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 sumofsquares 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 treering series.
Typically, a climatic
variable is defined as the predictand and treering variables from one or more sites are defined as
predictors.
The model is fit to a period – the
calibration period
– for which climatic and treering
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, treering 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 socalled
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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 Three '11
 Lee
 Regression Analysis, residuals, predictors, Geos

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