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In statistics, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.
Isotonic regression has applications in statistical inference. For example, one might use it to fit an isotonic curve to the means of some set of experimental results when an increase in those means according to some particular ordering is expected. A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing.
Another application is nonmetric multidimensional scaling,where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.
Isotonic regression is also used in probabilistic classification to calibrate the predicted probabilities of supervised machine learning models.
Isotonic regression for the simply ordered case with univariate has been applied to estimating continuous dose-response relationships in fields such as anesthesiology and toxicology. Narrowly speaking, isotonic regression only provides point estimates at observed values of Estimation of the complete dose-response curve without any additional assumptions is usually done via linear interpolation between the point estimates.
Software for computing isotone (monotonic) regression has been developed for R,Stata, and Python.
Let be a given set of observations, where the and the fall in some partially ordered set. For generality, each observation may be given a weight , although commonly for all .
Isotonic regression seeks a weighted least-squares fit for all , subject to the constraint that whenever . This gives the following quadratic program (QP) in the variables :
where specifies the partial ordering of the observed inputs (and may be regarded as the set of edges of some directed graph with vertices ). Problems of this form may be solved by generic quadratic programming techniques.
In the usual setting where the values fall in a totally ordered set such as , we may assume WLOG that the observations have been sorted so that , and take . In this case, a simple iterative algorithm for solving the quadratic program is the pool adjacent violators algorithm. Conversely, Best and Chakravarti studied the problem as an active set identification problem, and proposed a primal algorithm. These two algorithms can be seen as each other's dual, and both have a computational complexity of on already sorted data.
To complete the isotonic regression task, we may then choose any non-decreasing function such that for all i. Any such function obviously solves
and can be used to predict the values for new values of . A common choice when would be to interpolate linearly between the points , as illustrated in the figure, yielding a continuous piecewise linear function:
As this article's first figure shows, in the presence of monotonicity violations the resulting interpolated curve will have flat (constant) intervals. In dose-response applications it is usually known that is not only monotone but also smooth. The flat intervals are incompatible with 's assumed shape, and can be shown to be biased. A simple improvement for such applications, named centered isotonic regression (CIR), was developed by Oron and Flournoy and shown to substantially reduce estimation error for both dose-response and dose-finding applications. Both CIR and the standard isotonic regression for the univariate, simply ordered case, are implemented in the R package "cir". This package also provides analytical confidence-interval estimates.
Supervised learning is the machine learning task of learning a function that maps an input to an 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.
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In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani to blend properties of generalized linear models with additive models.
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A kernel smoother is a statistical technique to estimate a real valued function as the weighted average of neighboring observed data. The weight is defined by the kernel, such that closer points are given higher weights. The estimated function is smooth, and the level of smoothness is set by a single parameter.
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In statistics and machine learning, lasso is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term.
In statistics, projection pursuit regression (PPR) is a statistical model developed by Jerome H. Friedman and Werner Stuetzle which is an extension of additive models. This model adapts the additive models in that it first projects the data matrix of explanatory variables in the optimal direction before applying smoothing functions to these explanatory variables.
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In statistics, linear regression is a linear approach to modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.
Up-and-Down Designs (UDDs) are a family of statistical experiment designs used in dose-finding experiments in science, engineering, and medical research. Dose-finding experiments have binary responses: each individual outcome can be described as one of two possible values, such as success vs. failure or toxic vs. non-toxic. Mathematically the binary responses are coded as 1 and 0. The goal of dose-finding experiments is to estimate the strength of treatment (i.e., the 'dose') that would trigger the "1" response a pre-specified proportion of the time. This dose can be envisioned as a percentile of the distribution of response thresholds. An example where dose-finding is used: an experiment to estimate the LD50 of some toxic chemical with respect to mice.