Hi,
my
name
is
Colleen
McKendry,
and
I
am
a
senior
statistical
writer
at
JMP,
but
I
also
like
to
experiment
with
functional
data.
This
presentation
is
an
extension
of
when
Flash
was
inspired
by
a
presentation
that
was
originally
done
in
2020
titled
Measurement
Systems
Analysis
for
Curve
Data.
There
was
also
a
slightly
earlier
presentation
at
a
JSM
conference
in
2019.
My
talk
is
essentially
how
I
would
go
about
solving
the
problems
that
were
presented
in
those
original
papers.
I'll
discuss
them
a
little
bit
more
later
too.
First,
a
little
bit
of
background
on
measurement
systems
analysis.
MSA
studies
determine
how
well
a
process
can
be
measured
prior
to
studying
the
process
itself.
So
it
answers
the
question,
how
much
measurement
variation
is
contributing
to
the
overall
process
variation.
Specifically,
the
Gage
R&R
method,
which
I'll
be
using
later
in
this
analysis,
determines
how
much
variation
is
due
to
operation
variation
versus
measurement
variation.
These
types
of
studies
are
important
and
they're
often
required
to
be
performed
prior
to
any
type
of
statistical
process
control
or
design
of
experiments.
A
Gage
R&R
classical
MSA
model
is
shown
here.
For
a
given
measurement,
your
response,
Y
sub
I
K
is
the
Kth
measurement
on
the
Ith
part.
In
this
model,
you
have
a
mean
term
and
a
random
effect
that
corresponds
to
the
part
and
your
error term.
The
random
effect
in
the
error
term
are
normally
distributed
random
variables
with
mean
zero
and
some
corresponding
variance
components.
This
is
simply
a
random
effects
model,
and
we
can
use
that
model
to
estimate
the
variance
components,
and
then
use
those
variance
component
estimates
to
calculate
the
%
gage
R&R
using
the
formula
that's
shown
on
the
screen.
Here
we
have
the
same
model
but
the
crossed
version.
For
your
response,
Y
sub
IJK
,
that's
going
to
be
the
Kth
measurement
made
by
the
Jth
operator
on
the on
the
Ith
part.
A
gain,
we
have
a
mean
term,
a
random
effect
that
corresponds
to
the
part,
and
now
we
have
a
random
effect
that
corresponds
to
the
operator,
and
a
random
effect
that
corresponds
to
the
cross
term,
which
is
the
interaction
between
the
operator
and
the
part,
and
of
course,
our
error
term.
All
of
these
random
effects
are
normally
distributed
random
variables
with
mean
zero
and
some
corresponding
variance
component.
So
just
like
in
the
classical
model,
this
is
just
a
random
effects
model,
and
we
can
estimate
the
variance
components
and
use
them
to
calculate
the
%
gage
R&R.
In
both
of
the
models
that
I
just
described,
the
response
or
the
measurement
was
a
single
point.
But
what
happens
if
this
isn't
the
case?
What
if
your
measurement
is
something
like
a
curve
instead?
This
was
the
motivation
behind
those
initial
presentations
in
2019
and
2020
that
I
talked
about.
There
was
a
client
of
JMP
that
was
a
supplier
of
automotive
parts,
and
they
had
a
customer
that
specified
that
a
part
needed
to
have
a
specific
force
by
distance
curve.
Obviously,
the
client
wanted
to
match
their
customer
specified
curve,
and
so
they
wanted
to
run
a
functional
response
DOE
analysis
in
JMP
to
design
their
product
to
do
that.
However,
before
spending
the
money
on
this
type
of
experiment,
they
first
wanted
to
perform
an
MSA
on
their
ability
to
actually
measure
the
parts
force.
There's
a
lot
more
details
in
the
paper
noted
at
the
bottom,
so
if
you're
interested
in
more
of
the
background,
please
see
that.
This
is
what
the
data
looks
like.
We
have
force
on
the
Y
axis
and
distance
on
the
X
axis,
and
the
curves
are
colored
by
part.
It
looks
like
there
are
only
10
curves,
but
there
are
actually
250
curves
in
total.
It's
just
that
a
lot
of
the
curves
are
clustered
together.
In
the
data,
there
were
10
parts,
five
operators,
and
five
replications
per
part
operator
combination.
I
just
wanted
to
note
that
this
is
simulated
data,
and
it's
simulated
to
look
similar
to
the
actual
data,
but
that
we
aren't
sharing
any
proprietary
data
here.
A
few
function
characteristics
that
I
wanted
to
point
out,
the
functions
are
actually
all
different
lengths,
so
they
have
a
different
number
of
observations
in
their
curve.
Although
the
functions
were
collected
at
equally
spaced
time
intervals,
they
were
not
collected
at
equally
spaced
distances.
That
means
there's
no
true
replication
in
terms
of
distance.
When
this
project
was
first
presented,
one
of
the
original
ideas
thrown
out
was
whether
we
could
pick
a
set
of
distance
locations
and
do
a
standard
gage
R&R
MSA
at
each
of
those
locations
and
then
summarize
that
information
for
a
final
result.
The
problem
with
that
is
that
if
we
picked
a
specific
location,
there
wasn't
a
guarantee
that
there
would
be
an
observation
for
each
of
those
curves
because
there
wasn't
replication
for
the
distance.
Another
problem
that's
more
generalized
is
with
this
type
of
curve
data,
doing
point
wise
analysis
like
that
does
not
take
into
account
the
within
function
correlation.
Luckily,
there's
a
whole
field
of
statistics
dedicated
to
this
type
of
data
called
functional
data
analysis.
There
are
a
variety
of
techniques
to
handle
unequally
spaced
data.
A
lot
of
those
techniques
are
now
available
in
JMP
through
the
Functional
Data
Explorer
platform.
The
question
became,
can
functional
data
methods
be
combined
with
traditional
MSA
methods
to
perform
some
type
of
functional
measurement
systems
analysis?
This
was
the
solution
that
was
presented
in
the
older
papers
that
I
referenced.
This
is
just
going
to
be
a
little
bit
of
a
review
of
what
they
did.
First,
a
penalized
spline
model
was
fit
to
estimate
the
part
force
functions
and
so
there
were
10
functions
that
were
estimated,
averaged
over
operator
and
replicates.
Then
these
functions
were
subtracted
from
the
original
force
functions
to
obtain
a
set
of
residual
force
functions.
These
residual
functions
no
longer
contain
any
variation
due
to
the
part.
All
of
the
variation
in
those
residuals
were
due
to
the
operator
and
the
replicates.
They
then
fit
a
random
effects
model
to
the
residuals to
obtain
the
corresponding
variance
components
from
the
model.
A
graphical
method
was
then
used
to
find
the
smallest
part
variance
to
use
as
an
estimate
for
the
part
variance
component.
This
was
then
used
to
calculate
a
type
of
worst
case
scenario
%
gage
R&R.
This
method
they
worked
fairly
well.
They
got
results
that
made
sense
and
the
client
was
happy.
But
this
was
just
a
generalization
of
a
standard
MSA
with
some
functional
components
sprinkled
in.
When
I
looked
at
this
data
and
when
I
looked
at
the
problem,
I
wanted
to
try
to
take
a
more
traditional
functional
approach.
I
have
a
background
in
functional
data
analysis,
and
that
is
what
my
dissertation
was
on,
specifically
functional
mixed
models.
There
was
a
chapter
in
my
dissertation
dedicated
to
estimating
and
testing
the
variance
components
from
functional
mixed
models.
I
did
that
by
expanding
the
functional
model
using
eigen
function
or
basis
function
expansions
and
rewriting
it
as
a
mixed
model
and
then
using
known
techniques
to
estimate
those
variance
components.
I
started
to
think,
could
I
use
the
same
type
of
technique?
I
don't
need
a
full
mixed
model.
I
only
have
random
effects
here.
Can
I
create
a
functional
random
effects
model
for
the
part
and
operator
variance
components?
This
is
what
I
came
up
with
for
a
functional
MSA
across
models,
since
we
do
have
an
operator
term.
For
functional
models,
they're
set
up
a
little
bit
differently
because
they're
all
based
around
the
input.
In
this
case,
your
response,
Y
sub
IJK
is
the
Kth
replicate
made
by
the
Jth
operator
on
the
ith
part,
but
this
time
at
a
particular
distance,
D.
We
have
a
functional
mean
term,
a
functional
random
effect
that
corresponds
to
the
part,
a
functional
random
effect
that
corresponds
to
the
operator,
and
a
functional
random
effect
that
corresponds
to
the
cross
term
and
our
error
term.
In
this
method,
I
subtract
the
mean
term
over,
and
so
I'm
left
with
this
set
of
residuals,
and
that's
when
I'm
going
to
model.
This
here
represents
the
eigen
function
expansion
of
the
functional
model.
We're
going
to
have
capital
B
eigen
functions and
sum
all
of
those
parts
together
to
create
this
big
long
random
effects
model.
But
for
one
eigen
function,
what
is
shown
in
these
brackets
is
the
expansion.
For
each
functional
random
effect,
it's
split
into
two
parts.
We
have
a
functional
part
and
then
just
a
regular
part.
The
functional
part
is
taken
care
of
by
evaluating
those
eigen
functions.
Then
we
have
just
standard
random
terms
for
the
part,
the
operator,
and
the
cross
term.
Then
what
this
essentially
does
is
you
build
this
long
random
effects
model,
and
then
you
have
a
number
of
variance
components
for
each
term.
For
the
MSA,
there
will
now
be
three
sets
of
capital
B
variance
components.
There's
going
to
be
capital
B
part
variance
components,
capital
B
operator
variance
components,
and
capital
B
cross
term
variance
components.
Because
of
the
way
eigen
functions
are
structured,
they
are
known
to
be
independent
and
so
we
can
assume
that
based
on
how
we
structured
the
model,
all
of
these
variance
components
are
actually
independent
from
each
other
also.
That
means
we
can
sum
them
together
to
obtain
these
functional
variance
components.
Since
we're
just
summing
them
together,
we
can
also
substitute
them
into
the
formula
for
the
%
gage
R&R
and
compute
that
just
like
we
did
in
the
standard
models.
How
do
I
actually
do
this
in
JMP?
Well,
it's
a
multi
step
process.
I'm
going
to
briefly
outline
it
here,
and
then
I'm
going
to
do
a
demo
for
you.
First,
I'm
going
to
estimate
the
mean
curve
in
FDE
and
obtain
the
residual
curves.
I'm
then
going
to
model
those
residual
curves
serves
also
an
FDE
to
obtain
the
eigen
functions
needed
for
the
eigen
function
expansion.
I'm
going
to
save
those
eigen
functions
to
the
original
data
table
and
use
them
in
FitMix.
Using
FitMix,
I'm
going
to
fit
a
random
effects
model
to
the
original
data
using
nesting
and
the
eigen
functions
to
define
the
appropriate
model
specifications.
Hopefully,
that
all
makes
a
little
bit
more
sense
once
I
demo
it.
We're
going
to
exit
out
of
here.
Here
is
our
data,
and
we
have
a
column
for
the
ID
variable,
a
column
for
the
part
variable
that
defines
the
10
parts,
the
operator,
which
defines
the
five
operators,
our
distance
column,
and
our
force
column.
Just
as
a
reminder,
this
is
what
the
data
looked
like.
My
first
step
is
to
estimate
the
mean
function
of
the
force
curves
and
then
use
that
to
obtain
some
residuals.
To
do
that,
I'm
going
to
model
the
force
functions
in
FDE.
I'm
going
to
go
to
the
Analyze
menu,
Specialize
Modeling
and
select
Functional
Data
Explorer.
I'm
going
to
define
force
as
my
output,
distance
as
my
input.
Then
because
I
want
the
mean
function
averaged
over
all
of
the
IDs,
I'm
not
going
to
specify
an
ID
function
here.
I'm
going
to
click
Okay.
We
have
our
basic
intro
FDE
report.
But
I
want
to
fit
a
model,
so
I
can
go
to
the
red
triangle
menu,
models.
Technically,
you
could
fit
any
of
those
models.
I
just
chose
a
B
spline
because
it's
first,
it's
easy,
and
it'll
just
take
a
few
seconds
to
run
here.
Okay,
so
here's
our
model
fit.
There's
a
red
line
here
that's
pretty
hard
to
see,
but
that
is
what
is
the
mean
function,
the
estimated
mean
function.
I'm
going
to
give
you
a
better
picture
of
that
in
a
minute
or
so.
But
I
actually
want
to
save
the
functions
for
this
meme.
I
can
do
that
by
going
to
this
Function
Summaries
report.
I
can
click
the
red
triangle
menu
and
select
Customize
Function
Summaries.
I
only
want
the
same
formulas,
so
I'm
going
to
deselect
them
all
and
then
reselect
that
one
and
click
OK
and
Save.
I
get
a
new
data
table
with
what
appears
to
be
this
lonely
little
entry
here.
There
is
a
hidden
column,
so
I'm
going
to
unhide
that.
We
have
a
distance
column
and
then
we
have
this
force
mean
functional
formula.
I'm
going
to
take
a
look
at
that,
what
that
actually
looks
like.
When
we
look
at
the
formula
column,
we
can
see
that
this
is
a
function
of
distance.
For
any
value
of
distance,
this
is
going
to
be
evaluated
to
give
what
the
mean
function
is
at
that
distance.
This
formula
column
can
be
put
into
any
data
table
that
also
contains
a
distance
column.
That's
exactly
what
we're
going
to
do.
We're
going
to
make
sure
this
is
highlighted.
We're
going
to
right
click
and
select
copy
column
properties.
Then
we're
going
to
find
our
way
back
to
our
original
data
table.
Double
click
to
create
a
new
column.
I'm
going
to
right
click
here
and
do
paste
column
properties.
Now
we
have
the
mean
force
evaluated
at
every
level,
every
distance
value
in
our
data
table.
We
can
use
that
to
now
find
our
residual
function.
I'm
going
to
double
click
again
to
create
a
new
column
and
title
it
Force
Resids.
Now
I'm
going
to
create
my
own
formula
column
that
is
simply
going
to
be
force
minus
the
mean
force.
I'm
going
to
click
OK.
Now
we
have
our
set
of
residuals
and
this
is
what
it
looks
like.
In
the
top
graph,
these
light
gray
curves
are
the
original
functions
from
the
force
column.
This
red
line
is
the
same
red
line
that
I
tried
to
show
you
in
the
FDE
report
that
was
hard
to
see,
but
this
is
what
that
was.
That's
the
mean
function.
Then
the
bottom
graph
in
green
shows
the
residual
curves.
These
are
the
curves
that
I'm
going
to
use
to
proceed
with
my
analysis.
My
next
step
is
going
to
be
to
model
the
residual
curves
using
FDE
to
obtain
the
eigen
functions
that
I
need
for
the
model
expansion.
I'm
going
to
go
to
the
Analyze
menu
again,
select
Specialized
Modeling,
Functional
Data
Explorer.
This
time
I'm
going
to
specify
the
residuals
as
the
output,
distance
as
the
input,
and
I
am
going
to
specify
my
ID
column
this
time.
I'm
going
to
click
OK.
We
have
our
initial port
here,
and
now
I
want
to
fit
a
model
to
this
data.
I
go
to
the
red
triangle
menu
to
look
at
the
models.
Again,
technically,
you
can
fit
any
of
these
models.
In
my
experimentation,
I
found
that
these
top
three,
they
took
a
really
long
time
to
fit,
and
they
didn't
provide
super
great
fits
for
what
I
needed.
The
wavelets
models
and
the
direct
functional
PCA
were
much,
much
quicker
and
while
also
providing
better
fits.
The
caveat
with
those
two
models
is
that
they
require
the
data
to
be
on
an
evenly
spaced
grid.
A
s
I
mentioned,
when
I
introduced
the
data,
that's
not
the
case.
However,
in
FDE,
we
have
some
data
processing
steps
that
help
us
manipulate
our
data
a
little
bit.
We
can
go
to
the
clean
up,
reduce,
and
this
first
tab
is
what
we
want,
and
we
can
use
that.
Now
that
just
puts
it
so
that
every
distance
value
has
an
observation,
has
a
force
residuals
observation.
Now
we
can
go
ahead
and
fit
one
of
those
models.
I
just
chose
direct functional
PCA.
A
s
you
can
see,
it
was
very
quick.
The
fitting
was
super
fast.
This
functional
PCA
report
is
where
we're
going
to
get
all
of
our
information
that
we
need.
But
I
was
just
going
to
scroll
down
to
look
at
the
data
fit
a
little
bit.
We
can
see
that
these
look
pretty
good.
Then
if
we
look
at
the
diagnostic
plots,
these
are
on
the
diagonal,
the
residuals
look
good.
This
looks
like
a
pretty
good
fit,
and
I
can
use
this
information
from
the
Functional
PCA.
In
the
Functional
PCA
report,
we
have
a
table
of
eigen values
and
then
these
graphs
of
our
shape
functions.
The
shape
functions
are
actually
our
eigen
functions.
They're
just
called
shape
functions
in
JMP.
How
these
functions
work
is
that
your
original
input,
so
distance,
is
on
the
X
axis,
and
then
the
eigen
function
evaluation
is
on
the
Y
axis.
For
any
distance
D,
you're
going
to
have
an
evaluation
at
eigen
function 1
,
an
evaluation
at
eigen
function
2,
and
so
on.
You
can
use
the
eigen
functions
to
get...
You
can
use
a
linear
combination
of
the
eigen
functions
to
get
an
estimate
of
the
original
functions.
The
eigenvalues
table
gives
you
an
idea
of
how
much
%
of
the
overall
data
variation
you're
taking
into
account
when
you
use
a
certain
number
of
eigenvalue,
eigen
function
pairings.
In
this
case,
the
first
eigenvalue
and
eigen
function
pairing
actually
accounts
for
99.9
%
of
the
total
variation
in
the
data,
which
is
actually
pretty
incredible.
This
is
important
in
determining
how
many
eigen
functions
you're
going
to
use
for
the
basis
expansion.
Typically
when
you're
selecting
a
number
of
eigen
functions
to
use,
you
don't
actually
want
to
use
all
of
the
eigen
functions
that
you're
given.
You
want
to
use
the
least
number
of
eigen
functions
that
still
account
for
an
adequate
amount
of
variation
in
the
data.
This
is
because
the
more
eigen
functions
you
use,
the
bigger
your
random
effects
model
is
going
to
be,
and
it's
going
to
be
harder
to
estimate.
What
does
accounting
for
an
adequate
amount
of
variation
in
the
data
mean?
It
can
mean
different
things
to
different
people
in
different
fields.
Typically,
when
I'm
working
with
this,
I
have
used
in
the
past
about
90
%
as
a
cut
off.
If
I
was
just
doing
this
analysis
to
do
the
analysis,
I
might
only
take
this
first
eigen
function
since
it
explains
so
much,
and
run
with
that.
For
demonstration
purposes,
I'm
going
to
take
the
first
two
just
so
you
can
see
what
the
model
expansion
looks
like
using
two
eigen
functions.
To
save
these,
I'm
going
to
go
to
this
Function
Summaries
report
again,
click
on
the
red
triangle
menu
and
select
Customize
Function
Summaries.
I
just
mentioned
that
I'm
only
going
to
save
two
of
them,
so
I
want
to
enter
two
here.
I'm
going
to
deselect
these
all
again
and
only
save
the
formulas.
I
can
click
OK
and
Save
and
I
have
another
new
data
table
now.
The
things
that
I
need
are
actually
hidden.
We
want
to
look
for
these
force
resids
shape
functions
which
represent
our
eigen
functions.
We
can
unhide
those
so
that
they're
now
included
in
our
data
table.
We
can
take
a
look
at
these
formula
columns.
Just
like
with
our
mean
function,
this
is
simply
a
function
of
distance.
For
any
value
of
distance,
these
formulas
are
going
to
give
you
what
the
eigen
function
value
is
at
that
distance.
Also,
like
the
main
function,
we
can
put
these
formula
columns
into
any
data
table
that
also
contains
a
distance
column.
That's
what
we're
going
to
do
again.
We're
going
to
put
these
formula
columns
into
our
original
data
table.
We're
going
to
make
sure
both
of
these
are
highlighted
and
right
click
and
select
copy
column
properties.
Again,
find
our
way
back
to
our
original
data
table.
I
want
to
add
two
new
columns
to
my
data
table.
I'm
going
to
go
to
calls,
new
columns.
We're
just
going
to
title
them
E1
and
2
to
represent
the
eigen
functions.
I
want
to
add
two
columns
and
I
want
to
add
them
as
a
group.
I'll
click
okay.
Then
with
these
two
new
columns
highlighted,
I'm
going
to
right
click
and
select
paste
column
properties.
Now
we
have
our
eigen
functions
evaluated
for
every
distance.
Just
to
give
you
an
idea,
this
is
what
they
look
like
and
it's
the
same
graph.
It's
almost
the
same
graph
as
what
was
in
the
FDE
report.
We're
just
taking
what
was
graphically
there
and
now
we
have
the
numbers
for
everything,
for
every
value
here.
Now
we
want
to
do
the
eigen
function
expansion
and
expand
our
functional
model.
But
what
does
that
model
actually
look
like
when
you
have
two
eigen
functions?
I'm
going
to
hop
back
over
to
my
slides
real
quick
and
show
you
what
the
model
expansion
looks
like
when
capital
B
equals
2.
I
have
this
divided
into
a
section
for
the
part,
the
operator,
and
the
cross
term.
Then
within
each
of
these
sections,
we
see
that
we
have
a
term
that
involves
eigen
function
one
and
a
term
that
involves
eigen
function
two.
Essentially,
this
means
that
we're
going
to
have
two
variance
components
for
part,
two
variance
components
for
the
operator,
and
two
variance
components
for
the
cross
term.
Now
I'm
going
to
go
back
to
my
data
table.
I
want
to
fit
this
model
using
Fit
Model.
I'm
going
to
go
to
the
Analyze
menu
and
select
Fit
Model.
I
want
to
specify
my
personality
as
the
mixed
model.
Now
I'm
going
to
specify
the
residuals
as
my
Y.
Then
I'm
going
to
move
down
to
the
effects
section.
In
the
fixed
effects
tab,
I
don't
have
any
fixed
effects
and
I
also
don't
have
an
intercept
because
I
mean
term
over
originally.
Now
I'm
going
to
move
to
the
random
effects
tab.
Here
is
where
I'm
going
to
use
the
eigen
functions
and
the
part
and
operator
variables
and
nest
them
in
an
appropriate
way
to
define
the
model
that
I
just
showed
you.
We
can
add
both
of
these
eigen
functions
and
we're
going
to
select
these
and
also
select
part.
We're
going
to
nest
part
in
each
eigen
function.
Then
we're
going
to
do
the
same
thing
for
operator
and
the
same
thing
for
the
cross
term.
That's
how
we're
going
to
define
our
model.
The
last
thing
I
want
to
do
in
this
launch
window
is
deselect
the
Unbounded
Variance
Components
option.
When
this
is
selected,
it
means
that
you
can
have
negative
estimates
or
variance
components,
and
I
don't
want
that.
Any
negative
estimates
are
just
going
to
be
set
to
zero.
Now
I
can
run
this,
and
we
have
our
report
here.
This
is
the
table
that's
going
to
give
us
our
estimates
that
we
need
to
calculate
the
%
gage
R&R,
but
I'm
going
to
poke
around
the
report
a
little
bit
first.
This
actual
by
predicted
plot
looks
really
weird
at
first,
but
it
makes
sense
when
you
think
about
it.
Since
we
don't
have
any
fixed
effects
or
an
intercept,
when
we
don't
take
the
random
effects
into
account,
our
estimate
for
everything
is
just
zero.
When
we
do
take
those
random
effects
into
account,
we
see
that
the
actual
by
conditional
predicted
plot
looks
a
lot
better
and
that
these
observations
fall
pretty
well
along
this
diagonal.
We
can
also
take
a
look
at
the
conditional
residual
plots,
and
we
can
see
that
they're
pretty
small
and
they're
centered
around
zero.
We
do
have
some
deviation
from
this
line
here,
but
there's
nothing
super
crazy
about
the
residuals.
I
feel
okay
about
using
these
estimates
to
calculate
the
percdent
gage
R&R.
I
actually
pulled
this
table
and
put
it
back
into
my
slides.
I'm
going
to
go
back
to
my
slides
for
the
remainder
of
the
presentation.
Here's
that
data
table,
not
data
table,
the
report
table
that
was
just
there.
As
you
can
see,
you
have
a
variance
component,
S2
variance, components,
estimates
for
part,
two
for
operator,
and
two
for
the
cross
term.
A
s
I
mentioned
when
I
was
describing
the
model,
we
can
sum
these
together
to
calculate
the
functional
variance
components.
These
specific
numbers
aren't
as
important
in
this
analysis
as
what
you
get
when
you
put
them
into
the
formula
for
the
%
gage
R&R.
So
when
I
do
that,
we
get
a
%
gage
R&R
of
3.3030.
That
is
what
Baren
team
defines
as
an
acceptable
measurement
system.
If
this
had
been
my
project,
I
would
have
gone
back
to
the
client
and
said
that
it
seems
like
your
measurement
system is
accurate,
you
can
go
ahead
and
proceed
with
your
design
of
experiments.
That's
basically
it
for
this
analysis
for
this
particular
data.
Just
some
thoughts
that
I
had.
This
result
was
actually
very
similar
to
the
worst
case
scenario
%
gage
R&R
that
was
presented
in
the
2019
JSM
presentation.
It
was
higher
by
just
a
few
decimal
places.
It'd
be
really
interesting
to
compare
these
methods
and
other
data
sets
to
see
if
they
are
always
similar
or
if
this
was
just
a
happy
coincidence.
I
don't
have
much
experience
at
all
with
measurement
systems,
and
so
I
don't
have
any
other
data
to
play
around
with
or
even
really
know
how
to
obtain
it.
If
anybody
has
any
data
that
they
think
might
apply
to
this
type
of
project,
any
functional
data
that
also
they
might
want
to
do
an
MSA
on,
I'd
be
really
interested
in
hearing
about
it.
For
some
future
work,
some
thoughts
that
I
had
was,
the
first
one
was,
should
I
add
a
functional
random
effect
for
the
ID?
This
is
very
commonly
done
in
a
lot
of
functional
mixed
models,
at
least
in
the
fields
that
I
worked
in.
This
was
a
big
contribution
in
my
dissertation
was
the
use
of
this
functional
random
effect
for
ID.
Typically,
this
captures
the
within
function
correlation
across,
in
this
case,
distance.
I
played
around
with
this
random
effect
in
this
data,
and
every
model
I
used,
the
number
of
eigen
functions
I
used,
it
didn't
matter.
The
variance
component
associated
with
this
random
effect
always
came
out
to
be
zero,
and
that's
not
useful.
I
think
in
this
case,
once
you
took
into
account
the
variance
from
the
part
and
the
operator,
there
just
wasn't
any
variation
left
to
account
for.
I
don't
know
if
that's
true
for
all
functional
MSA
studies
or
if
this
was
just
true
for
this
particular
data.
If
I
was
ever
able
to
get
my
hands
on
some
different
data,
this
is
definitely
something
I
would
keep
in
mind
to
see
if
it
could
be
added
to
a
model
in
any
other
data
sets
more
successfully.
I
also
think
it
would
be
cool
if
we
could
calculate
a
confidence
interval
for
the
%
gage
R&R.
And
finally,
I
wanted
to
talk
about
the
one
thing
I
wasn't
super
happy
with
in
this
project,
which
was
the
residuals.
What
was
wrong
with
them?
These
are
graphs
of
some
different
models
and
the
residuals
for
each
one.
I'm
going
to
go
back
and
forth
between
this
slide
and
the
next
one.
So
yes,
the
residuals
are
relatively
small.
They're
centered
around
zero.
There's
no
crazy
spikes
or
outliers,
and
that's
good.
That's
what
we
want.
However,
in
all
the
models
I
fit,
I
just
still
didn't
love
how
they
looked
across
distance.
Looking
at
the
residuals
this
way
is
especially
important
when
working
with
functional
data.
This
is
because
it
can
really
show
when
you're
not
capturing
all
the
functional
parts
of
the
data.
A
lot
of
times
in
functional
data,
you
see
this
fanning
effect
where
the
residuals
are
really
good
in
the
beginning,
and
then
as
you
get
towards
the
end
of
your
domain,
they
fan
out
a
little
bit.
This
data
actually
had
almost
the
opposite
problem.
We
can
see
that
the
residuals
are
a
little
bit
wider
in
the
beginning
of
the
domain
and
get
closer
to
zero
as
distance
gets
larger.
There's
also
definitely
some
type
of
cyclical
pattern
in
these
residuals.
I
don't
think
it's
the
end
of
the
world.
I
think
they're
super
centered
around
zero,
but
you
can
see
in
these
graphs
that
there's
clearly
some
up
and
down
patterns.
Essentially
what
that
means
is
that
I'm
missing
something.
I'm
not
capturing
the
full
functional
nature
of
the
data,
and
I
don't
really
know
why
yet.
I'd
really
like
to
figure
that
out
and
fit
an
even
better
model,
whether
that's
possible
with
this
data
or
different
data
in
the
future.
I'm
not
sure,
but
it's
definitely
something
I
want
to
spend
a
little
more
time
on,
and
I'd
be
open
to
any
discussion
anyone
would
like
to
have
about
it.
That's
it
for
me.
Thanks
for
watching.
If
you
have
any
questions,
suggestions,
questions
or
feedback,
feel
free
to
email
me.
Thank
you.
