Statistical classification

Statistical classification is a statistical procedure in which individual items are placed into groups based on quantitative information on one or more characteristics inherent in the items (referred to as traits, variables, characters, etc) and based on a training set of previously labeled items.

Formally, the problem can be stated as follows: given training data $$\{(\mathbf{x_1},y_1),\dots,(\mathbf{x_n}, y_n)\}$$ produce a classifier $$h:\mathcal{X}\rightarrow\mathcal{Y}$$ which maps an object $$\mathbf{x} \in \mathcal{X}$$ to its classification label $$y \in \mathcal{Y}$$. For example, if the problem is filtering spam, then $$\mathbf{x_i}$$ is some representation of an email and $$y$$ is either "Spam" or "Non-Spam".

Statistical classification algorithms are typically used in pattern recognition systems.

Note: in community ecology, the term "classification" is synonymous with what is commonly known (in machine learning) as clustering. See that article for more information about purely unsupervised techniques.

Statistical classification techniques
While there are many methods for classification, they are solving one of three related mathematical problems

The first is to find a map of a feature space (which is typically a multi-dimensional vector space) to a set of labels. This is equivalent to partitioning the feature space into regions, then assigning a label to each region. Such algorithms (e.g., the nearest neighbour algorithm) typically do not yield confidence or class probabilities, unless post-processing is applied. Another set of algorithms to solve this problem first apply unsupervised clustering to the feature space, then attempt to label each of the clusters or regions.

The second problem is to consider classification as an estimation problem, where the goal is to estimate a function of the form
 * $$P({\rm class}|{\vec x}) = f\left(\vec x;\vec \theta\right)$$

where the feature vector input is $$\vec x$$, and the function f is typically parameterized by some parameters $$\vec \theta$$. In the Bayesian approach to this problem, instead of choosing a single parameter vector $$\vec \theta$$, the result is integrated over all possible thetas, with the thetas weighted by how likely they are given the training data D:
 * $$P({\rm class}|{\vec x}) = \int f\left(\vec x;\vec \theta\right)P(\vec \theta|D) d\vec \theta$$

The third problem is related to the second, but the problem is to estimate the class-conditional probabilities $$P(\vec x|{\rm class})$$ and then use Bayes' rule to produce the class probability as in the second problem.

Examples of classification algorithms include:
 * Linear classifiers
 * Fisher's linear discriminant
 * Logistic regression
 * Naive Bayes classifier
 * Perceptron
 * Quadratic classifiers
 * k-nearest neighbor
 * Boosting
 * Decision trees
 * Neural networks
 * Bayesian networks
 * Support vector machines
 * Hidden Markov models

An intriguing problem in pattern recognition yet to be solved is the relationship between the problem to be solved (data to be classified) and the performance of various pattern recognition algorithms (classifiers). Van der Walt and Barnard (see reference section) investigated very specific artificial data sets to determine conditions under which certain classifiers perform better and worse than others.

Classifier performance depend 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 emperical 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 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.

Application domains

 * Computer vision
 * Medical Imaging and Medical Image Analysis
 * Optical character recognition
 * Geostatistics
 * Speech recognition
 * Handwriting recognition
 * Biometric identification
 * Document classification
 * Internet search engines
 * Credit scoring