In
this
talk,
we're
going
to
describe
repeated
measures
degradation
and
its
implementation
in
JMP.
I'm
going
to
present
the
background,
motivation,
some
technical
ideas
and
examples.
Then
I'll
turn
it
over
to
Peng,
who
will
do
a
demonstration
showing
just
how
easy
it
is
to
apply
these
methods
in
JMP 17.
Here's
an
overview
of
my
talk.
I'm
going
to
start
out
with
some
motivating
examples,
and
then
explain
the
relationship
between
degradation
and
failure
and
the
advantages
of
using
degradation
modeling
in
certain
applications.
Then
I'll
describe
the
motivation
for
our
use
of
Bayesian
methods
to
do
the
estimation.
To
use
Bayes' methods
for
estimation,
you
need
to
have
prior
distributions.
I'll
talk
about
the
commonly
used,
noninformative
and
weekly
informative
prior
distributions.
Also
in
some
applications,
we
will
have
informative
prior
distributions
and
how
those
can
be
used.
Then
I'll
go
through
two
examples,
and
at
the
end
I'll
have
some
concluding
remarks.
Our
first example
is
crack
growth.
We
have
21
notched
specimens.
The
notches
were
.9
"
deep,
and
that's
like
a
starter
crack.
Then
the
specimens
are
subjected
to
cyclic
loading,
and
in
each
cycle,
the
crack
grows
a
little
bit.
When
the
crack
gets
to
be
1.6
"
long,
that's
the
definition
of
a
failure.
We
can
see
that
quite
a
few
of
the
cracks
have
already
exceeded
that
level,
but
many
of
them
have
not.
Traditionally,
you
could
treat
those
as
right
censored
observations.
But
if
you
have
the
degradation
information,
you
can
use
that
to
provide
additional
information
to
give
you
a
better
analysis
of
your
data.
The
basic
idea
is
to
fit
a
model
to
describe
the
degradation
paths,
and
then
to
use
that
model
to
induce
a
failure
time
distribution.
Our
second
example
is
what
we
call
Device
B,
a
radio
frequency
power
amplifier.
Over
time
the
power
output
will
decrease
because
of
an
internal
degradation
mechanism.
This
was
an
accelerated
test
where
units
were
subjected
to
higher
levels
of
temperature,
2
37, 195
and
150
degrees
C.
The
engineers
needed
information
about
the
reliability
of
this
device
so
that
they
could
determine
how
much
redundancy
to
build
into
the
satellite.
The
failure
definition
in
this
case
was
when
the
power
output
dropped
to
-.5
decibels.
We
can
see
that
all
of
the
units
had
already
failed
at
the
higher
levels
of
temperature,
but
at
150
degrees
C,
there
were
no
failures
yet.
But
there
is
lots
of
information
about
how
close
these
units
were
to
failure
by
looking
at
the
degradation
paths.
Again,
we
want
to
build
a
model
for
the
degradation
paths,
and
then
we
use
that
to
induce
a
failure
time
distribution.
In
this
case,
the
use
condition
is
80
degrees
C.
We
want
to
know
the
time
at
which
units
operating
at
80
degrees
C,
would
reach
this
failure
definition.
Once
again,
we
build
a
model
for
the
degradation
paths.
We
fit
that
model,
and
then
from
that
we
can
get
a
failure
time
distribution.
Many
failures
result
from
an
underlying
degradation
process.
In
some
applications,
degradation
is
the
natural
response.
In
those
situations,
it
makes
sense
to
fit
a
model
to
the
degradation
and
then
use
the
induced
failure
time
distribution.
In
such
applications,
once
we
have
a
definition
for
failure,
which
we
call
a
soft
failure
because
the
unit
doesn't
actually
stop
operating
when
it
reaches
that
level
of
degradation,
but
it's
close
enough
to
failures
that
engineers
say
we
would
like
to
replace
that
unit
at
that
point,
just
to
be
safe.
Now,
in
general,
there's
two
different
kinds
of
degradation
data,
repeated
measures
degradation,
like
the
two
examples
that
I've
shown
you,
and
destructive
degradation,
where
you
has
to
destroy
the
unit
in
order
to
make
the
degradation
measurement.
For
many
years,
JMP
has
had
very
good
tools
for
handling
degradation
data.
I'm
focused
in
this
talk
on
the
repeated
measures
degradation
methods
that
are
being
implemented
in
JMP 17.
There
are
many
other
applications
of
repeated
measures
degradation,
for
example,
LED
or
laser
output,
the
loss
of
gloss
in
an
automobile
coating
and
degradation
of
a
chemical
compound,
which
can
be
measured
with
techniques
such
as
FTIR
or
any
other
measured
quality
characteristic
that's
going
to
degrade
over
time.
There
are
many
applications
for
repeated
measures
degradation.
There
are
many
advantages
of
analyzing
degradation
data
if
you
can
get
them.
In
particular,
there
is
much
more
information
in
the
degradation
data
relative
to
turning
those
degradation
data
into
failure
time
data.
This
is
especially
true
if
this
heavy
censoring.
Indeed,
it's
possible
to
make
inferences
about
reliability
from
degradation
data
in
situations
where
there
aren't
any
failures
at
all.
Also,
direct
observation
of
the
degradation
process
allows
us
to
build
better
models
for
the
failure
time
distribution
because
we're
closer
to
the
physics
of
failure.
Now,
several
years
ago,
when
we
were
planning
the
second
edition
of
our
reliability
book,
we
made
a
decision
to
use
more
Bayesian
methods
in
many
different
areas
of
application.
One
of
those
is
repeated
measures
degradation.
Why
is
that?
What
was
the
motivation
for
using
Bayes'
methods
in
these
applications?
I
used
to
think
that
the
main
motivation
for
Bayesian
methods
was
to
bring
prior
information
into
the
analysis.
Sometimes
that's
true,
but
over
the
years,
I've
learned
that
there
are
many
other
reasons
why
we
want
to
use
Bayesian
methods.
For
example,
Bayesian
methods
do
not
rely
on
large
sample
theory
to
get
confidence
intervals.
It
relies
on
probability
theory.
Also,
it
turns
out
that
when
you
use
Bayes'
methods
with
carefully
chosen,
noninformative
or
weekly
informative
prior
distributions,
you
have
credible
interval
procedures
that
have
very
good
coverage
properties.
That
is,
if
you
ask
for
95
%
interval,
they
tend
to
cover
what
they're
supposed
to
cover
with
95%
probability.
In
many
applications,
there
are
many
non-Bayesian
approximations
that
can
be
used
to
set
confidence
intervals.
When
you
do
Bayesian
inference,
it's
very
straightforward.
There's
really
only
one
way
to
do
it.
Also,
Bayesian
methods
can
handle
with
relative
ease,
complicated
model
data
combinations
for
which
there's
no
maximum
likelihood
software
available.
For
example,
complicated
combinations
of
nonlinear
relationships,
random
parameters
and
sensor
data,
the
Bayes'
methods
are
relatively
straightforward
to
apply
in
these
complicated
situations.
Finally,
last
but
certainly
not
least,
Bayesian
methods
do
allow
an
analyst
to
incorporate
prior
information
into
the
data
analysis.
But
I
want
to
point
out
that
the
revolution
we've
had
in
the
world
of
data
analysis
to
use
more
Bayesian
methods,
most
analysts
are
not
bringing
informative
prior
information
into
their
analysis.
Instead,
they
use
weekly
informative
or
noninformative
priors
so
that
they
don't
have
to
defend
the
prior
distribution.
But
in
many
applications,
we
really
do
have
solid
prior
information
that
will
help
us
get
better
answers.
I
will
illustrate
that
in
one
of
the
examples.
Bayesian
methods
require
the
specification
of
a
prior
distribution.
As
I
said,
in
many
application,
analysts
do
not
want
to
bring
informative
prior
information
into
the
modeling
and
analysis.
What
that
requires
is
some
default prior
that's
noninformative
or
weekly
informative.
There's
been
a
large
amount
of
theoretical
research
on
this
subject
over
the
past
40
years,
leading
to
such
tools
as
reference
priors,
Jeffrey's
priors,
and
independent
Jeffrey's
priors
that
have
been
shown
to
have
good
frequentist
coverage
properties.
One
of
my
recent
and
current
research
areas
is
to
try
and
make
these
ideas
operational
in
practical
problems,
particularly
in
the
area
of
reliability.
A
simple
example
of
this
is
if
you
want
to
estimate
a
location
parameter
in
the
log
of
a
scale
parameter,
a
flat
prior
distribution
leads
to
credible
intervals
that
have
exact
coverage
properties.
That's
very
powerful.
Also,
flat
prior
distributions
can
be
well
approximated
by
a
normal
distribution
with
a
very
large
variance,
and
that
leads
to
weekly
informative
priors.
Again,
it's
somewhat
informative,
but
because
the
variance
is
very
large,
we
call it
weekly
informative.
The
approach
that
I've
been
taking
to
specify
prior
distributions
is
to
find
an
unconstrained
parameterization,
like
the
location
parameter
and
the
log
of
the
scale
parameter
that
I
mentioned
above,
and
then
use
a
noninformative
or
weekly
informative
flat
or
normal
distribution
with
very
large
variances
as
the
default
prior.
Then
it's
always
good
idea
to
use
some
sensitivity
analysis
to
make
sure
that
the
prior
are
approximately
noninformative.
That
is,
as
you
perturb
the
prior
parameters,
it
doesn't
affect
the
bottom
line
results.
JMP
uses
very
sensible,
well-performing
methods
to
specify
default
prior
distributions
that
are
roughly
in
line
with
what
I've
described
here.
Having
those
default
prior
distributions
makes
the
software
user-friendly,
because
then
the
user
only
has
to
specify
prior
distributions
where
they
have
informative
information
that
they
want
to
bring
in.
Here's
just
an
illustration
to
show
that
as
the
standard
deviation
of
a
normal
distribution
gets
larger,
you
approach
a
flat
prior
distribution.
Now,
as
I
said,
in
some
applications,
we
really
have
prior
information
that
we
want
to
bring
in
that
is
informative
prior
information.
When
we
have
such
information,
we
will
typically
describe
it
with
a
symmetric
distribution,
like
a
normal
distribution,
although
some
people
prefer
to
use
what
we
call
a
location-
scale
t
distribution
because
they
have
longer
tails.
In
most
applications
where
we
have
this
informative
prior
information,
it's
only
on
one
of
the
parameters.
Then
we're
going
to
use
noninformative,
or
weekly
informative
prior
distributions
for
all
of
the
other
parameters.
Let's
go
back
to
alloy
a.
What
we're
going
to
do
is
we're
going
to
fit
a
model
to
the
degradation
paths,
and
then
use
that
model
to
induce
a
failure
time
distribution.
Now
if
you
look
in
an
engineering
textbook
on
fatigue
or
materials
behavior,
you'll
learn
about
the
Paris
crack-g
rowth
model.
It's
always
nice
to
have
a
model
that
agrees
with
engineering
knowledge.
JMP
has
implemented
this
Paris crack-growth model.
Here's
the
way
it
would
appear
in
a
textbook.
Then
on
the
right
here
we
have
the
JMP
implementation
of
that.
It's
one
of
the
many
models
that
you
can
choose
to
fit
to
your
degradation
data.
Now,
c
and
m
here,
which
in
JMP
is
c
and
b2,
are
materials
parameters,
and
they
are
random
from
unit
to unit.
The
K
function
here
is
known
as
a
stress
intensity
function.
For
the
crack
we're
studying
here,
the
stress
intensity
function
has
this
representation.
Now
this
is
a
differential
equation,
because
we've
got
a
of
t
here,
and
also
a
here,
you
can
solve
that
differential
equation
and
get
this
nice
closed
form.
This
is
the
model
that's
being
fit
within
JMP.
Again
the
parameters
b1
and
b2
will
be
random
from
unit
to
unit.
Now
here's
the
specification
of
the
prior
distribution.
I've
illustrated
here
two
different
prior
distributions,
the
default
prior
in
JMP,
and
the
prior
distribution
that
we
used
in
the
second
edition
of
our
reliability
book,
which
we
call
SMRD2,
statistical
methods
for
reliability
addition
two.
Now
the
way
we
specify
prior
distributions
in
SMRD2
and
JMP
is
doing
this
as
well,
is
with
what
I
call
a
99
%
range.
For
example,
we
say
that
we're
going
to
describe
the
mean
of
the
b1
parameter
by
a
normal
distribution
that
has
99
%
of
the
probability
between
-15
and
22.
That's
a
huge
range.
That
is
weekly
informative.
Then
we
have
similar,
very
wide
ranges
for
the
other
mean
parameter
here.
Then
for
the
Sigma
parameter
JMP,
following
usual
procedures
for
these
uses
a
half
Cauchy distribution,
which
has
a
long
upper
tail,
and
therefore,
again,
is
weekly
informative.
Now
in
our
reliability
book,
we
used
much
tighter
ranges.
But
interestingly,
the
two
different
prior
distributions
here
give
just
about
the
same
answer,
because
both
are
weekly
informative.
That
is,
the
ranges
are
large
relative
to,
let
us
say,
the
confidence
interval
that
you
would
get
using
non-Bayes'
methods.
Now,
in
addition
to
specifying
the
prior
distributions,
which
again,
JMP
makes
very
easy
because
it
has
these
nice
default
priors,
you
also
have
the
ability
to
control
the
Markov chain Monte Carlo
algorithm.
The
only
default
that
I
would
change
here
is
typically
I
would
run
more
than
one
chain.
I
changed
the
one
here
to
four.
The
reason
for
doing
that
is
twofold.
First of
all,
in
most
setups,
including
JMP,
you
can
run
those
simultaneously,
so
it
doesn't
take
any
more
computer
time.
The
other
thing
is
we
want
to
compare
those
four
different
Markov
chains
to
make
sure
that
they're
giving
about
the
same
answers.
We
call
that
mixing
well.
If
you
see
a
situation
where
one
of
those
chains
is
different
from
the
others,
that's
an
indication
of
a
problem.
If
you
have
such
a
problem,
then
the
usual
remedy
is
to
increase
the
number
of
warmup
laps,
which
is
set
to
be
10
by
default,
but
you
can
increase
that.
What
that
does
it
allows
JMP
to
tune
the
MCMC
algorithm
to
the
particular
problem
so
that
it
will
sample
correctly
to
get
draws
from
the
joint
posterior
distribution.
In
all
of
my
experiences
using
JMP,
and
Peng
has
suggested
that
he's
had
similar
experiences
that
by
increasing
that
high
enough,
with
any
examples
that
we've
tried,
JMP
will
work
well.
But
with
10
for
most
applications,
that
is
a
sufficiently
large
number.
Here's
the
results.
Here's
a
table
of
the
parameter
estimates.
Well,
typically
in
reliability
applications,
we're
not
so
much
interested
in
the
estimates
themselves.
We're
going
to
be
interested
in
things
like
failure
distributions,
which
we
look
at
in
a
moment.
Then
in
this
plot,
we
have
estimates
of
the
sample
paths
for
each
of
the
cracks.
Again,
you
can
see
the
failure
definition
here.
As Peng
will
show
you,
JMP
makes
it
easy
to
look
at
MCMC
diagnostics.
It's
always
a
good
idea
to
look
at
diagnostics
to
make
sure
everything
turned
out
okay.
What
you
do
is
you
export
posterior
draws
from
JMP,
and
then
JMP
has
set
up
there
some
scripts
to
create
these
various
different
diagnostics.
For
example,
there's
a
script
to
make
a
trace
plot
or
a
time
series
plot.
I
always
like
to
compare
those
for
the
different
chains.
Then
there's
another
one
to
make
what
we
call
a
pairs
plot,
or
scatterplot
matrix
of
the
draws.
That's
what
we
see
here.
Then
as
I
said,
we
can
use
those
draws
to
generate
estimates
of
the
failure
time
distribution.
JMP
implements
that
by
using
the
distribution
profiler
here.
We
can
estimate
fraction
failing
is
a
function
of
time
for
any
given
number
of
cycles.
Now
let's
go
to
the
Device
B RF
Power A
mplifier,
again,
an
accelerated
repeated
measures
degradation
application.
We're
going
to
need
a
model
that
describes
the
shape
of
the
paths
and
the
relationship
the
temperature
has
on
the
rates
of
degradation.
Again,
the
use
condition
in
this
application
is
80
degrees
C,
and
we're
going
to
want
to
estimate
the
failure
time
distribution
at
80
degrees
C.
In
SMRD2,
this
is
the
way
we
would
describe
the
path
model
that
fits
device
b.
We
call
this
an
asymptotic
model,
because
as
time
gets
large,
we
eventually
reach
an
asymptote.
In
this
equation,
X
is
the
transformed
temperature.
We
call
it an
Uranus
transformation
of
temperature.
X
0
is
an
arbitrary
centering
value.
Beta
one
is
a
rate
constant
for
the
underlying
degradation
process.
Beta
three
is
the
random
asymptote.
Those
two
parameters,
the
rate
constant
and
the
asymptote,
are
random
from
unit
to
unit,
and
we're
going
to
describe
that
randomness
with
a
joint
or
bivariate
lognormal
distribution.
Beta
two,
on
the
other
hand,
is
a
fixed,
unknown
parameter
that
is
the
effect
of
activation
energy
that
controls
how
temperature
affects
the
degradation
rate.
This
is
where
the
X0
comes
in.
Typically
we
choose
X
0
to
be
somewhere
in
the
range
of
the
data
or
at
a
particular
temperature
of
interest,
because
beta
one
would
be
the
rate
constant
at
X
0.
Again,
there's
a
large
number
of
different
models
that
are
available.
Here
is
how
you
would
choose
this
particular
asymptotic
model.
This
corresponds
to
the
same
equation
we
have
in SMRD
2.
The
only
difference
is
that
JMP
uses
a
slightly
different
numbering
convention
for
the
parameters.
That
was
done
to
be
consistent
with
other
things
that
are
already
in
JMP
elsewhere.
Again,
we
have
to
specify
prior
distributions,
but
JMP
makes
that
easy
because
they
provide
these
defaults,
these
weekly,
informative
defaults.
Here
I
have
the
default
that
JMP
would
have
that
we're
going
to
use
if
we
did
not
have
any
prior
information
to
bring
in.
I'm
going
to
do
that
analysis,
but
I'm
also
going
to
bring
in
the
information
that
engineers
have.
In
particular,
we
only
have
information
for
b3
and
so
that's
being
specified
here.
But
all
the
other
entries
in
the
table
are
exactly
the
same
as
the
JMP
default,
again,
making
it
really
easy
to
implement
these
kinds
of
analyses.
Here's
the
results.
Once
again,
here
we
have
a
table
giving
the
parameter
estimates,
credible
intervals
and
so
forth.
In
this
plot,
again,
we
have
estimates
of
the
sample
paths
for
all
of
the
individual
units.
Again,
we
have
the
failure
definition
here,
but
what
we
really
want
are
estimates
of
the
failure
time
distribution
at
80
degrees
C.
Again,
we're
going
to
do
that
by
using
a
profiler.
On
the
left
here
we
have
the
estimate
of
fraction
failing
is
a
function
of
time
at
80
degrees
C
for
the
default
priors.
On
the
right
we
have
the
same
thing,
except
that
we've
used
the
informative
prior
distribution
for
B
3.
Immediately
you
can
see
that
prior
information
has
allowed
us
to
get
much
better
estimation
precision.
The
confidence
interval
is
much
more
narrow.
Interestingly,
the
point
estimate
of
fraction
failing
actually
increased
when
we
brought
in
that
prior
information.
In
this
case,
the
prior
information
would
allow
the
engineers
to
get
a
much
better
estimate
of
fraction
failing
as
a
function
of
time.
Then
to
make
that
important
decision
about
how
much
redundancy
to
build
into
the
satellite.
Let
me
end
with
some
concluding
remarks.
Repeated
measures
degradation
analysis
is
important
in
many
reliability
applications.
It
is
also
important
in
many
other
areas
of
application,
like
determining
expiration
dates
for
products
like
pharmaceuticals
and
foodstuffs.
When
will
the
quality
be
at
such
a
low
level
that
the
customers
no
longer
happy?
Also,
in
certain
circumstances,
we
can
bring
prior
information
into
the
analysis,
potentially
allowing
us
to
lower
cost
of
our
degradation
experiments.
JMP
17
has
powerful,
easy-
to-
use
methods
for
making
lifetime
inferences
from
repeated
measures
degradation
data.
Now
I'm
going
to
turn
it
over
to
Peng,
and
he's
going
to
illustrate
these
methods
to
you
using
a
demonstration
in
JMP.
Thank
you,
Professor.
Now,
the
demo
time.
The
purpose
of
a
demo
is
to
help
you
to
begin
exploring
the
stable
art
approach
to
analyze
repeated
measures
degradation
data.
I
will
show
you
how
to
locate
the
sample
data
tables
using
JMP,
how
the
information
is
organizing
in
the
report
and
highlight
some
important
information
that
you
need
to
know.
First,
there
are
three
repeated
measures
degradation
data
samples
among
JMP
sample
data
tables.
Alloy
a, d
evice
b,
are
two
examples
with
embedded
scripts.
The
s laser
does not
have an embedded script.
Alloy
a
is
an
example
without
an
x
variable.
Device
b
is
example
with
an
x
variable.
To
find
them
go
to
the
help
and
click
sample
index.
Try
to
find
the
outline
node
called
reliability/
survival.
Unfold it
and
should
see a
lloy
a
is
here,
and
device
b
is
here.
To
find
GAS
laser,
you
need
to
go
to
the
sample
data
folder
on
your
computer
by
clicking
this
button,
open
the
sample
data
folder.
Now
I'm
going
to
open
alloy a.
Then
I'm
going
to
analyze
menu,
reliability
and
survival,
choose
the
repeatedly
measure
degradation.
Now
we
see
the
launch
dialog.
Our
assign
length,
crack
lengths
goes
to
y,
specimen
goes
to
label
system
ID
and
million
cycles
go
to
times.
Then
I'm
going
to
click
okay.
This
is
the
initial
report.
It's
a
fifth linear
models
for
individual
batches.
I'm
going
to
select
the
third model
here
by
clicking
this
video
button.
This
fifth
initial
model
of
a
Paris
model
of
this
alloy a
data.
I'm
going
to
click
this,
go
to
Bayesian
estimation
button
and
generate
a
configuration
interface.
Here
we
see
the
model
formula.
Here
are
the
default
settings
for
the
priors.
Then
we
are
not
going
to
change
anything
right
now,
and
we
are
just
going
to
use
the
default
prior
to
fit
our
model.
Now
I'm
going
to
click
this
fit
model
button
here
and
let
it
run.
Then
I'm
going
to
explain
what
are
in
the
report.
In
the
end,
how
to
get
the
failure
distribution
profiler.
Now
I'll
click
the
button,
the
algorithm
start
run
and
the
progress
dialogue
said
that
the
first step
is
tuning.
The
underlying
algorithm
will
go
through
some
round
of
warm
up laps,
is
procedure.
The
algorithm
is
trying
to
learn
the
shape
of
the
posterior
distribution,
for
example,
where
the
peak,
how
wide
is
the
span,
et cetera.
In
the
end,
they
will
try
to
figure
out
what
is
a
good
thinning
value
to
draw
samples
from
the
posterior
distribution,
such
that
the
samples
are
as
little
auto
correlated
as
possible.
Then
the
algorithms
enter
the
second
step.
The
dialogue
says,
this
step
is
collecting
posterior
samples.
In
this
step,
an
automatic
thinning
of
80
is
applied.
The
dialogue
shows
how
much
time
in
total
the
algorithm
had
been
running.
In
the
second
stage,
the
dialogue
also
shows
expected
completion
time.
By
such,
I
hope
it
can
help
users
to
adjust
to
their
expectation
accordingly.
Sometimes
excessive,
long
expected
completion
time
is
a
sign
of
a
problem.
Then
we
wait
a
little
bit,
and
the
algorithm
should
finish
soon.
Okay,
now
the
algorithm
has
finished.
Let's
see
what's
in
the
report.
First is
the
completion
time.
If
we
left
your
computer
run
over
time,
you
may
want
to
know
in
the
morning
on
next
day.
The
second
is
a
copy
of
your
settings,
including
priors,
number
of
iterations,
random
seed
and
other
thing shows.
Third part
is
posterior
estimates.
Be it a
summary
or
the
posterior
samples,
there
are
two
links
on
the
site
to
allow
you
to
export
posterior
samples.
I'm
going
to
emphasize
the
first link.
One
purpose
of
using
these
first link
to
export
posterior
examples
is
to
inspect
potential
problems.
Two
main
concerns
are:
convergence
and
effective
sample
size.
Let's
look
at
it.
The
table
have
parameters
in
individual
columns.
Each row
a
posterior
sample.
There
are
several
embedded
scripts.
The
most
important
one
is
the
first one.
I'm
going
to
click
this
green
triangle
to
run the
script.
The script simply
run
time
series
on
individual
columns
and
show
their
time
series
plot.
In
the
context
of
MCMC,
this
plot
is
known
as
the
trace
plot.
What
do
we
see
here?
What
I
call
good
results.
The
series
are
stationary
and
no
significant
auto
correlation.
Loosely
speaking,
when
I
say
stationary
in
this
context,
I
specifically
mean
plots
looks
like
these.
They
are
straight
equal
with
band
of
random
dots.
Okay,
let
me
close
the
report
and
the
table.
We
are
seeing
good
results
here.
Also
the
data
and
a fitted model
also
shows
the
results
is
good.
Now
we
are
ready
to
ask
for
a fitted
time
distribution
profiler.
To
do
that,
go
to
the
report
outline
node
menu
and
select
show
like
distribution
profilers.
Most
entries
in
this
dialogue
has
sensible
default
values
and
then
we
only
need
to
supply
one
of
the failure
definition.
I'm
going
to
enter
1.6
to
this
upper
failure
definition.
Before
I'm
going
to
click okay,
I'm
going
to
reduce
this
number
of
SMRD2
realizations
to
5,000
to
save
me
some
time.
Then
I'm
click
on
okay.
This
is
also
a
computational
intensive
procedure,
but
not
as
expensive
as
MCMC
in
general.
It
should
finish
quickly.
You
can
use
the
profilers
to
get
the
failure
probability
and
the
quantile
estimates.
I'm
not
going
to
elaborate
further,
because
profiler
is
a
very
common
and
important
feature
in
JMP.
Okay,
this
is
the
end
of
the
demonstration,
and
you
are
ready
to
explore
by
yourself.
But
here
are
a
couple
of
tips
that
might
be
useful.
First,
before
you
save
the
script
to
the
table,
go
to
check
this
save
posterior
to
script
option
before
you
save
the
script
to
the
table.
By
this,
next
time
you
run
the
save
script,
software
will
bring
back
the
fitted
model
instead
of
going
through
the
lengthy
MCMC
procedure
once
again.
The
second
thing
that
I
want
to
bring
to
your
attention
is
we
have
seen
good
examples,
good
results,
but
there
are
bad
ones.
Here
are
some
bad
examples.
This bad
example
means
either
fail
to
converge,
or
there
are
high
auto
correlations.
To
address
them
my
first suggestion
is
to
increase
the
number
of
warm
up
laps.
Second
suggestion
where
we
turn
off
auto-thinning
and
apply
a
large
thinning
value,
a
thinning
number
manually.
If
those
suggestions
don't
work,
it's
likely
that
the
model
or
its
configurations
are
not
appropriate
for
the
data.
You
may
need
help.
Okay,
this are
all
we would
like
to
illustrate
this
time,
and
I
hope
you
can
start
to
use
this
information
to
explore
the
state
of
art
approach
to
analyze
repeated
measures
degradation
data.
Thank
you.
