Classification, clustering, and decisions

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Naive Bayes

Some background

A certain Reverend Thomas Bayes said a number of things about probability, used mostly in Bayesian inference in statistics, and in Naive Bayes classification.

Classification, in general, is usually based on a trained model (or sometimes hardcoded assumptions):

  • deciding on a set of distinct target classes (say, exclusives as 'spam' vs. 'not spam', or rough article subjects (preferably exclusive, or classification itself does not make that much sense, although the value-for-class assignments still may)
  • deciding some features to look at, that classification will be based on
  • learning the importance of features to each class (often using good examples for each class)

Given something unseen to classify, you extract its features the same way you did for the training set, and see which class compares best. Classification is regularly many fuzzy measures of fit followed by a hard choice of which seems best.

Naive Bayes refers to one of the simpler algorithms that does classification this way.

Its definition has a few notational conventions:

  • Each training class is referred to as c in the set of classes C (
  • We eventually want to judge an unseen document D
  • and what we want for D is is the class for which the probability the document fits that class is highest

You can express such Bayesian classification as:

the c for which P(c|D) is highest 

or, a bit more formally yet,

c = maxarg( P(ci|D) )

Bayes's rule

The P(c|D) above is not obvious to calculate, because it immediately asks us for a best choice for a document, and a thorough answer of that will probably depend on multiple pieces of information: document's features, all classes, and all documents.

So we need to break the problem down before we can give an estimation of the probability P(c|D), and the first step is to apply Bayes's rule (which pops up in various other types of probability calculations and estimations).

             P(D|c) * P(c)
P(c|D)  =  -----------------

...for each class c in C.

Looking at the parts:

P(c) is the base probability of each class, that is, the probability that a document is in class c given no other information.

This is sometimes assumed to be equal for all classes (in which case it falls away),
sometimes it is estimated, for example using the assumption that the amount of training examples in each class is a direct indication of how commonly an item falls into that class.

P(D) is the chance of finding the document, in general.

This is usually assumed to be independent of the classification problem.
because of that, and because it is hard to realistically estimate for an unseen document, it is often assumed to be equal for all documents (so falls away). P(D|c) is the interesting one. It asks for the probability that a particular document in a particular class.

This is still abstract, and not yet a directly calculable answer, but it is a smaller problem than we had a few paragraphs earlier: Unlike the earlier form (P(c|D)) which deals with an immediate choice between alternatives, we now deal mainly/only with an calculation/estimation of fit, of 'how well does a specific Document fit in every specific class?' .
This degree of fit for a document in a class can be done with any method that judges similarity, as long as it is (fairly) stable/consistent between documents, and it can be calculated (roughly) equally well for seen and unseen documents.

Long story somewhat shorter, applying Bayes' rule reduces the classification problem from

"Choose best class immediately"


"find the class for which our estimation of fit is highest" (by making an estimator that can judge individual documents, and can do so stably / comparably well for all documents)

Put another way, from:

maxarg( P(ci|D) )


maxarg( P(ci) * P(D|ci) )

or, assuming equal base-class probabilities, to:

maxarg( P(D|ci) )

Features, Words, and Naivety

So let's talk about an implementation.

If we want to calculate a class's fit based on features, we need to choose those features, and a method to calculate an arbitrary document's fit to those features.

It's nice if each feature is fairly robust in that it does not react in an unnecessarily noisy way, and productive for the classification task at hand.

In Naive Bayes introductions, the choice of features is often words (some introductions talk mainly of words, some talk more broadly about features), and the probabilities are based on the relative occurrence of each chosen word.

Words are a specific implementation choice, but are a nice introduction, in part because it speaks to the imagination in cases such as spam (even though it's limited for this, given that spammers know and try to defeat this model).

If you're programming such a toy example, then the amount of words is often reduced to maybe a thousand. The actual selection is interesting, in that they ought to be distinguishing within the training set ("the" may be common but not useful), but not be so unusual that they are rare within unseen documents.

There are cleverer ways of choosing words, and you probably want to consider the Zipf distribution of words, but to keep the example simple, let's say we take the most common 1000 in the training set, minus some obvious stopwords.

Naive Bayes is often explained by saying we want a probability based on n features in feature set F: P(c|f1..fn), which Bayes-inverses and denominator-eliminates to:

P(c) * P(f1..fn|c)

...which roughly means "how do features in this new document compare to the one for the class?"

Technically, P(f1..fn|c) should expand to a very long formula, as each feature depends on others. That's where the naivety of Naive Bayes comes in: the Naive Bayes assumption is simply that all features are independent of all others. This is rarely true, but is much simpler to work with, is evaluated much faster, and works better than you might expect. (this also effectively makes it a bag-of-words model)

It means calculation of the probability of the class based of the features simplifies to a simple multiplication of individual feature values / probabilities. Given there are n features:

P(c) * Πi=1 to n P(fi|c)

Note that the words-as-features setup is a #Bag_of_words.2Ffeatures_.28assumption.29 assumption: it plus the naivety means that word order is ignored. This means that potentially indicative phrases, collocations, and such are represented only weakly at best, by their words' independent probabilities being slightly higher. Word n-grams are one possible fix, but will likely run you into a sparseness problem since n-grams are implicitly rarer than single words. Even just bi-grams may be problematic - if you take all possible ones from a large set of documents, you'll find that most documents have a tiny fraction of the overall real use.

Bayesian classification as a procedure...

Naive Bayes training comes down to:

  • Deciding which set of features (F) you will use.
  • Train, which calculates characteristics for each class, namely:
    • P(c) for each class (see above), and more importantly,
    • all P(f|c): the probability of each feature f in each class c

Later, when classifying an unseen document, you calculate P(c) * P(f1..fn|c) for it:

  • for each c in C:
    • take P(c), times
    • the calculated probability of each feature in class c, i.e. each P(f|c) based on the document.
  • Choose the one with the highest probability, and you've got a classifier.

Choosing your features, and further discussion

Naive Bayes with simple word features works fairly well on various types of text, since it essentially notices differences in vocabulary use, the theory being that that is indicative of the document class. Common words can be useful when they aren't uniformly common between all classes, uncommon ones since they tend to up the probability of just a few classes.

You could simply use all words in all documents and end up with tens of thousands of features, although it would help speed to prune that a little. Which ones to leave is a choice you can be halfway smart about - although note that too much tweaking just means it will work better on the training data, with no guarantees for unseen data.

Also note that when you pre-process the training data, you need to process the unseen documents the same way. When choosing the features, you should consider what this may do to unseen documents.

A feature probability of zero is evil to calculations as it randomly destroys comparability in the both training and classification steps. It can happen fairly easily, such as when a word in the vocabulary doesn't occur in a document. A common solution is to cheat and pretend you did actually see it a little. There are a few different ways; some disturb the relative probabilities less than others.

You can add other features. The main property they should have is that their probabilities compare well, are in the same sort of scale. You could for example add important words a few times, a hackish way of weighing them more. You could for example add select word bigrams or add character n-grams.

You can alter the model further, observing that what you actually do is per-class strengthening or weakening of a probability that it will win. It doesn't even have to be a simple probability, even; you could for example include sentence length, which is not a direct probability, so you would have to e.g. remember the average sentence length in each class, calculate the average for the unseen document and define some metric of similarity. This isn't Naive Bayes anymore, but as a probability-based classifier it may work a little better if you choose your alterations well.


This example pseudocode uses only word features, and some basic add-one(verify) smoothing to avoid the zero problem.

Training (word features)

U = feature universe
D = document set (text,class pairs)
C = set of classes in D
foreach c in C:
  BaseProb(c) = |D with class c| / |D|
  Text = concatenation of documents in current class
  N    = |tokens in Text|
  foreach token t in U:
    P(t|c) = ( numoccurence(t in Text) + 1 ) / (numtokens(Text)+|U| )

Classifying (word features)

foreach c in C:
  foreach token in Document
    p = p*P(token|c)

...then collect each of these probabilities and choose the class with the highest probability.

In reality, the probabilities quickly become very small and you run into the problem that standard 32-bit or 64-bit floating point numbers cannot accurately contain them. A common workaround for this is to add the logs of the probabilities (yielding negative numbers) instead of multiplying the probabilities. These values are much less likely to scale out of control, and they are as just as monotonous as the probabilities.


  • Parzen classifier
  • Backpropagation classifier

Evaluating classifiers

Support Vector Machines

Support-vector regression

Kernel method



Clustering groups a few related sub-problems, including

cluster formation - organizing into clusters
cluster segmentation - dealing with boundaries (often using cluster centers)
labeling - assigning meaningful names
for the (relatively few) cases where this makes sense to estimate
deciding how many groups to have in the result
evaluation of a solution (possibly feeding back into the previous point)

Formally, the simplest clustering ca be described as:

  • you have a set of n data objects, call it D = { d1, ..., dn }
  • in its simplest shape, a clustering result is a disjoint partitioning of D
  • which makes clustering itself a function, mapping each datapoint to a cluster number/label that indicate membership of said cluster

The input data is often either

  • a set of a points in a many-dimensional space, usually a vector space, plus a metric to calculate distance between them, OR
  • a set of already-chosen distances
preferably complete, but depending on how it's made it might be sparse, and it may be easier to do some fuzzy statistical estimation than to ask people to complete it

The latter may well be in the form of a distance matrix / similarity matrix.

A bunch of methods given datapoints-plus-metric convert to that internally, but starting data-plus-metric is often a little more flexible up front - both for data massaging, and sometimes for implementation reasons.

That said, the choice of metric takes care, because there are many ways to accidentally put some bias into the metric.

Many methods look at element-to-element similarities/dissimilarities, while a few choose to be more involved with the data that comes from (e.g. some Maximum-likelihood-based methods common in bioinformatics)


Hard, soft, and fuzzy clustering

Hard clustering means each item should be assigned to a single group. This is essentially a partitioning.

Soft clustering means something can belong to more than one group.

Regularly used for data known to be too complex to be reduced cleanly with hard clustering, such as when there are closeby, overlapping, or ambiguous groups.

Soft clustering is generally understood as boolean soft clustering: something can belong to one or more clusters, but there are no degrees.

Fuzzy clustering is soft clustering plus degrees of membership.

This means intermediate results are effectively still moderately high-dimensional data, you often still have to make a decision about exclusion, thresholds or such (preferably within the algorithm, to have all information available).

If you don't make such a decision, the result more resembles dimensionality reduction.

Agglomerative versus divisive

Agglomerative clustering usually starts with each item in its own cluster and merges them where it seems a good idea.

Divisive clustering usually starts with every item in a single cluster and iteratively splits them as it sees fit.

The difference seems to lie largely in what side they err on in unclear cases.(verify)

Hierarchical clustering

Hierarchical clustering creates a tree of relations, often by an process where we keep tracks of how things join, rather than just assimilate things into a larger blob.

Hierarchical clusterers can be flexible, in that their results can partition into an arbitrary number of groups (by choosing the depth at which there are that amount of groups).

Depending on the data these results contain may also be useful as an approximation for fuzzy clustering. They may also be a little more helpful in cluster stability tests.

Some algorithms record and retain he distances of (/stress at) each such joint. These can be interesting to visualize (think dendrograms and such), and to effectively allow the amount-of-cluster choice to be made later (think threshold in a dendrogram).

Notes on....

Group number choice

Some algorithms try to decide on a suitable number of target groups, but many require you to choose an exact number before they get started.

This number is difficult to decide since there is usually is no well-defined, implicit, calculable best choice.

This is a problem particularly in hard clustering, because any decision of group membership is very final. The membership of bordercases may not be stable under even the slightest amount of (sample) noise.

Things you can do include:

  • use evaluation to measure the fitness of a solution (or sub-solutions while still clustering), based

Note that such a metric in itself is only a relative value in a distribution you don't know - you'll often have to calculate the fitness for many solutions to get a still-vague idea of fitness.

  • In the case of hard clustering, you can intentionally add some noise and see how much the membership of each item varies - and, say, report that as the confidence we have in a choice.
  • use some type of cross-validation

Inter-cluster and intra-cluster comparisons; susceptibilities

Depending on the algorithm, you often want to be able to compare

  • items to clusters (agglomerative and divisive decisions)
  • clusters to clusters (e.g. in hierarchical decisions)
  • items to items (e.g. for centroid/medoid decisions)

Cluster-to-cluster distances are most interesting to hierarchical clustering, and can be calculated in a number of ways (usually hardwired into the algorithm), including:

  • Single-link, a.k.a. minimum linkage or nearest linkage
the most similar combination (lowest distance) of possible comparisons
more susceptible to over-chaining than most other methods
  • Complete-link , a.k.a. max/farthest linkage
uses the least similar (largest distance) combination of possible comparisons
often gives a non-chained, more equally divided clusters than single-link
outliers may have disproportional influence
  • Average-link (a.k.a. group-average, a.k.a. Group-average (agglomerative) clustering (GAAC)) - average of distances between all inter-cluster pairs
less sensitive to outliers than complete-link, less sensitive to inversion than centroid approaches.
  • Centroid approaches use a calculated average for comparison to a cluster
quite susceptible to inversions (verify)
  • Medoid approaches try to use a representative item for comparison to a cluster
somewhat susceptible to inversions (verify)
  • Density-based methods care more about local density of items and less directly about the exact distances involved
  • Ward's method
based on Ward criterion, a.k.a. the Ward minimum variance criterion

Potential problems:

  • outliers
    • including a single outlier may drastically change comparisons to that group.
    • The timing of their inclustion can have significant effects on the result.
  • the chaining effect refers to algorithms doing a chain/string of assignments of to a group. The concept is clearest in
  • Inversions (sometimes 'reversal' or 'non-monotonicity') - describes when similarity values do not decrease monotonously in a series of iterations
    • easily happens when a process makes decisions based on centers that move in the process of clustering (such as in many centroid-style processes) particularly when combined with cases where there is no clear clustering solution.

See also:

TODO: read

on convergence
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Convergence is a nontrivial check in many algorithms.

You could check whether the group assignments have not changed, but this is sensitive to oscillations, resulting in a premature report of convergence and/or a failure to converge (depending somewhat on logic and data size).

A simple threshold is arbitrary since the error values often depend on the scale of the data values (which is not very trivial to correct for(verify)).

This is occasionally solved by error minimization criteria, for example minimization-of-the-sum-of-squares.

There are some details to this. For example, reallocating a point between clusters, various methods consider only the error decrease in the target cluster - while the solution's total error may increase. It usually still converges, but the total error decreases with a little more oscillation, which "no significant improvement in the last step" terminating criterion may be sensitive to (though arguably it's always more robust to check whether the error decrease is roughly asymptotic with the minimum you presume it'll get to).

The idea resembles Expectation Maximization (EM) methods in that it tries to maximize the probability of the clusters being the correct by minimizing the energy/error.

Purely random initial positioning may cause the local minimum problem. Smartly seeding the initial centroids helps and need not be too computationally expensive - and in fact helps convergence; see e.g. k-means++.

Alternatively, you could run many versions of the analysis, each with random initial placement, and see see whether (and/or to which degree) the results are stable, but this can be computationally expensive.

Robustness in hard clustering

Particularly hard clusterers are often not robust against even minor variations in the data. That is, separate data sets that are highly correlative may lead to significantly different results; areas in which membership is borderline flip-flip under the tiniest (sample) noise.

You can evaluating a solution for stress (or correlate distances it implies to the original data e.g. in hierarhical data), though in itself this is only a general thing. It is meaningful the same measre from other clusterings of the same data, meaning that you *can* roughly compare different solutions for expression of the original data, but only in a roughly converging way.

One trick is to cause the problem and test how varied results will be over mild variations over the data. You can for example repeat the clustering some amount of times with some noise, and record how often things change membership.

You can repeat the clustering omitting random pieces of data to lessen the effect of outliers - pieces of data that do not agree with the rest.

You can even aggregate the results from these runs and combine them into a sort of fuzzy cluster result that can show you instabilities, and/or converge on a clusters amount choice.

Silhouette coefficients
Davies-Bouldin index
Gap statistic

Implementation notes


The k-means problem is finding a cluster labelling for a given amount of clusters (k) with minimal error, where the error function is based on the the within-group sum of squares.

(For completeness, that means for all elements in a group, calculate the square of the euclidean distance to the centroid, and sum up all these squares, which gives per-cluster error values. Various convergence checks will want to know the sum of these errors)

Most implementations are iterative and look something like:

  • Position k cluster centroids (at random, or sometimes slightly more cleverly)
  • For each element, assign to the nearest centroid
  • Recalculate (affected) centroid means (and often the error/energy at the same time)
  • Check whether the moved centroids change the element assignments.
If so, iterate
If no change, we have converged and can stop

Of iterative clustering methods, k-means is the simplest and many others can be said to be based on it.

K-means gives better better results if the value for k is a good choice, representative for the data. (it's not unusual to try various k and test them)

The common 'nearest cluster' criterion will avoid attraction of multiple clusters, and the whole will converge to a decent solution for k groups.


  • results are sensitive to initial placement, and it is easy to get stuck in a local minimum.
It is not unusual to run the clustering various times with different starting clusters and see how stable the clustering is.
  • If k is not representative of the structure in the data the solution may not be satisfying at all. This is partly caused by, and partly independent of, the fact that there may be various possible stable clusterings.
  • the simple distance metric means we say the shape for inclusion is always a circle

Average-case runtime is decent because it's a fairly simple algorithm. Worst-case runtime, for fairly pathological datasets, is fairly quite high. There are faster approximations of k-means that you may wish to consider.

You can tweak k-means in various ways. For example, you can assign weights (based of frequency, importance, etc) to elements to affect the centroid mean recalculation.

See also:


  • ISODATA (Iterative Self-Organising Data Analysis Technique Algorithm) builds on k-means and tries to be smart in initial positions and in variations of k(verify).
  • H-means is a variation on k-means that recalculates the centroid only after a complete iteration over all the items, not after each reassignment.
Seen one way, it checks for error decrease less often, which makes it a smidge more sensitive to local minima and perhaps doesn't converge as nicely.
Although the difference in practice tends to be slight, k-means tends to be the slightly safer (and much more common) choice, even if the order it handles elements in is a different kind of bias that h-means avoids.
Canopy clustering

Bisecting k-means
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hard c-means
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UPGMA = Unweighed Pair Group Method using Arithmetic averaging

WPGMA = Weighed Pair Group Method using Arithmetic averaging

(These specific names/abbreviations come from Sneath and Sokal 1973)

UPGMA assigns equal weight to all distances:

D((u,v),w) = (nu*D(u,w) + nv*D(v,w)) / (nu+nu)

WPGMA uses:

D((u,v),w) = (nu*D(u,w) + nv*D(v,w)) / 2

In the unweighed variant, the two things being combined weigh equally, in the weighed variant, all leaves that are part of a cluster weigh in as much as the other part.

Bottom-up combiners working from a difference matrix, combining whatever leaf/cluster distance is minmal, then recalculating the difference matrix.

It's not too hard too argue this terminology is a little arbitrary

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Unweighed Pair Group Method using Centroids, and Weighed Pair Group Method using Centroids

(the specific names/abbreviations come from Sneath and Sokal 1973)

Fuzzy c-means (FCM)
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  • Type: fuzzy clustering (not really partitioning anymore)

Method: Like k-means, but weighs centroid recentering calculations by fuzzy distance to all data points

Fuzzy k-means
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Shell clustering
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Basic fuzzy c-means would include by a radius - a sphere.

There are various other options, including:

  • fuzzy c-quadric shells algorithm (FCQS) detects ellipsoids
  • fuzzy c-varieties algorithm (FCV) detects infinite lines (linear manifolds) in 2D
  • adaptive fuzzy c-varieties algorithm (AFC): detects line segments in 2D data
  • fuzzy c-shells algorithm (FCS) detects circles
  • fuzzy c-spherical shells algorithm (FCSS) detects circles
  • fuzzy c-rings algorithm (FCR) detects circles
  • fuzzy c-rectangular shells algorithm (FCRS) detects rectangles
  • Gath-Geva algorithm (GG) detects ellipsoids
  • Gustafson-Kessel algorithm (GK) detects ellipsoids of roughly the same size

Partitional, medoid-based


Partitional, medoid-based



AGglomerative NESting



DIvisie ANAlysis





BITCH (balanced iterative reducing and clustering using hierarchies)



Clustering Using REpresentatives



"Rock: A robust clustering algorithm for categorical attributes"


Hybrid, Hierarchical

"CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic Modeling"



The assumption that real objects will always be a dense cloud of points more easily rejects random points as noise/outliers (even if relatively close).

It can also deal decently with closeby nonlinear clusters, if separated cleanly.


Similar to DBSCAN


Fuzzy, density-based

Clustering by Committee (CBC)


Based on responsive elements (a comittee) voting on specific outcomes.

See also:

Principal Direction Divisive Partitioning (PDDP)
Information Bottleneck

See also:

Agglomerative Information Bottleneck

See also:

Expectation Maximization Clustering

Related fields

See also

Fuzzy coding, decisions, learning

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On bias

Methods / algorithms / searchers

Decision trees

Pruning (ID3, others)
Rule Post-pruning; C4.5

Instance-based learning

Bayesian learning

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

Bayesian learning is a general probablistic approach, mostly specifically used as a probablistic classifier.

Mathematically it is based on any observable attribute you can think of, and the math requires Bayesian inversion (see below).

Many basic implementations also use the Naive Bayes assumption (see below), because it saves a lot of computation time, and seems to work almost as well in most cases.

Bayesian classifier

Bayes Optimal Classifier
Naive Bayes Classifier

Bayesian (Belief) Network

Markov Models, Hidden Markov Models

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Something like (the simplest possible) Bayesian Belief Networks, but geared to streams of data. Can be seen as a state machine noting the likeliness of each next step based on a number of preceding steps.

The hidden variant only shows its output (and hides the model that produces it), the non-hidden one shows all of its state.

Simple ones are first-order