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										<sp count="6"/>PRECONDITIONING DATA</link>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Our example data is as follows:</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Given the oscillating nature of this data, try to fit a function of the form:</p>
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							<ml:apply>
								<ml:mult style="default"/>
								<ml:apply>
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								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:real>1</ml:real>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Last, make a guess for the unknown parameters:</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">guess</ml:id>
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						<ml:real>3</ml:real>
						<ml:real>5</ml:real>
						<ml:real>4</ml:real>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now determine the fit:</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
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						<ml:genfit method="levenberg-marquardt"/>
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			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Cfit</ml:id>
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						<ml:matrix rows="5" cols="1">
							<ml:real>5.2764145931945761</ml:real>
							<ml:real>3.1415926538342229</ml:real>
							<ml:real>0.16215800298317989</ml:real>
							<ml:real>4.2296035322230177</ml:real>
							<ml:real>6.646746196880926</ml:real>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Let's look at our fit:</p>
			</text>
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								</ml:apply>
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								<ml:mult style="default"/>
								<ml:apply>
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											<ml:real>3</ml:real>
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									</ml:apply>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
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							<ml:real>4</ml:real>
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				</ml:define>
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				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">tr</ml:id>
					<ml:range>
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							<ml:real>0.1</ml:real>
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						<ml:real>16</ml:real>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Not bad!<sp count="2"/>Now let's use the function definition to estimate some future values . . .</p>
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			<math optimize="false" disable-calc="false">
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					<ml:real>4</ml:real>
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				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
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						<ml:real>1</ml:real>
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				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
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					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:matrix rows="4" cols="1">
							<ml:real>11.762358812430573</ml:real>
							<ml:real>1.4264484018056249</ml:real>
							<ml:real>12.031858538476536</ml:real>
							<ml:real>1.2132893680724965</ml:real>
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					</result>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">For comparison, compute similar values with </c>
					<b>
						<c val="#000">predict</c>
					</b>
					<c val="#000">:</c>
				</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">P2</ml:id>
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							<ml:id xml:space="preserve">D</ml:id>
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			<math optimize="false" disable-calc="false">
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							<ml:real>12.909083381641842</ml:real>
							<ml:real>1.3505022593173846</ml:real>
							<ml:real>11.769513385698584</ml:real>
							<ml:real>1.243953382981722</ml:real>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Both approaches capture the periodic nature of the data,</p>
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		<region region-id="2186" left="36" top="1928.25" width="291" height="36" align-x="36" align-y="1938" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">which is not surprising since we have seen that </c>
					<b>
						<c val="#000">predict</c>
					</b>
					<c val="#000">
						<sp count="2"/>assumes periodic results, while in the case of </c>
					<b>
						<c val="#000">genfit</c>
					</b>
					<c val="#000"> we intentionally fitted a periodic function.</c>
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			</text>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Numerically the predicted values differ:</p>
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							<ml:real>12.031858538476536</ml:real>
							<ml:real>1.2132893680724965</ml:real>
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							<ml:real>1.3505022593173846</ml:real>
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							<ml:real>-1.1467245692112691</ml:real>
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							<ml:real>-0.030664014909225434</ml:real>
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					<c val="#000">We emphasize that there is no reason to expect the two methods to agree. They are both doing something that is essentially impossible: predicting the future. It is important to use whatever method best suits your knowledge of the data. If you have enough knowledge regarding the source of the data and can reasonably guess the form of a function to fit the data, use </c>
					<b>
						<c val="#000">genfit</c>
					</b>
					<c val="#000">. Otherwise, use </c>
					<b>
						<c val="#000">predict</c>
					</b>
					<c val="#000"> or some other method.</c>
				</p>
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