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Regression is the branch of Statistics in which a dependent variable of interest is Springer Undergraduate Mathematics Series Linear Models in Statistics.
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This is why we simplify the equation into:.

The method of least squares minimizes the square of the euclidian norm of the residuals. This is why the least squares method is based on the cost function defined by:. The least squares method minimizes the cost function :.

Regression (Linear Models in Statistics) - N.H. Bingham and John F. Fry.pdf

In the particular case where the deterministic function is linear with respect to the parameter , then the method reduces to the linear least squares. Let be the Jacobian matrix made of the partial derivatives of with respect to :. Let be a reference value of the parameter. Let us denote by the value of the Jacobian at the reference point.

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Since the function is, by hypothesis, linear, the Jacobian is independent of the point where it is evaluated. Since is linear, it is equal to its Taylor expansion:. The previous equations are the normal equations.

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Notice, however, that the previous linear system of equations is not implemented as is, i. Alternatively, various orthogonalization methods such as the QR or the SVD decomposition can be used to solve the linear least squares problem so that potential ill-conditionning of the normal equations is mitigated. This estimator can be proved to be the best linear unbiased estimator, the BLUE , that is, among the unbiased linear estimators, it is the one which minimizes the variance of the estimator. Therefore, the distribution of is:.

In the general case where the function is non linear with respect to the parameter , then the resolution involves a non linear least squares optimization algorithm. Instead of directly minimizing the squared euclidian norm of the residuals, most implementations rely on the residual vector, which lead to an improved accuracy.

The difficulty in the nonlinear least squares is that, compared to the linear situation, the theory does not provide the distribution of anymore. The bootstrap method is based on the following experiment.

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Provided that we can generate a set of input and output observations, we can compute the corresponding value of the parameter. Reproducing this sampling experiment a large number of times would allow to get the distribution of the estimated parameter. In practice, we only have one single sample of observations.

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  5. If this sample is large enough and correctly represents the variability of the observations, the bootstrap method allows to generate observations resamples, which, in turn, allow to get a sample of. An approximate distribution of can then be computed based on kernel smoothing, for example. In order to get a relatively accurate distribution of , the boostrap sample size must be large enough. Hence, this method requires to solve a number of optimization problems, which can be time consuming.

    Alternatively, we can linearize the function in the neighbourhood of the solution and use the gaussian distribution associated with the linear least squares. This method is efficient, but only accurate when the function is approximately linear with respect to in the neighbourhood of.

    A link between the method of least squares and the method of maximum likelihood can be done provided that two hypotheses are satisfied. The first hypothesis is that the random output observations are independent. The second hypothesis is that the random measurement error has the gaussian distribution. In this particular case, it can be shown that the solution of the least squares problem maximizes the likelihood. This is the reason why, after a least squares calibration has been performed, the distribution of the residuals may be interesting to analyze.

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    Regression: Linear Models in Statistics | Mathematical Association of America

    The use of registered names, trademarks, etc. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Preface The subject of regression, or of the linear model, is central to the subject of statistics. It concerns what can be said about some quantity of interest, which we may not be able to measure, starting from information about one or more other quantities, in which we may not be interested but which we can measure.

    We model our variable of interest as a linear combination of these variables called covariates , together with some error.

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    If only because regression is inherently a subject in two or more dimensions, it is not the first topic one studies in statistics. So this book should not be the first book in statistics that the student uses. That said, the statistical prerequisites we assume are modest, and will be covered by any first course on the subject: ideas of sample, population, variation and randomness; the basics of parameter estimation, hypothesis testing, p —values, confidence intervals etc.

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