Predicting Functional Responses using JMP Functional DOE and Curve DOE Modeling - (2023-US-30MP-1520)

Ryan_Gilmore

Functional or curved responses frequently occur in industry. Thanks to new features in JMP, we can now model and predict functional responses using key DOE or product design factors with JMP Functional DOE or Curve DOE modeling. A Functional DOE model is purely empirical. However, a Curve DOE model can incorporate mechanistic or expert knowledge on the functional form of the curved responses.  In this presentation, the methods and results of predicting functional responses using Functional DOE and Curve DOE modeling are compared using case studies from the consumer product industry.

Hello, my name is Fangy i Luo and I'm from Procter & Gamble.

Today I'm presenting with Chris Gotwalth from JMP.

We're going to talk about how to model data from designed experiments

when the response is functional curve.

Functional or curve responses occur very often in industry.

Thanks to the new development of JMP,

we can now model and predict functional responses

as a functional of key DOE or product design factors

using both functional DOE or curve DOE modeling.

A functional DOE model is purely empirical.

However, a curve DOE model can take into account mechanistic

or expert knowledge on the functional form of the curve responses.

In this presentation, the method and results of predicting

functional responses using functional DOE and curve DOE modeling will be compared

using case studies from the consumer product industry.

This is the outline of a talk.

We will break the talk into two parts.

In the first part, Chris will talk about

what are the functional data examples of functional data

and then he will help you with fundamental understanding

of the functional DOE modeling,

including functional principle component analysis

as well as curve DOE modeling.

In the second part,

I will use two examples from Procter & Gamble

and compare the results of functional DOE and curve DOE modeling

using these two examples.

The first example is Modeling Viscosity

over Time Data from F ormulation Experiment.

The second example is Modeling Absorption Volume over Time Data

From a Diaper Design of Experiment.

Then I will finish the talk with a brief summary and conclusion.

Thanks Fangy i.

Now I'm going to give a quick intro to functional and curve data analysis.

But first I want to point out

that there is a lot of this kind of data out there and JMP really has made

analyzing curve response data as fast, easy and accurate as possible.

If you haven't heard of functional data analysis before,

you have certainly seen it out there.

It's all over the place,

and I'll show you some examples to make that clear.

For example, here are annual home price indices

from 1992 -2021 for all 50 US states.

Each function has a beginning measurement

followed by a sequence of other measurements

and then a final measurement.

They all have a beginning, a middle and an end.

The functions don't have to all have the same start and endpoints

or measurements at the same times.

In a time series analysis, we are really interested in using data

to predict forward into the future using data observed from the past.

In a functional data analysis or a curve data analysis,

we are generally more interested

in explaining the variation internal to the functions

than predicting beyond the range of times we've observed.

In product and process improvement in industry,

we are often working on non-financial curves.

I'm going to show you some examples that our customers have shared with us.

Here we see a set of infrared spectra of gasoline samples

used to develop an inexpensive tool to measure octane and gasoline.

The green curves had high octane, and the red ones were low in octane.

The height of the left peak turned out to be critical

for predicting octane level.

Microbial growth curves

are a common type of functional data in the biotech industry.

Today, F angyi will be demonstrating two methods in JMP

that can be used for analyzing DOEs,

where the response is a set of measurements.

The first method is called functional DOE analysis

and is best for complicated response functions like spectra

when you need the model to really learn the curves and the data from scratch.

The second is a curve DOE analysis,

which is based on non-linear regression models.

When you can use the curve DOE analysis,

I found that you get more accurate results with it.

But if you can't get the curve DOE analysis to work,

you can always fall back on the functional DOE analysis,

as it's more general than curve DOE.

The critical step in functional data analysis

that will be new to most people

is called functional principle components analysis,

also called FPCA for short.

This is how we decompose the curves into shape components

that describe the typical patterns we see in the curves,

as well as weights that attribute how strongly each individual curve

correlates with those shape components.

It's a kind of dimension reduction and data compression technique

that reduces all the information in the curves

into the most compact representation possible.

To illustrate FPCA, take a look at the set of curves in the plot here.

What do they have in common?

How do they differ from one another?

What I see in common

is a set of peak shapes with one peak per curve,

and the shapes go to zero away from the peak.

They also appear to be symmetric around the center of the peak.

In terms of differences, I see variation in peak heights,

and there are clear horizontal shifts from left to right,

and some curves are also narrower than other ones.

In a functional data analysis,

the first thing we do is find a smoothing model

that converts or approximates the discrete measurements,

converting them into continuous functions.

There's a variety of smoothing models in FDE.

I don't really have a firm rule as to which one is the best in general,

but here are my observations about the most common ones.

Wavelets and splines have different strengths.

Wavelets are new in JMP Pro 17

and are very fast and are generally the best with complicated functions

such as spectra, as long as the X coordinates of the data are on a grid.

On the other hand, there are B and P splines,

which are slower computationally

but are better for data with irregularly- spaced X s,

and are also often better

when there are only a dozen or fewer measurements per function.

If the data aren't large, I would try both splines and wavelets

and see which one is giving us the best fit

by looking at the graphs.

The main graphs I use to make decisions about smoothing models

are actual by predicted plots

and you wanted the one that hugs the 45- degree line more closely.

In this case, I would choose the wavelets model on the right

over the spline model on the left,

because those points are tighter around that 45- degree line.

Immediately after JMP Pro fits a smoothing model to the data,

it decomposes the signals

into dominant characteristic shapes it found in the data.

In mathematical language, these shapes are called eigenfunctions,

but a better and more approachable name would be to call them shape components.

Here we see that JMP has found

that the overall mean function is a peak shape

and that there are three shape components

that explain 97% of the variation in the data.

The first shape component appears to correspond to a peak height.

I've learned to recognize that the second shape

is a type of left- right peak shift pattern and that the third shape component

is something that would control the peak width.

Remember that these are shapes learned from the data,

not something that I gave JMP outside of the data.

What has happened is the observed spectra in the data

has been decomposed into an additive combination

of the shape components

with unique weights for each individual curve.

The functional PCA is like reverse engineering the recipe of the curves

in terms of the shape components.

The mean function is the thing that they all have in common.

The shape components are the main ingredients.

And the weights are the amounts of the ingredients

in the individual curves.

The functional DOE analysis is the same mathematically

as extracting the scores or weights

and modeling them in fit model with the generalized regression platform.

Fortunately, there is a red triangle option

in the Functional Data Explorer that automates the modeling,

linking up the DOE models with the shape functions for you

and presenting you with a profiler

that connects the DOE models with the shape functions.

You can directly see how changing the DOE factors

leads to changes in the predicted curve or spectra.

There are many potential applications of functional DOE analysis,

some of which Fangyi will be presenting later in this talk.

There is another approach in JMP called curve DOE modeling.

This answers the same kind of question as functional DOE,

but it is nonlinear regression based rather than spline or wavelet based.

What that means is that if you have a good idea of a nonlinear model,

like a three- parameter logistic model, and if that model fits your data well,

you can get models and results

that generalize better than a functional DOE model,

because the general shape of the curve

doesn't have to be learned from scratch from the data using splines or wavelets.

The idea being that if you can make assumptions about your data

that reproduce the modeling effort needed,

your predictions will be more accurate, especially from small data sets.

Curve DOE analysis has a very similar workflow

to a functional DOE analysis,

except that you go through the Fit Curve platform

instead of the functional Data Explorer,

and instead of choosing wavelets or splines,

you chose a parametric model from the platform.

Just like in a functional DOE analysis,

you want to review the actual by predicted plot

to make sure that your nonlinear model is doing a good job of fitting the data.

A curve DOE analysis is the same as modeling

the nonlinear regression parameters

extracted from the curves using the generalized regression platform.

This is the same thing as what's going on with a functional DOE analysis

with the FPCA weights.

Fit Curve automates the modeling and visualization just as FDE does.

Once you know functional DOE analysis,

it's really not very hard at all to learn curve DOE analysis.

Now I'm going to hand it over to F angyi

who has some nice examples illustrating functional DOE and curve DOE.

Thanks Chris.

Next I'm going to talk about two examples from Procter & Gamble.

The first example is viscosity over time curves

collected from a number of historical formulation experiments

for the same type of liquid formulation.

There are six factors we would like to consider for the modeling.

They are all formulation ingredients and we call them factor one to factor six.

The goal of our modeling is to use these formulation factors

to predict or optimize viscosity over time curve.

The response of modeling is viscosity over time.

This slide showed you some viscosity over time data.

For majority of our formulations, the viscosity of the formulations

would increase first with time and then decrease later on.

Next, we're going to perform functional DOE analysis on viscosity over time data.

Before functional DOE analysis,

we need to perform functional principal component analysis

on the curves smooth using different method.

Here, we apply functional principal component analysis

to the curves first using B-s plines

and find five functional principal component

where they cumulatively explains about 100% of variation in the curves.

Each of the curve would express

as the sum of the mean function plus linear combination

of the five functional principal components

or eigen functions also called shape function.

We also apply direct functional principal component analysis to the data

where it find four functional principal components

that cumulatively explains

about 100% of variation across viscosity over time curves.

E ach curve will then be expressed as the mean function

plus linear combination of the four functional principal components.

This slide compares the functional principal component analysis model fit

using two different options.

The first one is using the B-s pline option

and the second one is using the direct functional PCA analysis.

As you can see using the B -spline option, the model fit seems to be smoother

as compared to the model fit using direct functional PCA analysis.

This slide showed you the diagnostic plots,

the observed versus predicted viscosity

from the functional principal component analysis

using two different options.

Using direct functional PCA analysis,

the points are closer to the 45- degree lines

as compared to the one using B-s pline option,

indicating that direct functional PCA analysis

fits the viscosity over time data

slightly better than the functional principal component analysis

using B-spline option.

After performing functional principal component analysis,

there's an option in JMP, you can perform functional DOE modeling

and get functional DOE profiler.

For functional DOE modeling,

basically it's combining the functional rincipal component analysis

with the model for the functional principal component scores

using formulation factors.

For this profiler we can predict the functional responses,

in our case, is viscosity over time curves using different formulation factors.

You can select a combination of the formulation factors

and it's able to predict the viscosity over time curve.

This slide shows you the diagnostic plots, the observed versus predictive viscosity

and also the residual plots from the functional DOE modeling.

As you can see that the residuals from the functional DOE modeling

are larger than the functional principal component analysis

before the functional DOE modeling.

Our colleagues at Procter & Gamble

actually they find that Gaussian Peak model would fit

individual viscocity curves very well.

This Gaussian Peak model has three parameters A, B, C,

and this A indicates the peak value of the viscosity over time curve

and B is a critical point,

which is a time when viscosity reaches maximum,

and C is a growth rate.

The rate of the viscosity increase during the initial phase.

This is the fitting of the viscosity over time curve

using the Gaussian Peak model

using a feature in JMP, called curve fitting.

These are the diagnostic plots

of the viscosity curve fitting using the Gaussian Peak models.

It looks like the model fitting are not too bad,

however, the arrows seems to be larger than the arrows from the fitting

using functional principal component analysis.

After curve DOE fitting using Gaussian P eak model,

there's option in JMP you can perform curve DOE modeling.

Basically, curve DOE model is combining

the parametric model for the curves, the Gaussian Peak model,

and the model for the parameters of the Gaussian Peak model

express the parameter as a function of formulation factors

using generalized regression models.

Then you get the curve DOE model

and this is a profiler of the curve DOE model.

Using this profiler you can predict the shape of the curve

by specifying combination of the formulation factors.

Actually, this profiler is somewhat different

from the functional DOE profiler we got previously.

These are the diagnostic plots from curve DOE model.

As you can see here that the curve DOE model

does not fit the data well and it's much worse than the functional DOE model.

These are the curve DOE model fit on the original data.

As you can see that for a number of formulations,

the curve DOE model does not fit the data well.

This is a comparison of the profilers

from functional DOE model and curve DOE model.

As you can see that the profilers, they look quite different.

This compares the diagnostic plots

from functional DOE model and curve DOE model.

As you can see that functional DOE model

fits the data much better than the curve DOE model

with a smaller root mean square error.

Now I'm going to show you the second example.

This example is from a diaper design of experiment

with four different product A, B, C, D

at three different stations labeled as S1, S2 and S3,

so it's a factorial design.

Diaper absorption volume was measured over time

for these four different product at three different stations.

The response is diaper absorption volume over time

and the goal is to understand the difference

in diaper absorption curves across different products and stations.

These are a few examples of diaper absorption volume over time curves

where the fitting lines are smoothing curves.

We performed functional principal component analysis

on the diaper absorption volume over time curves

and this functional principal component analysis

was able to find five functional principal component

where cumulatively,

they explains about almost 100% of variations among multiple curves.

These are the functional principal component analysis model fit.

As you can see, for almost all the curves,

the fitted curve plateaued after a certain time point.

Functional principal component analysis model fitted curves really well

as you can see from the diagnostic plots.

We performed functional DOE modeling

of the functional principal component analysis

and this is profiler of the functional DOE model.

This model allows us to evaluate shape of the curve

for different diaper products at different measuring stations.

The product comparison at station two seems to be different

from the product comparisons at station one and station three.

These are the diagnostic plots of the functional DOE model.

Next, we would like to perform curve DOE modeling.

Before curve DOE modeling,

we would like to find some parametric model

that fits the diaper absorption volume over time data well.

I found that there's a function in JMP called biexponential 4P model.

This model is a mixture of two experiential model

with four unknown parameters.

This model fits all the diaper absorption volume over time curves really well.

These are the diagnostic plots of the curve fitting and you can see

that the biexponential 4P model fits all the curves really well.

After fitting diaper absorption volume over time curves

using biexponential 4P model, we performed curve DOE modeling using JMP

and this is a profiler of the curve DOE model.

Using this profiler, you are able to see the shape of the curve

as a function of diaper product as well as a measuring station.

This is a profiler of product A at station two and then station three.

These are the diagnostic plots of the curve DOE model

and you can see that curve DOE model fits the data well,

except that at higher diaper absorption volume,

the residuals are getting larger.

These are the curve DOE model fit on the original data.

As you can see that for most of the curves,

this model fits the data really well.

This compels the model profiler

of the functional DOE model versus curve DOE model.

As you may notice that there's some difference

between these two profiler at the later time point.

The predicted diaper absorption volume at the later time point

tend to plateau from the functional DOE model,

but it continue to increase at later time point

using the curve DOE model.

This compares the diagnostic plots from the functional DOE model

versus curve DOE model using biexponential 4P model.

As you can see that both of these models fits the data really well,

with functional DOE being slightly better

with slightly small root mean square error.

Now, you have seen the comparison of functional DOE modeling

versus curve DOE modeling using two P&G examples

and this is our summary and conclusions.

Functional DOE modeling is always a good choice.

When the parametric model fits all the curve data well,

curve DOE modeling may perform really well.

However, if the parametric model does not fit the curve data well,

then the curve DOE modeling may perform poorly.

Functional DOE model is purely empirical.

However, curve DOE model

may take into account mechanistic understanding

or extrovert knowledge in the modeling, so it can be hybrid.

I t's good to try different method like different smoothing method

before functional principal component analysis.

In functional DOE modeling,

try functional DOE model versus curve DOE model

and see which one performs best.

This is end of our presentation.

Thank you all for your attention.