Chapter 2: Magpie and Statistical Descriptors

This chapter covers Magpie and Statistical Descriptors. You will learn essential concepts and techniques.

2.1 Magpie Descriptor Details

Design Philosophy of Ward et al. (2016)

The Magpie (Materials Agnostic Platform for Informatics and Exploration) descriptor is the definitive composition-based feature set published by Dr. Logan Ward and colleagues at Northwestern University in 2016. In their npj Computational Materials paper, Ward et al. (2016) proposed it as a "general-purpose framework for predicting material properties without structural information" (pp. 1-2).

The core design is based on three principles:

1. Physical interpretability: All features are based on physicochemical properties of elements
2. Scalability: Applicable to arbitrary chemical formulas (approximately 2-10 elements)
3. Information maximization: Comprehensive description of materials space with 132 dimensions from 22 elemental properties × 6 statistical measures

Structure of the 132-Dimensional Vector

Magpie descriptors have the following hierarchical structure:

Breakdown of dimensions:

• 22 elemental properties × minimum = 22 dimensions
• 22 elemental properties × maximum = 22 dimensions
• 22 elemental properties × range (max - min) = 22 dimensions
• 22 elemental properties × mean (fraction-weighted) = 22 dimensions
• 22 elemental properties × avg_dev (mean absolute deviation) = 22 dimensions
• 22 elemental properties × mode = 22 dimensions
• Total: 132 dimensions

Physical Meaning of Each Dimension

Each dimension of Magpie descriptors represents a physical quantity that directly affects material properties. For example:

2.2 Types of Elemental Properties

The 22 elemental properties of the magpie preset are fixed — they are exactly the list held in ElementProperty.from_preset("magpie").features, and they do not change between matminer releases. They fall into five natural groups.

Periodic Table Position (4 Types)

Where an element sits in the periodic table, which encodes chemical similarity:

1. Number (Atomic number, Z): Proton count, nuclear charge
2. MendeleevNumber (Mendeleev number): An ordering that places chemically similar elements next to each other
3. Column (Group number, 1-18): Chemical property periodicity
4. Row (Period number, 1-7): Electron shell count, atomic size

Atomic and Thermal Properties (4 Types)

Mass, size, bonding character, and thermal stability of the atom:

1. AtomicWeight (Atomic weight, g/mol): Affects mass and density
2. MeltingT (Melting point, K): Indicator of high-temperature stability
3. CovalentRadius (Covalent radius, pm): Determines bond lengths and lattice constants
4. Electronegativity (Electronegativity, Pauling scale): Ionic/covalent character of bonds

Valence Electron Counts (5 Types)

The number of valence electrons in each orbital type, plus their total:

1. NsValence (s-orbital valence electrons): Metallic bond strength
2. NpValence (p-orbital valence electrons): Semiconductor properties
3. NdValence (d-orbital valence electrons): Catalytic activity of transition metals
4. NfValence (f-orbital valence electrons): Magnetism in lanthanides and actinides
5. NValence (Total valence electrons): Overall bonding capacity and redox behaviour

Unfilled Orbital Counts (5 Types)

The number of vacancies in each orbital type — the counterpart of the valence counts, and a strong signal for reactivity:

1. NsUnfilled (Unfilled s states): Availability of s-type acceptor states
2. NpUnfilled (Unfilled p states): Anion character, p-band filling
3. NdUnfilled (Unfilled d states): Transition-metal reactivity and magnetism
4. NfUnfilled (Unfilled f states): f-electron behaviour in rare earths
5. NUnfilled (Total unfilled states): Overall electron-accepting capacity

Ground-State (DFT) Properties (4 Types)

Properties of the element's own ground-state crystal, tabulated from DFT calculations:

1. GSvolume_pa (Ground-state volume/atom, ų): Theoretical volume from DFT calculations
2. GSbandgap (Ground-state band gap, eV): Electrical properties of semiconductors/insulators
3. GSmagmom (Ground-state magnetic moment, μB): Properties of magnetic materials
4. SpaceGroupNumber (Space group number): Crystal symmetry of the elemental solid

Several quantities that feel like obvious candidates are not in these 22: ionization energy, electron affinity, density, boiling point, heat capacity, and ground-state energy per atom. If your problem needs them, use a different preset (matminer also ships deml, matminer, and magpie_oxid), pass an explicit property list to ElementProperty, or write a custom featurizer — Chapter 3 shows how.

2.3 Statistical Aggregation Methods

The Six Magpie Statistics

Each of the 22 elemental properties is collapsed into a single number six different ways, giving 22 × 6 = 132 columns named MagpieData {statistic} {property}. The six statistics are fixed: minimum, maximum, range, mean, avg_dev, mode.

Throughout, $f_i = n_i / \sum_j n_j$ is the atomic fraction of element $i$ (with $n_i$ its atom count), and $p_i$ is the property value of element $i$.

1. Minimum

Minimum property value in the composition. Represents the material's "bottleneck":

Example: minimum Electronegativity = min(Fe: 1.83, O: 3.44) = 1.83 (Fe)

2. Maximum

Maximum property value in the composition. Indicates the material's "peak performance":

Example: maximum MeltingT = max(Fe: 1811 K, O: 54 K) = 1811 K (Fe)

3. Range

Difference between maximum and minimum values. Represents property "spread":

Example: range Electronegativity = 3.44 - 1.83 = 1.61 (indicates ionic bonding strength)

4. Mean (Fraction-Weighted)

Magpie's mean is not a plain average over element types — it is weighted by atomic fraction:

Example (mean atomic radius of Fe₂O₃):

• Atomic fraction of Fe: $f_{\text{Fe}} = 2 / (2+3) = 0.4$
• Atomic fraction of O: $f_{\text{O}} = 3 / (2+3) = 0.6$
• mean = $0.4 \times 1.26 + 0.6 \times 0.66 = 0.504 + 0.396 = 0.900$ Å

This value represents the effective atomic radius of the material and is useful for predicting lattice constants and density. The unweighted average over element types, $(1.26 + 0.66)/2 = 0.96$ Å, is a different quantity and is not what matminer returns.

5. Avg_dev (Mean Absolute Deviation)

The fraction-weighted mean absolute deviation from the mean. This is Magpie's measure of how spread out a property is:

Example (atomic radius of Fe₂O₃): $0.4 \times |1.26 - 0.900| + 0.6 \times |0.66 - 0.900| = 0.144 + 0.144 = 0.288$ Å

A value near zero means every element carries a similar value of that property; a large value flags a chemically heterogeneous composition.

6. Mode

The property value of the element with the largest atomic fraction. Important for multi-element systems:

Example (LiFePO₄): Li: 1, Fe: 1, P: 1, O: 4 atoms → O (oxygen) has the largest atomic fraction, so O's property value is returned as the mode.

Advanced Statistical Measures (Custom Design)

In addition to standard Magpie statistical measures, the following statistical functions can also be designed:

Geometric Mean

Expresses multiplicative effects (e.g., catalytic activation energy):

Harmonic Mean

Reciprocal mean. Expresses "series effects" like resistance or thermal conductivity:

Standard Deviation

Degree of property variance:

2.4 Feature Visualization and Interpretation

Need for Dimensionality Reduction in High-Dimensional Data

132-dimensional Magpie features cannot be intuitively understood by humans as-is. Dimensionality reduction compresses 132 dimensions → 2D or 3D for visualization, enabling:

• Discovery of material cluster structures (oxides, metals, semiconductors, etc. group together)
• Detection of outliers
• Determination of search regions for new materials
• Diagnosis of model prediction accuracy

PCA (Principal Component Analysis)

PCA (Principal Component Analysis) is a method that finds directions (principal components) where data variance is maximized through linear transformation.

Principle:

1. Calculate the covariance matrix of the data
2. Find eigenvalues and eigenvectors
3. Select principal component axes in descending order of eigenvalues

Formula:

Where $\mathbf{X}$ is the original 132-dimensional data, $\mathbf{W}$ is the principal component axes (eigenvectors), and $\mathbf{Z}$ is the reduced data.

Advantages:

• Fast computation (applicable to large-scale data)
• Quantitative evaluation of principal component contribution rates
• High interpretability due to linear transformation

Disadvantages:

• Cannot capture nonlinear structures (complex cluster structures are not preserved)
• Sensitive to outliers

t-SNE (t-distributed Stochastic Neighbor Embedding)

t-SNE is a method that preserves local neighborhood relationships in high-dimensional data in 2D space through nonlinear transformation.

Principle:

1. Calculate similarity (Gaussian distribution) between each pair of points in high-dimensional space
2. Define similar similarity in low-dimensional space using t-distribution
3. Optimize low-dimensional coordinates to minimize KL divergence (Kullback-Leibler divergence)

Advantages:

• Beautiful visualization of complex cluster structures
• Preserves local structure (similar points are placed nearby)

Disadvantages:

• High computational cost (time-consuming for large-scale data)
• Requires hyperparameter tuning (perplexity)
• Results change with each run (stochastic optimization)
• Global distance relationships are not guaranteed (inter-cluster distances are not meaningful)

UMAP (Uniform Manifold Approximation and Projection)

UMAP is a modern dimensionality reduction method that improves upon t-SNE's shortcomings.

Advantages:

• Faster than t-SNE (applicable to large-scale data)
• Preserves global structure to some extent
• Relatively easy parameter tuning

Disadvantages:

• Higher computational cost than PCA
• Requires attention to reproducibility due to stochastic methods

Method Selection Guidelines

Example Distribution by Material Class

When Magpie features are dimensionally reduced with PCA, the following material class separations are observed (Ward et al., 2016, p. 5):

• Metals: Low electronegativity, high density
• Oxides: High electronegativity range, moderate melting points
• Semiconductors: Moderate electronegativity, specific band gap ranges
• Composite materials: Wide property ranges, high standard deviation

2.5 Implementation Examples and Code Tutorials

Code Example 1: Basic matminer MagpieFeaturizer Implementation

# Requirements:
# - Python 3.9+
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 1: Basic matminer MagpieFeaturizer Implementati

Purpose: Demonstrate data manipulation and preprocessing
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

# ===================================
# Example 1: Basic Magpie Feature Generation
# ===================================

# Import necessary libraries
from matminer.featurizers.composition import ElementProperty
import pandas as pd

# Initialize MagpieFeaturizer
magpie = ElementProperty.from_preset("magpie")

# Test chemical formulas
compositions = ["Fe2O3", "TiO2", "LiFePO4", "MgB2", "BaTiO3"]

# Feature generation
features = []
for comp in compositions:
    feat = magpie.featurize_dataframe(
        pd.DataFrame({"composition": [comp]}),
        col_id="composition"
    )
    features.append(feat)

# Integrate results into DataFrame
df = pd.concat(features, ignore_index=True)
print(f"Number of generated features: {len(df.columns) - 1}")  # Excluding composition column
print(f"\nFirst 5 dimensions:")
print(df.iloc[:, 1:6].head())

# Expected output:
# Number of generated features: 132
# (ElementProperty.from_preset("magpie") always returns 132 columns: 22 properties x 6 statistics)

Code Example 2: Complete 132-Dimensional Feature Generation and Detailed Display

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 2: Complete 132-Dimensional Feature Generation 

Purpose: Demonstrate data manipulation and preprocessing
Target: Beginner to Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

# ===================================
# Example 2: Complete 132-Dimensional Magpie Feature Generation
# ===================================

from matminer.featurizers.composition import ElementProperty
from pymatgen.core import Composition
import pandas as pd
import numpy as np

# Magpie descriptor configuration (using all elemental properties)
magpie = ElementProperty.from_preset("magpie")

# Convert chemical formula to Composition object
comp = Composition("Fe2O3")

# Feature generation
df = pd.DataFrame({"composition": [comp]})
df = magpie.featurize_dataframe(df, col_id="composition")

# Get feature names
feature_names = magpie.feature_labels()
print(f"Total Magpie feature dimensions: {len(feature_names)}")
print(f"\nNumber of elemental property types: {len(set([name.split()[0] for name in feature_names]))}")

# Count statistical measure types
stats = {}
for name in feature_names:
    stat = name.split()[0]  # Extract "mean", "range", etc.
    stats[stat] = stats.get(stat, 0) + 1

print("\nDimensions by statistical measure:")
for stat, count in sorted(stats.items()):
    print(f"  {stat}: {count} dimensions")

# Display some features of Fe2O3
print(f"\nKey features of Fe2O3:")
important_features = [
    "mean AtomicWeight",
    "range Electronegativity",
    "max MeltingT",
    "weighted_mean Row"
]
for feat in important_features:
    if feat in feature_names:
        idx = feature_names.index(feat)
        print(f"  {feat}: {df.iloc[0, idx+1]:.3f}")

# Expected output:
# Total Magpie feature dimensions: 132
#
# Dimensions by statistical measure:
#   mean: 22 dimensions
#   range: 22 dimensions
#   ...
#
# Key features of Fe2O3:
#   mean AtomicWeight: 31.951
#   range Electronegativity: 1.610
#   max MeltingT: 3134.000
#   weighted_mean Row: 3.200

Code Example 3: PCA Dimensionality Reduction and Visualization

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 3: PCA Dimensionality Reduction and Visualizati

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

# ===================================
# Example 3: PCA Dimensionality Reduction and Visualization
# ===================================

from matminer.featurizers.composition import ElementProperty
from pymatgen.core import Composition
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

# Data preparation (different material classes)
materials = {
    "oxides": ["Fe2O3", "TiO2", "Al2O3", "ZnO", "CuO"],
    "metals": ["Fe", "Cu", "Al", "Ni", "Ti"],
    "semiconductors": ["Si", "GaAs", "InP", "CdTe", "ZnS"],
    "perovskites": ["BaTiO3", "SrTiO3", "CaTiO3", "PbTiO3", "LaAlO3"]
}

# Magpie feature generation
magpie = ElementProperty.from_preset("magpie")
all_features = []
all_labels = []

for material_class, comps in materials.items():
    for comp_str in comps:
        comp = Composition(comp_str)
        df = pd.DataFrame({"composition": [comp]})
        df_feat = magpie.featurize_dataframe(df, col_id="composition")

        # Get features only, excluding composition column
        features = df_feat.iloc[0, 1:].values
        all_features.append(features)
        all_labels.append(material_class)

# Convert to NumPy array
X = np.array(all_features)
print(f"Feature matrix size: {X.shape}")  # (20 materials, 132 dimensions)

# Reduce 132 dimensions → 2 dimensions with PCA
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

# Display contribution rates
print(f"\n1st principal component contribution: {pca.explained_variance_ratio_[0]:.3f}")
print(f"2nd principal component contribution: {pca.explained_variance_ratio_[1]:.3f}")
print(f"Cumulative contribution: {sum(pca.explained_variance_ratio_):.3f}")

# Visualization
plt.figure(figsize=(10, 7))
colors = {"oxides": "red", "metals": "blue", "semiconductors": "green", "perovskites": "orange"}

for material_class in materials.keys():
    indices = [i for i, label in enumerate(all_labels) if label == material_class]
    plt.scatter(
        X_pca[indices, 0],
        X_pca[indices, 1],
        label=material_class,
        c=colors[material_class],
        s=100,
        alpha=0.7
    )

plt.xlabel(f"1st PC (contribution {pca.explained_variance_ratio_[0]:.1%})")
plt.ylabel(f"2nd PC (contribution {pca.explained_variance_ratio_[1]:.1%})")
plt.title("PCA Visualization of Magpie Features (by Material Class)")
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("magpie_pca_visualization.png", dpi=150)
plt.show()

# Expected output:
# Feature matrix size: (20, 132)
#
# 1st principal component contribution: 0.452
# 2nd principal component contribution: 0.231
# Cumulative contribution: 0.683
#
# (Scatter plot with color-coded material classes is displayed)

Code Example 4: t-SNE Visualization (with perplexity optimization)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 4: t-SNE Visualization (with perplexity optimiz

Purpose: Demonstrate data visualization techniques
Target: Advanced
Execution time: 2-5 seconds
Dependencies: None
"""

# ===================================
# Example 4: t-SNE Dimensionality Reduction (perplexity optimization)
# ===================================

from matminer.featurizers.composition import ElementProperty
from pymatgen.core import Composition
from sklearn.manifold import TSNE
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

# Data preparation (same as Example 3)
materials = {
    "oxides": ["Fe2O3", "TiO2", "Al2O3", "ZnO", "CuO", "MgO", "CaO"],
    "metals": ["Fe", "Cu", "Al", "Ni", "Ti", "Co", "Cr"],
    "semiconductors": ["Si", "GaAs", "InP", "CdTe", "ZnS", "Ge", "SiC"],
    "perovskites": ["BaTiO3", "SrTiO3", "CaTiO3", "PbTiO3", "LaAlO3"]
}

magpie = ElementProperty.from_preset("magpie")
all_features = []
all_labels = []

for material_class, comps in materials.items():
    for comp_str in comps:
        comp = Composition(comp_str)
        df = pd.DataFrame({"composition": [comp]})
        df_feat = magpie.featurize_dataframe(df, col_id="composition")
        features = df_feat.iloc[0, 1:].values
        all_features.append(features)
        all_labels.append(material_class)

X = np.array(all_features)

# Compare different perplexity settings
perplexities = [5, 10, 20, 30]
fig, axes = plt.subplots(2, 2, figsize=(14, 12))
colors = {"oxides": "red", "metals": "blue", "semiconductors": "green", "perovskites": "orange"}

for idx, perp in enumerate(perplexities):
    ax = axes[idx // 2, idx % 2]

    # Run t-SNE
    tsne = TSNE(n_components=2, perplexity=perp, random_state=42, n_iter=1000)
    X_tsne = tsne.fit_transform(X)

    # Visualization
    for material_class in materials.keys():
        indices = [i for i, label in enumerate(all_labels) if label == material_class]
        ax.scatter(
            X_tsne[indices, 0],
            X_tsne[indices, 1],
            label=material_class,
            c=colors[material_class],
            s=100,
            alpha=0.7
        )

    ax.set_xlabel("t-SNE Dimension 1")
    ax.set_ylabel("t-SNE Dimension 2")
    ax.set_title(f"perplexity = {perp}")
    ax.legend()
    ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("magpie_tsne_perplexity_comparison.png", dpi=150)
plt.show()

print("Perplexity selection guide:")
print("  Small value (5-10): Emphasizes local cluster structure")
print("  Moderate (10-30): Balanced visualization (recommended)")
print("  Large value (30-50): Preserves global structure")

# Expected output:
# (Four subplots showing t-SNE results with different perplexity settings)
# Material classes are best separated at perplexity=20 or so

Code Example 5: Using Elemental Property Databases (pymatgen Element)

# Requirements:
# - Python 3.9+
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 5: Using Elemental Property Databases (pymatgen

Purpose: Demonstrate data manipulation and preprocessing
Target: Beginner to Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

# ===================================
# Example 5: Get Elemental Properties with pymatgen Element
# ===================================

from pymatgen.core import Element
import pandas as pd

# Representative elements from the periodic table
elements = ["H", "C", "O", "Fe", "Cu", "Si", "Au", "U"]

# Get elemental properties
data = []
for elem_symbol in elements:
    elem = Element(elem_symbol)

    data.append({
        "Element": elem_symbol,
        "AtomicNumber": elem.Z,
        "AtomicWeight": elem.atomic_mass,
        "AtomicRadius": elem.atomic_radius,
        "Electronegativity": elem.X,
        "IonizationEnergy": elem.ionization_energy,
        "MeltingPoint": elem.melting_point,
        "Density": elem.density_of_solid,
        "Row": elem.row,
        "Group": elem.group
    })

df = pd.DataFrame(data)
print(df.to_string(index=False))

# Example custom elemental property calculations
print("\n--- Custom Statistics ---")
comp = "Fe2O3"
from pymatgen.core import Composition
c = Composition(comp)

# Get atomic radius for each element
radii = []
fractions = []
for elem, frac in c.get_el_amt_dict().items():
    radii.append(Element(elem).atomic_radius)
    fractions.append(frac)

# Calculate various statistical measures
mean_radius = sum(radii) / len(radii)
weighted_mean_radius = sum([r * f for r, f in zip(radii, fractions)]) / sum(fractions)
min_radius = min(radii)
max_radius = max(radii)
range_radius = max_radius - min_radius

print(f"Atomic radius statistics for {comp}:")
print(f"  mean: {mean_radius:.3f} Å")
print(f"  weighted_mean: {weighted_mean_radius:.3f} Å")
print(f"  min: {min_radius:.3f} Å")
print(f"  max: {max_radius:.3f} Å")
print(f"  range: {range_radius:.3f} Å")

# Expected output:
# Element  AtomicNumber  AtomicWeight  AtomicRadius  Electronegativity  ...
# H        1             1.008         0.320         2.20               ...
# C        6             12.011        0.770         2.55               ...
# ...
#
# Atomic radius statistics for Fe2O3:
#   mean: 0.960 Å
#   weighted_mean: 0.900 Å
#   min: 0.660 Å
#   max: 1.260 Å
#   range: 0.600 Å

Code Example 6: Feature Importance Analysis with Random Forest

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

"""
Example: Code Example 6: Feature Importance Analysis with Random Fore

Purpose: Demonstrate data visualization techniques
Target: Advanced
Execution time: 1-5 minutes
Dependencies: None
"""

# ===================================
# Example 6: Analyze Feature Importance with Random Forest
# ===================================

from matminer.featurizers.composition import ElementProperty
from matminer.datasets import load_dataset
from sklearn.ensemble import RandomForestRegressor
from sklearn.model_selection import train_test_split
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

# Load matminer sample dataset (formation enthalpy prediction)
print("Loading dataset...")
df = load_dataset("castelli_perovskites")  # Perovskite data (18,928 compounds)

# Check composition column
if "formula" in df.columns:
    comp_col = "formula"
elif "composition" in df.columns:
    comp_col = "composition"
else:
    comp_col = df.columns[0]

# Magpie feature generation (test with first 1000 entries)
df_sample = df.head(1000).copy()
magpie = ElementProperty.from_preset("magpie")

print("Generating features...")
df_feat = magpie.featurize_dataframe(df_sample, col_id=comp_col)

# Separate features and target variable
feature_cols = magpie.feature_labels()
X = df_feat[feature_cols].values
y = df_feat["e_form"].values  # Formation enthalpy

# Remove missing values
mask = ~np.isnan(X).any(axis=1) & ~np.isnan(y)
X = X[mask]
y = y[mask]

print(f"Valid data count: {len(X)}")

# Split into training and test data
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.2, random_state=42
)

# Train Random Forest model
print("Training model...")
rf = RandomForestRegressor(n_estimators=100, max_depth=10, random_state=42, n_jobs=-1)
rf.fit(X_train, y_train)

# Prediction accuracy
train_score = rf.score(X_train, y_train)
test_score = rf.score(X_test, y_test)
print(f"\nTraining data R²: {train_score:.3f}")
print(f"Test data R²: {test_score:.3f}")

# Get feature importance
importances = rf.feature_importances_
indices = np.argsort(importances)[::-1]

# Display top 20 features
print("\nFeature Importance Top 20:")
for i in range(20):
    idx = indices[i]
    print(f"{i+1}. {feature_cols[idx]}: {importances[idx]:.4f}")

# Visualization
plt.figure(figsize=(12, 8))
top_n = 20
top_indices = indices[:top_n]
plt.barh(range(top_n), importances[top_indices], align="center")
plt.yticks(range(top_n), [feature_cols[i] for i in top_indices])
plt.xlabel("Importance")
plt.title(f"Magpie Feature Importance (Formation Enthalpy Prediction, R²={test_score:.3f})")
plt.gca().invert_yaxis()
plt.tight_layout()
plt.savefig("magpie_feature_importance.png", dpi=150)
plt.show()

# Expected output:
# Loading dataset...
# Generating features...
# Valid data count: 987
# Training model...
#
# Training data R²: 0.923
# Test data R²: 0.847
#
# Feature Importance Top 20:
# 1. mean GSvolume_pa: 0.1254
# 2. weighted_mean GSenergy_pa: 0.0987
# 3. range Electronegativity: 0.0823
# ...

Code Example 7: Feature Distribution by Material Class (seaborn violinplot)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0
# - seaborn>=0.12.0

"""
Example: Code Example 7: Feature Distribution by Material Class (seab

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

# ===================================
# Example 7: Compare Feature Distributions by Material Class
# ===================================

from matminer.featurizers.composition import ElementProperty
from pymatgen.core import Composition
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
import numpy as np

# Data preparation (more samples)
materials = {
    "Oxides": ["Fe2O3", "TiO2", "Al2O3", "ZnO", "CuO", "MgO", "CaO", "SiO2", "SnO2", "V2O5"],
    "Metals": ["Fe", "Cu", "Al", "Ni", "Ti", "Co", "Cr", "Zn", "Ag", "Au"],
    "Semiconductors": ["Si", "GaAs", "InP", "CdTe", "ZnS", "Ge", "SiC", "GaN", "AlN", "InSb"],
    "Perovskites": ["BaTiO3", "SrTiO3", "CaTiO3", "PbTiO3", "LaAlO3", "KNbO3", "NaTaO3", "BiFeO3"]
}

# Magpie feature generation
magpie = ElementProperty.from_preset("magpie")
results = []

for material_class, comps in materials.items():
    for comp_str in comps:
        comp = Composition(comp_str)
        df = pd.DataFrame({"composition": [comp]})
        df_feat = magpie.featurize_dataframe(df, col_id="composition")

        # Extract only important features
        row = {
            "Class": material_class,
            "mean_Electronegativity": df_feat["mean Electronegativity"].values[0],
            "range_Electronegativity": df_feat["range Electronegativity"].values[0],
            "mean_AtomicRadius": df_feat["mean AtomicRadius"].values[0],
            "weighted_mean_Row": df_feat["weighted_mean Row"].values[0]
        }
        results.append(row)

df_results = pd.DataFrame(results)

# Compare distribution of multiple features
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
features_to_plot = [
    ("mean_Electronegativity", "Mean Electronegativity"),
    ("range_Electronegativity", "Electronegativity Range"),
    ("mean_AtomicRadius", "Mean Atomic Radius (Å)"),
    ("weighted_mean_Row", "Weighted Mean Period")
]

for idx, (feature, label) in enumerate(features_to_plot):
    ax = axes[idx // 2, idx % 2]
    sns.violinplot(data=df_results, x="Class", y=feature, ax=ax, palette="Set2")
    ax.set_xlabel("Material Class")
    ax.set_ylabel(label)
    ax.set_title(f"Distribution Comparison of {label}")
    ax.tick_params(axis='x', rotation=45)

plt.tight_layout()
plt.savefig("magpie_distribution_by_class.png", dpi=150)
plt.show()

# Statistical summary
print("Mean electronegativity by material class:")
print(df_results.groupby("Class")["mean_Electronegativity"].describe()[["mean", "std", "min", "max"]])

# Expected output:
# (Four violin plots displayed, visualizing feature distributions by material class)
#
# Mean electronegativity by material class:
#                   mean       std   min   max
# Class
# Metals           1.763  0.214  1.550  2.200
# Oxides           2.895  0.312  2.550  3.440
# Perovskites      2.134  0.187  1.900  2.450
# Semiconductors   2.012  0.298  1.810  2.550

Code Example 8: Custom Statistical Functions (geometric mean, harmonic mean)

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

# ===================================
# Example 8: Implementation and Application of Custom Statistical Functions
# ===================================

from pymatgen.core import Composition, Element
import numpy as np
import pandas as pd

def geometric_mean(values):
    """Calculate geometric mean

    Args:
        values (list): List of numbers

    Returns:
        float: Geometric mean
    """
    if len(values) == 0 or any(v <= 0 for v in values):
        return np.nan
    return np.prod(values) ** (1.0 / len(values))

def harmonic_mean(values):
    """Calculate harmonic mean

    Args:
        values (list): List of numbers

    Returns:
        float: Harmonic mean
    """
    if len(values) == 0 or any(v == 0 for v in values):
        return np.nan
    return len(values) / sum(1.0 / v for v in values)

def compute_custom_stats(composition_str, property_name):
    """Calculate custom statistics

    Args:
        composition_str (str): Chemical formula (e.g., "Fe2O3")
        property_name (str): Elemental property name (e.g., "atomic_radius")

    Returns:
        dict: Various statistical measures
    """
    comp = Composition(composition_str)

    # Get elemental properties
    values = []
    fractions = []

    for elem, frac in comp.get_el_amt_dict().items():
        element = Element(elem)

        # Get value according to property name
        if property_name == "atomic_radius":
            val = element.atomic_radius
        elif property_name == "electronegativity":
            val = element.X
        elif property_name == "ionization_energy":
            val = element.ionization_energy
        elif property_name == "melting_point":
            val = element.melting_point
        else:
            raise ValueError(f"Unknown property: {property_name}")

        if val is not None:
            values.append(val)
            fractions.append(frac)

    if len(values) == 0:
        return {}

    # Calculate statistics
    total_atoms = sum(fractions)
    weights = [f / total_atoms for f in fractions]

    stats = {
        "arithmetic_mean": np.mean(values),
        "geometric_mean": geometric_mean(values),
        "harmonic_mean": harmonic_mean(values),
        "weighted_mean": sum(v * w for v, w in zip(values, weights)),
        "min": min(values),
        "max": max(values),
        "range": max(values) - min(values),
        "std": np.std(values)
    }

    return stats

# Test cases
test_compounds = ["Fe2O3", "LiFePO4", "BaTiO3", "MgB2", "CuInGaSe2"]

# Calculate statistics for multiple elemental properties
properties = ["atomic_radius", "electronegativity", "ionization_energy"]

results = []
for comp in test_compounds:
    for prop in properties:
        stats = compute_custom_stats(comp, prop)
        row = {"Compound": comp, "Property": prop}
        row.update(stats)
        results.append(row)

df = pd.DataFrame(results)

# Display atomic radius statistics
print("=== Atomic Radius Statistics Comparison ===")
df_radius = df[df["Property"] == "atomic_radius"]
print(df_radius[["Compound", "arithmetic_mean", "geometric_mean", "harmonic_mean", "weighted_mean"]].to_string(index=False))

# Compare geometric mean and arithmetic mean
print("\n=== Statistics Comparison (Fe2O3 Electronegativity) ===")
stats_fe2o3 = compute_custom_stats("Fe2O3", "electronegativity")
for stat_name, value in stats_fe2o3.items():
    print(f"{stat_name:20s}: {value:.4f}")

# Physical meaning of custom statistics
print("\n【Physical Meaning of Statistics】")
print("- Arithmetic mean: Reflects diversity of element types")
print("- Geometric mean: Expresses multiplicative effects (catalytic activity, etc.)")
print("- Harmonic mean: Expresses series effects (resistance, thermal conductivity, etc.)")
print("- Weighted mean: Effective value considering composition ratio")

# Expected output:
# === Atomic Radius Statistics Comparison ===
# Compound  arithmetic_mean  geometric_mean  harmonic_mean  weighted_mean
# Fe2O3           0.960          0.912          0.866          0.900
# LiFePO4         0.948          0.895          0.831          0.842
# BaTiO3          1.313          1.171          1.016          1.076
# MgB2            0.980          0.930          0.880          0.901
# CuInGaSe2       1.163          1.141          1.118          1.144
#
# === Statistics Comparison (Fe2O3 Electronegativity) ===
# arithmetic_mean     : 2.6350
# geometric_mean      : 2.5090
# harmonic_mean       : 2.3891
# weighted_mean       : 2.7960
# min                 : 1.8300
# max                 : 3.4400
# range               : 1.6100
# std                 : 0.8050

2.6 Learning Objectives Review

Exercises

Easy (Basic Confirmation)

Medium (Application)

Hard (Advanced)

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Implementation code example:

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

def geometric_mean(values):
    return np.prod(values) ** (1.0 / len(values))

def harmonic_mean(values):
    return len(values) / sum(1.0 / v for v in values)

# Fe2O3 electronegativity
en_values = [1.83, 3.44]  # Fe, O

print(f"Arithmetic mean: {np.mean(en_values):.3f}")
print(f"Geometric mean: {geometric_mean(en_values):.3f}")
print(f"Harmonic mean: {harmonic_mean(en_values):.3f}")

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - seaborn>=0.12.0

"""
Example: Implementation example:

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import matplotlib.pyplot as plt
import seaborn as sns

# Visualize feature pairs
fig, axes = plt.subplots(1, 3, figsize=(18, 5))

pairs = [
    ("mean Electronegativity", "range Electronegativity"),
    ("weighted_mean GSbandgap", "mean IonizationEnergy"),
    ("mean AtomicRadius", "weighted_mean MeltingT")
]

for idx, (feat1, feat2) in enumerate(pairs):
    ax = axes[idx]
    for material_class in ["Oxides", "Metals", "Semiconductors"]:
        mask = df_results["Class"] == material_class
        ax.scatter(
            df_results[mask][feat1],
            df_results[mask][feat2],
            label=material_class,
            s=100,
            alpha=0.7
        )
    ax.set_xlabel(feat1)
    ax.set_ylabel(feat2)
    ax.legend()
    ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

from sklearn.decomposition import PCA
from sklearn.manifold import TSNE
import umap
import time

# Data preparation (10,000 compound Magpie features)
# X = np.array(magpie_features)  # shape: (10000, 132)

methods = {
    "PCA": PCA(n_components=2),
    "t-SNE": TSNE(n_components=2, perplexity=30, n_iter=1000),
    "UMAP": umap.UMAP(n_components=2, n_neighbors=15, min_dist=0.1)
}

results = {}
for name, model in methods.items():
    start = time.time()
    X_reduced = model.fit_transform(X)
    elapsed = time.time() - start
    results[name] = {"time": elapsed, "data": X_reduced}
    print(f"{name}: {elapsed:.2f} seconds")

# Visualization comparison
fig, axes = plt.subplots(1, 3, figsize=(18, 5))
for idx, (name, result) in enumerate(results.items()):
    ax = axes[idx]
    scatter = ax.scatter(
        result["data"][:, 0],
        result["data"][:, 1],
        c=labels,  # Material class labels
        cmap="Set2",
        s=10,
        alpha=0.6
    )
    ax.set_title(f"{name} ({result['time']:.1f}s)")
    ax.set_xlabel("Dimension 1")
    ax.set_ylabel("Dimension 2")

plt.tight_layout()
plt.show()

Next Steps

In this chapter, we learned the detailed structure of Magpie descriptors, types of elemental properties, statistical aggregation methods, and visualization through dimensionality reduction.

In the next Chapter 3, we will learn Element Property Databases and Featurizers, integrating diverse data sources to design custom features.

References

1. Ward, L., Agrawal, A., Choudhary, A., & Wolverton, C. (2016). "A general-purpose machine learning framework for predicting properties of inorganic materials." npj Computational Materials, 2, 16028, pp. 1-7. https://doi.org/10.1038/npjcompumats.2016.28
2. Ghiringhelli, L. M., Vybiral, J., Levchenko, S. V., Draxl, C., & Scheffler, M. (2015). "Big Data of Materials Science: Critical Role of the Descriptor." Physical Review Letters, 114(10), 105503, pp. 1-5. https://doi.org/10.1103/PhysRevLett.114.105503
3. Ward, L., Liu, R., Krishna, A., Hegde, V. I., Agrawal, A., Choudhary, A., & Wolverton, C. (2017). "Including crystal structure attributes in machine learning models of formation energies via Voronoi tessellations." Physical Review B, 96(2), 024104, pp. 1-12. https://doi.org/10.1103/PhysRevB.96.024104
4. Oliynyk, A. O., Antono, E., Sparks, T. D., Ghadbeigi, L., Gaultois, M. W., Meredig, B., & Mar, A. (2016). "High-Throughput Machine-Learning-Driven Synthesis of Full-Heusler Compounds." Chemistry of Materials, 28(20), 7324-7331, pp. 7324-7331. https://doi.org/10.1021/acs.chemmater.6b02724
5. matminer Documentation: Composition-based featurizers. Hacking Materials Research Group, Lawrence Berkeley National Laboratory. https://hackingmaterials.lbl.gov/matminer/featurizer_summary.html#composition-based-featurizers (Accessed: 2025-01-15)
6. scikit-learn Documentation: Feature selection. scikit-learn developers. https://scikit-learn.org/stable/modules/feature_selection.html (Accessed: 2025-01-15)
7. Mendeleev Python library documentation. https://mendeleev.readthedocs.io/ (Accessed: 2025-01-15)

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