt("BAD_HERE", pfa, bad.pfaValue(clusterNames))
>>> t.assignAt("METRIC_HERE", pfa, good.metric.pfa("datum", "clusterCenter"))
>>> t.assignAt("METRIC_HERE_2", pfa, good.metric.pfa("input", "closestGoodCluster"))
>>> t.assignAt("METRIC_HERE_3", pfa, good.metric.pfa("input", "closestBadCluster"))

Et cetera, ad infinitum.

Adding structure to the clusters

In the simple case, the cluster records contain only two pieces of information: a center point and an id label. You can add anything you want to it and use that information in the scoring procedure. For instance, you could even attach a whole secondary model to each cluster and evaluate the secondary model corresponding to the cluster (geometry-based segmentation of models).

Here is a common case: reporting the training population associated with the matching cluster. This requires information from the training process, so you need to get it from the KMeans object.

>>> pfa = titus.prettypfa.jsonNode("""
... types:
...   record(Cluster,
...          id: string,
...          center: array(double),
...          population: int)        // note the new field
... input: array(double)
... output: int
... cells:
...  clusters(array(Cluster)) = CLUSTERS_HERE
... action:
...   model.cluster.closest(
...     input,
...     clusters,
...     fcn(datum: array(double), clusterCenter: array(double) -> double)
...       METRIC_HERE)["population"]
... """, check=False)
>>> 
>>> t.assignAt("CLUSTERS_HERE", pfa, kmeans.pfaValue(clusterNames, populations=True))
>>> t.assignAt("METRIC_HERE", pfa, kmeans.metric.pfa("datum", "clusterCenter"))
>>> 
>>> t.look(t.find(t.Min(init=t.Start()), pfa)["init"], indexWidth=10, inlineDepth=1)
index      data
--------------------
           [
0            {
0,popul...     "population": 50,
0,id           "id": "c1",
0,center       "center": [5.0060000000000002, 3.4180000000000001, 1.464, 0.24399999999999999]
             },
1            {
1,popul...     "population": 62,
1,id           "id": "c2",
1,center       "center": [5.9016129032258062, 2.7483870967741937, 4.3935483870967742, 1.4338709677419355]
             },
2            {
2,popul...     "population": 38,
2,id           "id": "c3",
2,center       "center": [6.8500000000000005, 3.0736842105263156, 5.742105263157895, 2.0710526315789473]
             }
           ]

In general, you’ll get data from other sources, such as labeling in semi-supervised clustering. The output of model.cluster.closest is a record that you define, so you can do whatever you want.