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										<sp count="6"/>PRECONDITIONING DATA</link>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">So, for example, if you wish to scale data to be displayed in the picture operator, you can set your range from 0 to 255:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If the data were unscaled, the image would appear black. Try changing the argument name to see this.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Some algorithms are sensitive to large or small scale data, or may not display sensibly for certain scales of data. It may be useful to scale the data to a reasonable range before applying a polynomial regression, or displaying a histogram, for example, to get sensible bin markers.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A scaling may be appropriate before attempting a Least-Squares <b>
						<u>
							<link href="./regression.xmcd" title="Links to section" popup="false">regression</link>
						</u>
					</b>. All solvers for this type of problem eventually calculate a matrix product. It's worthwhile to check the determinant of M<f size="12">
						<sup>T</sup>
					</f>*M where M*c=y. When the determinant goes to 0 or infinity, your answers from a numerical solver are highly suspect, if the solver converged at all. Rescaling the data can help this problem.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Transforming data</p>
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				<phrase>transforming data</phrase>
				<phrase>data, transforming</phrase>
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					<c val="#000">It may be worthwhile to transform data to a different functional form to apply simpler, more robust parametric </c>
					<b>
						<u>
							<link href="./linfit.xmcd" title="Links to section" popup="false">linear fit</link>
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					<c val="#000"> routines. For example, </c>
					<c val="#000">data following the polynomial form</c>
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					<c val="#000">can be transformed into linearly related data by using logarithms</c>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Carrying this through with the example data</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To return to the original data form, do the inverse transformation on the coefficients.</p>
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					<c val="#000">Other types of transformations can be applied to data whose error terms are not normally distributed with a constant variance. It's important to remember, of course, that </c>
					<b>
						<u>
							<link href="./regression.xmcd" title="Links to section" popup="false">regression</link>
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					<c val="#000"> results will be based on the transformed variables, rather than on the originals.</c>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Stabilizing the Error Variance</p>
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					<c val="#000">The </c>
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							<link href="./regression.xmcd" title="Links to section" popup="false">regression</link>
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					<c val="#000"> model</c>
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								<ml:real>...</ml:real>
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					<c val="#000">assumes that values of the independent variables are fixed, and the only random component is the error term </c>
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						<c val="#000">e</c>
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					<c val="#000">. Violations concerning </c>
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						<c val="#000">e</c>
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					<c val="#000"> may sometimes be corrected by transforming the dependent variable. In theory, if the </c>
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							<c val="#000">error variance </c>
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					<c val="#000"> is a function of the mean of Y</c>
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					<c val="#000">then the transformation satisfying </c>
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					<c val="#000">where c</c>
					<c val="#000"> is any constant, will produce an error term with a nearly constant variance. Integrating the preceding equation gives T(y) as</c>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Here are three of the more common transformations, in order of increasing severity. </p>
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				<phrase>square root transformation</phrase>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">When</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">that is,</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation T(y) is defined by </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Taking </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation can be simplified to</p>
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				<phrase>logarithm transformation</phrase>
				<phrase>transformation, logarithm</phrase>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">When</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">that is,</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation T(y) is defined by </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Taking </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation can be simplified to</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Reciprocal Transformation</p>
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				<phrase>reciprocal transformation</phrase>
				<phrase>transformation, reciprocal</phrase>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">When</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">that is,</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation T(y) is defined by </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Taking </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">the transformation can be simplified to</p>
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					<c val="#000">In practice, however, the functional relationship between </c>
					<f family="Symbol" charset="2">
						<inlineAttr font-weight="normal" line-through="false">
							<c val="#000">s</c>
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					</f>
					<c val="#000">
						<sup>2</sup>
					</c>
					<c val="#000"> and the mean of the dependent variable</c>
					<c val="#000"> isn't always evident. Finding the right transformation to correct a particular violation may be more a matter of trial and error.</c>
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