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Concept# Linear regression

Summary

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent 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.
In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the con

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Laya Ghodrati, Victor Panaretos

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval. Our regression model is based on the theory of optimal transportation, and links the conditional Frechet mean of the response to the covariate via an optimal transport map. We define a Frechet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observations of the regression pairs. Computation of the estimator is shown to reduce to a standard convex optimization problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data.

Polynomial autoregressions have been most of the time discarded as being unrealistic. Indeed, for such processes to be stationary, strong assumptions on the parameters and on the noise are necessary. For example, the distribution of the latter has to have finite support. Nevertheless, the use of polynomials has been advocated by a few authors: Cox (1991), Cobb and Zacks (1988) and Chan and Tong (1994). From a model-free perspective, that is using parametric families of functions (e.g. Fourier series, wavelets, neural networks, MARS, ...) as approximators of the optimal predictor, there is no more concern about stationarity. Still, polynomials have not been used within this framework; the reason is the now well known "curse of dimensionality". Indeed, when high lag orders are used to forecast, a common situation in time series, polynomial autoregressions imply too many parameters to estimate. We will introduce a new family of predictors based on polynomials and on a projection scheme. A censoring procedure allows to solve the instability inherent to polynomials. The forecasting method is parsimonious (that is non-linearity is introduced with a small amount of parameters) and thus allows to forecast noisy time series of short to moderate length.

1996This paper describes inferences based on linear predictors for stationary time series. These methods are flexible, since relatively few assumptions are needed to fit a linear predictor. A confidence interval for the resulting predicted value, which takes account of the variance of the estimated parameters, is discussed. The possible non-parsimony of the linear prediction compared to the classical ARMA forecasting method is a drawback often mentioned in the literature. On the other hand, as we show in a small simulation study, the usual predictive inference based on an ARMA modelling is overoptimistic in small samples, whereas the coverage rate of our confidence interval is close to the nominal value even for small series.

1993