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In machine learning, Platt scaling or Platt calibration is a way of transforming the outputs of a classification model into a probability distribution over classes. The method was invented by John Platt in the context of support vector machines, [1] replacing an earlier method by Vapnik, but can be applied to other classification models. [2] Platt scaling works by fitting a logistic regression model to a classifier's scores.
Consider the problem of binary classification: for inputs x, we want to determine whether they belong to one of two classes, arbitrarily labeled +1 and −1. We assume that the classification problem will be solved by a real-valued function f, by predicting a class label y = sign(f(x)). [lower-alpha 1] For many problems, it is convenient to get a probability , i.e. a classification that not only gives an answer, but also a degree of certainty about the answer. Some classification models do not provide such a probability, or give poor probability estimates.
Platt scaling is an algorithm to solve the aforementioned problem. It produces probability estimates
i.e., a logistic transformation of the classifier scores f(x), where A and B are two scalar parameters that are learned by the algorithm. Note that predictions can now be made according to if the probability estimates contain a correction compared to the old decision function y = sign(f(x)). [3]
The parameters A and B are estimated using a maximum likelihood method that optimizes on the same training set as that for the original classifier f. To avoid overfitting to this set, a held-out calibration set or cross-validation can be used, but Platt additionally suggests transforming the labels y to target probabilities
Here, N+ and N− are the number of positive and negative samples, respectively. This transformation follows by applying Bayes' rule to a model of out-of-sample data that has a uniform prior over the labels. [1] The constants 1 and 2, on the numerator and denominator respectively, are derived from the application of Laplace smoothing.
Platt himself suggested using the Levenberg–Marquardt algorithm to optimize the parameters, but a Newton algorithm was later proposed that should be more numerically stable. [4]
Platt scaling has been shown to be effective for SVMs as well as other types of classification models, including boosted models and even naive Bayes classifiers, which produce distorted probability distributions. It is particularly effective for max-margin methods such as SVMs and boosted trees, which show sigmoidal distortions in their predicted probabilities, but has less of an effect with well-calibrated models such as logistic regression, multilayer perceptrons, and random forests. [2]
An alternative approach to probability calibration is to fit an isotonic regression model to an ill-calibrated probability model. This has been shown to work better than Platt scaling, in particular when enough training data is available. [2]
Supervised learning (SL) is a machine learning paradigm for problems where the available data consists of labelled examples, meaning that each data point contains features (covariates) and an associated label. The goal of supervised learning algorithms is learning a function that maps feature vectors (inputs) to labels (output), based on example input-output pairs. It infers a function from labeled training data consisting of a set of training examples. In supervised learning, each example is a pair consisting of an input object and a desired output value. A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used for mapping new examples. An optimal scenario will allow for the algorithm to correctly determine the class labels for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way. This statistical quality of an algorithm is measured through the so-called generalization error.
In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier. SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. New examples are then mapped into that same space and predicted to belong to a category based on which side of the gap they fall.
In statistics, naive Bayes classifiers are a family of simple "probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features. They are among the simplest Bayesian network models, but coupled with kernel density estimation, they can achieve high accuracy levels.
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Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Pattern recognition has its origins in statistics and engineering; some modern approaches to pattern recognition include the use of machine learning, due to the increased availability of big data and a new abundance of processing power. These activities can be viewed as two facets of the same field of application, and they have undergone substantial development over the past few decades.
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In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity of a set of functions that can be learned by a statistical binary classification algorithm. It is defined as the cardinality of the largest set of points that the algorithm can shatter, which means the algorithm can always learn a perfect classifier for any labeling of at least one configuration of those data points. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis.
There are two main uses of the term calibration in statistics that denote special types of statistical inference problems. "Calibration" can mean
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics.
In statistical classification, two main approaches are called the generative approach and the discriminative approach. These compute classifiers by different approaches, differing in the degree of statistical modelling. Terminology is inconsistent, but three major types can be distinguished, following Jebara (2004):
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In information theory, the cross-entropy between two probability distributions and over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution , rather than the true distribution .
In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. It is used for classification and regression. In both cases, the input consists of the k closest training examples in a data set. The output depends on whether k-NN is used for classification or regression:
In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification. The RVM has an identical functional form to the support vector machine, but provides probabilistic classification.
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables.
Discriminative models, also referred to as conditional models, are a class of logistical models used for classification or regression. They distinguish decision boundaries through observed data, such as pass/fail, win/lose, alive/dead or healthy/sick.
Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods.
In machine learning, a probabilistic classifier is a classifier that is able to predict, given an observation of an input, a probability distribution over a set of classes, rather than only outputting the most likely class that the observation should belong to. Probabilistic classifiers provide classification that can be useful in its own right or when combining classifiers into ensembles.
In machine learning, local case-control sampling is an algorithm used to reduce the complexity of training a logistic regression classifier. The algorithm reduces the training complexity by selecting a small subsample of the original dataset for training. It assumes the availability of a (unreliable) pilot estimation of the parameters. It then performs a single pass over the entire dataset using the pilot estimation to identify the most "surprising" samples. In practice, the pilot may come from prior knowledge or training using a subsample of the dataset. The algorithm is most effective when the underlying dataset is imbalanced. It exploits the structures of conditional imbalanced datasets more efficiently than alternative methods, such as case control sampling and weighted case control sampling.
The following outline is provided as an overview of and topical guide to machine learning. Machine learning is a subfield of soft computing within computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence. In 1959, Arthur Samuel defined machine learning as a "field of study that gives computers the ability to learn without being explicitly programmed". Machine learning explores the study and construction of algorithms that can learn from and make predictions on data. Such algorithms operate by building a model from an example training set of input observations in order to make data-driven predictions or decisions expressed as outputs, rather than following strictly static program instructions.
