Data modeling, restructuring, analysis, fuzzy cases, learning

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This is more for overview of my own than for teaching or exercise.

Overview of the areas

Arithmetic · 'elementary mathematics' and similar concepts
Set theory, Category theory
Geometry and its relatives · Topology
Elementary algebra - Linear algebra - Abstract algebra
Calculus and analysis
 : Information theory · Number theory · Decision theory, game theory · Recreational mathematics · Dynamical systems · Unsorted or hard to sort

Math on data:

  • Statistics as a field
some introduction · areas of statistics
types of data · on random variables, distributions
Virtues and shortcomings of...
on sampling · probability
glossary · references, unsorted
Footnotes on various analyses

Other data analysis, data summarization, learning

  • Data modeling, restructuring, analysis, fuzzy cases, learning
Statistical modeling · Classification, clustering, decisions · dimensionality reduction · Optimization theory, control theory
Connectionism, neural nets · Evolutionary computing

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Overall concepts, Glossary

Types of problems

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The types of tasks/problems are often broadly sorted into some distinct types. ...which says little about how they are solved.

  • Regression
predicts a continuous variable
  • Clustering
yields groups of (mutual) similarity, and dissimilarity from other groups
clustering may not deal well with future data of the same sort, unless some care has been taken, so isn't necessarily a learner/predicter
  • Vector quantization
Discretely dividing continuous space into various areas/shapes
which itself can be used for decision problems, labeling, clustering, etc.
  • Dimensionality reduction
what: projecting attributes into lower-dimensional data
with the idea that the resulting data is (hopefully) comparably predictive/correlative (compared to the original)
The reason is often to eliminate attributes/data that may be irrelevant or too informationally sparse
see also #Ordination.2C_Dimensionality_reduction.2C_Factor_Analysis.2C_Multivariate_analysis
  • Feature extraction: discovering (a few nice) attributes from (many) old ones, or just from data in general.
Often has a good amount of overlap with dimensionality reduction
  • Control optimization problem - finding the best action of control in every (known) state of a system

Slightly more specifically:

  • Reinforcement learning has a few different takes.
You can see it as any task where actor learns how to do things in an environment
You can also see it as implementations that do so in more specific reward systems, which is an idea that comes back in a handful of different learning algorithms

  • Structured prediction refer to prediction problems that predict things more complex than single values
specifically those that want multiple labels (amount potentially varying with the input), and their prediction relies on intermediate workings of the others
which actually addresses implementation a little more than most others, in that running independent classifiers will give an answer but not be a structured prediction

  • collaborative filtering is a relatively new one, from the internet era
consider e.g. people categorizing music, movies, products
the contribution from each user is very sparse

  • others...

Related fields

Descriptors of learning systems

Model-based versus model-free systems

When you create a model of what to learn and where from, this can save a lot of computation, but will often not be adaptive to things that don't fit that model well.

Model-free solutions avoid assumptions and their bias and can learn more, but may be infeasible because of not enough information to stably learn from, or requiring too much time and computation to optimize parameters well enough.

Supervised , unsupervised, and more

Classically we split things into supervised and unsupervised, and perhaps call thoes paradigms.

Supervised usually refers to the training processes that work (only) using already-annotated training data.

For example many document classification methods, least-squares fitting, basic neural network back-propagation
This often means manually generating training data, which is often a slow process, and not always

Unsupervised refers to processes that work without intervention, often meaning they work without annotation.

For example, clustering documents based on similarity, self-organizing maps

We've thought up other terms since then to refer to variations.

Semi/lightly supervised usually means there is an iterative process, which needs only minimal human intervention, e.g. to require manual intervention only for unclear cases.

Self-supervised learning often refers to cases where you can automatically refine a solution, e.g. by automatically creating labels for further training.

This tends to work better for relatively narrowly defined tasks, say, learning to segment humans, within a piece of video at a time.

Note that self-supervised can variably be seen as

applying automatic supervision to an unsupervised question
a flavour of lightly supervised, as it can be used to label a bunch of data more quickly

Reinforcementt learning could be another entry in that list, in that it's a fairly differerent category of expressing problems and solutions.

Problem-wise, RL is those where an actor learns how to do things in an environment.

There generally is no labeled data, but there is a way of saying what is preferable and what fails to work. Not in the form of "this was the correct immediate response (input/output pair)", and more that the eventual outcome was good or bad.

Classically, RL it was a bit narrower, often expressed in things like like a markov decision process, in part because that was a basic and clear way to express what to reward and how (often solved via dynamic programming), and it might give a a more complete description than some other methods, which is something you'ld want around control optimization problems.

It is also used in a wider sense of any sort of reward-based learning, i.e. "have an idea of outcome, punish the bad, reward the good"

Reward itself can be

looser way (e.g. genetic algorithm just seeing how far things get)
more direct (e.g. control optimization calculating reward per unit time within a horizon)

This is related to more finer distinctions with in RL, like

  • episodic RL - episodic tasks have recognizable chunks of events to learn from, say each game of chess
  • continuous RL - continuous tasks never, meaning you reward on the shorter term

Being so broad, it often lies somewhere between supervised and unsupervised, roughly because you're telling it what you want, but not how to get there.

Implementation-wise, that comes back in various methods, in different forms - see e.g.

e.g. Markov Decision Process
evolutionary/genetic algorithms
arguably a bunch of NN

...but note that there are quantifiable, sometimes huge differences between many of those, e.g. NNs (and problems for which we prefer NN) have a huge state space compared to MDPs.

More on RL

See also:

Inductive versus deductive systems

Transfer learning

Transfer learning is the idea that you could use knowledge from other systems.

State observers

State observers / state estimation will estimate the internal state of a real system, measuring what they easily can, and estimating what they need to. Optimization_theory,_control_theory#State_observers_.2F_state_estimation_.28and_filters_in_this_sense.29

This typically a singular state in control theory without much state, but can also be useful to present data to a learning system.

Related are the (introduced by Kalman)

the idea that the states of a system can be reasonably inferred from knowledge of its outputs.
(which seems to include that all states the system can be in are known)
in practical terms, that a good set of sensors lets you build a model that covers all its important variables.(verify)
the idea that you can actually guide a system through all states.(verify)

Both are related to building something that is likely to be stable in a knowable way.

Again, concepts more related to control theory, but more widely applicable.

Properties and behaviours of learners

Underfitting and overfitting (learners)

Underfitting is when a model is too simple to be good at describing all the patterns in the data.

Underfitted models and learners may still generalize very well, and that can be intentional, e.g. to describe just the most major patterns.

It may be hard to quantify how crude is too crude, though.

Overfitting often means the model is allowed to be so complex that a part of it describes all the patterns there are, meaning the rest ends up describing just noise, or insignificant variance or random errors in the training set.

A little overfitting is not disruptive, but a lot of it often is, distorting or drowning out the parts that are actually modeling the major relationships.

Put another way, overfitting it is the (mistaken) assumption that convergence in the training data means convergence in all data.

There are a few useful tests to evaluate overfitting and underfitting.


More concepts, some shared tools

On the theory / math side

Stochastic processes, deterministic processes, random fields

A deterministic process deals with possible determined cases, with no unknowns or random variables.

A stochastic process (a.k.a. random process) allows indeterminacy, typically by working with probability distributions.

A lot of data is stochastically modeled, often because you only need partial data (and often only have partial data to start with).

Hybrid models in this context means mixing deterministic and stochastic processes

A random field is random function over an arbitrary domain.

When that domain was time, we would probably call it a random process, but since this is basically a generalization that says it's not necessarily necessarily time, or one-dimensional, or real-valued, we think of it as space instead, and field becomes the more apt description. (verify)

See also:

Markov property

the Markov property is essentially that there is no memory, only direct response: that response of a process is determined entirely by its current state, and current input (if you don't already define that as part of the state).

More formally, "The environment's response (s,r) at time t+1 depends only on the Markov state s and action a at time t" [1]

There are many general concepts that you can make stateless, and thereby Markovian:

  • A Markov chain refers to a Markov process with finite, countable states [3]
  • A Markov random field [4]
  • A Markov logic network [5]
  • A Markov Decision Process (MDP) is a decision process that satisfies the Markov property
  • ..etc.

Markov Decision Process

Belief network

Log-Linear and Maximum Entropy

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

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, often 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

Curse of dimensionality

The curse of dimensionality is, roughly, the idea that each dimension you add to your model makes life a little harder.

Intuitively, this is largely because a lot of these dimensions may be noninformative, but will still contribute to everything you do.

A common example is distance calculations. As you add dimensions, the contributing value of any one decays, meaning the bulk of less-expressive are going to overpower the few good ones. It's very likely to drown any good signal in a lot more noise (unless your method explicitly learns this instead of using a standard metric).

And things that build on those distances, like clustering, are going to have a harder time.

Kernel method

Statistical modeling

Regression analysis

Linear regression
Logistic regression

Feature selection

Graph based statistical modeling


Dimensionality reduction

As a wide concept

Dimensionality reduction can be seen in a very wide sense, of creating a simpler variant of the data that focuses on the more interesting and removes the less interesting.

Note that in such a wide sense a lot of learning is useful as dimensionality reduction, just because the output is a useful and smaller thing, be it clustering, neural networks, whatever.

But in most cases it's also specific output, minimal output for that purpose.

Dimensionality reduction in practice is much more mellow version of that, reducing data to a more manageable form. It historically often referred to things like factor analysis and multivariate analysis, i.e. separating out what seem to be structural patterns, but not doing much with them yet, often acknowledging that we usually still have entangled surface effects in the observations given, and our direct output is probably still a poor view of the actual dependent variables that created the observations we look at. (Whether that's an issue or not depends largely on what it's being used for)

Reducing the amount of dimensions in highly dimensional data is often done to alleviate the curse of dimensionality.[6]

Said curse comes mainly from the fact that for every dimension you add, the implied volume increases quicker and quicker.

So anything that wants to be exhausitve is now in trouble. Statistics has an exponentially harder job of proving significance (at least without exponential amounts more data), Machine learning needs as much more data to train well, optimization needs to consider all combinations of dimensions as variables, etc.

Distance metrics are funnier, in that the influence of any one dimension becomes smaller, so differences in metrics smaller - and more error-prone in things like clustering. (verify)

It's not all bad, but certainly a thing to consider.

Ordination, Factor Analysis, Multivariate analysis

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)

(See also Statistics#Multivariate_statistics)

Ordination widely means (re)ordering objects so that similar objects are near each other, and dissimilar objects are further away.

Ordination methods are often a step in something else, e.g. be good data massage before clustering.

It can become more relevant to higher-dimensional data (lots of variables), in that ordination is something you watch (so sort of an implicit side effect) in the process of dimensionality reduction methods.

Of course, different methods have their own goals and focus, different requirements, and their own conventions along the way.

Typical methods include PCA, SVD, and

Factor Analysis, Principal Component Analysis (PCA), and variants

Correspondence Analysis (CA)

Conceptually similar to PCA, but uses a Chi-square distance, to be more appicable to nominal data (where PCA applies to continuous data).

See also:

Multi-dimensionsional scaling (MDS)

Refers to a group of similar analyses, with varying properties and methods, but which all focus on ordination

Commonly mentioned types of MDS include:

  • Classical multidimensional scaling, a.k.a. Torgerson Scaling, Torgerson–Gower scaling.
  • Metric multidimensional scaling
  • Non-metric multidimensional scaling

It is regularly used to provide a proximity visualization, so the target dimensions may be two or three simply because this is easier to plot.

Depending on how you count, there are somewhere between three and a dozen different MDS algorithms.

Some MDS methods closely resemble things like PCA, SVD, and others in how they change the data. Some more generic MDS variants are more on the descriptive side, so can be solved with PCA, SVD, and such.

A ordination, most try to not change the relative distances, but do change the coordinate system in the process.


Input to many method is a similarity matrix - a square symmetric matrix often based on a similarity metric. Note some similar methods may be based on dissimilarity instead.

At a very pragmatic level, you may get

  • a ready-made similarity matrix
  • items plus a method to compare them
  • a table of items versus features (such as coordinates, preferences), with a method to compare them
  • perceived similarities between each item (e.g in market research)

There is little difference, except in some assumption like whether the feature values are Euclidean, independent, orthogonal, and whatnot.

Result evaluation

MDS solutions tend to be fairly optimal, in that for the amount of target dimensions you will get the solution's distances correlating to the original data's distance's as well as they can.

There are still a number of things that help or hinder accuracy, primarily:

  • the choice of input values
  • the choice of target dimensions (since too few lead to ill placement choices)
  • (to some degree) the type of MDS
  • methods that have cluster-like implementations may be sensitive to initial state

You probably want to see how good a solution is.

The simplest method is probably calculating the correlation coefficient between input (dis)similarity and resulting data (dis)similarity, to show how much the MDS result fits the variation in the original. By rule of thumb, values below 0.6 mean a bad solution, and values above 0.8 or 0.9 are pretty good solutions (depending on the accuracy you want, but also varying with the of MDS).

Other methods include Kruskal's Stress, split data tests data stability tests (i.e., eliminating one item, see if result is similar) test-retest reliability [7]


Note that in general, correlation can be complete (if k point in k-1 dimensions, or distances between any two items are equal whichever way they are combined, e.g. if distances are those in euclidean space), but usually is not. The output's choice of axes is generally not particularly meaningful.

Probably classically the most common example is is principal coordinates analysis (PCO, PCoA), also known as Classical multidimensional scaling, Torgerson Scaling and Torgerson-Gower scaling, which is a single calculation step so does not require iteration or convergence testing.

See also (Torgerson, 1952), (Torgerson, 1958), (Gower, 1966), CMDSCALE in R

(Note: PCO (Principle Coordinate analysis) should not be confused with PCA (Principle Component Analysis) as it is not the same method, although apparently equivalent when the PCA kernel function is isotropic, e.g. is working on Euclidean coordinates/distance)(verify)

The first dimension in the result should capture the most variability (first principal coordinate), the second the second most (second principal coordinate), etc. The eigenvectors of the input distance matrix yield the principal coordinates, and the eigenvalues give proportion of variance accounted for. As such, eigenvalue decomposition (or the more general singular value decomposition) can be used for this MDS method. (The degree to which distances are violated can be estimated by how many small or negative eigenvalues. If there are none (...up to a given amount of dimensions...) then the analysis is probably reasonable), and you can use the eigenvalues to calculate how much of the total variance is accounted for(verify) - and you have the option of choosing afterwards how many dimensions you want to use (which you can't do in non-metric).

Metric (multidimensional) scaling: a class of MDS that assumes dissimilarities are distances (and thereby also that they are symmetric). May be used to indicate PCO, but is often meant to indicate a class based on something of a generalization, in that the stress function is more adjustable. The optimization method used is often Stress Majorization (see also SMACOF [8] (Scaling by Majorizing A COmplicated Function)).

On iterative MDS methods: minimize stress no unique solution (so starting position may matter)

  • stress-based MDS methods
  • may be little more than non-parametric version of PCO(verify)

Non-metric multidimensional scaling can be a broad name, but generally find a (non-parametric) monotonic relationship between [the dissimilarities in the item-item matrix and the Euclidean distance between items] and [the location of each item in the low-dimensional space].

The optimization method is usually something like isotonic regression (which is due to monotonicity constraints). Methods regularly have both metric and non-metric parts, and non-metric scaling in the broad sense can describe quite varying methods (see e.g. Sammon's NLM).

Note that the the monotonic detail means that ranking of items ends up as more important than the (dis)similarities. This may be a more appropriate way of dealing with certain data, such as psychometric data, e.g. ratings of different items on an arbitrary scale.

Non-metric MDS may give somewhat lower-stress solutions than metric MDS in the same amount of dimensions.(verify)

Also, certain implementations may deal better with non-normality or varying error distributions (often by not making those assumptions).


  • Kruskal's non-metric MDS(verify)
  • Shepard-Kruskal Scaling(verify) (and (verify) whether that isn't the same thing as the last)
  • Sammon non-linear mapping [9]

Variations of algorithms can be described as:

  • Replicated MDS: evaluates multiple matrices simultaneously
  • Three-way scaling
  • Multidimensional Unfolding
  • Restricted MDS

Multidimensional scaling (MDS) [10]

  • classical MDS (quite similar to PCA under some conditions, apparently when you use euclidean distance?)
    • (Difference matrix -> n dimensional coordintes (vectors))
  • Kruskal's non-metric MDS (R: isoMDS)
  • Sammon's non-linear mapping (R: sammon)
See also
  • WS Torgerson (1958) Theory and Methods of Scaling
  • JB Kruskal, and M Wish (1978) Multidimensional Scaling
  • I Borg and P Groenen (2005) Modern Multidimensional Scaling: theory and applications
  • TF Cox and MAA Cox (1994) Multidimensional Scaling

Generalizized MDS (GMDS)

A generalization of metric MDS where the target domain is non-Euclidean.

See also:

Sammon’s (non-linear) mapping

Singular value decomposition (SVD)

See also:

Nonlinear dimensionality reduction / Manifold learning

Expectation Maximisation (EM)

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)

A broadly applicable idea/method that iteratively uses the Maximum Likelihood (ML) idea and its estimation (MLE).

Decisions, fuzzy coding

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On bias

Methods / algorithms / searchers

Decision trees

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

Instance-based learning


Markov Models, Hidden Markov Models

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)

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


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

Combining classifiers

Support Vector Machines

Support-vector regression

Using the 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, stopping either when some goal is satisfied, or sometimes until everything is split.

Differences include:

  • what side they err on in unclear cases.(verify)
  • divisive by nature makes its decision on more data, so can be more accurate
e.g. early combination decisions in agglomerative may be less sensible but cannot be undone without making the method more complex
  • divisive is often more efficient to calculate
  • divisive is a little more supervised, in that you may need to explecitly give it a way to split that makes sense of the data

In general, divisive clusters seems better at finding large clusters, agglomerative clustering is good at finding small clusters.(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
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)

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


Clustering as neighbour search
Hamming embedding

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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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)

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

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)

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)
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)
  • 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
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)

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


Hierarchical, divisive

DIvisive ANAlysis - the idea is that at every step, the most homogenous cluster is split in two





BIRCH (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

State observers / state estimation (and filters in this sense)

State observers / state estimation will estimate the internal state of a real system, measuring what they easily can, and estimating what they need to,

Good knowledge of a system is useful to various control problems.

Bayes estimator, Bayes filter

alpha-beta filter

An alpha beta filter (a.k.a. f-g filter, g-h filter)

Kalman filter

Multi hypothesis tracking

Particle filter

Data fusion

Sensor fusion

Optimization theory, control theory


System analysis

Problem theory

Types of problems

Some controllers

In terms of the (near) future:

  • greedy control doesn't really look ahead.
  • PID can be tuned for some basic tendencies
  • MPC tries to minimize mistakes in predicted future

  • For example, take a HVAC system that actively heats but passively cools. This effectively means you should be very careful of overshooting. You would make the system sluggish -- which also reduces performance because it lengthens the time of effects and settling


  • HVAC

Greedy controllers

Doesn't look ahead, just minimizes for the current step.

Tends not to be stable. Can be stable enough for certain cases, in particular very slow systems where slow control is fine, and accuracy not so important.

For example, water boilers have such large water volume, which has high heat capacity, that even a bang-bang controller (turn a few-hundred-watt heating element fully on or off according to temperature threshold) will keep the water within a few degrees of that threshold, simply because the water's heat capacity is large in relation to the heating element you'ld probably use (see also thermal inertia).

If on the other hand the boiler's volume is small or the heater too powerful, it will spike much too warm, and average will be noticeably above the setpoint.

A proportional controller (e.g. for the boiler case, even slow PWM of that heating element) would probably be good enough.

....but more generally, greedy control tends to not be stable when you want control to be fast. You easily run into the issue that actuation may be too fast for the measurement, easily causing feedback and oscillations as well.

Hysteresis (behaviour)

Hysteresis control (type)

Map-based controller (type)

PID controller

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)

PID is a fairly simple and generic control-loop system, still widely used in industrial control systems.

It is useful in systems that have to deal with some delay between and/or in actuation and sensing, where they can typically be tuned to work better than greedy proportional-only controllers (and also be tuned to work worse - which you want to avoid).

Compared to some other cleverer methods, PID is computationally very cheap (a few adds and multiplies per step).


  • there is no easily quantified way to describe or guarantee optimality stability
you have to tune them,
and learn how to tune them
  • tuning is complex in that it depends on
how fast the actuation works
how fast you sample
how fast the system change in reaction / settles
  • doesn't deal well with long time delays
  • derivative component is sensitive to noise, so filtering may be a good idea
  • has trouble controlling complex systems
more complex systems should probably look to MPC or similar.
  • linear at heart (assumes measurement and actuation are relatively linear)
so doesn't perform so well in non-linear systems
  • symmetric at heart, so not necessarily well-suited to non-symmetric actuation
consider e.g. a HVAC system -- which would oscillate around its target by alternately heating and cooling.
It is much more power efficient to do one passively, e.g. active heating and passive cooling (if it's cold outside), or active cooling and passive heating (if it's warmer outside)
means it's easier to overshoot, and more likely to stick off-setpoint on the passive side, so on average be on one side
You could make the system sluggish -- in this case it reduces the speed at which it reaches the setpoint, but that is probably acceptable to you.
in other words: sluggish system and/or a bias to one side

Some definition
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)

The idea is to adjust the control based on some function of the error, and a Proportional–Integral–Derivative (PID) controller combines the three components it names, each tweaked with their own weight (gain).

The very short version is that

  • P adjusts according to the proportional error
  • I adjusts according to the integrated error
  • D adjusts according to the derivative error

It can be summarized as:

PID formula.png


  • e(t) is the error
  • P, I, and D are scalar weights controlling how much effect each component has

Some intuition
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)

So how do you tune it?

MPC (Model Predictive Control)

FLC (Fuzzy Logic Control)



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)

Q-learning is a specific type of #reinforcement learning.

Q-learning models discretized (state,action) mapping, and learns the values for each mapping and implicitly the optimal action in each state, by trial and error.

You can see it as a state-by-action table where each cell stores a value that is increased on success, decreased on failure (often with an EWMA(verify) to control learning rate?).

Like reward-based learning in general, Q-learning can be supported by something as simple or complex as you wish.

The discretized approach mainly makes when there is a relatively small set of possible states and actions, but the larger that space is, the longer it takes takes very long to actually learn enough for each pair.

Deep Q-learning is a variant that outputs the value for all actions from a network instead, which is often a more compact way to store what can works out as a very large state space, and may be m

and can be more robust(verify)

See also


Structured prediction

Mistake Bound learning

PAC theory

Evolutionary/genetic computation

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)

For context:

Part of a wider category of evolutionary computation, which is is inspired by the observation that biological evolution tends to adapt well for a specific task.

Evolutionary computation is a wider field including concepts like optimizations imitating ant colonies or particle swarms, to gene expression, to self-organization, to agent-based modeling, to genetic programming.

Applications include include classification, curve fitting and other statistical/data modelling, feature selection.

Perhaps the largest applications lie around gaming, research, as people don't seem to like the opaqueness of the method for 'real' applications. But now that we're okay with equally opaque neural networks, perhaps that has ended? (and yes, things like NEAT apply for both and works decently).

Technically, genetic algorithms focus more on the evolutionary concepts and is more a method that barely gets into possible implementations, while genetic programming on its application specifically to generating machine programs, but border between the two is sometimes vague and used interchangably.

Genetic programming is specifically concerned with generating a program fit for a specific task by trial and error, with the largest components being a fitness function and introducing some randomness in a group (population) of candidate programs.

The representation that is altered can be anything productive, from trees of operators to machine code to neural nets.

Practically, you have to deal with aspects like

  • generating an initial population
  • evaluate fitness of individuals
  • deciding how much and how to
    • reproduce (copy as-is)
    • cross over (combine into offspring)
    • and mutate (applying a genetic variation)
  • deciding when to stop