Detailed walkthrough of Competitive Regression using Fig 2C as an example.
<p>(A) Competitive Regression first regresses a terminal vertex <i>Y</i> on and estimates the coefficients in Line 1. Any variant with a <i>ground truth</i> non-zero coefficient in , such as and <i>S</i><sub><i>l</i></sub> in the exam...
محفوظ في:
| المؤلف الرئيسي: | |
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| مؤلفون آخرون: | |
| منشور في: |
2025
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إضافة وسم
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| الملخص: | <p>(A) Competitive Regression first regresses a terminal vertex <i>Y</i> on and estimates the coefficients in Line 1. Any variant with a <i>ground truth</i> non-zero coefficient in , such as and <i>S</i><sub><i>l</i></sub> in the example, is a cause of <i>Y</i>. (B) The algorithm next regresses <i>Y</i> on and the gene expression levels to estimate the coefficients in Line 2. Any variant with a ground truth non-zero coefficient in is a direct cause of <i>Y</i>, such as <i>S</i><sub><i>l</i></sub> in the example. As a result, we have whenever <i>S</i><sub><i>l</i></sub> directly causes <i>Y</i>. (C) We unfortunately do not have access to the ground truth values but must estimate the coefficients and from data and set an appropriate threshold to identify non-zero coefficients. Identifying an accurate threshold <i>ε</i> is difficult, so CR avoids this issue by setting . In particular, if <i>S</i><sub><i>l</i></sub> is truly a direct cause of <i>Y</i>, then in the ground truth because <i>S</i><sub><i>l</i></sub> does not predict any member of conditional on . As a result, we have the inequality , where acts as a data-driven threshold strictly greater than zero in the finite sample setting. TWRCI thus annotates <i>S</i><sub><i>l</i></sub> to <i>Y</i> in Line 3 when the inequality holds. Note that if both the left and right hand side of are zero in the population setting, then <i>S</i><sub><i>l</i></sub> is not a direct cause of any member of . Assigning <i>S</i><sub><i>l</i></sub> to this still yields a superset of the direct causes of <i>Y</i>. Finally, we can substitute <i>Y</i> in the above argument with any , so long as is a terminal vertex.</p> |
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