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Biostatistics-Lecture 14 Generalized Additive Models

Biostatistics-Lecture 14 Generalized Additive Models. Ruibin Xi Peking University School of Mathematical Sciences. Generalized Additive models. Gillibrand et al . (2007) studied the bioluminescence-depth relationship. Generalized Additive models.

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Biostatistics-Lecture 14 Generalized Additive Models

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  1. Biostatistics-Lecture 14Generalized Additive Models Ruibin Xi Peking University School of Mathematical Sciences

  2. Generalized Additive models • Gillibrand et al. (2007) studied the bioluminescence-depth relationship

  3. Generalized Additive models • Gillibrand et al. (2007) studied the bioluminescence-depth relationship • We may first consider the model

  4. Generalized Additive models • Gillibrand et al. (2007) studied the bioluminescence-depth relationship • We may first consider the model

  5. Generalized Additive models • Gillibrand et al. (2007) studied the bioluminescence-depth relationship • We may first consider the model

  6. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based

  7. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based

  8. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based

  9. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based

  10. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based • Spline-based

  11. Generalized Additive models • LOESS (locally weighted scatterplot smoothing)-based • Spline-based

  12. Generalized Additive models • Suppose that the model is • As discussed before, we may approximate the unknown function f with polynomials • But the problem is this representation is not very flexible

  13. Generalized Additive models

  14. Generalized Additive models • What if the data looks like

  15. Generalized Additive models • One way to overcome this difficulty is to divide the range of the x variable to a few segment and perform regression on each of the segments

  16. Generalized Additive models • One way to overcome this difficulty is to divide the range of the x variable to a few segment and perform regression on each of the segments

  17. Generalized Additive models • One way to overcome this difficulty is to divide the range of the x variable to a few segment and perform regression on each of the segments • The cubic spline ensures the line looks smooth

  18. Generalized Additive models • Cubic spline bases (assuming 0<x<1) • With this bases, the model may be approximated by knots

  19. Generalized Additive models • Cubic spline bases (assuming 0<x<1) • With this bases, the model may be approximated by knots

  20. Generalized Additive models • How to determine the number of knots? • Use model selection methods? • But this is problematic • Instead we may use the penalized regression spline • Rather than minimizing • We could minimize

  21. Generalized Additive models • Because f is linear in the parameters , the penalty can be written as • The first two rows and columns are 0 • The penalized regression spline can be written as

  22. Generalized Additive models • Because f is linear in the parameters , the penalty can be written as • The first two rows and columns are 0 • The penalized regression spline can be written as

  23. Generalized Additive models • How to choose λ? • Ordinary cross-validation (OCV) • We my try to minimize • Instead, we may minimize

  24. Generalized Additive models • It is inefficient to use the OCV by leaving out one datum at a time • But, it can be shown that • Generalized cross-validation (GCV)

  25. Generalized Additive models • It is inefficient to use the OCV by leaving out one datum at a time • But, it can be shown that • Generalized cross-validation (GCV)

  26. Generalized Additive models • Additive models with multiple explanatory variables • We may also consider the model like

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