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11/23/2009 10:40:54 AM
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Title
Techniques of Weather-Resources Investigations of the USGS Book 4, Chapter A1
Date
1/1/1968
Floodplain - Doc Type
Educational/Technical/Reference Information
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<br />10 <br /> <br />Y:=a+bX <br /> <br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS <br /> <br />. <br /> <br />f7 <br /> <br />I <br /> <br />y <br /> <br /> <br />x <br /> <br />x <br /> <br />! Y=a+bX+bIX2+b2X3 <br /> <br />yl <br /> <br /> <br />x <br /> <br />Y=a+bX+cZ <br /> <br />Y=a+bX+cZ+dXZ <br />:\"v <br />'~ <br /> <br />y~t <br /> <br />x <br /> <br />x <br /> <br />Figure 1 C,-Equations and graphs of some common regression <br />models. <br /> <br />extended to include additional independent <br />variables. <br />Having selected a suitable model, the coef- <br />ficients in that model equation are computed <br />from sample data by the method of least <br />squares as described subsequently. <br />Note that although two of the graphs in <br />figure 10 are curved, all of the model equations <br />are in linear form. This linearity of the model <br />equation i~ a requirement for direct least- <br />squares solution. Linearity can sometimes be <br />attained by transforming the variables. <br /> <br />Transformations <br /> <br />There are two principal reasons for trans- <br />forming data before analysis: (1) to obtain a <br />linear regression model, and (2) to achieve equal <br />variance about the regression line throughout <br />the range. <br />We have seen from figure 10 that certain <br />two-variable regressions may be linearized <br />without transforming the variables. The method <br /> <br />is known as polynomial regression in which <br />additional variables in successively higher <br />powers of the independent variable are added <br />to the model. But suppose we know or postulate <br />that a relation should be of the form <br /> <br />Y=aX'. <br /> <br />By taking logarithms of both sides of the equa- <br />tion the resulting linear equation is obtained: <br /> <br />log Y=log a+b log X, <br /> <br />in which log a and b are constants which can be <br />computed by a least-squares analysis using the <br />variables log Y and log X. Likewise the relation <br /> <br />Y=abX <br /> <br />can be transformed to <br /> <br />log Y= log a+Xlog b, <br /> <br />where log a and log b are the constants and <br />log Y and X are the variables. Other transforma- <br />tions are sometimes used, but the logarithmic <br />transformation is by far the most common. <br />The second reason for transforming data, <br />and the more important one, is to achieve <br />equal variance about the regression line. One <br />of the assumptions basic to the regression <br />method is that the distribution of errors about <br />the regression line is normal and constant <br />throughout the range (fig. 8). Again a log <br />transformation is often used. For example, <br />the graph on the left of figure 11 (from L'.S. <br />Geological Survey, 1949, p. 488) shows in- <br />creasing scatter of points with increasing <br />rainfall. But when the variables are plotted <br />on the log chart (right graph, fig. 11), the <br />scatter of points is almost uniform throughout <br />the range. Thus, if it had been desired to <br />carry the analysis beyond the graphical pres- <br />entation, a transformation of the variables <br />should have been made. (Ordinarily the vari- <br />ables would be reversed on the chart if " <br />regression were to be made because runoff is <br />the dependent variable.) <br />Other reasons for transforming data are to <br />introduce additivity to the model and to <br />"chieve normality. The use of transformations <br />is discussed by Acton (1959,1'.219-223). <br /> <br />. <br /> <br />. <br />
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