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# regression - 11 Multiple Linear Regression Multiple linear...

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Notes_11, GEOS 585A, Spring 2009 1 11 Multiple Linear Regression Multiple linear regression (MLR) is a method used to model the linear relationship between a dependent variable and one or more independent variables. The dependent variable is sometimes also called the predictand, and the independent variables the predictors. MLR is based on least squares: the model is fit such that the sum-of-squares of differences of observed and predicted values is minimized. MLR is probably the most widely used method in dendroclimatology for developing models to reconstruct climate variables from tree-ring series. Typically, a climatic variable is defined as the predictand and tree-ring variables from one or more sites are defined as predictors. The model is fit to a period – the calibration period – for which climatic and tree-ring data overlap. In the process of fitting, or estimating, the model, statistics are computed that summarize the accuracy of the regression model for the calibration period. The performance of the model on data not used to fit the model is usually checked in some way by a process called validation . Finally, tree-ring data from before the calibration period are substituted into the prediction equation to get a reconstruction of the predictand. The reconstruction is a “prediction” in the sense that the regression model is applied to generate estimates of the predictand variable outside the period used to fit the data. The uncertainty in the reconstruction is summarized by confidence intervals, which can be computed by various alternative ways. Regression has long been used in dendroclimatology for reconstructing climate variables from tree rings. A few examples of dendroclimatic studies using linear regression are reconstruction of annual precipitation in the Pacific Northwest (Graumlich 1987), reconstruction of runoff of the White River, Arkansas (Cleaveland and Stahle 1989), reconstruction of an index of the El Nino Southern Oscillation (Michaelsen 1989), and reconstruction of a drought index for Iowa (Cleaveland and Duvick 1992). MLR is not strictly a “time series” method. The most important point in application to time series is that observations are typically not independent of one another. As a consequence, special attention must be paid to a regression assumption about the independence of the residuals. The predictors in any regression problem might be intercorrelated. This so-called multicolinearity does not preclude the use of regression, but can make it impossible or difficult to assess the relative importance of individual predictors from the estimated coefficients of the regression equation. It is ironic that the most interesting periods of dendroclimatic reconstructions derived from regression are often the periods for which application of the regression model is most problematical -- periods whose climatic anomalies are most unlike those of today. The reconstruction for those periods is likely to be more uncertain than implied by regression statistics

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regression - 11 Multiple Linear Regression Multiple linear...

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