1. Mouse Livers: Derivatives and Functional Linear Models

2. How does cholesterol get metabolized in the liver?

3. What questions can we ask of the data? What does the real, “smooth” process look like? What does the real, “smooth” process look like? Do shapes differ among groups? Do shapes differ among groups? Do rates of change differ among groups? Do rates of change differ among groups?

4. What do the flow curves look like as functional objects? Took the derivative of the smoothed curves. Took the derivative of the smoothed curves. Still retain curve-to- curve variability, but now much smoother. Still retain curve-to- curve variability, but now much smoother.

5. How can I graphically explore the data? Have: Have: flow curves x(t). rate of change of flow curves Dx(t). Plot Dx(t) vs x(t). No longer an explicit function of time! Plot Dx(t) vs x(t). No longer an explicit function of time! Overlay time points on the curve for interpretation. Overlay time points on the curve for interpretation. Gives information about how function is linked with its derivative. Gives information about how function is linked with its derivative. Phase-Plane Plots

6. What do we see in these phase-plane plots? Difference in curves between receptors and no receptors Difference in curves between receptors and no receptors Cusps or ‘change- points’ when there are receptors Cusps or ‘change- points’ when there are receptors Minute 9 for Receptor A; Minute 15 for Receptor B Minute 9 for Receptor A; Minute 15 for Receptor B Minute 9 for Both Receptors: Interactive Effect? Minute 9 for Both Receptors: Interactive Effect?

7. What is the relationship between the covariates and response curves? Functional response; Scalar predictors. Functional response; Scalar predictors. Regression coefficients are functional. Regression coefficients are functional. Use basis expansion methods. Use basis expansion methods. Functional Linear Models X(t) = β 0 (t) + β 1 (t)A + β 2 (t)B + β 3 (t)A*B + ε(t),

8. Receptors affect steady state. Receptors affect steady state. B stronger than A. B stronger than A. Effects strongest after minute 9. Effects strongest after minute 9. A and B have inhibitory relationship after minute 9. A and B have inhibitory relationship after minute 9.

9. Receptors affect steady state. Receptors affect steady state. B stronger than A. B stronger than A. Effects strongest after minute 9. Effects strongest after minute 9. A and B have inhibitory relationship after minute 9. A and B have inhibitory relationship after minute 9.

11. Can also do a functional linear model for derivative (rate of change): FDA allows us to work with derivatives – which are closer to the mechanisms of the process dX/dt = β 0 (t) + β 1 (t)A + β 2 (t)B + β 3 (t)A*B + ε(t),

12. A “kicks in” earlier than does B. A “kicks in” earlier than does B. A kicks in at minute 9, B at minute 15. A kicks in at minute 9, B at minute 15. When together, see push only at minute 9 (from A?) When together, see push only at minute 9 (from A?)

15. What have we learned? Creating a functional object Creating a functional object Smoothing with basis expansions to reduce noise Smoothing with basis expansions to reduce noise Examining derivatives graphically Examining derivatives graphically Phase-plane plots Phase-plane plots Building functional linear models Building functional linear models Functional regression coefficients Functional regression coefficients Derivatives helpful here, too Derivatives helpful here, too