Full and Fractional Factorial Designs¶

BoFire can be used to setup full (two level) and fractional factorial designs (https://en.wikipedia.org/wiki/Fractional_factorial_design). This tutorial notebook shows how.

Imports and helper functions¶

In [ ]:

import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns

import bofire.strategies.api as strategies
from bofire.data_models.domain.api import Domain
from bofire.data_models.features.api import ContinuousInput
from bofire.data_models.strategies.api import FractionalFactorialStrategy
from bofire.utils.doe import get_alias_structure, get_confounding_matrix, get_generator


def plot_design(design: pd.DataFrame):
    # we do a plot with three subplots in one row in which the three degrees of freedom (temperature, time and ph) are plotted
    _, axs = plt.subplots(1, 3, figsize=(15, 5))
    axs[0].scatter(design["temperature"], design["time"])
    axs[0].set_xlabel("Temperature")
    axs[0].set_ylabel("Time")
    axs[1].scatter(design["temperature"], design["ph"])
    axs[1].set_xlabel("Temperature")
    axs[1].set_ylabel("pH")
    axs[2].scatter(design["time"], design["ph"])
    axs[2].set_xlabel("Time")
    axs[2].set_ylabel("pH")
    plt.show()

import matplotlib.pyplot as plt import pandas as pd import seaborn as sns import bofire.strategies.api as strategies from bofire.data_models.domain.api import Domain from bofire.data_models.features.api import ContinuousInput from bofire.data_models.strategies.api import FractionalFactorialStrategy from bofire.utils.doe import get_alias_structure, get_confounding_matrix, get_generator def plot_design(design: pd.DataFrame): # we do a plot with three subplots in one row in which the three degrees of freedom (temperature, time and ph) are plotted _, axs = plt.subplots(1, 3, figsize=(15, 5)) axs[0].scatter(design["temperature"], design["time"]) axs[0].set_xlabel("Temperature") axs[0].set_ylabel("Time") axs[1].scatter(design["temperature"], design["ph"]) axs[1].set_xlabel("Temperature") axs[1].set_ylabel("pH") axs[2].scatter(design["time"], design["ph"]) axs[2].set_xlabel("Time") axs[2].set_ylabel("pH") plt.show()

Setup the problem domain¶

The designs are generated for a simple three dimensional problem comprised of three continuous factors/features.

In [ ]:

domain = Domain(
    inputs=[
        ContinuousInput(key="temperature", bounds=(20, 80)),
        ContinuousInput(key="time", bounds=(60, 120)),
        ContinuousInput(key="ph", bounds=(7, 13)),
    ],
)

domain = Domain( inputs=[ ContinuousInput(key="temperature", bounds=(20, 80)), ContinuousInput(key="time", bounds=(60, 120)), ContinuousInput(key="ph", bounds=(7, 13)), ], )

Setup a full factorial design¶

Here we setup a full two-level factorial design including a center point and plot it.

In [ ]:

strategy_data = FractionalFactorialStrategy(
    domain=domain,
    n_center=1,  # number of center points
    n_repetitions=1,  # number of repetitions, we do only one round here
)
strategy = strategies.map(strategy_data)
design = strategy.ask()
display(design)

plot_design(design=design)

strategy_data = FractionalFactorialStrategy( domain=domain, n_center=1, # number of center points n_repetitions=1, # number of repetitions, we do only one round here ) strategy = strategies.map(strategy_data) design = strategy.ask() display(design) plot_design(design=design)

The confounding structure is shown below, as expected for a full factorial design, no confound is present.

In [ ]:

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2])

sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f")
plt.show()

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2]) sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f") plt.show()

Setup a fractional factorial design¶

Here a fractional factorial design of the form $2^{3-1}$ is setup by specifying the number of generators (here 1). In comparison to the full factorial design with 9 candidates, it features only 5 experiments.

In [ ]:

strategy_data = FractionalFactorialStrategy(
    domain=domain,
    n_center=1,  # number of center points
    n_repetitions=1,  # number of repetitions, we do only one round here
    n_generators=1,  # number of generators, ie number of reducing factors
)
strategy = strategies.map(strategy_data)
design = strategy.ask()
display(design)

strategy_data = FractionalFactorialStrategy( domain=domain, n_center=1, # number of center points n_repetitions=1, # number of repetitions, we do only one round here n_generators=1, # number of generators, ie number of reducing factors ) strategy = strategies.map(strategy_data) design = strategy.ask() display(design)

The generator string is automatically generated by making use of the method get_generator and specifying the total number of factors (here 3) and the number of generators (here 1).

In [ ]:

get_generator(n_factors=3, n_generators=1)

get_generator(n_factors=3, n_generators=1)

As expected for a type III design the main effects are confounded with the two factor interactions:

In [ ]:

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2])

sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f")
plt.show()

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2]) sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f") plt.show()

This can also be expressed by the so called alias structure that can be calculated as following:

In [ ]:

get_alias_structure("a b ab")

get_alias_structure("a b ab")

Here again a fractional factorial design of the form $2^{3-1}$ is setup by providing the complete generator string of the form a b -ab explicitly to the strategy.

In [ ]:

strategy_data = FractionalFactorialStrategy(
    domain=domain,
    n_center=1,  # number of center points
    n_repetitions=1,  # number of repetitions, we do only one round here
    generator="a b -ab",  # the exact generator
)
strategy = strategies.map(strategy_data)
design = strategy.ask()
display(design)

strategy_data = FractionalFactorialStrategy( domain=domain, n_center=1, # number of center points n_repetitions=1, # number of repetitions, we do only one round here generator="a b -ab", # the exact generator ) strategy = strategies.map(strategy_data) design = strategy.ask() display(design)

The last two designs differ only in the last feature time, since the generator strings are different. In the first one it holds time=ph x temperature whereas in the second it holds time=-ph x temperature, which is also reflected in the confounding structure.

In [ ]:

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2])

sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f")
plt.show()

m = get_confounding_matrix(domain.inputs, design=design, interactions=[2]) sns.heatmap(m, annot=True, annot_kws={"fontsize": 7}, fmt="2.1f") plt.show()