FIM Scaling

[bbefort]

11/22 To Do: (@bbefort, @jialuw96, @adowling2 )

• [x] @bbefort verifies theta is dimensionless
• [x] @bbefort computes sigma^-2 for best model for all systems; chooses one number for uncertainty for all FIM calculations and checks if consistent with KU work
• [x] @jialuw96 computes FIM, posts to github, gets feedback on order of magnitude for D-, A-, and E- optimality and eigen decomposition, checks theta scaling in Pyomo.Doe
• [x] @jialuw96 recomputes FIM for all cases

[bbefort]

Note Nov 22, 2022.pdf

[bbefort]

@jialuw96 @adowling2 Here are the sigma^2 values across systems for the same model:

R32/emim (PR-6): 6.129898371002546e-09 (PR-4): 8.37460197746672e-09

R32/bmim (PR-6): 9.278203566112187e-08 (PR-4): 1.5644966239299104e-07

R125/emim (PR-6): 1.4171234680259503e-07 (PR-4): 1.2511693644269783e-07

See here for details of the calculation: https://github.com/dowlinglab/extractive-distillation2/blob/mbdoe/modsel/paramest/emimtf2n/R32/Final_Results/Data/Fits/Uncertainty-residuals-calc.ipynb

These seem small and vary by orders of magnitude between systems. What do you think?

[bbefort]

Also, KU uses +/- 0.0008 MPa uncertainty in the pressure value.

[adowling2]

@bbefort What is the square root of the variance of the residuals converted to MPa? Is this standard deviation consistent with the +/- value from KU?

[bbefort]

@adowling2 Sorry, I forgot to do this. The square roots in MPa (approximately) for the example systems above:

R32/emim (PR-6): 0.00008 (PR-4): 0.00009

R32/bmim (PR-6): 0.0003 (PR-4): 0.0004

R125/emim (PR-6): 0.0004 (PR-4): 0.0004

So they are generally on the same order of magnitude as the KU values. I propose we use the KU value as it is easily defensible. We can comment that we calculated the residuals using the method I used above and they were comparable. Let me know what you think. Thanks.

[adowling2]

Let's go with the KU value of +/- 0.0008 MPa (standard deviation, sigma) as @bbefort suggests. Moreover, let's report the values posted here in an SI table for completeness.

[adowling2]

@jialuw96 Post here when you have an update or if you have any questions.

[jialuw96]

Please see the following for the heatmaps for R32/emim system, PR_quad model.

• Parameter scaling:
• Results before: scale_parameter=True, scale_constant = 10**(-5) (Pa --> bar)
• I reran results with: scale_parameter=False, scale_unit = 10**(-6) (Pa --> MPa).

Q: do we want to scale with parameter values? Results before, we generated all results (Prior info, heatmaps) with scale_parameter=True. Currently, I only updated this option for PR_quad model.

• The following heatmaps are based on the following computation:

[adowling2]

@jialuw96 Thank you for your update.

I am a little confused. Are you scaling the results or the parameters? The text of our post suggests you are scaling the results (the numerator of the sensitivities Q) by the syntax suggests you are scaling the unknown parameters (denominator of the sensitivities Q). What is the difference between scale_constant, scale_unit, and scale_parameter?

I looked at the interface for Pyomo.DoE (https://github.com/jialuw96/pyomo/blob/e3a2976e10cbccb7d720e748fe8b1c6e311cfb95/pyomo/contrib/doe/fim_doe.py#L366) which did not clarify my question.

[jialuw96]

@adowling2 scale_parameter scales Q by the parameters (denominator of Q), scale_unit == scale_constant scales Q by the measurement unit (numerator of Q). The settings are as the following (I change scale_unit to be scale_constant here to avoid confusion, they are the same option):

• Results before: scale_parameter=True, scale_constant = 10**(-5) (Pa --> bar)
• I reran results with: scale_parameter=False, scale_constant = 10**(-6) (Pa --> MPa).

With scale_constant changing from E-5 to E-6, we are changing the measurement unit from bar to MPa. I am wondering if we keep scaling Q with parameters, or not, or either is ok as along as we keep this option consistent? Thank you!

[jialuw96]

• [x] make sure scaling matches the unit of measurements and the unit of variance;
• [x] Try with and without parameter scaling.
• [x] Change the scale_constant = 1, analyze with measurement STD in Pascal. Should give same answers.

[jialuw96]

Change the unit to Pascal, reproduce the heatmap above: (scale_parameter = False, scale_unit = 1, STD = 90 Pascal)

They are consistent with the results when we use the unit MPa.

Try setting: scale_parameter = True (scale_parameter = True, scale_unit = 10**(-6), STD = 0.00009 MPa)

[bbefort]

@adowling2 just to confirm, we want to use the plots where Jialu scales the parameters (final three in above post)?

[bbefort]

@jialuw96 Could you please remake these plots?

[adowling2]

@bbefort Yes, we now have consistency with data/predictions in MPa or Pa. I like using parameter scaling. We'll need to describe that in the methods (just a sentence is fine).

Let's go with black circles for the data.

[jialuw96]

@adowling2 When you say "go with black circles for the data", do you mean we use black circles for the 7th, and 8th experiments too? In the above figure, the first 6 exps are black circles, the 7th is star, and the 8th is triangle. Thank you!

[adowling2]

The new figures you posted earlier have grey circles

[jialuw96]

@bbefort Please see the following figures:

[6 experiments prior, exp 2, 5, 12, 19, 23, 26 ] a

b

c

[7 exps, add 16] d

e

f

[8 exps, add 3] g

h

i

[bbefort]

@jialuw96 The FIM values changed for this, correct? If so, where do I find the new/updated FIM?

Thanks!

[jialuw96]

@bbefort Please see the updated figures, with STD= 0.00008 MPa:

[6 experiments prior, exp 2, 5, 12, 19, 23, 26 ]

[7 exps, add 16]

[8 exps, add 3]