View source for Data modeling, restructuring, analysis, fuzzy cases, learning

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[[Category:Math on data]]


==Concepts, Glossary, and some shared tools==

===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 in every (visited) state of a system



Slightly more specifically:
* '''Reinforcement learning''' has a few different takes
: problem-wise, it's those where an actor learns how to do things in an environment
:: where often your only data is trying and failing
:: which may well be formulated as a control process, e.g. a [[Markov Decision Process]]
:: and was classically implemented via dynamic programming
: implementation-wise, it is sometimes used to mean reward-based learning, i.e. "have an idea of outcome, punish the bad, reward the good"
:: which comes back in many learner systems - see evolutionary/genetic algorithms, and arguably a bunch of NN -- but there are quantifiable differences in the approach taken, e.g. NNs (and problems for which we want to pick NN) have a huge state space compared to e.g. MDP, and the learners differ accordingly
: paradigm-wise, it is often contrasted as to supervised learning and unsupervised learning


* '''Structured prediction''' refer to prediction problems 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

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* '''Data fusion''' means merging data from various sources, e.g. various sensors. 
: This often implies some sort of modelling, though this ''can'' be learned
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* others...

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https://cs.stackexchange.com/questions/58131/are-neural-networks-a-type-of-reinforcement-learning-or-are-they-different
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===Related fields===
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Parts of machine learning are just applied statistics (and a few specific parts of math), other parts are very much their own thing.

You can argue that statistics is more of a game of description and verification,
whereas machine learning is more prediction, which takes more assumptions and more engineering.


'''Data mining''' can apply statistics and learning,
but often has a more specific purpose, like confirming a theory or detecting patterns to explain,
but not the ''prediction'' that ML is often aimed at.

Similarly, there is statistics and ML in optimization theory and control theory,
and while the methods vary wildly, their application is guided by "what do I do right now?".


Even plain detection can involve prediction - say, counting people may track objects over
time and discount them by "that no longer matches how I predict a human can move".


You can argue that a lot of the techniques are just applied in different ways
by different communities, to the point that they don't even know they've reinvented the wheel under different names (see e.g. how many )



These days '''AI''' sometimes gets assumed to be all deep learning NN, 
but even today a lot of "AI" is actually much more classical methods just because they fit the problem more easily, if sometimes updated and mixed.




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===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.
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(See e.g. moderately fancier neural net training easily being GPU-weeks (dozens of kilowatt-hours).
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Intuitively, model-free is

A model's assumptions are often initially fast, and eventually limiting.
Note there are some methods that try to consider this, e.g. 


Model-free often makes for a lot more possibilities 

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====supervised versus unsupervised (learners)====

'''Supervised''' usually means the training process is suggested or somehow (dis)approved.
Usually it refers to having annotated training data, sometimes to altering it. Example: Classification of documents, basic neural network back-propagation, least-squares fitting, operator cloning


'''Unsupervised''' refers to processes that work without intervention. For example, self-organizing maps, or clustering documents based on similarity needs no annotation.


'''Semi/lightly supervised''' usually means there is an iterative process which needs only minimal human intervention, be it to deal with a few unclear cases, for information that may be useful, or such.

====Reinforcement learning====

There is also an argument for a split between supervised, unsupervised, and '''reinforcement learning'''.


Reinforcement learning 
is knowing what sort of outcome you want and somehow rewarding that, and steps towards it.

It lies somewhere between supervised and unsupervised, in that 
there generally is no labeled data, 
there is feedback but it is not in the form of "this was the correct input/output pair", or that specific decisions were, 
but only that the eventual outcome was good or bad.

The concept of using rewards and penalties for an (adaptive, often model-free) system's estimation of rules and/or parameters of its environment.

A loosely supervised method that works by trial and error, and can eventually be fairly exhaustive (depending on the case).



There are many distinct ways you can do that.

It seems this was classically often expressed in things like like a markov decision process{{verify}}, 
as that makes it clearer ''what'' to reward and how, and it might give a a more complete description,
which is something you'ld want around control optimization problems.

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


RL is useful when there is no easy way to do a complete, closed closed analysis, e.g. because there are a lot of states, or the underlying structure is not simple. 

...because if it ''is'' simple, you can often use dynamic programming to e.g. do an exhaustive(-enough) search.

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Often means steering such an agent, and often to learn to imitate something from near-scratch, or perhaps train a ''simulation''.


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See also:
* http://en.wikipedia.org/wiki/Reinforcement_learning

* LP Kaelbing, ML Littman, AW Moore.  (1996) "{{search|Reinforcement learning: A survey."}}
* RS Sutton, AG Barto (1998) ''{{search|Reinforcement Learning: An Introduction}}''




=====Q-learning=====

Q-learning learns optimal action in every state via trial and error.

It is explained as a discretized (state,action) mapping to a value that is increased on success, decreased on failure (often with an EWMA{{verify}} to control learning rate?), from either simulation or real world tests.


Like reward-based learning in general, Q-learning can be supported by something as simple or complex as you wish. 
More recently there are NN implementations, in part because it can be a more compact way to store a very large state space, than the table that stores the state,action->value in many examples. {{verify}}


https://en.wikipedia.org/wiki/Q-learning

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https://web.mst.edu/~gosavia/neural_networks_RL.pdf

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====Inductive versus deductive systems (learners)====

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This one is actually fuzzier than is seems.

The reference is to relatively formal logic (in philosophy), where
* [[inductive inference]] is going from smaller facts/observations to "probably true in general" statements
* [[deductive inference]] is going from universal truths to more specific statements (and often more applied ones) 


'''Inductive learning''' would then refers to learning that takes data and generalizes from those examples,
to be able to deal with similar unseen cases.


But then, what would '''deductive learning''' mean? (I cannot find a lot of words but roughly zero examples)


A teacher deductively creating example cases from an already-known generalized model, e.g. as a means to do transfer learning? 

Maybe, but then you're just transferring the list of ''specific'' things a teacher knew already.

Because if you're doing more, that would probably still be learning general knowledge - and in the "probably in general", i.e. inductive way, rather than the 'always in specific' that deductive does. 



(Or were we just pointing out that some learners tend to have inductive steps that model from data, and deductive steps that try to predict new data?)



Deductive learning would probably refer to a system trying to find ''descriptive'' truths,
based on general rules, the means to break those into parts, create a space of all possible hypotheses, and see which are true.

Say, it could learn from a whole bunch of weather data that there ''have'' to be clouds for there to be rain. 


Or take existing hypotheses and see if you can make it narrower and stronger but narrower.

Less likely than inductive to [[overfitting]] - but probably often [[underfitting]] instead.


But even in that hypothesis space it would be tempting to generalize, and use probabilities.

---

H Madala (1991), "{{search|Comparison of Inductive Versus Deductive Learning Networks}}"

https://content.wolfram.com/uploads/sites/13/2018/02/05-2-6.pdf


---


Some inductive learning can sometimes be usefully seen, and approached, as a search in a space of possible hypotheses.

The hypothesis space can finitely include all possible hypotheses about the training data, and choose one so that it hypothesizes correctly for every single one. Of course, this is overfitting: It would just be a description of all of the data. 

FIND-S is a simple algorithm to work a little better. It comes down to:
* looking for the most specific hypothesis (or hypotheses; consider that these could ignore parameters) consistent with the trainign data. 
* Based on conjunction of constraints
* If the training data converges, this should find a simple hypothesis to apply to unseen data with the same attributes.
* Problem: It finds only one, and not necessarily the best for anything but the training data 
* Noise will easily disturb into too general hypothesis
* Introduces '''restriction bias''' in that we can have hypothesises with only discrete-valued attributes and can only specify a specific value or "don't care," not a subset of possible values, in a single hypothesis.

This can be generalized into also having disjunction, negation and more, but this ends up in a mess: Only the training data could be unambiguously classified, because it is our training data. Unseen data can be classified with a plethora of hypotheses, each of which are as valid as the other. The only vaguely viable thing would be to take the subset of all valid hypotheses and use that somehow, but this is not a good solution. 


This teaches that we in fact need bias to be able to generalize, usually through something refered to as '''inductive bias''', which is basically the `inate' set of assumptions a learner uses. Algorithms will usually differ in types of bias, so when this bias matches that of the data the algorithm will work better for that data.

It also means that the more accurate prior knowledge you have, the higher the quality of the learning - which makes intuitive sense. On the other hand, embedding too much prior knowledge will dismiss what is seen by it to be unusual data.






In this context you could also put some explanation on 
'''transductive learning''' which instead of generalization uses one specific example to handle another.

and even abductive learning.
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====Transfer learning====

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

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In some cases, particularly model-based systems, this might be straightforward parameters, or otherwise fairly obvious.


In other, e.g. model-free ones, it may be far from obvious how exactly the system does what it does, and the best you can do is e.g. reinforcement learning until the new system answers similarly to the last.


Neural nets tend to not allow much transfer learning unless specifically designed to do so.


: this doesn't really work on classical neural nets
: or RNNs or their variations, basically because what it learned is entangled
: CNNs may allow it a bit more if well designed, e.g. when your change really just implies changing the last one or two layers (e.g. classification) but not the rest (learning of what kind of features are interesting to base that on)
* when transfer learning ''doesn't'' work, you basically need new training data





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====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)
* [https://en.wikipedia.org/wiki/Observability observability] criterion 
: 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}}

* [https://en.wikipedia.org/wiki/Controllability controllability] criterion
: 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. 

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via cross validation can be a useful test, as it will reveal predictions that aren't even true in other parts of the training data.
It may also help estimate where a good balance lies.


Notes:
* The [https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff Bias–variance angle] 
: see e.g. http://datascience.stackexchange.com/questions/361/when-is-a-model-underfitted





For learners
* a pathological case would be to have a model that is an exactly rule-based description of all the training data, with exactly the same complexity as that data


* overfitted learners often means very little generalization

* ...and so has a limited ability to deal with cases that don't look very much like ''specific'' cases in the training set.


Good ways to avoid overfitting depend a little on the type of learner you're using.
If you're halfway involved in the analysis you're doing, and not fully automating analysis,
applied gut feeling tends to avoid most of it already.


See also:
* http://en.wikipedia.org/wiki/Overfitting
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====Interpretability====
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'''Explainable AI''' is mostly this applied to NNs, 
as free-form NNs tend to not be interpretable, and designed ones are only somewhat better.



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===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''' basically describes the modelling generalization when parameters (dependent variables) are not necessarily time, or one-dimensional, or real-valued. {{verify}}


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A '''conditional random field''' turns up in 

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See also:
* http://en.wikipedia.org/wiki/Stochastic_process
* http://en.wikipedia.org/wiki/Deterministic_system
* http://en.wikipedia.org/wiki/Random_field
* http://en.wikipedia.org/wiki/Conditional_random_field

====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 {{comment|(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''" [http://en.wikipedia.org/wiki/Markov_property]


There are many general concepts that you can make stateless, and thereby Markovian:
* A '''Markov model''' is a [[stochastic model]] with the Markov property

* A '''Markov process''' is a [[stochastic process]] with the Markov property [http://en.wikipedia.org/wiki/Markov_process]

* A '''Markov chain''' refers to a Markov process with finite, countable states [http://en.wikipedia.org/wiki/Markov_chain]

* A '''Markov random field''' [http://en.wikipedia.org/wiki/Markov_random_field]

* A '''Markov logic network'''  [http://en.wikipedia.org/wiki/Markov_logic_network]

* A '''Markov Decision Process (MDP)''' is a decision process that satisfies the Markov property 

* ..etc.

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There are different variants of markov models:
* decision making processes, versus describing system behaviour
* having a complete model of the relevant apsects, or only a partial one.

Basically, describing with a complete model is a markov chain, 
describing with a partial model is a hidden markov model.


The markov chain is the simplest variant.




A state machine is often deterministic 



A markov model is 
sorta-deterministic, 
in that while it says something about state at time t+1 based on state at time t,
it does so with probablistic estimation.

In other words, it aims to make a good guess, does not pretend to be perfect.


Yes, you can make Markov-model implementations of a purely deterministic
state machine, but this mostly proves
* compatibility in that both are discrete-time 
* that the specific cases where the interaction is already known can be modeled by something that
can also be more fuzzy


Similarly, a Markov ''chain'' can be modelled by a finite state machine



Marchov chain is a discrete-time stochastic process



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====Markov Decision Process====
<!--
Consists of 
* S: a number of states
* A: a number of actions
* R(s,a): a reward function 
* T: a table of effect of each action in each state
:: it's markovian in that this table assumes the effect depends only on the most recent state


T can be 
* deterministic
: (state,action) leads to a new state

* stochastic
: (state,action) has estimated probabilities for the likeliness of each new state

Implicitly assumes observability, in that the new state must exist already.


https://www.cs.rice.edu/~vardi/dag01/givan1.pdf

'''Partially Observable MDP'''

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====Belief network====
<!--
A belief network, a.k.a. Bayesian network



https://artint.info/html/ArtInt_148.html
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====Log-Linear and Maximum Entropy====
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I general, we often want to make independence assumptions to make a problem more tractable.

That will come at the cost of the amount the model will fit.


Log-linear models make no independence assumptions{{verify}}.



Maximum Entropy uses log-linear and tries to maximize the likelihood of the training data.

The idea seems to 
be a model that is most like uniform distribution
''and'' making the least independence assumptions. 

It is, however, still inductive learning, so it easily has problems like [[overfitting]].
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===Curse of dimensionality===
The '''[http://en.wikipedia.org/wiki/Curse_of_dimensionality 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.


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...unless each problem happens to need most dimensions, which isn't going to happen unless you did a ''lot of very clever'' work up front.

It's certainly a reason we do dimensionality reduction, but that needs to be done in an informed way to actually help.




For example
* you need proportionally more data for decent training of that model.

* things like distances may becomes less meaningful

* anything that relies on distances (e.g. clustering) will have issues as a result

* a good amount of samples and examples for to train each dimension will be (dimensions) times bigger, so take more storage, and time to process


Since the volume increases so fast, you will probably have sparsity problems.



'''Curse or blessing?'''

There is an argument for the '''blessing of dimensionality'''.

The curse argument assumes you have some abstract n-dimensional space where you have no idea how they relate, but it is likely that for any one problem, there are a few good dimensions and the rest are noise.

It assumes you will ''use'' them equally, like a distance using all dimensions will. There are some alleviations to that

But more than that, if you're paying some attention to what you add, then a lot of added dimensions tend ''not'' to be entirely distinct and independent.  They tends to agree with others, in ways that various algorithms will pick up on.



: local minima fall away more than a clear global maximum {{verify}}



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===Kernel method===
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The kernel method seems to have meen introduced by SVM, but is a more widely useful idea. 

Others that use it include Linear Discriminant Analysis (LDA), Principal Components Analysis (PCA), ridge regression, spectral clustering - and you could apply it anywhere where you know it's useful for the input data.


The point is to turn any linear model to something that deals with nonlinear data, 
using a mapping (often selected to be suited for the data) into a different space that makes it easier to work with -- often a higher-dimensional one.

In the case of SVM typically for for linear separability.


The linear classifier in the higher-dimensional space is effectively a non-linear classifier in the original space, in a way that depends on how you mapped the data.

Intuitively, any case where you can think of a function that will warp the data so that there is a straight decision boundary.


For example, if you want to detect a 2D circle around the origin, you could create dimensions based on x^2 and y^2, which means what is radius in the original space (and effectively a round thing) is a distance in this new space, which can be split with a line/plane (linear).



That higher-dimensional space is often only ''implied'' by the kernel function, 
but in the case of various implementations, including SVM, the coordinates in that higher-dimensional space are not explicitly calculated, because it turns out to be easier (and often computationally cheaper)
to calculate the distances between data pairs directly, rather than mapping to coordinates and then using a generic metric.

This also makes it somewhat easier to deal with data that is not vectors.



(note that NN doesn't really need this, as multiple layers learn nonlinear in their own way{{verify}}, but it can still help avoid a layer or two and training time)

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https://en.wikipedia.org/wiki/Kernel_method

=Some methods=


==Use of linear regression==
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Applying linear regression between two variables assumes the relation is linear, and finds the best-fitting line.

While it sounds perhaps too basic, it is a better-than-nothing fit and already helps to suggest whether two things may be correlated.


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==Fuzzy coding, decisions, learning==
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Basic '''machine learning''' is concerned with approximating data into rules, usually as simple as possible and as automatic as workable. 
The point is usually to handle unseen data as correctly as can be expected, according to the given task. 


More formally, it is interested in:
* Maximizing or improving measurable performance
* At a specific task
* Given some sort of experience


Statistics-based versus model-based.



-->

<!--
...which is to say, those are headers under which it is easy enough to elaborate.
For example, the assumption is generally that experience can be interpolated and/or extrapolated, that the task has to be defined enough to give the sorts of answers you want, that the given experience (and the bits that are considered important) will converge on something interesting, and that that and experience will control what will be actually learned. The way performance is measured will too when it has effect on a next learning run.

This is a rough model of human learning. This doesn't mean that 'performance' and 'knowledge aquisition' as you might attribute them to humans will mean the same thing or be modeled even roughly the same - both are a lot more liquid for humans, and draw on a lot more seemingly casual knowledge.


While no artificial intelligence, machine learning is often more than just a few statistics or probabilities, in that machine learning locates and focuses on structure in data, at least ideally.
This to one of various ends, for example reproducing behaviour, categorizing data, creating an AI-like agent, though time is not usually a part of ML algorithms.




There is, naturally, overlap with and application in data mining, natural language processing, character or other sorts of image recognition, artificial intelligence and other areas that have to deal with changing or inexact environments.



'''Learning versus adaptation'''

You may notice that learners are similar to agents (and their design) in artificial intelligence. This is not unintentional, but may suggest more autonomy than will often be achieved. AI often focuses on learners that are robust to (can maintain performance under) change in the environment, and even to changes the controlled system (and possibly even to the learning process itself). 

Such systems are more complex. One of the issues is that there is a feedback loop which can have issues of its own, for example oscillating feedback loops (note that feedback can both have useful converging effects and annoying ones such as oscillation -- depending on what?{{verify}}). 

Adaptive controllers are necessary given the complexity of systems that interact. For example, the more sensors and actuators, the more prone to error design by parameter estimation is, and the more that adaptive learning can simplify this for the initial, but also unseen tasks.

-->


===On bias===
<!--
Bias: Experience, Defining a Hypothesis Space, and the Target Function


One thing to realize is that many different decisions can bias your results in that it can cut off types of answers, skew away from things not commonly experienced (in a training set), and other things. Sometimes this is the point - and that's one of the reasons this is an issue. Consider that simplification of a document into a class can be the entire use of a system, but often enough there are decisions that will limit your system's expressiveness, which you should realize when you use it.


=====Experience=====
Central questions to ask are what experience actually consists of, how it should be represented, what parts of it are useful features, what should be learned, and even whether the data will conform to the restrictions on the algorithm(s) you'll use - in classification, for example, you need to know whether something only fits intside a single class, and how noise-robust the algorithm is and how noisy the data is.

Models will usually allow you to embed or seed with prior knowledge as well as learn from experience data. Sometimes that distintion isn't even that large.


<small>Lightly sidestepping into AI for a second, you could define a task of playing checkers:
* T: Playing chekers
* P: Games won (this is a fairly general, long-term and loose goal, yes)
* E: Games played against itself

An interesting point to be made here is that training by example games wouldn't necessarily be so interesting option because that would likely focus learning on roughly copying legal moves, which is unlikely to be exhaustive or even tactical in the term of a few or even a single move - depending, of course, on how much prior data and domain theory you introduce.</small>


=====Target function=====
Mathematically, you can see many algorithms-for-problems as a combination of:
* Input data (the unseen data)
* A target function (trained with experience)
* A desired output mapped to by that target function

In slightly more general terms, you canbe said to be looking for a a useful hypothesis (function) among many possible ones. Sometimes the hypothesis space (the possible ones) is finite and even computationally tractable, in which case you can calculate the best one. Often, however, you can't, and you will have to approximate the ideal target function.

The type of value that the target function produces is representative of what sort of algorithm you are dealing with. Discrete values will often be used to classify (and boolean often to classify into relevant or not, in eg. concept learning), whereas real values usually point to correlation and regression tasks.

You definition of the target function introduces '''Preference bias''' in that specifically your output will have limited range, and therefore be limited to specific values. More exactly, you will (have to) internally choose a specific representation for data. 

Think of ways you would choose a target function for playing a move in chess. There are various ways to define this target function and what you would actually value when choosing a best. Consider for example that you can map a board position to a best move, or to that of the possible board positions that we like best. They sound the same, but will probably lead to different focus, and different direct evaluation.
In fact, direct evaluation this in an integral part of he function: What do you actually prefer? Would you just value having more pieces? That and threatened pieces? That and the amount you threaten? All of that for all the types of your pieces? Doing just that will focus your tactics entirely on those aspects and nothing else -- basically because that will be the universe in terms of evaluation.

To come back to learning: choosing a nonoptimal representation isn't actually fatal. For example, in the chess playing setup just suggested, what you would probably actually decide to learn is how important each of those aspects are. In the long term your algorithm would probably learn that pawns can be quite useful, queens perhaps not so much as the prior knowledge you may put in there may suggest, and it may even discover that evaluation aspects you introduced are almost entirely irrelevant.

====Experience====
To learn with experience without supervision, things like iterative methods are used, which are seen to have at least the elements of experience and that of measuring performance.

That's a couple of new terms. The amount of supervision (supervised, unsupervised, semi-supervised) refers to how much human interaction is needed to make the system work at all, though of course interactive supervision will often lead to tweaking as well.

Iterative learning, or in more AI-like terms, reinforcement learning, is simply a step towards the unsupervised. It learns particularly what interactions are effective. Note that most basic ML isn't directly that advanced.





===Fuzzy logic===
Fuzzy logic is itself fuzzily defined.

It is mostly a formalization of working with degrees of applicability, or rather, degree of membership to a group, according to fuzzy set theory[http://en.wikipedia.org/wiki/Fuzzy_set]. 

Superficially, partial membership is the same as probability, but there are differences you can care about.


Fuzzy logic is often contrasted with boolean logic, which in itself can be a misleading way to introduce it.

Fuzzy logic based decisions / control can often just as easily be described a conditional logic based with a few degrees (seemingly particularly so for the more basic, linear fuzzy set membership functions) and thresholds that are in themselves nothing special particularly in programming, which is where most fuzzy logic is applied, though they can be used to keep such a system simple, or keep it more numeric, abstract and/or apart from main code.



http://en.wikipedia.org/wiki/Fuzzy_logic
http://en.wikipedia.org/wiki/Fuzzy_set
-->




===Methods / algorithms / searchers===
<!--
There are a lot of algorithms, which can usually be said to model in specific ways. Usually these model specific types of structures well and others not so well.


-->

===Decision trees===
<!--
Decision trees are used to classify: to decide.


They are, in some ways, no more than a codification of already-known rules. 
In themselves they are memorization, not learning.

You can create such decision trees, but this requires fairly clean data.




Decision trees are essentially acyclic graphs where each edge/transition is the answer to the node it originates in.
For example, the root note could ask "Weather", from which "Overcast" leads to "Sad", and "Sunny" leads to "Happy."

Of course, such decision nodes can follow each other, which also puts them in the context of the information given by the decisions of the previous nodes. As such, a set of given attributes is a walk in the tree. 

To be exact:
* Node: attribute
* Branch (edge): attribute value (discrete)
* leaf: classification


Such a tree can contain all training data in a dense tree, but that isn't very useful. Interestingly, on no-noise data it can do so fairly compactly. The value of this is the structure and the possible transformations on it. Pruning  branches and making a decision is analogous to making a generalisation: for example, when all leafs after 'Sunny' lead to 'Happy', you can replace that subtree with a single leaf 'Happy' and have exactly the same data encoded.

Similarly, when you notice that ''most'' of them do, you can '''prune''' the tree and do the same simplification at the cost of only a little accuracy on unseen data - or such is the assumption when you assume training data is representative.


It is also interesting to make the most important/clearest decisions first. Among other things, it ensures pruning lower nodes will have minimal effect.

The usefulness of a decision can be measured by information gain, basically a measure of usefulness to the decision tree based on the amount of entropy an attribute causes.

-->
=====ID3=====
<!--
In fact, the ID3 algorithm does just what is described above. It constructs a decision tree from scratch by locally using the node that adds the most value / information gain to the decision tree.

More exactly, it measures the entropy of each attribute value and weighs it by the fraction of cases it represents {{verify}}. The gain of information is the current node's entropy minus all such values. It comes down to that more distinctive decisions will be chosen first, less ones later, meaning that the important decisions come first.

It also means this effect is purely local optimality and ignores depthwise considerations, meaning it is not guaranteed to be optimal, only a fair guess. 


ID3 searches (and encodes) the entire hypothesis space. It is also statistically based, so can handle training data that is a little noisy: The information gain will be lower, but not necessarily disturn the process or choice.


Now note that instead of restruction bias (limiting the range of outcomes or attributes) we have preference bias: we decided we prefer 'useful' decisions based on some measure. 

However, it is convenient. It is in the spirit of Occam's razor in that it tries to explain everything it sees with the shortest hypothesis. If such a short one exists, it is more likely to be structure in the data than coincidence.


=====Overfitting=====
In the context of a decision tree, overfitting means that adding too many levels of deep tree detail will be too specific for the training data and be invalid for unseen data and probably the entire general universe of possible data (not that that one's measurable).

-->
=====Pruning (ID3, others)=====
<!--
Given a decision tree that contains all hypotheses is quite likely to be overfitting, so you probably want to generalize by pruning. 

You could in theory do this during, e.g. figure that if your best dicision isn't statistically very strong you do not want to add a brancj.

More generally, you could post-prune trees. When you have less informative information about the construction of the tree, you can prune do so by a sort of trial and error based on a training set of data, for example by seeing which tree minimizes (treesize + misclassifications).

Pruning will often mean an accuracy increase because it will work for more general cases, but only a little because it will exclude other cases in the process.


=====Tree effects/extensions=====
You can limitedly use continuous values by using a threshold to map them into a boolean or a set of discrete values.

Attributes with many values will tend to be chosen for infogain when it isn't really.

There are alternatives to simple infogain

You can deal with missing values in various ways.

-->

=====Rule Post-pruning; C4.5=====
<!--
{{stub}}

One thing you can do with a tree is turn it into equivalent rules - each path, after all, is a conjunction of conditions, so can be made into an if-then structure. You can then prune arguments in each rule's expression, based one something such as information gain, and sort the rules for application.

Apparently this is the C4.5 method. I'll have to look up the details.

http://en.wikipedia.org/wiki/C4.5_algorithm

https://octaviansima.wordpress.com/2011/03/25/decision-trees-c4-5/

-->

===Instance-based learning===
<!--
"Just Store all data in memory" and find likely answer by analogy: 
* Define distance measure between two 
* Use distance measure for unseen-any training data member.
* Calcualte answer from most likely entry's target value

The nearest neighbour(s) based on euclidian distance in n-dimensional space -- when instances canbe mapped to such a space, and only when sensibly (as dimensions also interact).

Lazy method (decision on the spot), vs. eager (generalize/learn before using).

====k-Nearest Neighbor (k-NN)====
Use the k nearest neighbours, or just one.

Weighed variant: Weigh closer neighbours more, or possibly even all.

-->


===Bayesian learning===
{{stub}}

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=====
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A Bayes Optimal Classifier considers all possible hypotheses in the hypothesis space, and as such its solution is unbeatable on average.

However, it is intractable for all but the simplest problems, so it is not often mentioned outside of theory.

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=====Naive Bayes Classifier=====
<!--
Simple, tractable for pretty complex problems.

'Naive' in that it assumes all attributes are independent.
This saves a lot of computation, and does not remove as much accuracy as you might think.

See [[Naive Bayes]]

-->

====Bayesian (Belief) Network ====
<!--
Names include
: (Bayesian) belief network
: Bayesian model
: Bayesian network
: Bayes network
: Bayes(ian) model 
: probabilistic directed acyclic graphical model


An acyclic graph where each node conditionally depends only on its direct predecessors, and the existence of the edges define (in)dependence between variables.

Allows selective inclusion of prior knowledge.

Not entirely unlike neural networks, but bayesian networks are more directly about relation of variables, whereas NNs can be structured in a more flexible way.


[https://en.wikipedia.org/wiki/Deep_belief_network Deep belief network]
-->

===Markov Models, Hidden Markov Models===
{{stub}}

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 

<!--
Expectation-maximization (EM) is regularly used.
(EM: Iterative process based on maximum likelihood of untrustable (hidden) values)
-->

==Classification==


===Naive Bayes===
====Some background====
A certain Reverend [http://en.wikipedia.org/wiki/Thomas_Bayes 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 {{comment|(or sometimes hardcoded assumptions)}}:
* deciding on a set of distinct target classes (say, exclusives as 'spam' vs. 'not spam', or rough article subjects {{comment|(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 '''<tt>c</tt>''' in the set of classes '''<tt>C</tt>''' {{comment|(<tt>c<sub>1</sub></tt>...<tt>c<sub>i</sub></tt>)}}
* We eventually want to judge an unseen document '''<tt>D</tt>'''
* and what we want for '''<tt>D</tt>''' 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 <tt>P(c|D)</tt> is highest 

or, a bit more formally yet,
 c = maxarg( P(c<sub>i</sub>|D) )

====Bayes's rule====

The <tt>P(c|D)</tt> 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 <tt>P(c|D)</tt>,
and the first step is to apply '''Bayes's rule''' {{comment|(which pops up in various other types of probability calculations and estimations)}}.
              P(D|c) * P(c)
 P(c|D)  =  -----------------
                  P(D)
...for each class <tt>c</tt> in <tt>C</tt>.


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 {{comment|(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).

...so '''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 {{comment|('''P(c&#x7c;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 '''D'''ocument fit in every specific '''c'''lass?' ''.

: 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"
to
: "find the class for which our estimation of fit is highest" {{comment|(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(c<sub>i</sub>|D) )
to:
 maxarg( P(c<sub>i</sub>) * P(D|c<sub>i</sub>) )
or, assuming equal base-class probabilities, to:
 maxarg( P(D|c<sub>i</sub>) )

====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 {{comment|(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 {{comment|(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 [http://en.wikipedia.org/wiki/Zipf_distribution 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.

<!--
In fact, spam filters may be bayesian but will have more specific features, such as 'apparently sent to self', 'sent to dozens of people', 'uses many non-dictionary words', some estimation of 'uses misspellings likely to be read over but avoid exact matchers', presence of common spam-phrases (e.g. based on regexp matches), trust or distrust of certain mail networks, etc.
-->


Naive Bayes is often explained by saying we want a probability based on <tt>n</tt> features in feature set F: <tt>P(c|f<sub>1</sub>..f<sub>n</sub>)</tt>, which Bayes-inverses and denominator-eliminates to:
 P(c) * P(f<sub>1</sub>..f<sub>n</sub>|c)
...which roughly means "how do features in this new document compare to the one for the class?"


Technically, <tt>P(f<sub>1</sub>..f<sub>n</sub>|c)</tt> 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) * <span style="font-size:210%">&#928;</span><sub>i=1 to n</sub> P(f<sub>i</sub>|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 <tt>P(c) * P(f<sub>1</sub>..f<sub>n</sub>|c)</tt> 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.


====Pseudocode====
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:
     <tt>P(t|c) = ( '''numoccurence'''(t in Text) + 1 ) / ('''numtokens'''(Text)+|U| )</tt>


'''Classifying''' (word features)
 foreach c in C:
   p=P(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.



===Semi-sorted===
* [[#Bayesian classifier]]

* Parzen classifier

* Backpropagation classifier

<!--

Applications of more general techniques:
* PCA
* SVM
* Any clustering in a feature space, e.g. kNN
-->


====Evaluating classifiers====
<!--
Precision/recall, accuracy, variant/hybrid methods (, valuing precision over recall.)


Classification can be tested. Usually, there is a set of annotated data. When it is large enough, it can be split into a training set and an evaluation set. The idea is that you cannot evaluate based on the training set iself, as that should give (nearly) perfect results, and would not at all be a test of how it would work on unseen data.

When you have enough data, you could split into training, evaluation and a completely separate evaluation set that you use for nothing else and see as your unseen data. Usually this will contain some bias, that of the fact it comes from the same data collection, but it is a pretty good test of how a system tweaked to training data works with data it hasn't seen before, and should reveal if it is unflexible.

When data is scarce, you can do an n-fold crossvalidation. This means you split your data into n (nearly-)equal-sized buckets, and use n-1 for training and the last one for testing, in all possible combinations, and averaging the accuracy. It is essentially pretending that you have more training data. On the upside, it will use all your data for testing and thus make it less likely that your test won't accidentally be skewed by a few odd cases.



-->

====Combining classifiers====
<!--
Combining classifiers can be useful when you think some may have imperfections that sometimes leads to strange answers for certain data. 

Averaging can sometimes make sense. There is also focus on:
* Voting: let each model predict, and look for consensus: use the most common answer.

* Bagging: Voting, but use same learner on different training data, to the end of hoping to lessen [[overfitting]].

* Boosting: Focus on misclassified data by weighing it more the next iteration


Also: 
* Use classification to self-teach or 
* co-teach by generating corpora yourself. May easily delude itself.
* Add a human for difficult cases only (like co-training)
-->

====Support Vector Machines====

<!--
Supervised, trained from samples in one of two categories.
(Sometimes trained from unlabeled data via clustering)


By themselves, they do binary, linear classification.

SVM is generally introduced as linear SVM, the moving on to uses of the kernel method.
(basically something that maps a complex thing into something more linearly separable for SVM to deal with).



One of the ideas behind SVM is that you can try to minimize overfitting by trying to pick a set of most useful points out of the training data. You can see it as ignoring outliers, or generalizing as part of the learning process.


The idea is that each datapoint is one of:
* Important (a.k.a. a 'Support Vector')
* Wrong    
* Ignorable (correct, but not strong)

The best results usually come from widening the margin, maximizing the distance from the SV's to the hyperplane.

This makes it a constraint problem, which is you can always deal well with using the Lagrangian equation. So they say:)


Given n attributes to use, SVM takes place in n-dimensional space, constructing a hyperplane to categorize the data into two groups, or because of the margin, more accurately a hypertube. (...closest human-likable concept given the margin is also n-dimensional)





-->

=====Support-vector regression=====

====Using the kernel method====
<!--
See also [[Kernel method]]

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==Clustering==

===Intro===

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)


<!--

Clustering methods vary in in intermediate definitions, e.g. 
* method of calculating the similarity/distance (the metric)
* intermediate definitions, such as agglomerate distances in hierarchical clustering
* thresholding or other means of inclusion/exclusion decision per cluster, possibly a target amount of clusters.


It is generally not hard to find a reasonable solution for nice data, as each strong-enough cluster in the data attracts a cluster center into it / makes for clear division / makes for clear local density.


For some algorithms there are visual interpretations.
For example, assuming 2D data, simpler nearest-cluster-by-distance methods resemble [[Voronoi]]-like division.



-->

'''Formally''', the simplest clustering ca be described as:
* you have a set of n data objects, call it  D = { d<sub>1</sub>, ..., d<sub>n</sub> }
* 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 {{comment|(e.g. some Maximum-likelihood-based methods common in bioinformatics)}}



====Variations====
=====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.
<!--
This starts to resemble other things, such as probability-based classifiers, and some other methods mentioned here.
-->

=====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).


http://gepas.bioinfo.cipf.es/cgi-bin/docs/clusterhelp

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=====Graph clustering=====

If you have a graph (in the mathematical network sense) you could cluster it
 
Markov Cluster Algorithm
[http://www.micans.org/mcl/]
-->

====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.
<!--
You could even create a dendrogram from a (separate) hierarchical clustering and decide based on that, which is computationally more expensive than a single run, but often cheaper than running the clustering for all possible number of groups.
-->

* 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


<!--
* Clusters into a fixed number of groups that is either manually set, or sometimes based on the data. 
* input: n coordinates in many-dimensional space, and where applicable a target number of groups
* potential problems: sensitive to outliers, sensitive to unrepresentative choices of k (whether given or automatic)
* complexity: Generally, each iteration is O(n*k). For most data sets there will be less than n iterations, though there are pathological cases
-->

=====Inter-cluster and intra-cluster comparisons; susceptibilities=====

<!--
Each element can be and often is represented by multidimensional data. The distance measures simplify this to a (usually symmetric) difference matrix to work on - though various methods needs to also define the distance between clusters.
-->


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''' {{comment|(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 [http://en.wikipedia.org/wiki/Ward's_method#The_minimum_variance_criterion 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 <!-- hierarchical cases that make for a very deep tree with one leaf at each depth. In flattened form {{comment|(and some non-hierarchical cases)}} it means one very large group and a few tiny ones, rather than a more balanced clustering. Sometimes contrasted with a symmetry effect{{verify}}-->

* '''Inversions''' {{comment|(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.<!--
** They lead to items which show a larger distance in the result
** They are not necessarily a problem - small inversions can indicate areas where there is no clear clustering solution. However, there are also situations where 

* '''Non-uniqueness'''
** In agglomerative hierarchical clustering regularly wants to join two clusters, and may find that where distances between different clusters are equal, so equally valid at that level. An algorithm will often choose an arbitrary one. Those different solutions may be different at more detailed levels.

-->

<!--
Non-uniqueness and inversions are sometimes underestimated.
-->


See also:
* http://en.wikipedia.org/wiki/Hierarchical_clustering

TODO: read
* http://people.revoledu.com/kardi/tutorial/Similarity/index.html
* http://www.google.com/search?hl=en&q=Mahalanobis+distance

=====on convergence=====
{{stub}}

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.

=====Evaluation=====

<!--
Evaluating the accuracy of clustering (and dimensionality reduction) is by nature an imperfect thing. 
You always reduce information - the question is how useful that is to you, which is subjective.


In the case of hierarchical clustering, the cophenetic correlation coefficient (CPCC or CCC) is common, and makes the cluster-based metric taxa-based.

You could test how stable a result is: add noise to your data, see how often and how much the clusterings disagree.

There are also some metrics of clusterings. For example, the silhouette value can measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation).

For dimensionality reduction, the most basic method is probably to have some metric, and and see how well these distances correlate before and after.

Note that it matters whether a clustering is complete or not. The ability to ignore points as outliers


In evalauation, there are various concept that com eup:
* '''cohesion''' of/within a cluster
* '''separation''' between clusters is relevant.

* '''entropy'''
* '''purity'''

-->

======Silhouette coefficients======
<!--
Uses cohesion and separation, and can be evaluation for points, clusters, and the whole clustering.


https://cs.fit.edu/~pkc/classes/ml-internet/silhouette.pdf
http://scikit-learn.org/stable/modules/generated/sklearn.metrics.silhouette_score.html

-->

======Davies-Bouldin index======

======Gap statistic======
<!--

https://web.stanford.edu/~hastie/Papers/gap.pdf

https://en.wikipedia.org/wiki/Silhouette_%28clustering%29

http://scikit-learn.org/stable/auto_examples/cluster/plot_kmeans_silhouette_analysis.html


http://stats.stackexchange.com/questions/10540/how-to-interpret-mean-of-silhouette-plot
-->


======Unsorted======

<!--
SSE ?


http://www.cs.kent.edu/~jin/DM08/ClusterValidation.pdf


https://cgi.csc.liv.ac.uk/~katie/adma10.pdf

-->

===Implementation notes===

=====k-means=====
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.

{{comment|(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. 


Limitations: 
* 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:
* [http://en.wikipedia.org/wiki/K-means_algorithm Wikipedia: K-means algorithm] 
* Arthur & Vassilvitskii, [http://ilpubs.stanford.edu:8090/778/1/2006-13.pdf ''k-means++ The Advantages of Careful Seeding'']
* JA Lloyd (1957),  ''Least Squares Quantization in PCM''
* EW Forgy (1965), ''Cluster analysis of multivariate data: efficiency vs interpretability of classifications''
* MacQueen (1967), ''Some methods for classification and analysis of multivariate observations.''
* Hartigan and Wong (1979), ''A K-means clustering algorithm''
* Kanungo, Mount, Netanyahu, Piatko, Silverman, and Wu, [http://www.kanungo.us/pubs/tpami02-kmeans.pdf ''An efficient k-means clustering algorithm: Analysis and implementation'']
* Bottou, Bengio, [http://citeseer.ist.psu.edu/bottou95convergence.html ''Convergence Properties of the K-Means Algorithms'']



Variations:
* '''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=====
<!--
In part the idea that you can deal with more data if you first do rough and fast clustering within windows, then merge that, and continue with more accurate clustering (e.g. hierarchical or k-means).

...where those windows can be quickly calculated. The windows are intended to overlap, so a bunch of data plays in more than one such pre-clustering.
-->
https://en.wikipedia.org/wiki/Canopy_clustering_algorithm

=====Bisecting k-means=====
{{stub}}


=====hard c-means=====
{{stub}}


=====UPGMA, WPGMA=====
{{stub}}

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) = (n<sub>u</sub>*D(u,w) + n<sub>v</sub>*D(v,w)) / (n<sub>u</sub>+n<sub>u</sub>)

WPGMA uses:
 D((u,v),w) = (n<sub>u</sub>*D(u,w) + n<sub>v</sub>*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

=====UPGMC, WPGMC=====
{{stub}}

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)=====
{{stub}}

* Type: '''fuzzy''' clustering (not really partitioning anymore)

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

http://www.elet.polimi.it/upload/matteucc/Clustering/tutorial_html/cmeans.html

http://www.scholarpedia.org/article/Fuzzy_C-Means_Cluster_Analysis

=====Fuzzy k-means=====
{{stub}}

http://www.know-center.tugraz.at/forschung/wissenserschliessung/downloads_demos/fuzzy_k_means_and_k_means_clustering_demo

=====Shell clustering=====
{{stub}}

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

=====PAM=====
Partitional, medoid-based

=====CLARA=====
Partitional, medoid-based


=====AGNES=====
Hierarchical

AGglomerative NESting

=====DIANA=====
Hierarchical

DIvisie ANAlysis

=====Buckshot=====
Hybrid

=====Birch=====
Hybrid

BITCH (balanced iterative reducing and clustering using hierarchies)

https://en.wikipedia.org/wiki/BIRCH

=====Cure=====
Hybrid

Clustering Using REpresentatives

https://en.wikipedia.org/wiki/CURE_algorithm

=====Rock=====
Hybrid


[http://theory.stanford.edu/~sudipto/mypapers/categorical.pdf "Rock: A robust clustering algorithm for categorical attributes"]

=====Chameleon=====
Hybrid, Hierarchical


[https://www-users.cs.umn.edu/~hanxx023/dmclass/chameleon.pdf "CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic Modeling"]

=====DBSCAN=====

Density-based.

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.



http://en.wikipedia.org/wiki/DBSCAN

=====OPTICS=====
Similar to DBSCAN

https://en.wikipedia.org/wiki/OPTICS_algorithm

=====FLAME=====
Fuzzy, density-based

http://en.wikipedia.org/wiki/FLAME_Clustering



=====Clustering by Committee (CBC)=====
Hybrid


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


See also:
* http://www.aclweb.org/aclwiki/index.php?title=CBC
* PA Pantel (2003) ''Clustering by Committee'' [http://scholar.google.com/scholar?q=pantel+%22Clustering+by+Committee%22]


=====Principal Direction Divisive Partitioning (PDDP)=====


=====Information Bottleneck=====


See also:
* https://en.wikipedia.org/wiki/Information_bottleneck_method
* N Slonim, "{{search|The Information Bottleneck: Theory and Applications}}"


=====Agglomerative Information Bottleneck=====

See also:
* N Slonim, N Tishby (1999) "{{search|Slonim Tishby Agglomerative Information Bottleneck|Agglomerative Information Bottleneck}}"

=====Expectation Maximization Clustering=====

===Related fields===

<!--


Particularly bio-informatics and AI have seen specific research into finding relations in data, taxonomic overviews, and change over time - phylogenic relations. This may overlap somewhat with computational linguistics.

Bio-informatics is also one of the driving forces between concepts like:
* '''Phenetics''' are a group of techniques that mostly concerns itself with relations and separation in taxonomies, and attempts to classify by direct and overall similarity and differences, specifically the observable ones, rather than dealing with evolutionary (phylogenetic) relations. Its main developers are probably Sokal and Sneath.

* '''Cladistics''' are a set of (computationally hungier) techniques that focuses more on phylogenetic relations.  It places a few stricter requirements on the data.  Its focus on clades is somewhat specific to evolutionary biology, so phenetics is also still used.  

* Dendrograms, phylograms (from phylogenetics) and cladograms (from cladistics) are fairly closely related -- arguably the largest difference is that the name suggests what type of data they were based on{{verify}}, and how they can be visualised.

-->
===See also===
* http://www.statsoft.com/textbook/stcluan.html
* http://www.cs.joensuu.fi/pages/franti/vq/algorithms.htm




=Semi-sorted=

===Structured prediction===
<!--
: yes, you could approached these questions as a series of independent classifications, but there is often reason that considering them together gives better answers
: we use the name when this consideration is part of our problem, e.g. when our scoring function considers this
: For example
:: taking written letters and detecting words will probably consider whether the series forms a word
:: assigning lexical categories to words in a sentence, which will often include sentence structure (e.g. via dependency-tree analysis)
:: figuring out what objects an image contains may consider which objects make sense together
: the output doesn't need to be highly structured itself

https://www.youtube.com/watch?v=e5sPNlgbZAE

https://pystruct.github.io/intro.html

https://en.wikipedia.org/wiki/Structured_prediction
-->

===Mistake Bound learning===
<!--

Basically any where the algoritm predicts a classification of an unlabeled example,
and is later told the correct answer.

With just that, it sounds like supervised learning you could do in any way,
but the name points at an approach that learns the logical conjunction of features,
so learning whether each presumed features has positive, negative, or no contribution.



http://pages.cs.wisc.edu/~yliang/cs760_fall17/slides/lecture16-learning-theory-2.pdf


-->

===PAC theory===
<!--
Probably Approximately Correct theory is an analysis that 
e.g. think about the lower bounds of generalization error of a generatization function.



https://www.cs.utexas.edu/~klivans/f06lec2.pdf

-->


[[Category:Algorithms]]
[[Category:Math on data]]
[[Category:Machine Learning]]

===Kernel method===

[[Classification,_clustering,_and_decisions#Kernel_method]]

=Unsorted=
<!--



Domain adaptation 


Zero-Shot Learning (ZSL) is when a learner is expected to learn a new class 
without having explicitly been shown examples. It is an extreme form of form of domain adaptation.

One-shot learning 

Few-show learning

...where classically we have large dataset of human-labeled 


Zero-shot learning still typically counts on auxiliary information, and knowledge transfer.
Consider how as a human, you might be able to visually identify a platypus, never having seen one,
but only descriptions.






(k-)Nearest Neighbour, a.k.a. kNN

Not the same as k-means. TODO: See [http://www.ir.iit.edu/~dagr/DataMiningCourse/Spring2001/Notes/Clustering.pdf] for distinction.

TODO: See [http://www.google.com/search?q=%22k-nearest%22%20neighbor%20%22k-means%22], [http://people.revoledu.com/kardi/tutorial/KNN/index.html], [http://www.autonlab.org/autonweb/software/10408.html], [http://www.dbs.informatik.uni-muenchen.de/~boehm/publications/dexa03-final.pdf]





* various at http://crd.lbl.gov/~cding/, http://ecology.msu.montana.edu/labdsv/R/, http://www.statsoft.com/textbook/stcluan.html

* [http://video.google.com/videoplay?docid=2546753955666121345 Google Video: Clustering aggregation (Aristides Gionis)]

* VQ / LVQ ((Learning) Vector Quantization)
** (apparently like kNN and SOM)

* Curvilinear Component Analysis (CCA)

* [http://en.wikipedia.org/wiki/Expectation-maximization_algorithm Expectation Maximization] 

* Quality Threshold (QT)
** Doesn't cluster elements deemed low quality


* Fuzzy ([[http://public.fh-wolfenbuettel.de/~hoeppnef/clustering.html]], [http://documents.wolfram.com/applications/fuzzylogic/Manual/12.html], [http://fuzzy.iau.dtu.dk/tutor/ddfm_files/v3_document.htm])


* Spectral Clustering http://crd.lbl.gov/~cding/Spectral/notes.html


* BSAS, see maybe [http://www-staff.it.uts.edu.au/~paulk/DMBio.cgi/pubs/paul-clustering.pdf]

* Artificial neural network (ANN)
* Self-organizing map (SOM) (sometimes Kohonen Map, or Self Organizing Feature Map) [http://www.sims.berkeley.edu/~hearst/irbook/10/node5.html]
** Usually used for reducing dimensionality 

* Shi-Malik
* Meila-Shi


* Discriminant Function Analysis (DFA, DA)


Trained Structure
Classifier


Adaptive Resonance Theory


Linear programming [http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html#Q1] [http://www.google.com/search?q=Integer%20Linear%20Programming]


*RIPPER



Tile coding: simple formalism to have learning affect similar states {{verify}}








* Jain, ''Murty, Data Clustering: A Review,'' (1999), [http://citeseer.csail.mit.edu/jain99data.html]
* [http://en.wikipedia.org/wiki/Data_clustering Wikipedia: Data clustering]
* http://www.kddresearch.org/Courses/Fall-2001/CIS732/Lectures/

* http://www.autonlab.org/tutorials/index.html


Software:
* [http://bips.u-strasbg.fr/fr/Documentation/ClustalX/ ClustalX]
* [http://evolution.genetics.washington.edu/phylip.html Phylip]


Unsorted:
http://citeseer.csail.mit.edu/722717.html

http://citeseer.csail.mit.edu/718017.html

http://citeseer.csail.mit.edu/guha98cure.html

http://citeseer.csail.mit.edu/chu02incremental.html

http://citeseer.csail.mit.edu/steinbach00comparison.html

http://citeseer.csail.mit.edu/karypis99chameleon.html

http://citeseer.csail.mit.edu/fred03robust.html

http://citeseer.csail.mit.edu/712324.html

http://citeseer.csail.mit.edu/topchy03combining.html

http://citeseer.csail.mit.edu/711453.html

http://citeseer.csail.mit.edu/706677.html

http://citeseer.csail.mit.edu/jain01structure.html

http://citeseer.csail.mit.edu/719978.html

http://citeseer.csail.mit.edu/714801.html

http://citeseer.csail.mit.edu/guha00rock.html

http://citeseer.csail.mit.edu/na01performance.html

http://citeseer.csail.mit.edu/260668.html

http://citeseer.csail.mit.edu/ng94efficient.html

http://citeseer.csail.mit.edu/cutting92scattergather.html

http://citeseer.csail.mit.edu/steinbach00comparison.html

http://citeseer.csail.mit.edu/fred02data.html

http://citeseer.csail.mit.edu/law04simultaneous.html

http://fuzzy.iau.dtu.dk/tutor/ddfm_files/v3_document.htm

http://citeseer.csail.mit.edu/masulli02soft.html


-->