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<p>The mathematical tool “Parametrical Identity Mapping (PIM)” is presented in detail. Its tasks are to effectively assists (1) in choosing the best form of a measurement model, e.g., for the transient hot-wire method and (2) in completely adjusting the selected model by two global correction factors. These factors control the amplitude and time-response of the adjusted model in a way that this form of the model optimally predicts the experimental data points. The goodness of the adjusted model can easily be determined by a simple statistic, the level of similarity, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10765_2024_3499_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(R^{2} \le 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10765_2024_3499_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(R^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is known as the coefficient of determination. Here, it describes the proportion of the variation in the observed values that is predictable from the estimated values. PIM is able to create an adjusted model of a level of similarity that comes very close to unity, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10765_2024_3499_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(R^{2} \approx 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>≈</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Parametrical Identity Mapping: An Evaluation Procedure for THW Signals and Other Time-Series Methods

  • 期刊论文

  • 发表日期：

    2025年1月15日

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  • U. Hammerschmidt

摘要

The mathematical tool “Parametrical Identity Mapping (PIM)” is presented in detail. Its tasks are to effectively assists (1) in choosing the best form of a measurement model, e.g., for the transient hot-wire method and (2) in completely adjusting the selected model by two global correction factors. These factors control the amplitude and time-response of the adjusted model in a way that this form of the model optimally predicts the experimental data points. The goodness of the adjusted model can easily be determined by a simple statistic, the level of similarity, \(R^{2} \le 1\) R 2 ≤ 1 . \(R^{2}\) R 2 is known as the coefficient of determination. Here, it describes the proportion of the variation in the observed values that is predictable from the estimated values. PIM is able to create an adjusted model of a level of similarity that comes very close to unity, \(R^{2} \approx 1\) R 2 ≈ 1 .