Hi, I'm 론 Kenett.
This is a joint talk with Chris Gotwalt.
The talk is on Functional 데이터 분석 and Nonlinear Regression 모델.
And in 주문 to examine the options
and what we get out of this 유형 of analysis,
we will take an 정보 quality perspective.
In a sense, this is a follow up to a talk we gave 마지막 year
at the same Discovery Summit.
So I will start with simple examples
to introduce FDA and Nonlinear Regression.
And then Chris will cover a complex and more
substantially more complex example of optimization,
which includes mixture experiments designed to match a 참조 profile.
So the 이야기 starts with data on tablets that are dissolved
and measurements are done at different 시간 포인트,
five minutes, ten minutes, 15 minutes, every five minutes,
and then 20 minutes, ten minutes later,
30, and then 15 minutes later, 45 minutes.
We have 12 tablets that are our 제품
and 12 tablets that are the reference.
Our goal is to have a 제품 that matches the reference.
And in this 유형 of data, we have a profile,
and we consider two options,
FDA and NLR.
In Chris' example, we'll talk about something called the F 2,
which is a third option for analyzing this 유형 of data.
So here's what it looks like with the graph 빌더.
On the left, we have the 참조 profiles.
On the right, we have the tablets and the test.
We can see this is an example from my 책 on modern industrial statistics,
the 책 with Shelley Zacks, which is now in its third edition.
So on the left you can see there is a 태블릿 that seems
a 비트 different.
It's labeled T 5R.
And if we 실행 a functional data 분석 of this data,
T 5R does look different.
We see that the growth part is different.
It has a 느린 growth but consistent growth.
It does not have the shape that we see in the other dissolution curves.
This was done with a Quadratic B-spline with 1 knot,
and the quadratic was in this case,
fitting the data better than the QB.
This is a 비트 of an unusual situation.
So because of the shapes, the Quadratic B-spline was a better fit.
If we look at T 1R,
the first 태블릿 that has still different shape,
it shoots up and then it stays.
So basically, the 태블릿 has dissolved.
Obviously, beyond a high number of dissolution,
there's not much left for the solution.
So T 1R and T 5R, they seem different.
T 5R stands out more than T 1R.
So, 예, T 5R on the cluster 분석
on the functional principal components does stand out.
So here we see how functional principal components
scatter plot of the first two functional principal components 포인트,
what we observe visually.
And T1R, which is 다음 to T 2R,
is a different cluster.
We can proceed with a nonlinear regression approach.
Here we are fitting a Gompertz 세 parameter 모델
with 세 parameters,
the asymptote, the growth rate, and the inflection 포인트.
This is the 모델 and when we fit the profiles,
we again see that T 5R stands out.
So we have the same qualitative impression that we had with FDA.
Now we have these 세 parameters listed
and we can 실행 a profiler on the 모델 because we now have a model.
This is where T 1R stands.
So by running the profile on the different tablets,
we can also see how similar or different they look like.
This is the 테이블 that 지도 out the parameters of this asymptotes.
so T 1R growth rate .21,
T5R, this 태블릿 that stood out.
Growth rate .075,
very 느린 growth rate,
consistent but slow.
The inflection 포인트 is 11.5, way on the right.
So we can see through this parameter values the difference.
We can also pick up two tablets that stand out for growth rates.
T2R 1.77,
and T8R, almost no growth.
We'll get 뒤로 to T 2 and T 8 in a minute.
If we take the principal components of these 세 parameters 공간.
So we conceal the parameters as if these are the measurements
and we 실행 a multivariate control shot.
We can see T 1, this is the first 한
and we can see T 5, this is the 블루 one, the fifth one.
This we already saw.
They are within the control limits of the T 광장
multivariate statistical distance control chart.
And T 2 and T 8 that I highlighted before now stand out
and we can see qualitatively why.
This is the 모델 degradation approach in the guidance documents
that is used for 모델링 dissolution curves.
In running such 분석 with an 정보 quality perspective,
the first question is asking what is the goal of the analysis?
And then we can consider the method of 분석.
Here we're using nonlinear regression and functional data 분석.
Chris will get into how this is combined
with data derived from experimental design.
We have a 유틸리티 function and the 정보 quality
is the 유틸리티 of applying a method F on data X conditio n of the goal.
It is evaluated with eight dimensions.
And here Chris again we'll talk about data resolution and data 구조.
So Chris, the floor is yours.
Thanks Ron.
Now I'm going to give an example that is a little 비트 more complicated
than the first one.
In Ron's example,
he was comparing the dissolution curves of test tablets
to those from a set of 참조 tablets.
In that situation, the expectation is that
the curves should generally be following the same path.
And he showed how to find anomalous curves
that deviate from the rest of the population.
In this second example, we also have a 참조 dissolution curve.
But we are analyzing data from a designed experiment
where the goal is to find a formulation in two polymer additives
and the amount of force used in the 태블릿 production process
that leads to a close match
to the 참조 splashes the solution curve.
The graph you see here shows
the data from the 참조 curve that we want to match.
To do this, I'm going to demonstrate 세 분석 of the data
that 사용 different methods and 모델
to find factor 설정 that match the 참조 that lead to a...
To do this, I'm going to demonstrate 세 분석 of this data
that 사용 different methods and 모델 to find factor 설정
that will best match the 참조 curve.
In the first 분석,
I'm going to summarize each of the DoE curves
다운 to a single metric called F2
that is typically used in dissolution curve 분석,
a measure of agreement with the 참조 match.
There, I'll 사용 standard DoE methods to 모델 that F2 response
and then find the factor 설정 that are predicted
to best agree with the reference.
In the second analysis,
I'll 사용 a functional DoE 모델링 approach
where I 모델 the curves using these blinds,
추출 functional principle component stores, and 모델 them.
I'll 로드 the 참조 batch as a target function
in the Functional 데이터 Explorer 플랫폼
and then 사용 the FDoE profiler
to find the closest match recommended by that model.
These first two approaches 사용 little subject matter 정보
about these types of tablets.
In the third analysis,
I'll 모델 the curves using a nonlinear 모델
that was known to fit this 유형 of 태블릿 웰
and 사용 the Curve DoE option in the fit Curve 플랫폼
to 모델 the relationship between the DoE factors
and the shape of the curve.
I want to credit Clay Barker for adding this capability to JMP Pro 16.
I think it has a lot of promise for 모델링 curves
whose general shape can be assumed to be known in advance
to come from 한 of the 지원됨 nonlinear models.
At the end, verification batches were made
using the recommended formulation 설정 for each of the 세 analyses,
and we compared them to a new 참조 batch.
What we found was that the nonlinear regression-based approach
led to the closest match to the reference.
What we see here is a scatter plot matrix of the four factors
in the designed experiment.
There was a mixture constraint between the two polymers,
as 웰 as a constraint on the total amount of polymer
and the proportions of the individual polymers.
Here's a look at some of the 원시 data from the experiment.
At the top of the 테이블, we have data from the 참조 that we wish to match.
There are 16 DoE formulations or batches in the experiment.
We can only see data from two of them in this 사진, though.
There were six tablets per formulation.
There were four dissolution measurements per tablet.
Here we see plots of the dissolution curves for each of the 16 DoE formulations
with the dissolution curve of the 참조 batch here at the lower right.
Now, I'm going to do a quick preliminary 정보 quality assessment
using the questions that you'll find in the spreadsheet
that you can 다운로드 from the JMP 사용자 커뮤니티 페이지.
The first part of the assessment is related to the data resolution.
In this case, I think we're looking pretty good.
The data 스케일 is 웰 aligned
with the stated goal because it's a design experiment.
The measuring devices seem to be reliable and precise,
and the data 분석 is definitely going to be suitable
for the data aggregation level,
and we'll be illustrating different kinds of data aggregation
as we 추출 features from these dissolution curves.
As far as the data structure goes, we're in pretty 좋은 shape.
The data is certainly aligned with the stated goal,
we don't have any 문제 with outliers or missing values,
and the 분석 methods are 모두 suitable for the data structure,
although we do see some variation
in the quality of the 결과 depending on the 유형 of 분석 we do.
As far as data integration goes, this is a pretty simple 분석.
We have multiple responses,
and we're 탐색 different ways of combining them into extracted features.
So there's a 공통 workflow to 모두 세 of the 분석 I'm going to be showing.
First, we have to get the data into a 양식
that is analyzable by the 플랫폼 that we're using.
Then there's around a feature extraction.
Then we 모델 those features.
That's where there's a lot of difference between the methods,
and then we 사용 the profiler in different ways
to find a formulation that closely matches the reference.
First, I'm going to go over the F2 분석.
F 2 is a standard measure of agreement of a dissolution curve
relative to a 참조 dissolution curve.
In the formula,
the Rs are the means of the 참조 curve at each 시간 포인트,
and the Ts are the means of the non-reference curve.
The convention is to say that the two curves are equivalent
when F2 is greater than or equal to 50.
It's important to 포인트 out
that I'm including this F 2 based 분석
not just as an example of a dissolution DoE analysis,
but more broadly as an example of how reducing a response
that is inherently a curve 다운 to a single number
leads to a much lower quality 분석
and 결과 at the 끝
than a procedure that treats curves as first-class citizens.
So now I'm going to share the F2 분석 of the dissolution DoE data.
The first thing we have to do is calculate the batch means
of the dissolution curves at the different 시간 포인트.
Then we create a formula column
that calculates the F2 dissolution curve agreement statistic
for each of these curves relative to the 참조 batch,
and we modeled the F2 using the DoE factors as inputs
and 사용 the profiler to find the factor 설정 that match the reference.
Before the analysis,
we 사용 the table's summary feature
to calculate the means of the dissolution measurements
by batch and across each of the times.
We can save ourselves a little 비트 of work by using 모두 of the DoE factors here
as 그룹화 variables so they'll be carried through
into the subsequent 테이블.
Now we have a 17-row data set
and we hide and exclude the 참조 batch.
Now take note of the values of the dissolution means for the 참조
because we're going to 사용 those
when we create a formula column that calculates the F2 agreement metric
for each of the batches relative to the 참조 batch
and now we're going to be able to 사용 this F2 formula column
as a response to be modeled.
We 사용 the 모델 script created by the DoE 플랫폼
to set up our 모델 for us.
We place F2 as our response variable
and we're going to 분석 this data today
using the generalized regression 플랫폼 in JMP Pro.
When we get into the 플랫폼
we see that it has automatically done a standard lease squares 분석
because it found that there were enough degrees of freedom
in the data for it to do so
and it's given us an AICc of 155.6.
I'm going to see if we can do better by trying
a best subsets reduction of the 모델
and when we do that we see
that the AICc of that best subsets goes 다운 to 136
smaller is better with the AICc
and the difference of 20 is pretty substantial
so I would conclude that the normal best subset is a better 모델
than the standardly squares one.
I'm going to try 한 more thing, though,
and fit a log normal distribution with best subsets to the data.
When I do that, the AICc goes 다운 a little 비트 further to 130.6.
That's a modest difference, but it's 좋은 enough
that I'm going to conclude that we are going to work
with the Lo gNormal,
especially because we know that we're working with a strictly positive response
and the LogNormal distribution fits data that is strictly positive.
From there, the 분석 is pretty straightforward,
so I'm going to jump straight ahead to using the profiler.
F2 is an agreement metric that we want to maximize.
So we get into the profiler,
we turn on Desirability functions and have them set to maximize,
and then maximize desirability to find the combination of factor 설정
that this 모델 says gives us the closest match to the 참조,
and that would be at this combination of factor 설정 that we see here.
Now the F2 분석 is complete,
and we're going to go into the second analysis,
the functional DoE 분석.
For this analysis,
we're going to work with the data in a stacked format
where 모두 of the dissolution measurements have been combined into a single column,
and we have a 시간 column as 웰.
The first thing we do is go into the Functional 데이터 Explorer 플랫폼.
In the 플랫폼 launch,
we put dissolution as our response, 시간 is our X,
the batch column as our ID,
and we supply the four DoE factors as supplementary variables.
Once we're in the 플랫폼,
we take a look at the data using the initial data plot.
This particular data set doesn't need any clean up or alignment options,
but we are going to go ahead and 로드 the 참조 dissolution curve
as a target function.
For relatively simple functions like these,
I typically 사용 B -Splines for my functional model.
When we do that, we see our B-Spline 모델 fit,
and the initial fit that has come up is a cubic 모델 that is behaving poorly.
It's interpolating the data 포인트 웰,
but kind of doing crazy things in between them.
So I'm going to change from the default recommended 모델
over to a quadratic Spline 모델 instead of the cubic one.
We do that by simply clicking on Quadratic over here
in the right of the B-Spline 모델 fit.
We'll see that this quadratic 모델 fits the data 웰.
A functional principal components 분석 is automatically calculated,
and we see that the Functional 데이터 Explorer 플랫폼 has found
세 functional principal components.
The leading 한 is very dominant,
explaining 97. 9% of the functional variation.
And it looks like this is a level set up or 다운 kind of shape component.
The second 한 looks like a rate component,
and the third 한 almost looks like a quadratic.
Looking a little closer at this quadratic B-Spline 모델 fit,
we see that this 모델 is fitting the individual dissolution curves
pretty 웰.
So now we're ready to do our functional DoE 분석.
Each of our individual dissolution curves has been approximated now
by an underlying mean function 공통 to 모두 the batches
plus a batch dependent FPC score times the first eigenfunction,
plus another batch dependent FPC score times the second eigenfunction,
and so on with the third one.
What we're going to do is set up individual DoE 모델
for each of these functional principal component scores
as responses using our DoE factors as inputs.
The Functional 데이터 Explorer 플랫폼, of course,
makes 모두 this simple and kind of ties it up into a bow for us.
And when I say that,
it ties it up in a bow for us, what I really mean is the FDoE profiler.
So this pane here shows our predicted trajectory
of dissolution as a function of 시간,
and then we can see how that trajectory would change
by altering the DoE factors.
That relationship with the DoE factors
comes from these 세 generalized regression 모델
for each of our functional principal component scores.
If we want, we can open those up
and we can look at the relationship
between the DoE factors and that functional principal component score,
and we could even alter the 모델
by moving around to other ones in the solution path.
I just want to 포인트 out that it's possible to change the DoE 모델
for an FPC score.
In the interest of 시간,
I'm just going to have to 이동 on and not demonstrate that, though.
We have diagnostic plots,
the most important 한 probably being the Actual by Predicted Plot.
This has our plotted dissolution measurements on the Y-axis
and the predicted dissolution values using the functional DoE model.
And as always, we want to see that plot
have data 포인트 꽉 along the 45 degree line.
And in this case, I think this 모델 looks pretty good.
We don't want to see any 패턴 in our residuals,
and I'm not seeing any 나쁜 ones here.
So this 모델 looks pretty 좋은, and we're going to work with it.
So I've already explained how this pane right here
represents the predicted dissolution curve as a function of 시간
and the individual DoE factors.
Now, these other two rows here are because
we've loaded the 참조 as a target function.
So this 행 is the difference
of the predicted dissolution curve from the target 참조 curve.
And then the bottom pane here is the integrated distance
of the predicted curve from the target.
When we maximize desirability in this profiler,
it gives us the combination of factor 설정
that minimize this integrated distance from the target.
So I'm going to do that by bringing up maximize desirability.
And now we see the 결과 of the functional DoE analysis,
where we have identified .725 of polymer A
275년 polymer B,
a total polymer of about .17,
and a compression force of about 1700년
minimize the distance between our predicted curve and the reference.
Now, we've done two analyses.
Both of those 분석 have recommended that we go
to the lowest 설정 of polymer A and the highest 설정 of polymer B.
They differ in their recommendations
for what total polymer amount to 사용 and how much compression force to use.
The third 분석 I'm going to do is the Curve DoE 분석.
This is going to be structured pretty similar
in some ways to the functional DoE analysis,
in that we're going to 사용 the same 버전 of the data
where dissolution measurements are 모두 in 한 column
and we have a 시간 column.
But we don't have a built-in target function option
in the 피트 Curve 플랫폼 yet.
So the first thing we have to do is fit just the 참조 batch
and save its prediction formula 뒤로 to the 테이블.
Then we do a Curve DoE analysis,
which is largely similar to a functional DoE 분석
in that we're extracting features from the curves
모델링 the curves.
Then we go to the graph profile to find 설정 that best match the reference.
The nonlinear 모델 that we're going to be using is
a 세 parameter Weibull Growth Curve,
which has a 긴 역사 in the 분석 of dissolution curves.
Weibull Growth Curve have an asymptote parameter A
that represents the value as 시간 goes to infinity.
There's what's called the inflection 포인트 parameter
that I see is a scaling factor that kind of stretches out
or squeezes in the entirety of the curve.
And then there's also a growth rate parameter
that dictates the shape of the curve.
What I think is really valuable about using this 모델
relative to the functional DoE 모델 or the F2 유형 분석
is that we're going to be 모델링 features extracted from the data
that have real 과학 meaning,
especially the asymptote and inflection 포인트 parameters.
Now Curve DoE 분석 doesn't have a target 매칭 capability
like the Functional 데이터 Explorer.
So we begin the 분석 by excluding 모두 of the DoE rows in the data 테이블.
These are represented with the set column equal to A.
So I 선택 a 셀 there,
선택 매칭 셀, and then hide and exclude those rows
so that I only have the 참조 batch not excluded.
Then I go to the 피트 Curve 플랫폼,
로드 it up, get in there, fit the Weibull growth model,
and then I save that prediction formula 뒤로 to the 테이블.
Once we complete the Curve DoE analysis,
we're going to compare the Curve DoE prediction formula
to this 참조 predictor
to find combinations of the factor 설정
that get us as close to this curve as possible.
So now we unhide and exclude the DoE batches,
go 뒤로 into the 피트 Curve 플랫폼,
just like the Functional 데이터 Explorer 플랫폼.
We're going to 로드 up the DoE factors as supplementary variables.
Now that we're in the 플랫폼, we can fit our Weibull growth model.
The initial fit here looks pretty good.
Looks like we're capturing the shape of the dissolution curves.
한 thing I like to do 다음 is to make a parameter 테이블.
This creates a data 테이블 with our fitted 다운 the near regression parameters.
I like to look at these in the distribution 플랫폼
to see if there are outliers in there or anything unusual.
I also like to look at the 패턴 I see in the multivariate 플랫폼
just gives you a better sense of what's going on
with the nonlinear 모델 fit.
Once we know that everything is looking pretty 좋은,
we can do our curve DoE 분석
and this looks very much like the functional DoE 분석 from before.
We have a profiler that shows
the relationship between dissolution and 시간
and how that relationship changes as a function of our DoE factors.
And then we also have a generalized regression 모델
for each of those 세 parameters that we can take a look at individually.
The first thing I would do before trying to 사용 the 모델 in any way
is look at the Actual by Predicted Plot,
so that's what we see here.
This is the predicted values incorporating
both the 시간 모델 from the mean function and the eigen functions,
as 웰 as the DoE 모델 on the nonlinear regression parameters.
This looks pretty 좋은
because there is a fairly Easy interpretation
for the viable growth 모델 parameters.,
It can be useful and interesting to open up
the individual 모델 fits for these parameters.
For example, here are the coefficients for the inflection 포인트 model.
Because inflection 포인트 is a strictly positive quantity,
a LogNormal best subsets 모델 has been fixed with the data
by the generalized regression 플랫폼.
We see that the mixture manifests have been forced in
and that the compression forced by polymer A interaction
is the only other term in the model.
What this means is that
if we hold the polymer proportions constant
and increase the compression force,
we would expect a larger value of the inflection 포인트.
한 would observe this as a 태블릿 that takes longer to dissolve,
which is exactly what we would expect to have happen.
We can save the curve DoE prediction formula 뒤로 to the 테이블
and we can see in 모두 its 고리 detail
how the 모델 for asymptote inflection 포인트
and growth rate are combined with 시간 to come up with our overall prediction
for the dissolution curve based on.
Fortunately, with 정크 we don't have to look at the formula too closely though,
because we have profilers that
let us see the relationships in a visual way
rather than an algebraic one.
To solve our 문제 of finding the combination of factors
that give us the dissolution curve
that would be closest to the 참조,
I created a formula column that calculated
the percentage difference of the predicted curve,
taking into consideration the DoE factors from the reference.
The 마지막 step of the 분석 is
to bring up this percent difference response
in the profiler that is under the graph menu,
being sure to check the Expand Intermediate formulas option.
This led to a profiler where we're able to see the percent difference
from the 참조 as a function of 시간 and the DoE factors,
I've shared the region where the difference
is less than one percent in green.
By manually adjusting the factors,
I was able to find 설정 where the predicted curve
is less than one percent from the 참조 across 모두 시간 values.
This looks really 좋은, but in practice I bet that this is overly optimistic.
Here we see the optimal values of the factor 설정 for 모두 세 analyses.
The curve DoE 분석 is in the interior of the range for the polymers.
The optimal value for total polymer is. 16
which is close to the functional DoE 분석 결과,
and compression force is in between the optimal values
recommended by the F2 분석 and the functional DoE 분석.
After this, we made new formulations based on the recommended factor 설정
from each of these 모델 and measured their dissolution curves
as 웰 as took a new set of measurements from the reference.
Here we see a summary of the final 결과 from the verification runs.
The new 참조 distribution curve is in 블랙,
and the curve DoE in green is the closest curve to it,
followed by the F DoE curve in blue.
The 결과 of 모델링 F2 is in red, and it did the poorest overall.
This should perhaps not be too surprising.
The F2 metric was the simplest,
reducing the data 다운 to a single metric and did the poorest.
The functional DoE 모델 had to empirically
derive the shapes of the curves
and then 모델 세 features of those shapes,
essentially using more of the 정보 in the data.
The curve DoE led to the best formulation
because it used the data efficiently via some prior knowledge
about the parametric 양식 of the dissolution curves.
We see that the 결과 of the F2 based 분석
are not equivalent with the new 참조 패치,
while the approaches that treated curves as first 클래스 objects are equivalent.
What this means is that the approach would have required
at least another round of DoE runs,
and so an inefficient 분석 leads to an inefficient 사용
of 시간 and resources.
I'm going to close the presentation
with a retrospective 정보 Q Assessment of the Results.
Overall, we found that the curved DoE prediction
generalized the best to new data,
but was the most difficult 분석 to perform.
I want to note that if we didn't have a known nonlinear 모델
to work with that fit the data 웰, we could not have done that 분석.
The functional DoE 분석 and the F 2 based approach
can be used more broadly in other situations.
The profiler leads to excellent 커뮤니케이션 scores
for 모두 세 analyses.
The ability to see how the shape of the dissolution curve changes
with the DoE factors in the functional and curve-based approaches
leads me to give them a better 커뮤니케이션 score.
I see the curve DoE approach
having the highest 커뮤니케이션 score by a little 비트
because we're 직접 모델링 more meaningful parameters
than the functional DoE approach.
That's 모두 we have for you today.
I want to thank you for your 시간, interest, and attention.
