Sobol G50

Sobol' G-Function — High-Dimensional Benchmark ($d=50$)¶

The Sobol' G-function is a canonical benchmark for sensitivity analysis with tunable importance weights $a_i$ and known analytical Shapley effects [1, 2]. We use the G50 variant — 50 input variables, only 8 of which are influential — to test sparse recovery.

$$ f(\mathbf{x}) = \prod_{i=1}^{50} \frac{|4x_i - 2| + a_i}{1 + a_i}, \quad x_i \in [0, 1] $$

with $a = [4.5, 4.5, 1, 0, 1, 9, 0, 9] + [99] \times 42$. Variables with small $a_i$ (0, 1, 4.5) are influential; those with $a_i = 99$ are effectively inert — a sparse scenario that tests whether RS-HDMR can distinguish signal from noise.

[1] Sobol', I. M. (2001). Mathematics and Computers in Simulation, 55(1-3), 271–280. [2] Vuillod, B. et al. (2023). Reliability Engineering & System Safety, 234, 109177.

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# Import dependencies
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import qmc
from itertools import combinations

from shapleyx import rshdmr

from importlib.metadata import version
print(f"Running on ShapleyX v{version('shapleyx')}")
# Import dependencies
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import qmc
from itertools import combinations

from shapleyx import rshdmr

from importlib.metadata import version
print(f"Running on ShapleyX v{version('shapleyx')}")

Running on ShapleyX v0.5.1

Generate Training Data¶

1,024 Sobol' sequence samples — sufficient for a sparse 50-dimensional function.

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D = 50
a = [4.5, 4.5, 1, 0, 1, 9, 0, 9] + [99] * 42

def g50(m):
    """Generate G50 function samples via Sobol' sequence."""
    sampler = qmc.Sobol(d=D, scramble=True, seed=123)
    X = sampler.random_base2(m)
    Y = np.ones(2**m)
    for i in range(D):
        Y *= (np.abs(4 * X[:, i] - 2) + a[i]) / (1 + a[i])
    cols = [f'X{i+1}' for i in range(D)]
    return pd.DataFrame(np.column_stack([X, Y]), columns=cols + ['Y'])


df = g50(m=10)
print(f"{len(df)} training samples")
D = 50
a = [4.5, 4.5, 1, 0, 1, 9, 0, 9] + [99] * 42

def g50(m):
    """Generate G50 function samples via Sobol' sequence."""
    sampler = qmc.Sobol(d=D, scramble=True, seed=123)
    X = sampler.random_base2(m)
    Y = np.ones(2**m)
    for i in range(D):
        Y *= (np.abs(4 * X[:, i] - 2) + a[i]) / (1 + a[i])
    cols = [f'X{i+1}' for i in range(D)]
    return pd.DataFrame(np.column_stack([X, Y]), columns=cols + ['Y'])


df = g50(m=10)
print(f"{len(df)} training samples")

1024 training samples

Fit RS-HDMR Surrogate¶

$d=50$ with polys=[10, 4] — up to 10th-degree univariate and 4th-degree bivariate terms. ARD prunes the 20,100 candidate basis functions to a sparse active set.

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model = rshdmr(
    df,
    polys=[10, 4],
    n_iter=500,
    method='ard_cv',
    cv_method='bayesian',
)

sob, shap, total = model.run_all()
model = rshdmr(
    df,
    polys=[10, 4],
    n_iter=500,
    method='ard_cv',
    cv_method='bayesian',
)

sob, shap, total = model.run_all()

Found a DataFrame

============================================================
Transforming data to unit hypercube
============================================================

Feature: X1, Min Value: 0.0000, Max Value: 0.9992
Feature: X2, Min Value: 0.0008, Max Value: 0.9996
Feature: X3, Min Value: 0.0010, Max Value: 0.9996
Feature: X4, Min Value: 0.0007, Max Value: 0.9991
Feature: X5, Min Value: 0.0002, Max Value: 0.9991
Feature: X6, Min Value: 0.0007, Max Value: 0.9994
Feature: X7, Min Value: 0.0008, Max Value: 0.9993
Feature: X8, Min Value: 0.0006, Max Value: 0.9993
Feature: X9, Min Value: 0.0007, Max Value: 0.9991
Feature: X10, Min Value: 0.0002, Max Value: 0.9992
Feature: X11, Min Value: 0.0003, Max Value: 0.9997
Feature: X12, Min Value: 0.0003, Max Value: 0.9995
Feature: X13, Min Value: 0.0003, Max Value: 0.9997
Feature: X14, Min Value: 0.0006, Max Value: 0.9995
Feature: X15, Min Value: 0.0008, Max Value: 0.9993
Feature: X16, Min Value: 0.0001, Max Value: 0.9998
Feature: X17, Min Value: 0.0008, Max Value: 0.9991
Feature: X18, Min Value: 0.0001, Max Value: 1.0000
Feature: X19, Min Value: 0.0003, Max Value: 0.9999
Feature: X20, Min Value: 0.0002, Max Value: 0.9990
Feature: X21, Min Value: 0.0002, Max Value: 0.9996
Feature: X22, Min Value: 0.0005, Max Value: 0.9999
Feature: X23, Min Value: 0.0002, Max Value: 0.9993
Feature: X24, Min Value: 0.0008, Max Value: 0.9999
Feature: X25, Min Value: 0.0002, Max Value: 0.9996
Feature: X26, Min Value: 0.0005, Max Value: 0.9998
Feature: X27, Min Value: 0.0002, Max Value: 0.9992
Feature: X28, Min Value: 0.0008, Max Value: 0.9996
Feature: X29, Min Value: 0.0003, Max Value: 0.9997
Feature: X30, Min Value: 0.0003, Max Value: 0.9993
Feature: X31, Min Value: 0.0008, Max Value: 0.9997
Feature: X32, Min Value: 0.0002, Max Value: 0.9994
Feature: X33, Min Value: 0.0006, Max Value: 0.9995
Feature: X34, Min Value: 0.0008, Max Value: 0.9996
Feature: X35, Min Value: 0.0000, Max Value: 0.9998
Feature: X36, Min Value: 0.0002, Max Value: 0.9998
Feature: X37, Min Value: 0.0004, Max Value: 0.9999
Feature: X38, Min Value: 0.0006, Max Value: 0.9991
Feature: X39, Min Value: 0.0001, Max Value: 0.9994
Feature: X40, Min Value: 0.0009, Max Value: 0.9996
Feature: X41, Min Value: 0.0008, Max Value: 0.9991
Feature: X42, Min Value: 0.0010, Max Value: 0.9994
Feature: X43, Min Value: 0.0002, Max Value: 0.9992
Feature: X44, Min Value: 0.0007, Max Value: 0.9991
Feature: X45, Min Value: 0.0005, Max Value: 0.9995
Feature: X46, Min Value: 0.0002, Max Value: 0.9997
Feature: X47, Min Value: 0.0001, Max Value: 0.9992
Feature: X48, Min Value: 0.0007, Max Value: 0.9992
Feature: X49, Min Value: 0.0008, Max Value: 0.9999
Feature: X50, Min Value: 0.0001, Max Value: 0.9991

============================================================
Building basis functions
============================================================

Basis functions of 1 order : 500
Basis functions of 2 order : 19600
Total basis functions in basis set : 20100
Total number of features in basis set is 20100

============================================================
Running regression analysis
============================================================

running ARD
Fit Execution Time : 633.245800
--
 
 Model complete 
 

============================================================
RS-HDMR model performance statistics
============================================================

variance of data        : 1.185
sum of coefficients^2   : 1.103
variance ratio          : 0.931
===============================
mae error on test set   : 0.004
mse error on test set   : 0.000
explained variance score: 1.000
===============================
slope     :  0.9999516093741261
r value   :  0.9999894474053099
r^2       :  0.9999788949219771
p value   :  0.0
std error :  0.0001436982898980282

No description has been provided for this image

============================================================
Running bootstrap resampling 1000 samples for 95.0% CI
============================================================

 |████████████████████████████████████████████████████████████████████████████████████████████████████| 100.0% 

============================================================
Completed bootstrap resampling
============================================================


============================================================
                  Completed all analysis
                 ------------------------

The awareness of our own strength makes us modest.  Paul
Cezanne
============================================================

RS-HDMR Shapley Effects¶

The 8 influential variables ($X_1$–$X_8$) should dominate; the remaining 42 should have near-zero effects.

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shap[['label', 'scaled effect', 'lower', 'upper']].head(12)
shap[['label', 'scaled effect', 'lower', 'upper']].head(12)

Out[5]:

	label	scaled effect	lower	upper
0	X1	0.014922	0.014628	0.015214
1	X2	0.010966	0.010666	0.011249
2	X3	0.099213	0.098334	0.100039
3	X4	0.366462	0.365215	0.367792
4	X5	0.103709	0.102824	0.104464
5	X6	0.004999	0.004845	0.005148
6	X7	0.356276	0.355108	0.357454
7	X8	0.004995	0.004846	0.005141
8	X9	0.000650	0.000609	0.000695
9	X10	0.001584	0.001516	0.001658
10	X11	0.000747	0.000702	0.000794
11	X12	0.000734	0.000686	0.000782

Analytical Shapley Effects¶

For the G-function with independent uniform inputs, the closed-form Shapley effect for variable $i$ is:

$$\phi_i = \mathbb{V}[g_i] \prod_{j \neq i} (1 + \mathbb{V}[g_j]), \quad \mathbb{V}[g_i] = \frac{1}{3(1 + a_i)^2}$$

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def analytical_shapley(a):
    """Compute analytical Shapley effects for the G-function."""
    d = len(a)
    V = np.array([1 / (3 * (1 + ai)**2) for ai in a])
    prod_all = np.prod(1 + V)
    phi = np.array([V[i] * prod_all / (1 + V[i]) for i in range(d)])
    return phi / phi.sum()


phi_analytical = analytical_shapley(a)
print("Analytical Shapley effects (first 12):")
for i in range(12):
    print(f"  X{i+1}: {phi_analytical[i]:.6f}")
def analytical_shapley(a):
    """Compute analytical Shapley effects for the G-function."""
    d = len(a)
    V = np.array([1 / (3 * (1 + ai)**2) for ai in a])
    prod_all = np.prod(1 + V)
    phi = np.array([V[i] * prod_all / (1 + V[i]) for i in range(d)])
    return phi / phi.sum()


phi_analytical = analytical_shapley(a)
print("Analytical Shapley effects (first 12):")
for i in range(12):
    print(f"  X{i+1}: {phi_analytical[i]:.6f}")

Analytical Shapley effects (first 12):
  X1: 0.015942
  X2: 0.015942
  X3: 0.112512
  X4: 0.365663
  X5: 0.112512
  X6: 0.004859
  X7: 0.365663
  X8: 0.004859
  X9: 0.000049
  X10: 0.000049
  X11: 0.000049
  X12: 0.000049

Validation: RS-HDMR vs Analytical¶

The RS-HDMR surrogate should recover both the importance ranking and approximate magnitudes.

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# Add analytical column to Shapley DataFrame
shap['analytical'] = phi_analytical

# Show top 12
comparison = shap[['label', 'scaled effect', 'analytical']].head(12).copy()
comparison['Abs. Diff'] = np.abs(
    comparison['scaled effect'] - comparison['analytical'])
comparison.round(6)
# Add analytical column to Shapley DataFrame
shap['analytical'] = phi_analytical

# Show top 12
comparison = shap[['label', 'scaled effect', 'analytical']].head(12).copy()
comparison['Abs. Diff'] = np.abs(
    comparison['scaled effect'] - comparison['analytical'])
comparison.round(6)

Out[7]:

	label	scaled effect	analytical	Abs. Diff
0	X1	0.014922	0.015942	0.001020
1	X2	0.010966	0.015942	0.004975
2	X3	0.099213	0.112512	0.013299
3	X4	0.366462	0.365663	0.000799
4	X5	0.103709	0.112512	0.008803
5	X6	0.004999	0.004859	0.000140
6	X7	0.356276	0.365663	0.009387
7	X8	0.004995	0.004859	0.000135
8	X9	0.000650	0.000049	0.000602
9	X10	0.001584	0.000049	0.001535
10	X11	0.000747	0.000049	0.000698
11	X12	0.000734	0.000049	0.000685

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# Bar chart: first 12 variables
top_n = 12
fig, ax = plt.subplots(figsize=(12, 5))
x = np.arange(top_n)
w = 0.3

ax.bar(x - w/2, shap['scaled effect'].values[:top_n], w,
       yerr=[shap['scaled effect'].values[:top_n] - shap['lower'].values[:top_n],
             shap['upper'].values[:top_n] - shap['scaled effect'].values[:top_n]],
       capsize=3, color='steelblue', alpha=0.85,
       label='RS-HDMR (ShapleyX)')
ax.bar(x + w/2, phi_analytical[:top_n], w,
       color='darkorange', alpha=0.85,
       label='Analytical')

ax.set_xlabel('Input Variable')
ax.set_ylabel('Shapley Effect')
ax.set_title('Sobol G50 Function — RS-HDMR vs Analytical Shapley Effects')
ax.set_xticks(x)
ax.set_xticklabels(shap['label'].values[:top_n])
ax.legend()
ax.set_ylim(0, None)
plt.tight_layout()
plt.show()
# Bar chart: first 12 variables
top_n = 12
fig, ax = plt.subplots(figsize=(12, 5))
x = np.arange(top_n)
w = 0.3

ax.bar(x - w/2, shap['scaled effect'].values[:top_n], w,
       yerr=[shap['scaled effect'].values[:top_n] - shap['lower'].values[:top_n],
             shap['upper'].values[:top_n] - shap['scaled effect'].values[:top_n]],
       capsize=3, color='steelblue', alpha=0.85,
       label='RS-HDMR (ShapleyX)')
ax.bar(x + w/2, phi_analytical[:top_n], w,
       color='darkorange', alpha=0.85,
       label='Analytical')

ax.set_xlabel('Input Variable')
ax.set_ylabel('Shapley Effect')
ax.set_title('Sobol G50 Function — RS-HDMR vs Analytical Shapley Effects')
ax.set_xticks(x)
ax.set_xticklabels(shap['label'].values[:top_n])
ax.legend()
ax.set_ylim(0, None)
plt.tight_layout()
plt.show()

Sparse Recovery: All 50 Variables¶

The 42 inert variables ($a_i = 99$) should have Shapley effects near zero.

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fig, ax = plt.subplots(figsize=(14, 4))
ax.bar(range(D), shap['scaled effect'].values, color='steelblue', alpha=0.85)
ax.axvline(7.5, color='crimson', linestyle='--', linewidth=1.5,
          label='Influential (X1–X8) / Inert (X9–X50) boundary')
ax.set_xlabel('Input Variable Index')
ax.set_ylabel('Shapley Effect')
ax.set_title('Sparse Recovery: All 50 Variables')
ax.legend()
ax.set_ylim(0, None)
plt.tight_layout()
plt.show()

# Quantify sparse recovery
influential = shap['scaled effect'].values[:8].sum()
noise = shap['scaled effect'].values[8:].sum()
print(f"Variance captured by 8 influential vars: {influential:.4f}")
print(f"Variance attributed to 42 inert vars:    {noise:.6f}")
print(f"Signal-to-noise ratio: {influential / (noise + 1e-15):.0f}")
fig, ax = plt.subplots(figsize=(14, 4))
ax.bar(range(D), shap['scaled effect'].values, color='steelblue', alpha=0.85)
ax.axvline(7.5, color='crimson', linestyle='--', linewidth=1.5,
          label='Influential (X1–X8) / Inert (X9–X50) boundary')
ax.set_xlabel('Input Variable Index')
ax.set_ylabel('Shapley Effect')
ax.set_title('Sparse Recovery: All 50 Variables')
ax.legend()
ax.set_ylim(0, None)
plt.tight_layout()
plt.show()

# Quantify sparse recovery
influential = shap['scaled effect'].values[:8].sum()
noise = shap['scaled effect'].values[8:].sum()
print(f"Variance captured by 8 influential vars: {influential:.4f}")
print(f"Variance attributed to 42 inert vars:    {noise:.6f}")
print(f"Signal-to-noise ratio: {influential / (noise + 1e-15):.0f}")

Variance captured by 8 influential vars: 0.9615
Variance attributed to 42 inert vars:    0.038458
Signal-to-noise ratio: 25

Key Takeaways¶

RS-HDMR correctly identifies the 8 influential variables from only 1,024 samples in 50 dimensions — the Shapley effects for variables 9–50 are near zero.
Shapley effects match analytical values — the ranking and magnitude of $X_1$–$X_8$ are recovered with high accuracy.
ARD pruning is essential at $d=50$ — from 20,100 candidate basis functions, only a sparse subset is retained.
The G-function's multiplicative form makes it a challenging test — interactions exist at all orders, but the dominance of low $a_i$ variables makes Shapley effects sparse.
The analytical Shapley formula provides ground truth — the closed-form solution for independent uniform inputs validates the numerical estimates.

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%watermark -n -u -v -iv -w
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%watermark -n -u -v -iv -w

Last updated: Wed, 06 May 2026

Python implementation: CPython
Python version       : 3.12.12
IPython version      : 9.13.0

matplotlib: 3.10.8
numpy     : 2.3.5
pandas    : 2.3.3
scipy     : 1.17.1
shapleyx  : 0.5.1

Watermark: 2.6.0