Statistical classification

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.

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.

Early work on statistical classification was undertaken by Fisher,[2][3] 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.[4] 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.[4][5] Later work for the multivariate normal distribution allowed the classifier to be nonlinear:[6] 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.

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.[7] 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.[8]

Some Bayesian procedures involve the calculation of group membership probabilities: these provide a more informative outcome than a simple attribution of a single group-label to each new observation.

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

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 features may or may not be statistically independent). Features may variously be binary (e.g. "on" or "off"); 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).

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:

${\displaystyle \operatorname {score} (\mathbf {X} _{i},k)={\boldsymbol {\beta }}_{k}\cdot \mathbf {X} _{i},}$

where X_i is the feature vector for instance i, β_k is the vector of weights corresponding to category k, and score(X_i, 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

• Logistic regression and Multinomial logistic regression
• Probit regression
• The perceptron algorithm
• Support vector machines
• Linear discriminant analysis.

In unsupervised learning, classifiers form the backbone of cluster analysis and in 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.[10]

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:[11]

• Linear classifiers
  • Fisher's linear discriminant
  • Logistic regression
  • Naive Bayes classifier
  • Perceptron
• Support vector machines
  • Least squares support vector machines
• Quadratic classifiers
• Kernel estimation
  • k-nearest neighbor
• Boosting (meta-algorithm)
• Decision trees
  • Random forests
• Neural networks
• Learning vector quantization

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

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 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