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In [[machine learning]] and [[statistics]], '''classification''' is the problem of identifying to which of a set of [[categorical data|categories]] (sub-populations) a new [[observation]] belongs, on the basis of a [[training set]] of data containing observations (or instances) whose category membership is known.  Examples are assigning a given email to the [[Spam filtering|"spam" or "non-spam"]] class, and assigning a diagnosis to a given patient based on observed characteristics of the patient (sex, blood pressure, presence or absence of certain symptoms, etc.).  Classification is an example of [[pattern recognition]].

In the terminology of machine learning,<ref>{{cite book|last=Alpaydin|first=Ethem|title=Introduction to Machine Learning|date=2010|publisher=MIT Press|isbn=978-0-262-01243-0|page=9|url=https://books.google.com/books?id=7f5bBAAAQBAJ&printsec=frontcover#v=onepage&q=classification&f=false}}</ref> classification is considered an instance of [[supervised learning]], i.e., learning where a training set of correctly identified observations is available.  The corresponding [[unsupervised learning|unsupervised]] procedure is known as [[cluster analysis|clustering]], and involves grouping data into categories based on some measure of inherent similarity or [[distance]].

Often, the individual observations are analyzed into a set of quantifiable properties, known variously as [[explanatory variables]] or ''features''.  These properties may variously be [[categorical data|categorical]] (e.g. "A", "B", "AB" or "O", for [[blood type]]), [[ordinal data|ordinal]] (e.g. "large", "medium" or "small"), [[integer|integer-valued]] (e.g. the number of occurrences of a particular word in an [[email]]) or [[real number|real-valued]] (e.g. a measurement of [[blood pressure]]). Other classifiers work by comparing observations to previous observations by means of a [[similarity function|similarity]] or [[metric (mathematics)|distance]] function.

An [[algorithm]] that implements classification, especially in a concrete implementation, is known as a '''classifier'''.  The term "classifier" sometimes also refers to the mathematical [[function (mathematics)|function]], implemented by a classification algorithm, that maps input data to a category.

Terminology across fields is quite varied. In [[statistics]], where classification is often done with [[logistic regression]] or a similar procedure, the properties of observations are termed [[explanatory variable]]s (or [[independent variable]]s, regressors, etc.), and the categories to be predicted are known as outcomes, which are considered to be possible values of the [[dependent variable]].  In [[machine learning]], the observations are often known as ''instances'', the explanatory variables are termed ''features'' (grouped into a [[feature vector]]), and the possible categories to be predicted are ''classes''.  Other fields may use different terminology: e.g. in [[community ecology]], the term "classification" normally refers to [[cluster analysis]], i.e., a type of [[unsupervised learning]], rather than the supervised learning described in this article.

==Relation to other problems==
Classification and clustering are examples of the more general problem of [[pattern recognition]], which is the assignment of some sort of output value to a given input value.  Other examples are [[regression analysis|regression]], which assigns a real-valued output to each input; [[sequence labeling]], which assigns a class to each member of a sequence of values (for example, [[part of speech tagging]], which assigns a [[part of speech]] to each word in an input sentence); [[parsing]], which assigns a [[parse tree]] to an input sentence, describing the [[syntactic structure]] of the sentence; etc.

A common subclass of classification is [[probabilistic classification]].  Algorithms of this nature use [[statistical inference]] to find the best class for a given instance.  Unlike other algorithms, which simply output a "best" class, probabilistic algorithms output a [[probability]] of the instance being a member of each of the possible classes.  The best class is normally then selected as the one with the highest probability.  However, such an algorithm has numerous advantages over non-probabilistic classifiers:
*It can output a confidence value associated with its choice (in general, a classifier that can do this is known as a ''confidence-weighted classifier'').
*Correspondingly, it can ''abstain'' when its confidence of choosing any particular output is too low.
*Because of the probabilities which are generated, probabilistic classifiers can be more effectively incorporated into larger machine-learning tasks, in a way that partially or completely avoids the problem of ''error propagation''.

==Frequentist procedures==

Early work on statistical classification was undertaken by [[Ronald Fisher|Fisher]],<ref>[[R. A. Fisher|Fisher R.A.]] (1936) " [https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1469-1809.1936.tb02137.x The use of multiple measurements in taxonomic problems]", ''Annals of Eugenics'', 7, 179&ndash;188</ref><ref>Fisher R.A. (1938) " [https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1469-1809.1938.tb02189.x The statistical utilization of multiple measurements]", ''Annals of Eugenics'', 8, 376&ndash;386</ref> in the context of two-group problems, leading to [[Fisher's linear discriminant]] function as the rule for assigning a group to a new observation.<ref name=G1977>Gnanadesikan, R. (1977) ''Methods for Statistical Data Analysis of Multivariate Observations'', Wiley. {{ISBN|0-471-30845-5}} (p. 83&ndash;86)</ref> This early work assumed that data-values within each of the two groups had a [[multivariate normal distribution]]. The extension of this same context to more than two-groups has also been considered with a restriction imposed that the classification rule should be [[linear]].<ref name=G1977/><ref>[[C. R. Rao|Rao, C.R.]] (1952) ''Advanced Statistical Methods in Multivariate Analysis'', Wiley. (Section 9c)</ref> Later work for the multivariate normal distribution allowed the classifier to be [[nonlinear]]:<ref>[[T. W. Anderson|Anderson, T.W.]] (1958) ''An Introduction to Multivariate Statistical Analysis'', Wiley.</ref> several classification rules can be derived based on different adjustments of the [[Mahalanobis distance]], with a new observation being assigned to the group whose centre has the lowest adjusted distance from the observation.

==Bayesian procedures==

Unlike frequentist procedures, Bayesian classification procedures provide a natural way of taking into account any available information about the relative sizes of the different groups within the overall population.<ref>Binder, D.A. (1978) "Bayesian cluster analysis", ''[[Biometrika]]'', 65, 31&ndash;38.</ref> Bayesian procedures tend to be computationally expensive and, in the days before [[Markov chain Monte Carlo]] computations were developed, approximations for Bayesian clustering rules were devised.<ref>Binder, D.A. (1981) "[https://academic.oup.com/biomet/article-abstract/68/1/275/237691 Approximations to Bayesian clustering rules]", ''[[Biometrika]]'', 68, 275&ndash;285.</ref>

Some Bayesian procedures involve the calculation of  [[class membership probabilities|group membership probabilities]]: these can be viewed as providing a more informative outcome of a data analysis than a simple attribution of a single group-label to each new observation.

==Binary and multiclass classification==
Classification can be thought of as two separate problems – [[binary classification]] and [[multiclass classification]]. In binary classification, a better understood task, only two classes are involved, whereas multiclass classification involves assigning an object to one of several classes.<ref>[[Sariel Har-Peled|Har-Peled, S.]], Roth, D., Zimak, D. (2003) "Constraint Classification for Multiclass Classification and Ranking." In: Becker, B., [[Sebastian Thrun|Thrun, S.]], Obermayer, K. (Eds) ''Advances in Neural Information Processing Systems 15: Proceedings of the 2002 Conference'', MIT Press. {{ISBN|0-262-02550-7}}</ref> Since many classification methods have been developed specifically for binary classification, multiclass classification often requires the combined use of multiple binary classifiers.

== Feature vectors ==
Most algorithms describe an individual instance whose category is to be predicted using a [[feature vector]] of individual, measurable properties of the instance.  Each property is termed a [[feature (pattern recognition)|feature]], also known in statistics as an [[explanatory variable]] (or [[independent variable]], although features may or may not be [[statistically independent]]).  Features may variously be [[binary data|binary]] (e.g. "on" or "off"); [[categorical data|categorical]] (e.g. "A", "B", "AB" or "O", for [[blood type]]); [[ordinal data|ordinal]] (e.g. "large", "medium" or "small"); [[integer|integer-valued]] (e.g. the number of occurrences of a particular word in an email); or [[real number|real-valued]] (e.g. a measurement of blood pressure).  If the instance is an image, the feature values might correspond to the pixels of an image; if the instance is a piece of text, the feature values might be occurrence frequencies of different words.  Some algorithms work only in terms of discrete data and require that real-valued or integer-valued data be ''discretized'' into groups (e.g. less than 5, between 5 and 10, or greater than 10).

== Linear classifiers ==
{{main|Linear classifier}}
A large number of [[algorithm]]s for classification can be phrased in terms of a [[linear function]] that assigns a score to each possible category ''k'' by [[linear combination|combining]] the feature vector of an instance with a vector of weights, using a [[dot product]].  The predicted category is the one with the highest score.  This type of score function is known as a [[linear predictor function]] and has the following general form:

:<math>\operatorname{score}(\mathbf{X}_i,k) = \boldsymbol\beta_k \cdot \mathbf{X}_i,</math>

where '''X'''<sub>''i''</sub> is the feature vector for instance ''i'', '''&beta;'''<sub>''k''</sub> is the vector of weights corresponding to category ''k'', and score('''X'''<sub>''i''</sub>, ''k'') is the score associated with assigning instance ''i'' to category ''k''.  In [[discrete choice]] theory, where instances represent people and categories represent choices, the score is considered the [[utility]] associated with person ''i'' choosing category ''k''.

Algorithms with this basic setup are known as [[linear classifier]]s.  What distinguishes them is the procedure for determining (training) the optimal weights/coefficients and the way that the score is interpreted.

Examples of such algorithms are
*[[Logistic regression]] and [[Multinomial logistic regression]]
*[[Probit regression]]
*The [[perceptron]] algorithm
*[[Support vector machine]]s
*[[Linear discriminant analysis]].

== Algorithms ==

In [[unsupervised learning]], classifiers form the backbone of cluster analysis and in [[Supervised learning|supervised]] or semi-supervised learning, classifiers are how the system characterizes and evaluates unlabeled data. In all cases though, classifiers have a specific set of dynamic rules, which includes an interpretation procedure to handle vague or unknown values, all tailored to the type of inputs being examined.<ref>{{Cite web|url=https://deepai.org/machine-learning-glossary-and-terms/classifier|title=What is a Classifier in Machine Learning?}}</ref>

Since no single form of classification is appropriate for all data sets, a large toolkit of classification algorithms have been developed. The most commonly used include:<ref>{{Cite news|url=https://builtin.com/data-science/tour-top-10-algorithms-machine-learning-newbies|title=A Tour of The Top 10 Algorithms for Machine Learning Newbies|date=2018-01-20|work=Built In|access-date=2019-06-10}}</ref>

* [[Linear classifier]]s
** [[Fisher's linear discriminant]]
** [[Logistic regression]]
** [[Naive Bayes classifier]]
** [[Perceptron]]
*[[Support vector machine]]s
**[[Least squares support vector machine]]s
* [[Quadratic classifier]]s
* [[Variable kernel density estimation#Use for statistical classification|Kernel estimation]]
** [[k-nearest neighbor algorithm|k-nearest neighbor]]
* [[Boosting (meta-algorithm)]]
* [[Decision tree learning|Decision tree]]s
** [[Random forest]]s
* [[Artificial neural networks|Neural network]]s
* [[Learning vector quantization]]

== Evaluation ==
Classifier performance depends greatly on the characteristics of the data to be classified. There is no single classifier that works best on all given problems (a phenomenon that may be explained by the [[No free lunch in search and optimization|no-free-lunch theorem]]). Various empirical tests have been performed to compare classifier performance and to find the characteristics of data that determine classifier performance. Determining a suitable classifier for a given problem is however still more an art than a science.

The measures [[precision and recall]] are popular metrics used to evaluate the quality of a classification system. More recently, [[receiver operating characteristic]] (ROC) curves have been used to evaluate the tradeoff between true- and false-positive rates of classification algorithms.

As a performance metric, the [[uncertainty coefficient]] has the advantage over simple [[accuracy]] in that it is not affected by the relative sizes of the different classes.
<ref name="Mills2010">
{{Cite journal
 | author = Peter Mills
 | title = Efficient statistical classification of satellite measurements
 | journal = International Journal of Remote Sensing
 | volume = 32
 | issue = 21
 | pages = 6109–6132
 | doi= 10.1080/01431161.2010.507795
 | year = 2011
| arxiv = 1202.2194
 }}</ref>
Further, it will not penalize an algorithm for simply ''rearranging'' the classes.

==Application domains==
{{see also|Cluster analysis#Applications}}
Classification has many applications. In some of these it is employed as a [[data mining]] procedure, while in others more detailed statistical modeling is undertaken.

* [[Computer vision]]
** [[Medical imaging]] and medical image analysis
** [[Optical character recognition]]
** [[Video tracking]]
* [[Drug discovery]] and [[Drug development|development]]
** [[Toxicogenomics]]
** [[Quantitative structure-activity relationship]]
* [[Geostatistics]]
* [[Speech recognition]]
* [[Handwriting recognition]]
* [[Biometric]] identification
*[[Biological classification]]
* [[Statistical natural language processing]]
* [[Document classification]]
* Internet [[search engines]]
* [[Credit scoring]]
* [[Pattern recognition]]
* [[Recommender system]]
*Micro-array classification

{{More footnotes|date=January 2010}}

== See also ==
{{Commons category}}
{{Portal|Mathematics}}
{{colbegin}}
* [[Artificial intelligence]]
* [[Binary classification]]
* [[Class membership probabilities]]
* [[Classification rule]]
* [[Compound term processing]]
* [[Data mining]]
* [[Data warehouse]]
* [[Fuzzy logic]]
* [[Information retrieval]]
* [[List of datasets for machine learning research]]
* [[Machine learning]]
* [[Recommender system]]
{{colend}}

==References==
{{Reflist}}

{{Statistics|analysis||state=expanded}}

{{DEFAULTSORT:Statistical Classification}}
[[Category:Machine learning]]
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[[Category:Statistical classification| ]]