Modeling a Mixture in Traditional Representation¶
When modeling mixtures, we are often faced with a large set of ingredients to choose from. A common way to formalize this type of selection problem is to assign each ingredient its own numerical parameter representing the amount of the ingredient in the mixture. A sum constraint imposed on all parameters then ensures that the total amount of ingredients in the mix is always 100%. In addition, there could be other constraints, for instance, to impose further restrictions on individual subgroups of ingredients. In BayBE’s language, we call this the traditional mixture representation.
In this example, we demonstrate how to create a search space in this representation, using a simple mixture of up to six components, which are divided into three subgroups: solvents, bases and phase agents.
Slot-based Representation
For an alternative way to describe mixtures, see our slot-based representation.
Imports¶
import numpy as np
import pandas as pd
from baybe.constraints import ContinuousLinearConstraint
from baybe.parameters import NumericalContinuousParameter
from baybe.recommenders import RandomRecommender
from baybe.searchspace import SearchSpace
Parameter Setup¶
We start by creating lists containing our substance labels according to their subgroups:
g1 = ["Solvent1", "Solvent2"]
g2 = ["Base1", "Base2"]
g3 = ["PhaseAgent1", "PhaseAgent2"]
Next, we create continuous parameters describing the substance amounts for each group. Here, the maximum amount for each substance depends on its group, i.e. we allow adding more of a solvent compared to a base or a phase agent:
p_g1_amounts = [
NumericalContinuousParameter(name=f"{name}", bounds=(0, 80)) for name in g1
]
p_g2_amounts = [
NumericalContinuousParameter(name=f"{name}", bounds=(0, 20)) for name in g2
]
p_g3_amounts = [
NumericalContinuousParameter(name=f"{name}", bounds=(0, 5)) for name in g3
]
Constraints Setup¶
Now, we set up our constraints. We start with the overall mixture constraint, ensuring the total of all ingredients is 100%:
c_total_sum = ContinuousLinearConstraint(
parameters=g1 + g2 + g3,
operator="=",
coefficients=[1] * len(g1 + g2 + g3),
rhs=100,
)
Additionally, we require bases make up at least 10% of the mixture:
c_g2_min = ContinuousLinearConstraint(
parameters=g2,
operator=">=",
coefficients=[1] * len(g2),
rhs=10,
)
By contrast, phase agents should make up no more than 5%:
c_g3_max = ContinuousLinearConstraint(
parameters=g3,
operator="<=",
coefficients=[1] * len(g3),
rhs=5,
)
Search Space Creation¶
Having both parameter and constraint definitions at hand, we can create our search space:
searchspace = SearchSpace.from_product(
parameters=[*p_g1_amounts, *p_g2_amounts, *p_g3_amounts],
constraints=[c_total_sum, c_g2_min, c_g3_max],
)
Verification of Constraints¶
To verify that the constraints imposed above are fulfilled, let us draw some random points from the search space:
recommendations = RandomRecommender().recommend(batch_size=10, searchspace=searchspace)
print(recommendations)
Base1 Base2 PhaseAgent1 PhaseAgent2 Solvent1 Solvent2
0 10.312686 10.855187 2.100364 1.181167 27.213585 48.337011
1 15.883004 2.333435 0.623992 3.766257 69.160952 8.232359
2 11.574172 0.386536 1.442716 3.344695 10.809507 72.442375
3 14.356882 7.136506 0.640561 2.947057 34.179075 40.739919
4 9.082438 16.948816 3.060132 0.973393 59.960873 9.974347
5 19.326918 1.033319 1.739571 2.614461 52.858459 22.427272
6 17.394761 10.804006 1.180754 0.370260 44.692004 25.558216
7 11.411518 4.761565 3.257237 1.003068 74.278940 5.287671
8 6.467253 4.494095 2.812083 1.363578 65.989564 18.873426
9 4.138126 19.163808 4.266978 0.340309 44.219631 27.871148
Computing the respective row sums reveals the expected result:
stats = pd.DataFrame(
{
"Total": recommendations.sum(axis=1),
"Total_Bases": recommendations[g2].sum(axis=1),
"Total_Phase_Agents": recommendations[g3].sum(axis=1),
}
)
print(stats)
Total Total_Bases Total_Phase_Agents
0 100.0 21.167873 3.281532
1 100.0 18.216439 4.390249
2 100.0 11.960708 4.787411
3 100.0 21.493388 3.587618
4 100.0 26.031254 4.033525
5 100.0 20.360237 4.354032
6 100.0 28.198767 1.551014
7 100.0 16.173083 4.260306
8 100.0 10.961348 4.175662
9 100.0 23.301934 4.607287
assert np.allclose(stats["Total"], 100)
assert (stats["Total_Bases"] >= 10).all()
assert (stats["Total_Phase_Agents"] <= 5).all()