Retrieving Arbitrary True Part Distributions from Measured Distributions and Gauge Characteristics - (2023-US-30MP-1402)

JerryFish

Jerry Fish, Systems Engineer, JMP

 

Part distributions are easy to measure: parts are built, an operator measures the parts with a gauge, and the results are assembled into a measured part distribution (MPD). 

 

But the resulting distribution is contaminated by errors associated with the measurement system. Random errors, gauge bias, and linearity problems all contribute to inaccuracies in measuring the true part values, so the individual values can never be truly known. 

 

However, if we had a way to estimate the true part distribution (TPD), we could compare it to the MPD and calculate the impact (cost) associated with using the imperfect gauge in terms of Type 1 and Type 2 errors.

It is trivial to estimate the TPD from an MPD if the gauge creates simple normally distributed errors around a normally distributed TPD (i.e., simply subtract variance of gauge from MPD variance to get TPD variance). But what if the gauge has linearity problems? Or what if the TPD has a non-normal shape?

 

This paper describes a new JSL script for determining an arbitrary (i.e., non-parametric) TPD from an arbitrary MPD and associated gauge performance characteristics. The resulting TPD can then be fed to a second script to determine production costs associated with the imperfect gauge and setting guardbands to optimize economics of the gauge errors. Performance of the estimation routine is evaluated, in terms of shape of TPD, various gauge characteristics, and resolution of distributions.

 

 

Hi,  I'm  Jerry  Fish.

I  work  for  JMP.

I  support  our  customers  in  the Central  Region  of  the  United  States.

Today,  I'd  like  to  talk  to  you  about an  add-in  that  I've  developed.

The  title  of  the  paper  is

Retrieving A rbitrary True Part  Distributions

from M easured  Part  Distributions and  the  Gauge  Characteristics

that  go  along  with  the  measurement.

Today's  agenda.

First  we're  going  to  talk  about, of  course,  what  does this  talk  address?

Why  is  this  so  important?

Why  can't  we  just  subtract  variances to  get  our   True Part Distribution?

A  little  bit  about  what's  behind our  estimation  computations.

I'll  demo  the  add-in,

including  some  test  results and  add  some  troubleshooting  tips.

I'll  tell  you  where  you  can  find the  add-in,

and  then  we'll  share  with  you  how  you  can  give  me  feedback

on  what's  good   and  what  you  don't  like  about  the  add-in,

areas  for  improvements,  and  so  forth.

What  are  we  addressing  here?

Well,  we're  talking  about  an  add-in that  determines  a   True Part Distribution

if  you  give  it   a  measured  part  distribution

and  if  you  describe  your gauge  performance  characteristics.

It's  pretty  easy  to  conceptualize   if  we  start  with  a   True Part Distribution.

Here's  our  true  part  value versus  our  percentage  of  parts.

Then  we  run  that  through  a  gauge,

an  imperfect  gauge  that  has  some variance  and  bias  characteristics.

We  will  get  a  measured  part distribution  out  of  that.

We  don't  know,  though, what  our   True Part Distribution  is.

What  we're  talking  about is  swapping  those  positions

where  we  start  with  a  Measured P art  Distribution,

we  subtract  out our  gauge  performance  characteristics,

and  we  end  up with  a   True Part Distribution.

That's  pretty  simple  to  understand, but  it  gets  more  complicated

if  we  have  a  Measured  Part  Distribution that  is  not  normally  distributed

and/or  we  have  a  gage  that  performs in  non-standard  ways,  you  might  say.

Perhaps  our  standard  deviation  shows curvature  with  the  measured  part  value,

or  maybe  it  has  bias   that  linearly  changes,

or  maybe  it  has  curvature  as  well.

How  can  we  take  these  quantities, the  Measured  Part  Distribution

and an  arbitrary  gauge  performance  curve,

and  come  up  with   what  the  True  Part  Value  was

that  must  have  caused this  Measured  Part  Value?

Why  is  it  important?

Well,  we  all  know  that  all  gages are  imperfect.

We'd  like  to  get  an  idea of  this   True Part Distribution.

You'll  see  it  referred  to  as,   we  go  along  as  TPD.

We  can  understand  our  type  1 and  type  2  errors.

A  type  1  error  means  our  gauge is  throwing  away  good  parts.

A  type  2  error  means  our gauge  is  accepting  bad  parts.

Both  of  these,  particularly   in  a  manufacturing  environment,

are  bad  things  to  happen.

If  we're  throwing  away  good  parts, then  that's  waste.

We  don't  want  to  have waste  in  our  process.

That's  just  a  straight  bottom  line deduction  from  our  profit  statement.

We  also  don't  want  to  accept  bad  parts.

If  we  do  that,  we  ship  the  bad  parts out  to  a  customer.

We're  likely  going  to  get  complaints, we're  going  to  get  returns,  reworks,

it's  going  to  damage  our company's  reputation.

We  don't  want  to  have  either one  of  those  types  of  errors.

They  both  hurt  our  company.

If  we  knew  the   True Part Distribution,

we  could  estimate  the  costs associated  with  these  errors.

That  particular  subject   is  addressed  in  another  paper

being  presented  here  at  Discovery  2023.

With  this  title  and  this  paper  number, I  encourage  you  to  look  it  up.

I  co-authored  this  paper  with  two

of  my  colleagues,  Brady  Brady, and  Jason  Wiggins.

We  need  that   True Part Distribution to  make  this  assessment.

Why  can't  we  just  subtract  the  variances?

Well,  you  can.

If  your   Measured Part Distribution  is  normally

and  your  gauge  has  constant  variance and  bias  across  the  measurement  range,

then  you  can  get to  your   True Part Distribution.

You  don't  know  the  True Part  values of  individual  parts.

You're  never  going  to  know  that, but  you  can  get  to  the  distribution.

It's  simply,  under  these  constraints, under  these  assumptions,

the  variance  of  the   True Part Distribution is  just  the  difference  in  the  variances

of  your  Measured Part Distribution and  your  gauge  variance.

You  subtract  those  two,

you  take  the  square  root,  and  you  get the  standard  deviation  of  your  gauge.

T he  average,  I'm  sorry,  standard  deviation of  your   True Part Distribution.

The  average  where  your True  Part  Distribution  is  centered

is  simply  wherever your  Measured  Part  Distribution

is  centered  minus  the  bias  of  the  gauge.

O f  course,  the  question  is,  what  do  you  do

if  your   Measured Part Distribution  is  not  normal

or  if  your  gauge has  unusual  characteristics?

This  is  how  we  can  conceptualize  inputting these  values,

and  I'll  show  you the  add-in  in  just  a  second.

We  can  have  any  arbitrary  input.

Here  we've  got   for  our   Measured Part Distribution,

this  is  our   Measured Part value.

T he  counts  for  however   parts   that  we  have  measured.

Maybe  this  looks  like  a  combination of  two  normal  distributions.

Maybe  it's  something a  little  different  than  that.

The  point  is,  you  can  put  in  any  input that  you'd  like,

any  input  shape for  the  Measured  Part  Distribution.

Then  we  described  the  gauge using  quadratic  functions

for  the  sigma  and  for  the  bias  of  the  gauge.

Normally,  all  you're  going  to  have is   a  standard  gauge

would  be  just  these  constants out  here  in  front,

C0  and  D0,  and  D0  may  be  zero if you  don't  have  any  gauge  bias.

I f  your  sigma  changes  linearly   with  part  value,

we  allow  you  to  put  in  C1 and  D1  if  your  bias  changes  linearly.

If  there's  any  curvature, we  allow  you  to  put  in  a  C2  and  a  D2.

When  you  set  up  your  gauge  equation, if  you  put  in  all  of  these  values,

it's  possible  to  generate negative  standard  deviations

within  the  measurement  range.

Don't  do  that.

If  you  can  avoid  it,  don't  do  that.

There  may  be  unexpected  results with  the  add-in

if  you  have negative  standard  deviations.

Just  beware  of  that.

What's  behind  the  estimation  computations?

Well,  we  start  with,  of  course,

the  actual  measure  part  distribution and  the  gauge  characteristics.

We  choose  an  estimated   True Part Distribution,

which  seems  like  a  good  idea.

We'll  start  with the actual  Measured  Part D istribution.

Then  we  put   that  estimated  True Part D istribution

through  a  transformation  that  represents the  gauge  characteristics,

and  that  yields an  estimated  Measured  Part  Distribution.

We  can  then  compare   the  estimated  Measured Pa rt  Distribution

with  the  actual  Measured P art  Distribution on  a  bin-by-bin  basis

and  get  a  Residual  Sum  of  Squares  error for  that  comparison.

Then  if  we  go  back and  adjust  the  amplitudes

of  the  True  Part D istribution  estimation,

we can  adjust  those  until  we  get the  estimated  Measure  Part  Distribution

to  agree  as  closely  as  possible with the  actual  measure  part  distribution.

We  do  that  using a  JSL Minimize  function

to  try  to  minimize the  Residual  Sum  of  Squares.

All  right,  let's  take  a  look   at  the  add-in.

Once  you  install  the  add-in,

it  will  come  in  under  Gauge  Study  tools   and  TPD  estimation.

This  is  what  the  add-in currently  looks  like,  version  1.0.

We  start  off  with  the  ability  to  choose

what  type  of  input Measured Part Distribution do  you  have?

Now  that  varies,  I'll  come  back to  the  arbitrary  shape  in  a  minute.

We  also  have  normal,

you  input  the  average  and  standard deviation,  LogNormal,  Weibull,

Exponential,  Gamma,   and  two-mixture  normal  distribution.

We  can  set  these  up  to  be parametric  if  we  want.

If  you  know  that  you  have   a  Weibull  distribution,  for  example,

you  can use  that  as  your  input  distribution.

Let's  start  with  normal. Let's  just  make  it  simple.

Here  we  have  a  normal  distribution that has  a  mean  of  zero

and  a  standard  deviation  of  three.

T hat's  shown  in  this  panel  here  in  this  little  graph.

Let's  use  a  gauge, that's  a  very  simple  gauge,

has  a  standard  deviation of  one  and  a  bias  of  zero.

Click  Next.

Here,  much  like  above,  we  get  to  choose the   True Part Distribution  shape.

We  could  say  that's  going  to  be  arbitrary, or  it  could  have  a  normal  distribution,

or  it  could  have  a  lognormal, all  the  same  distributions  here.

Or  down  here  at  the  bottom,

we  give  you  the  option  to  fit all  of  the  distributions  above.

Let's  again  start  with  a  simple  example of  a  normal  distribution,

and  we'll  calculate  those  results.

We  present  two  output  plots.

This  plot  shows  the True Part Distribution  in  blue,

the  estimated   True Part Distribution versus  the  Measured  Part  Distribution,

the  actual  Measured  Part  Distribution   in  red.

As  you  would  expect  in  this  case,

we've  got  a  normal  distribution  for  the measured,  and  we've  got  a  simple  gauge.

This  is  one  of  those  that  we  could solve by  hand  if  we  wanted  to.

We  end  up  with  a  slightly  narrower True Part Distribution  than  the  Measured.

If  we  then  go  to  do  a  check  on  that,

we  can  take  that   True Part Distribution, put  it  through  the  gauge,

and  we  end  up  with  an  estimated  Measured Distribution  versus  the  actual.

That's  what  we  get  down  here.

It  looks  like  we've  got  very  good agreement  in  this  particular  case.

Let's  go  back  up to  our  gauge  definition.

We'll  keep  the  same measured  part  distribution.

This  time  we'll  put  in  a  bias  of  two.

We'll  solve  again,  assuming  that

our True Part Distribution is  normally  distributed.

We  get  this  out.

Pretty  easy  to  conceptualize.

Everything  is  just  shifted over  by  two  units.

Here's  our   True Part Distribution and  our  Measured  Part  Distribution.

If  we  put  that  through  that  gauge with  the  bias,

we  get  back to  very  good  agreement  between

the  actual   Measured Part Distribution and the  estimated  Measure  Part  Distribution.

Third  example,  let's  say  that...

Let's  come  back  up  here  and  I'll  turn bias  off  since  we've  demonstrated  that.

Let's  say  we've  got  the  same  input, we've  got  the  same  simple  gauge,

but  now  let's  say  maybe  we've got  a  Gamma  distribution  here.

There's  Gamma,  and  we  want  to  fit  that.

We'll  hit  calculate.

This  is  the  best- fit  Gamma  distribution for  that  input  normal  distribution.

You  can  see  it  doesn't  fit  quite  as  well.

Our   True Part Distribution  is  a  little  bit skewed,  which  is  characteristic  of  Gammas.

If  you  put  that  through  our  gauge, we  end  up  with  this  agreement

between  the  Measured  Part  Distribution, actual  and  the  estimated.

It's  not  as  good  a  fit.

A  summary  of  those  is given  here  in  this  table.

This  shows  us  that  the  first  time  we  ran this,  we  did  a  normal  distribution  input,

which  with  two  parameters, we  did  a  normal  fit  on  the  output

with  two  parameters,  and  we  got this  sum  of  squares  error.

The  second  time  was  with  a  bias,

and  we  got  the  same  sum  of  squares  error   in  the  end,  as  you  might  expect.

Then  with  the  Gamma,  our  sum of  squares  error  was  a  little  bit  higher.

We  get  a  quick  little summary  in  this  table.

There  are  two  other

JMP  data  tables  that  are  built  that  have all  of  this  information,

the  original  distribution   and  the  output  distributions

and  all  the  gauge  characteristics.

All  of  those  are  summarized in  these  other  two  tables

to  allow  you  to  go  through and  make  your  own  plots  if  you  want  to.

They  are  also  there.

Let's  do  one  that's a  little  bit  different.

Let's  come  back  up  to  the  top and  let's  choose  a  user-defined  shape.

The  data  table  is  simply a  two-column  data  table.

The  first  column  is  assumed to  be  the  centers  of  your  part  values,

your  bin  centers, if  you  will,  in  that  histogram.

The  second  column  represents the  amplitudes  as  you  go  across.

Those  amplitudes  can  be  actual  part  counts,  they  can  be  percentages.

Anything  that  each  bin  height

or  each  histogram  bar  height is  relative  to  the  other  heights.

I  scale  everything  to  make the  sum  of  all  those  add  up

to  one  within  the  program  anyway.

As  long  as  the  relative  heights are  the  same,

doesn't  matter  what the  actual  amplitudes  are.

I  give  the  option  to  open  a  data  table   if  it's  not  already  opened,

or  if  it's  already  opened  within  JMP,

then  we  can  just  say, select  the  already  opened  data  table.

Here's  an  example with  a  square  wave

or  a  uniform  distribution  for  our  input, Measured  Part  Distribution.

Now,  this  is  a  tough  distribution  to  have.

If  you  think  about  this,  if  you've  got  a  gauge

that's  making  normally  distributed  errors at  any  point,

it's  going  to  be  really  hard  to  make

something  that's  nice  and  sharp  and  crisp like  this  distribution  on  the  output.

Let's  give  it  a  try.

Let's  say  here  I've  got a  pretty  wide  variation.

This  goes  from  zero  to  30,  I  think.

Let's  say  we've  got  a  gage  that  has  a standard  deviation  of  five  with  no  bias.

Let's  say  we  want  to  fit   a  normal  distribution  to  that,

and  now  we'll  calculate  those  results.

Here  we've  got   the  best  fit  normal  distribution

for  a   True Part Distribution  that's  going to  run  through  this  gauge

with  a  standard  deviation  of  five   to  try  to  give  us  this  square  wave

for  our M easured  Part  Distribution.

How  well  did  we  do?

Well,  it's  here.

It's  not  a  real  great  shape, and  you  probably  wouldn't  expect  it

to  be  a  great  fit  given  that  we're  trying to  use  a  normal  distribution

to  fit  a  square, an  I-sharp  square  function.

If  we  wanted  to  do  an  arbitrary  function, let's  say  this  one  here.

This  one  I  just  made  up  some  data.

I'll  show  you  a  little  bit more  about  what  it  is.

Maybe  this  looks  like  a  normal

two mixtures,  two  normal distributions  mixed  together.

Let's  check  that  out.

Let's  see  if  we  can  fit  this to  a  two-mixture  normal,

and  that  option  is  down  here, and  we'll  calculate  those  results.

Here  we  go.

Let  me  run  that  one  more  time.

I  don't  want  my  standard  deviation to  be  that  big.

Let's  take  a  smaller  standard  deviation.

We'll  talk  about  that  thing  in  a  minute.

Everything  else  the  same, calculate  results.

Sometimes  it  takes a  few  seconds  to  come  back.

It  just  depends  on the way  the  routine  is  fitting  things.

Here  is  our  fitted  True Part Distribution

compared to  the  Measured  Part  Distribution.

Assuming  that  our   True Part Distribution

is  two  normal  distributions mixed  together.

If  you  run  that  through  the  gauge, it  ends  up  looking  like  this.

This  is  the  attempt  to  match  that

and  what  our  Measured  Part D istribution   would  have  been.

That's  not  too  bad.

That's  the  way  that  the  add-in  works.

Now,  there  is  another  option  here,

and  that  is  when  you  fit, you  can  choose  whatever  inputs  you  want

for  your  Measured  Part  Distribution, your  gauge  characteristics.

Then  when  you  fit  down  here, you  can  also  fit  an  arbitrary  shape.

Now,  that  takes  on  my  PC maybe  a  minute  to  run.

I'm  going  to  spare  you  that  and  just  show you  the  outputs  within  a  PowerPoint  slide.

Here  we  are  back  in  PowerPoint.

This  is  one  other  example  that  I've  got before  I  get  to  the  arbitrary  inputs.

This  one  has  a  bias  that  I've  expressed as  one  plus  0.03  times  the  part  value.

I've  got  a   linearly  changing  bias  across   the  measurement  range.

I  have  a  normal  distribution  for  my  input,

and  I  want  to  fit  a  normal  distribution   to  the  output.

As  it  turns  out,  this  is  my   True Part Distribution,

and  this  is  my  Measured  Part  Distribution.

If  I  run  those  through  this  gauge,

even  though  it's  got this  linearly  changing  bias

across  the  measurement  range,  I  get very  good  agreement  between  the  two.

This  is  what  happens if  I  take  that  square  wave

and  say,  "Hey  JMP,  go  fit  whatever   True Part Distribution  you want

and  run  it  through  a  gauge  that  has a  sigma  of  two  and  a  bias  of  zero,

and  tell  me  what  that  distribution might  look  like.

What  you  get  out,  the  red  curve  again

is  the  measured  part  distribution, that  square  wave.

You  get  this  crazy-looking  thing  with

all   different  peaks  and  valleys in  it  as  the   True Part Distribution.

Well,  that  doesn't  look  like  any True Part  Distribution  that  I  would  have,

but  if  you  look  down  here, when  you  run  that  through  the  gauge,

it  does  a  pretty  good  job  of  simulating this  square  wave  distribution.

This is  uniform  distribution.

I  believe  it's  working.

Now,  there  are  reasons  that  it  might come up  with  something  like  this,

probably  associated  with  the  resolution of  the  gauge  that  you  have.

The  gauge  just  may  not  resolve  enough elements  across  the  measurement  range

as what  you  need  to  get a  nice  smooth  distribution  over  here.

That's  the  idea   that  you  can  do  this  with  this  gauge.

A  little  about  troubleshooting.

There  are  problems if  your  gauge  standard  deviation

is  too  large  in  comparison to  the  Measured  Part  resolution.

If  you  have  a  Measured  Part  Distribution,

let's  say  it's  normally  distributed with  a  standard  deviation  of  three,

and  you  tell  this  add-in  that  my  gauge has a  standard  deviation  of  five.

Well,  there's  no  way  to  get, even  if  you  have  the  same  true  part

that  you  measure  over  and  over  and  over again,  you're  going  to  get  a  spread

that  has  a  standard  deviation of  five,  it  can't  fit.

You'll  get  some  an  error  when  this  occurs.

It's  up  to  you  to  figure  that  out  that,

"Hey,  my  standard  deviation for  my  gauge  is  way  too  large."

If  it's  simply  on  the  verge of  being  too  large.

Let's  say  your  standard  deviation of  your  measured  distribution  is  3,

and  your  gauge  is  2.8,

then  again,  it's  going  to  try to  give  you  a  very  narrow

True Part Distribution  to  support  that,

and  that  can  lead   to  some  strange  results.

There  are  some  odd  combinations   that  I've  run  across

that  can  cause  these  things, I  call  them  untrappable  errors.

When  you  go  into   the  JMP  Minimize  function,

it  does  its  own  thing and  then  comes  back  with  an  answer.

If  it  runs  into  a  problem, it  will  throw  an  error.

Maybe  like  this.

I've  seen  two  or  three  different  ones, this  is  one  of  them.

I  don't  have  a  way  to  trap  for  those.

If  you  get  an  error  like  this, the  add-in  will  continue  to  run,

but  you'll  need  to  look  into   what  conditions  have  you  put  in  here

that  JMP  doesn't  like   that  it's  having  trouble  solving  for.

Chunky  Measured  Part  Distributions.

If  you  have  a  Measured  Part  Distribution,

in  fact,  this  one  here, this  might  be  pretty  chunky.

By  chunky,  I  mean  there  aren't  very  many bins  across  the  measurement  range.

It's  related  to  your  gauge  in  the  end. It's  how  much  can  your  gaumakege  resolve.

You  want  to  have a  lot  of  bins  across  here.

More  bins  is  better.

Fewer  bins  makes  the  true  part distribution  very  difficult  to  estimate.

Then  I  mentioned  or  alluded  to  earlier that  you  can  have  long  convergence  times,

particularly  when you're  trying  to  solve  for  these

arbitrary  True Part  Distributions.

On  my  PC,  it's  not  uncommon  to  go  a  minute or  a  little  bit  more.

Just  hang  in  there. The  add-in  has  always  come  back  for  me.

It  doesn't  hang. It  just  takes  a  while  for  some  solutions.

Add-in  availability  should  be  attached to  this  particular  recording,

and  you  should  also  be  able  to  find  it

in  the  JMP  Community  File  Exchange under  TPD  estimation.

If  you  have  comments  or  questions, you  can  post  them  either  below  this  video

or  on  the  File  Exchange,  and  please   put  in  there  any  suggestions  that  you  have

for  improved  graphical  User I nterface, any  changes  in  the  outputs.

We  didn't  talk  about  the  data  tables that  I  built,

but  if  you  see  those  and  you  decide, "Hey,  I  wish  it  would  be  in  this  format."

Let  me  know.

Those  are  things I  can  change  fairly  easily.

Suggestions  for  more  Parametric, Measured,  or   True Part Distributions.

These  are  the  normal,  the  lognormal,

the  Weibull,  the  Gamma, all  of  those  functions.

If  you  have  more that  you  want  to  add  to  that,

let  me  know  and  I'll  see   if  I  can  incorporate  those.

Then,  of  course,  problems  encountered.

If  you  can  include  a  description of  the  problem,  how  it  occurred,

that  will  help  me  in  debugging.

If  possible,  include   a  non-confidential  sample  input file

that  I  can  use   to  help  replicate  the  problem.

And  wherever  you  post  these  comments,

please  include  @JerryFish  in  your comments,  so  I'll  get  a  notification.

Thank  you  very  much  for  listening to  this  recording.

Don't  forget  to  check out  the  accompanying  Discovery  paper,

News Flash,  Gauges  aren't  perfect, okay,  you  know  that.

But  how  much  is  it  costing  your  business?

Under  this  particular  paper  number?

Thank  you  very  much  for  your  time.

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