Mouse Livers:
Derivatives and Functional Linear
Models
How does cholesterol get
metabolized in the liver?
What questions can we ask of the
data?
What does the
real, “smooth”
process look like?
Do shapes differ
among groups?
Do rates of
change differ
among groups?
What do the flow curves look like
as functional objects?
Took the derivative
of the smoothed
curves.
Still retain curve-to-
curve variability, but
now much
smoother.
How can I graphically explore the data?
Phase-Plane Plots 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!
Overlay time points
on the curve for
interpretation.
Gives information
about how function is
linked with its
derivative.
What do we see in these
phase-plane plots?
Difference in
curves between
receptors and no
receptors
Cusps or „change-
points‟ when there
are receptors
Minute 9 for
Receptor A; Minute
15 for Receptor B
Minute 9 for Both
Receptors:
Interactive Effect?
What is the relationship between
the covariates and response
curves?
Functional Linear Models
Functional response; Scalar predictors.
Regression coefficients are functional.
X(t) = β0(t) + β1(t)A + β2(t)B + β3(t)A*B + ε(t),
Use basis expansion methods.
Receptors affect
steady state.
B stronger than
A.
Effects
strongest after
minute 9.
A and B have
inhibitory
relationship
after minute 9.
Receptors affect
steady state.
B stronger than
A.
Effects
strongest after
minute 9.
A and B have
inhibitory
relationship
after minute 9.
Can also do a functional linear model for
derivative (rate of change):
dX/dt = β0(t) + β1(t)A + β2(t)B + β3(t)A*B + ε(t),
FDA allows us to work with derivatives –
which are closer to the mechanisms of the
process
A “kicks in”
earlier than
does B.
A kicks in at
minute 9, B
at minute 15.
When
together, see
push only at
minute 9
(from A?)
What have we learned?
Creating a functional object
Smoothing with basis expansions to reduce
noise
Examining derivatives graphically
Phase-plane plots
Building functional linear models
Functional regression coefficients
Derivatives helpful here, too