Model with conditional interaction

In this example, we show the heterogeneity of the global effects, using PDP, ALE, and RHALE, on a model with conditional interactions. We will use the following model:

f (x_{1}, x_{2}, x_{3}) = - x_{1}^{2} 1_{x_{2} < 0} + x_{1}^{2} 1_{x_{2} \geq 0} + e^{x_{3}}

where the features $x_{1}, x_{2}, x_{3}$ are independent and uniformly distributed in the interval $[- 1, 1]$ . The model has an interaction between $x_{1}$ and $x_{2}$ caused by the terms: $f_{1, 2} (x_{1}, x_{2}) = - x_{1}^{2} 1_{x_{2} < 0} + x_{1}^{2} 1_{x_{2} \geq 0}$ . This means that the effect of $x_{1}$ on the output $y$ depends on the value of $x_{2}$ and vice versa. Terms like this introduce heterogeneity. Each global effect method has a different formula for qunatifying such heterogeneity; below, we will see how PDP, ALE, and RHALE handles it.

In contrast, $x_{3}$ does not interact with any other feature, so its global effect has zero heterogeneity.

import numpy as np
import matplotlib.pyplot as plt
import effector

np.random.seed(21)

model = effector.models.ConditionalInteraction()
dataset = effector.datasets.IndependentUniform(dim=3, low=-1, high=1)
x = dataset.generate_data(10_000)

PDP

Effector

Let's see below the PDP heterogeneity for each feature, using effector.

pdp = effector.PDP(x, model.predict, dataset.axis_limits)
pdp.fit(features="all", centering=True)
for feature in [0, 1, 2]:
    pdp.plot(feature=feature, centering=True, heterogeneity=True, y_limits=[-2, 2])

for feature in [0, 1, 2]:
    pdp.plot(feature=feature, centering=True, heterogeneity="ice", y_limits=[-2, 2])

pdp = effector.PDP(x, model.predict, dataset.axis_limits, nof_instances="all")
pdp.fit(features="all", centering=True)
heter_per_feat = []
for feature in [0, 1, 2]:
    y_mean, y_var = pdp.eval(feature=feature, xs=np.linspace(-1, 1, 100), centering=True, heterogeneity=True)
    print(f"Heterogeneity of x_{feature}: {y_var.mean():.3f}")
    heter_per_feat.append(y_var.mean())

Heterogeneity of x_0: 0.093
Heterogeneity of x_1: 0.088
Heterogeneity of x_2: 0.000

Conclusions:

The global effect of $x_{1}$ arises from heterogenous local effects, as $h (x_{1}) > 0$ for all $x_{1}$ . The std margins (red area of height $\pm h (x_{1})$ around the global effect) musleadingly suggest that the heterogeneity is minimized at $x_{1} = \pm \frac{2}{3}$ . ICE provide a clearer picture; they reveal two groups of effects, $- x_{1}^{2} + c_{1}$ and $x_{1}^{2} + c_{2}$ . The heterogeneity as a scalar value is $H_{x_{1}} \approx 0.9$ .
Similar to $x_{1}$ , the global effect of $x_{2}$ arises from heterogeneous local effects. However, unlike $x_{1}$ , both std margins and ICE plots indicate a constant heterogeneity along all the axis, i.e., $h (x_{2}) \approx 0.9 \forall x_{2}$ . ICE plots further show a smooth range of varying local effects around the global effect, without distinct groups. The heterogeneity as a scalar is $ H_{x_2} \approx 0.9 $, which is identical to $ H_{x_1} $. This is consistent, as heterogeneity measures the interaction between a feature and all other features. In this case, since only $ x_1 $ and $ x_2 $ interact, their heterogeneity values should be the same.
$x_{3}$ shows no heteregeneity; all local effects align perfectly with the global effect.

Derivations

How PDP (and effector) reached such hetergoneity functions?

For $x_{1}$ :

The average effect is $f^{P D P} (x_{1}) = 0$ . ICE plots are: $-x_1^2 + \frac{1}{3} $ when $x_{2}^{i} < 0$ and $x_1^2 - \frac{1}{3} $ when $x_{2}^{i} \geq 0$ . Due to the square, they both create the same deviation from the average effect: $ \left ( x_1^2 - \frac{1}{3} \right )^2 $.

\begin{aligned} h (x_{1}) & = \frac{1}{N} \sum_{i}^{N} {(f_{c}^{I C E, i} (x_{1}) - f_{c}^{P D P} (x_{1}))}^{2} \\ = \frac{1}{N} \sum_{i}^{N} {(x_{1}^{2} - \frac{1}{3})}^{2} \\ = x_{1}^{4} - \frac{2}{3} x_{1}^{2} + \frac{1}{9} \end{aligned}

The heterogeneity as a scalar is simply the mean of the heterogeneity function:

\begin{aligned} H_{x_{1}} & = \frac{\int_{- 1}^{1} (x_{1}^{4} - \frac{2}{3} x_{1}^{2} + \frac{1}{9}) \partial x_{1}}{2} = \frac{4}{45} \approx 0.9 \end{aligned}

For $x_{2}$ :

The average effect is $f^{P D P} (x_{2}) = - \frac{1}{3} 1_{x_{2} < 0} + \frac{1}{3} 1_{x_{2} \geq 0}$ and the ICE plots are $f^{I C E, i} (x_{2}) = - (x_{1}^{i})^{2} 1_{x_{2} < 0} + (x_{1}^{i})^{2} 1_{x_{2} \geq 0}$ .

\begin{aligned} h (x_{2}) & = \frac{1}{N} \sum_{i}^{N} {(f_{c}^{I C E, i} (x_{2}) - f_{c}^{P D P} (x_{2}))}^{2} \\ = \frac{1}{N} \sum_{i}^{N} (1_{x_{2} < 0} {(- \frac{1}{3} + (x_{1}^{i})^{2})}^{2} + 1_{x_{2} \geq 0} {(- \frac{1}{3} + (x_{1}^{i})^{2})}^{2}) \\ = \frac{1}{N} \sum_{i}^{N} {(- \frac{1}{3} + (x_{1}^{i})^{2})}^{2} \\ = \frac{4}{45} \approx 0.9 \end{aligned}

The heterogeneity as a scalar is simply the mean of the heterogeneity function, so:

$ $H_{x_{2}} \approx 0.9$ $.

For $x_{3}$ , there is no heterogeneity so $h (x_{3}) = 0$ and $H_{x_{3}} = 0$ .

Conclusions

PDP heterogeneity provides intuitive insights:

The global effects of $x_{1}$ and $x_{2}$ arise from heterogeneous local effects, while $x_{3}$ shows no heterogeneity.
The heterogeneity of $x_{1}$ and $x_{2}$ is quantified at the same level (0.99), which makes sense since only these two features interact.
However, the heterogeneity of $x_{1}$ appears misleading when centering ICE plots, as it falsely suggests minimized heterogeneity at $\pm \frac{2}{3}$ , which is not accurate.

def pdp_ground_truth(feature, xs):
    if feature == 0:
        ff = lambda x: x**4 - 2/3*x**2 + 1/9
        return ff(xs)
    elif feature == 1:
        ff = lambda x: np.ones_like(x) * 0.088
        return ff(xs)
    elif feature == 2:
        ff = lambda x: np.zeros_like(x)
        return ff(xs)

# make a test
xx = np.linspace(-1, 1, 100)
for feature in [0, 1, 2]:
    pdp_mean, pdp_heter = pdp.eval(feature=feature, xs=xx, centering=True, heterogeneity=True)
    y_heter = pdp_ground_truth(feature, xx)
    np.testing.assert_allclose(pdp_heter, y_heter, atol=1e-1)

ALE

Effector

Let's see below the PDP effects for each feature, using effector.

ale = effector.ALE(x, model.predict, axis_limits=dataset.axis_limits)
ale.fit(features="all", centering=True, binning_method=effector.axis_partitioning.Fixed(nof_bins=31))

for feature in [0, 1, 2]:
    ale.plot(feature=feature, centering=True, heterogeneity=True, y_limits=[-2, 2])

ALE states that:

Feature $x_{1}$ :
The heterogeneity varies across all values of $x_{1}$ . It starts large at $x_{1} = - 1$ , decreases until it becomes zero at $x_{1} = 0$ , and then increases again until $x_{1} = 1$ .
This behavior contrasts with the heterogeneity observed in the PDP, which has two zero-points at $x_{1} = \pm \frac{2}{3}$ .
Feature $x_{2}$ :
Heterogeneity is observed only around $x_{2} = 0$ . This behavior also contrasts PDP's heterogeneity which is constant for all values of $x_{2}$
Feature $x_{3}$ :
No heterogeneity is present for this feature.

ale.feature_effect["feature_1"]

{'limits': array([-1.        , -0.93548387, -0.87096774, -0.80645161, -0.74193548,
        -0.67741935, -0.61290323, -0.5483871 , -0.48387097, -0.41935484,
        -0.35483871, -0.29032258, -0.22580645, -0.16129032, -0.09677419,
        -0.03225806,  0.03225806,  0.09677419,  0.16129032,  0.22580645,
         0.29032258,  0.35483871,  0.41935484,  0.48387097,  0.5483871 ,
         0.61290323,  0.67741935,  0.74193548,  0.80645161,  0.87096774,
         0.93548387,  1.00000001]),
 'dx': array([0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451613, 0.06451613, 0.06451613, 0.06451613, 0.06451613,
        0.06451614]),
 'points_per_bin': array([320, 339, 318, 345, 313, 312, 331, 311, 293, 326, 309, 294, 314,
        312, 335, 318, 354, 328, 312, 331, 315, 307, 341, 320, 330, 336,
        312, 368, 325, 294, 337]),
 'bin_effect': array([0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        , 0.        ,
        9.93884099, 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        ]),
 'bin_variance': array([ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
        82.51292593,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ]),
 'alg_params': 'fixed',
 'norm_const': np.float64(0.3206077740261283)}

Derivations

For x_1:

The $x_{1}$ -axis is divided into $K$ equal bins, indexed by $k = 1, \dots, K$ , with the center of the $k$ -th bin denoted as $c_{k}$ . In each bin: if $x_{2} < 0$ , the local effects is $- 2 c_{k}$ , and if $x_{2} \geq 0$ the local effect is $2 c_{k}$ . This gives a variance of $4 c_{k}^{2}$ within each bin. Therefore, $h (x_{1}) = 4 c_{k}^{2}$ where $k$ is the index of the bin that contains $x_{1}$ .

\begin{aligned} h (x_{1}) & = \frac{1}{| S_{k} |} \sum_{x^{i} \in S_{k}} {(\frac{f (z_{k}, x_{2}^{i}, x_{3}^{i}) - f (z_{k - 1}, x_{2}^{i}, x_{3}^{i})}{z_{k} - z_{k - 1}} - {\hat{μ}}_{k}^{A L E})}^{2} \\ = \frac{1}{| S_{k} |} \sum_{x^{i} \in S_{k}} {(\frac{- 2 c_{k} (z_{k} - z_{k - 1}) 1_{x_{2}^{i} < 0} + 2 c_{k} (z_{k} - z_{k - 1}) 1_{x_{2}^{i} \geq 0}}{z_{k} - z_{k - 1}})}^{2} \\ = \frac{1}{| S_{k} |} \sum_{x^{i} \in S_{k}} 4 c_{k}^{2} \\ = 4 c_{k}^{2} \end{aligned}

For $x_{2}$ :

In all bins except the central one, the local effects are zero. In the central bin, however, the local effects are $ 2(x_1^i)^2 $, which introduces some heterogeneity. So, if $ x_2 $ is not in the central bin $ k = K/2 $, $h (x_{2}) = 0$ . If $ x_2 $ is in the central bin $ k = K/2 $:

\begin{aligned} h (x_{2}) & = \frac{1}{| S_{k} |} \sum_{x^{i} \in S_{k}} {(\frac{f (z_{k}, x_{2}^{i}, x_{3}^{i}) - f (z_{k - 1}, x_{2}^{i}, x_{3}^{i})}{z_{k} - z_{k - 1}} - {\hat{μ}}_{k}^{A L E})}^{2} \\ = \frac{2}{z_{k} - z_{k - 1}} \frac{1}{| S_{k} |} \sum_{x^{i} \in S_{k}} (x_{1}^{i})^{2} \\ = \frac{2}{3} (z_{k} - z_{k - 1}) \\ \approx 10 for K = 31 \end{aligned}

So: $\begin{aligned} h (x_{2}) & = {\begin{cases} 0, & if x_{2} is not in the central bin (k = K / 2), \frac{31}{3} \approx 10 & if x_{2} is in the central bin (k = K / 2) . \end{cases} \end{aligned}$

For $x_{3}$ :

The effect is zero everywehere.

def ale_ground_truth(feature):
    K = 31
    bin_centers = np.linspace(-1 + 1/K, 1 - 1/K, K)
    if feature == 0:
        return 4*bin_centers**2
    elif feature == 1:
        y = np.zeros_like(bin_centers)
        y[15] = None
        return y
    elif feature == 2:
        return np.zeros_like(bin_centers)

# make a test
K = 31
bin_centers = np.linspace(-1 + 1/K, 1 - 1/K, K)
for feature in [0, 1, 2]:
    bin_var = ale.feature_effect[f"feature_{feature}"]["bin_variance"]
    gt_var = ale_ground_truth(feature)
    mask = ~np.isnan(gt_var)
    np.testing.assert_allclose(bin_var[mask], gt_var[mask], atol=1e-1)

Conclusions

Is the heterogeneity implied by the ALE plots meaningful? It is

RHALE

Effector

Let's see below the RHALE effects for each feature, using effector.

rhale = effector.RHALE(x, model.predict, model.jacobian, axis_limits=dataset.axis_limits)
rhale.fit(features="all", centering=True)

for feature in [0, 1, 2]:
    rhale.plot(feature=feature, centering=True, heterogeneity=True, y_limits=[-2, 2])

RHALE states that:

Feature $x_{1}$ : As in ALE, the heterogeneity varies across all values of $x_{1}$ . It starts large at $x_{1} = - 1$ , decreases until it becomes zero at $x_{1} = 0$ , and then increases again until $x_{1} = 1$ .
Feature $x_{2}$ :
No heterogeneity is present for this feature.
Feature $x_{3}$ :
No heterogeneity is present for this feature.

Derivations

\begin{aligned} H (x_{1}) & \approx \end{aligned}

\begin{aligned} H (x_{2}) & \approx \end{aligned}

\begin{aligned} H (x_{3}) & \approx \end{aligned}

Conclusions

Are the RHALE effects intuitive?