Statistical classification

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In machine learning and statistics, classification is the problem of identifying to which of a set of 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. The individual observations are analyzed into a set of quantifiable properties, known as various explanatory variables, features, etc. These properties may variously be categorical (e.g. "A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a part word in an email) or real-valued (e.g. a measurement of blood pressure). 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). An example would be assigning a given email into "spam" or "non-spam" classes or assigning a diagnosis to a given patient as described by observed characteristics of the patient (gender, blood pressure, presence or absence of certain symptoms, etc.).

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, implemented by a classification algorithm, that maps input data to a category.

In the terminology of machine learning,[1] classification is considered an instance of supervised learning, i.e. learning where a training set of correctly identified observations is available. The corresponding unsupervised procedure is known as clustering or cluster analysis, and involves grouping data into categories based on some measure of inherent similarity (e.g. the distance between instances, considered as vectors in a multi-dimensional vector space).

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 variables (or independent variables, 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. There is also some argument over whether classification methods that do not involve a statistical model can be considered "statistical". 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[edit]

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, 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.

The language usage does vary widely. Usually 'classification' is viewed a something distinct (exclusive) from regression, but sometimes regression is seen as an important special case of classification, as it predicts over a large count of classes that happen to be ordered (i.e., "continuous classes"). [2]

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[edit]

Early work on statistical classification was undertaken by Fisher,[3][4] 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.[5] 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.[5][6] Later work for the multivariate normal distribution allowed the classifier to be nonlinear:[7] several classification rules can be derived based on slight 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[edit]

Unlike frequentist procedures, Bayesian classification procedures provide a natural way of taking into account any available information about the relative sizes of the sub-populations associated with the different groups within the overall population.[8] 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.[9]

Some Bayesian procedures involve the calculation of 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[edit]

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.[10] Since many classification methods have been developed specifically for binary classification, multiclass classification often requires the combined use of multiple binary classifiers.

Feature vectors[edit]

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, also known in statistics as an explanatory variable (or independent variable, although in general different features may or may not be statistically independent). Features may variously be binary ("male" or "female"); categorical (e.g. "A", "B", "AB" or "O", for blood type); ordinal (e.g. "large", "medium" or "small"); integer-valued (e.g. the number of occurrences of a particular word in an email); or 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).

The vector space associated with these vectors is often called the feature space. In order to reduce the dimensionality of the feature space, a number of dimensionality reduction techniques can be employed.

Linear classifiers[edit]

A large number of algorithms for classification can be phrased in terms of a linear function that assigns a score to each possible category k by 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:

\operatorname{score}(\mathbf{X}_i,k) = \boldsymbol\beta_k \cdot \mathbf{X}_i,

where Xi is the feature vector for instance i, βk is the vector of weights corresponding to category k, and score(Xi, 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 classifiers. 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

Algorithms[edit]

The most widely used classifiers are the neural network (multi-layer perceptron), support vector machines, k-nearest neighbours, Gaussian mixture model, Gaussian, naive Bayes, decision tree and RBF classifiers.[citation needed]

Examples of classification algorithms include:

Evaluation[edit]

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 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. [11] Further, it will not penalize an algorithm for simply rearranging the classes.

Application domains[edit]

Classification problems has many applications. In some of these it is employed as a data mining procedure, while in others more detailed statistical modeling is undertaken.

See also[edit]

References[edit]

  1. ^ Alpaydin, Ethem (2010). Introduction to Machine Learning. MIT Press. p. 9. ISBN 978-0-262-01243-0. 
  2. ^ Hall, Frank, Holdmes, Pfahringer, Reutemann, Witten (2099). "The WEKA Data Mining Software: An Update". SIGKDD. 
  3. ^ Fisher R.A. (1936) " The use of multiple measurements in taxonomic problems", Annals of Eugenics, 7, 179–188
  4. ^ Fisher R.A. (1938) " The statistical utilization of multiple measurements", Annals of Eugenics, 8, 376–386
  5. ^ a b Gnanadesikan, R. (1977) Methods for Statistical Data Analysis of Multivariate Observations, Wiley. ISBN 0-471-30845-5 (p. 83–86)
  6. ^ Rao, C.R. (1952) Advanced Statistical Methods in Multivariate Analysis, Wiley. (Section 9c)
  7. ^ Anderson,T.W. (1958) An Introduction to Multivariate Statistical Analysis, Wiley.
  8. ^ Binder, D.A. (1978) "Bayesian cluster analysis", Biometrika, 65, 31–38.
  9. ^ Binder, D.A. (1981) "Approximations to Bayesian clustering rules", Biometrika, 68, 275–285.
  10. ^ Har-Peled, S., Roth, D., Zimak, D. (2003) "Constraint Classification for Multiclass Classification and Ranking." In: Becker, B., Thrun, S., Obermayer, K. (Eds) Advances in Neural Information Processing Systems 15: Proceedings of the 2002 Conference, MIT Press. ISBN 0-262-02550-7
  11. ^ Peter Mills (2011). "Efficient statistical classification of satellite measurements". International Journal of Remote Sensing. doi:10.1080/01431161.2010.507795. 

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