pyaerial 0.2.4

An implementation of the Aerial neurosymbolic association rule mining algorithm from tabular datasets.

pyaerial: scalable association rule mining

---

This is a Python implementation of the Aerial scalable neurosymbolic association rule miner for tabular data.

Table of Contents

• Introduction
• Installation
• Usage Examples
  • Association rule mining from categorical tabular data
  • Specifying item constraints
  • Setting Aerial parameters
  • Fine-tuning Autoencoder architecture and dimensions
  • Running Aerial for numerical values
  • Frequent itemset mining with Aerial
  • Using Aerial for rule-based classification for interpretable inference
  • Fine-tuning the training parameters
  • Setting the log levels
• How Aerial works?
• Functions Overview
• Citation
• Contact
• Contributing

---

Introduction

Aerial is a scalable Neurosymbolic association rule mining (ARM) method for tabular data. It aims to address the rule explosion and execution time problems in ARM and it is fully compatible with the existing solutions. Aerial first creates a neural representation of a given tabular data using an Autoencoder, and then extracts association rules from the neural representation.

See How Aerial works? section or our paper for the details of Autoencoder architecture, training and rule extraction algorithm Neurosymbolic Association Rule Mining from Tabular Data. If you use Aerial in your work, please cite our paper.

---

Installation

You can easily install pyaerial using pip:

pip install pyaerial

Usage Examples

This section exemplifies the usage of Aerial with and without hyperparameter tuning.

1. Association rule mining from categorical tabular data

from aerial import model, rule_extraction, rule_quality
from ucimlrepo import fetch_ucirepo

# load a categorical tabular dataset from the UCI ML repository
breast_cancer = fetch_ucirepo(id=14).data.features

# train an autoencoder on the loaded table
trained_autoencoder = model.train(breast_cancer)

# extract association rules from the autoencoder
association_rules = rule_extraction.generate_rules(trained_autoencoder)

# calculate rule quality statistics (support, confidence, zhangs metric) for each rule
if len(association_rules) > 0:
    stats, association_rules = rule_quality.calculate_rule_stats(association_rules, trained_autoencoder.input_vectors)
    print(stats, association_rules[:1])

Following is the partial output of above code:

>>> Output:
breast_cancer dataset:
     age menopause tumor-size inv-nodes  ... deg-malig  breast breast-quad irradiat
0  30-39   premeno      30-34       0-2  ...         3    left    left_low       no
1  40-49   premeno      20-24       0-2  ...         2   right    right_up       no
2  40-49   premeno      20-24       0-2  ...         2    left    left_low       no
                                         ...

Overall rule quality statistics: {
   "rule_count":15,
   "average_support":  0.448,
   "average_confidence": 0.881,
   "average_coverage": 0.860,
   "average_zhangs_metric": 0.318
}

Sample rule:
{
   "antecedents":[
      "inv-nodes__0-2" # meaning column "inv-nodes" has the value between "0-2"
   ],
   "consequent":"node-caps__no", # meaing column "node-caps" has the value "no"
   "support": 0.702,
   "confidence": 0.943,
   "zhangs_metric": 0.69
}

2. Specifying item constraints

Instead of performing rule extraction on all features, Aerial allows you to extract rules only for features of interest. This is called ARM with item constraints.

In ARM with item constraints, the antecedent side of the rules will contain the items of interest. However, the consequent side of the rules may still contain other feature values.

features_of_interest parameter of generate_rules() can be used to do that (also valid for generate_frequent_itemsets(), see below).

from aerial import model, rule_extraction
from ucimlrepo import fetch_ucirepo

# categorical tabular dataset
breast_cancer = fetch_ucirepo(id=14).data.features

trained_autoencoder = model.train(breast_cancer)

# features of interest, either a feature with its all values (e.g., "age") or with its certain values (e.g., premeno value of menopause feature is the only feature value of interest)
features_of_interest = ["age", {"menopause": 'premeno'}, 'tumor-size', 'inv-nodes', {"node-caps": "yes"}]

association_rules = rule_extraction.generate_rules(trained_autoencoder, features_of_interest, cons_similarity=0.5)

The output rules will only contain features of interest on the antecedent side:

>>> Output:
association_rules: [
   {
      "antecedents":[
         "menopause__premeno"
      ],
      "consequent":"node-caps__no",
      ...
   },
   {
      "antecedents":[
         "menopause__premeno"
      ],
      "consequent":"breast__right",
      ...
   },
   ...
]

3. Setting Aerial parameters

Aerial has 3 key parameters; antecedent and consequent similarity threshold, and antecedent length.

As shown in the paper, higher antecedent thresholds results in lower number of higher support rules, while higher consequent thresholds results in lower number of higher confidence rules.

These 3 parameters can be set using the generate_rules function:

import pandas as pd
from aerial import model, rule_extraction, rule_quality
from ucimlrepo import fetch_ucirepo

breast_cancer = fetch_ucirepo(id=14).data.features

trained_autoencoder = model.train(table_with_labels)

# hyperparameters of aerial can be set using the generate_rules function
association_rules = rule_extraction.generate_rules(trained_autoencoder, ant_similarity=0.5, cons_similarity=0.8, max_antecedents=2)
...

4. Fine-tuning Autoencoder architecture and dimensions

Aerial uses an under-complete Autoencoder and in default, it decides automatically how many layers to use and the dimensions of each layer (see Functions Overview, Autoencoder).

Alternatively, you can specify the number of layers and dimensions in the train method to improve performance.

from aerial import model, rule_extraction, rule_quality

...
# layer_dims=[2] specifies that there is gonna be 1 hidden layer with a dimension of 2
trained_autoencoder = model.train(breast_cancer, layer_dims=[2]) 
...

5. Running Aerial for numerical values

Discretizing numerical values is required before running Aerial. We provide 2 discretization methods as part of the discretization.py script; equal-frequency and equal-width discretization. However, Aerial can work with any other discretization method of your choice as well.

from aerial import model, rule_extraction, rule_quality, discretization
from ucimlrepo import fetch_ucirepo

# load a numerical tabular data
iris = fetch_ucirepo(id=53).data.features

# find and discretize numerical columns 
iris_discretized = discretization.equal_frequency_discretization(iris, n_bins=10)

trained_autoencoder = model.train(iris_discretized, layer_dims=[19], epochs=5)

association_rules = rule_extraction.generate_rules(trained_autoencoder, ant_similarity=0.1, cons_similarity=0.5)

Following is the partial iris dataset content before and after the discretization:

>>> Output:
# before discretization
   sepal length  sepal width  petal length  petal width
0           5.1          3.5           1.4          0.2
1           4.9          3.0           1.4          0.2
...

# after discretization
  sepal length  sepal width  petal length   petal width
0  (5.0, 5.27]  (3.4, 3.61]  (0.999, 1.4]  (0.099, 0.2]
1   (4.8, 5.0]   (2.8, 3.0]  (0.999, 1.4]  (0.099, 0.2]
...

6. Frequent itemset mining with Aerial

Aerial can also be used for frequent itemset mining besides association rules.

from aerial import model, rule_extraction, rule_quality
from ucimlrepo import fetch_ucirepo

# categorical tabular dataset
breast_cancer = fetch_ucirepo(id=14).data.features
trained_autoencoder = model.train(breast_cancer, epochs=5, lr=1e-3)

# extract frequent itemsets
frequent_itemsets = rule_extraction.generate_frequent_itemsets(trained_autoencoder)

# calculate support values of the frequent itemsets
support_values, average_support = rule_quality.calculate_freq_item_support(frequent_itemsets, breast_cancer)

Note that we pass the original dataset (breast_cancer) to the calculate_freq_item_support() in this case. The following is a sample output:

>>> Output:

Frequent itemsets: 
{('menopause__premeno',): 0.524, ('menopause__ge40',): 0.451, ... }

Average support: 0.295

7. Using Aerial for rule-based classification for interpretable inference

Aerial can be used to learn rules with a class label on the consequent side, which can later be used for inference either by themselves or as part of rule list or rule set classifiers (e.g., from imodels repository).

This is done by setting target_class parameter of the generate_rules function. This parameter refers to the class label column of the tabular data.

import pandas as pd
from aerial import model, rule_extraction, rule_quality
from ucimlrepo import fetch_ucirepo

# categorical tabular dataset
breast_cancer = fetch_ucirepo(id=14)
labels = breast_cancer.data.targets
breast_cancer = breast_cancer.data.features

# merge labels column with the actual table 
table_with_labels = pd.concat([breast_cancer, labels], axis=1)

trained_autoencoder = model.train(table_with_labels)

# generate rules with a target class, this learns rules that has the "target_class" column (in this case this column is called "Class") on the consequent side
association_rules = rule_extraction.generate_rules(trained_autoencoder, target_class="Class", cons_similarity=0.5)

if len(association_rules) > 0:
    stats, association_rules = rule_quality.calculate_rule_stats(association_rules, trained_autoencoder.input_vectors)

Sample output showing rules with class labels on the right hand side:

>>> Output:

{
   "antecedents":[
      "menopause__premeno"
   ],
   "consequent":"Class__no-recurrence-events", # consequent has the class label (column) named "Class" with the value "no-recurrence-events"
   "support":np.float64(0.35664335664335667),
   "confidence":np.float64(0.68),
   "zhangs_metric":np.float64(-0.06585858585858577)
}

8. Fine-tuning the training parameters

The train() function allows programmers to specify various training parameters:

• autoencoder: You can implement your own Autoencoder and use it for ARM as part of Aerial, as long as the last layer matches the original version (see our paper or the source code, model.py)
• feature_value_indices: if the data is already one-hot encoded using _one_hot_encoding_with_feature_tracking() function which returns feature_value_indices that tracks features' indices in a vector, passing feature_value_indices to this function won't run the one-hot encoding again
• noise_factor default=0.5: amount of random noise (+-) added to each neuron of the denoising Autoencoder before the training process
• lr default=5e-3: learning rate
• epochs default=1: number of training epochs
• batch_size default=2: number of batches to train
• loss_function default=torch.nn.BCELoss(): loss function
• num_workers default=1: number of workers for parallel execution

from aerial import model, rule_extraction, rule_quality, discretization
from ucimlrepo import fetch_ucirepo

# a categorical tabular dataset
breast_cancer = fetch_ucirepo(id=14).data.features

# increasing epochs to 5, note that longer training may lead to overfitting which results in rules with low association strength (zhangs' metric)
trained_autoencoder = model.train(breast_cancer, epochs=5, lr=1e-3)

association_rules = rule_extraction.generate_rules(trained_autoencoder)
if len(association_rules) > 0:
    stats, association_rules = rule_quality.calculate_rule_stats(association_rules, trained_autoencoder.input_vectors)

9. Setting the log levels

Aerial source code prints extra debug statements notifying the beginning and ending of major functions such as the training process or rule extraction. The log levels can be changed as follows:

import logging
import aerial

# setting the log levels to DEBUG level
aerial.setup_logging(logging.DEBUG)
...

How Aerial works?

The figure below shows the pipeline of operations for Aerial in 3 main stages.

1. Data preparation.

   1. Tabular data is first one-hot encoded. This is done using data_preparation.py:_one_hot_encoding_with_feature_tracking().
   2. One-hot encoded value are then converted to vector format in the model.py:train().
   3. If the tabular data contains numerical columns, they are pre-discretized as exemplified in Running Aerial for numerical values.

2. Training stage.

   1. An under-complete Autoencoder with either default automatically-picked number of layers and dimension (based on the dataset size and dimension) is constructed, or user-specified layers and dimension. ( see Autoencoder)
   2. All the training parameters can be customized including number of epochs, batch size, learning rate etc. ( see train())
   3. An Autoencoder is then trained with a denoising mechanism to learn associations between input features. The full Autoencoder architecture is given in our paper.

3. Rule extraction stage.

   1. Association rules are then extracted from the trained Autoencoder using Aerial's rule extraction algorithm ( see rule_extraction.py:generate_rules()). Below figure shows an example rule extraction process.

   2. Example. Assume $weather$ and $beverage$ are features with categories {cold, warm} and {tea, coffee, soda} respectively. The first step is to initialize a test vector of size 5 corresponding to 5 possible categories with equal probabilities per feature, [0.5, 0.5, 0.33, 0.33, 0.33]. Then we mark $weather(warm)$ by assigning 1 to warm and 0 to cold, [1, 0, 0.33, 0.33, 0.33], and call the resulting vector a test vector.

      Assume that after a forward run, [0.7, 0.3, 0.04, 0.1, 0.86] is received as the output probabilities. Since the probability of $p_{weather(warm)} = 0.7$ is bigger than the given antecedent similarity threshold ($\tau_a = 0.5$), and $p_{beverage(soda)} = 0.86$ probability is higher than the consequent similarity threshold ($\tau_c = 0.8$), we conclude with $weather(warm) \rightarrow beverage(soda)$.

   

   3. Frequent itemsets (instead of rules) can also be extracted (rule_extraction.py:generate_frequent_itemsets()).
   4. Finally various quality criteria is then calculated for each rule as well as an overall average values, .e.g, support, confidence, coverage, association strength (zhangs' metric). This is done in rule_quality.py. See calculate_rule_stats() and calculate_basic_rule_stats()

Functions overview

This section lists the important classes and functions as part of the Aerial package.

AutoEncoder(input_dimension, feature_count, layer_dims=None)

Part of the model.py script. Constructs an autoencoder designed for association rule mining on tabular data, based on the Neurosymbolic Association Rule Mining method.

Parameters:

• input_dimension (int): Number of input features after one-hot encoding.

• feature_count (int): Original number of categorical features in the dataset.

• layer_dims (list of int, optional): User-specified hidden layer dimensions. If not provided, the model calculates a default architecture using a logarithmic reduction strategy (base 16).

Behavior:

• Automatically builds an under-complete autoencoder with a bottleneck at the original feature count.

• If no layer_dims are provided, the architecture is determined by reducing the input dimension using a geometric progression and creates log₁₆(input_dimension) layers in total.

• Uses Xavier initialization for weights and sets all biases to zero.

• Applies Tanh activation functions between layers, except the final encoder and decoder layers.

train function

train(
    transactions,
    autoencoder=None,
    noise_factor=0.5,
    lr=5e-3,
    epochs=1,
    batch_size=2,
    loss_function=torch.nn.BCELoss(),
    num_workers=1,
    layer_dims=None
)

Part of the model.py script. Trains the AutoEncoder model using one-hot encoded tabular transaction data.

Parameters:

• transactions (pd.DataFrame): Tabular input data for training.

• autoencoder (AutoEncoder, optional): A preconstructed autoencoder instance. If not provided, one is created automatically.

• noise_factor (float): Controls the amount of Gaussian noise added to inputs during training (denoising effect).

• lr (float): Learning rate for the Adam optimizer.

• epochs (int): Number of training epochs.

• batch_size (int): Number of samples per training batch.

• loss_function (torch.nn.Module): Loss function to apply (default is BCELoss).

• num_workers (int): Number of subprocesses used for data loading.

• layer_dims (list of int, optional): Custom hidden layer dimensions for autoencoder construction (if applicable).

Returns: A trained instance of the AutoEncoder.

generate_rules

generate_rules(
    autoencoder,
    features_of_interest=None,
    ant_similarity=0.5,
    cons_similarity=0.8,
    max_antecedents=2,
    target_class=None
)

Part of the rule_extraction.py script. Extracts association rules from a trained AutoEncoder using the Aerial algorithm.

Parameters:

• autoencoder (AutoEncoder): A trained autoencoder instance.

• features_of_interest (list): only look for rules that have these features of interest on the antecedent side accepted form ["feature1", "feature2", {"feature3": "value1}, ...], either a feature name as str, or specific value of a feature in object form

• ant_similarity (float): Minimum similarity threshold for an antecedent to be considered frequent.

• cons_similarity (float): Minimum probability threshold for a feature to qualify as a rule consequent.

• max_antecedents (int): Maximum number of features allowed in the rule antecedent.

• target_class (str, optional): When set, restricts rule consequents to the specified class (constraint-based rule mining).

Returns:

A list of extracted rules in the form:

[
    {"antecedents": [...], "consequent": ...},
    ...
]

generate_frequent_itemsets

generate_frequent_itemsets(
    autoencoder,
    features_of_interest=None,
    similarity=0.5,
    max_length=2
)

Part of the rule_extraction.py script. Generates frequent itemsets from a trained AutoEncoder using the same Aerial+ mechanism.

Parameters:

• autoencoder (AutoEncoder): A trained autoencoder instance.

• features_of_interest (list): only look for rules that have these features of interest on the antecedent side accepted form ["feature1", "feature2", {"feature3": "value1}, ...], either a feature name as str, or specific value of a feature in object form

• similarity (float): Minimum similarity threshold for an itemset to be considered frequent.

• max_length (int): Maximum number of items in each itemset.

Returns:

A list of frequent itemsets, where each itemset is a list of string features:

[
    [...],  # e.g., ['gender=Male', 'income=High']
    ...
]

equal_frequency_discretization

equal_frequency_discretization(df: pd.DataFrame, n_bins=10)

Discretizes all numerical columns into equal-frequency bins and encodes the resulting intervals as string labels.

Parameters:

• df: A pandas DataFrame containing tabular data.

• n_bins: Number of intervals (bins) to create.

Returns: A modified DataFrame with numerical columns replaced by string-encoded interval bins.

equal_width_discretization

equal_width_discretization(df: pd.DataFrame, n_bins=10)

Discretizes all numerical columns into equal-width bins and encodes the resulting intervals as string labels.

Parameters:

• df: A pandas DataFrame containing tabular data.

• n_bins: Number of intervals (bins) to create.

Returns: A modified DataFrame with numerical columns replaced by string-encoded interval bins.

calculate_basic_rule_stats

calculate_basic_rule_stats(rules, transactions)

Computes support and confidence for a list of rules using parallel processing.

Parameters:

• rules: List of rule dictionaries with 'antecedents' and 'consequent'.

• transactions: A pandas DataFrame of one-hot encoded transactions.

Returns: A list of rules enriched with support and confidence values.

calculate_freq_item_support

calculate_freq_item_support(freq_items, transactions)

Calculates the support for a list of frequent itemsets.

Parameters:

• freq_items: List of itemsets (list of strings in "feature__value" format).

• transactions: A pandas DataFrame of categorical data.

Returns: A dictionary of itemset supports and their average support.

calculate_rule_stats

calculate_rule_stats(rules, transactions, max_workers=1)

Evaluates rules with extended metrics including: Support, Confidence, Zhang’s Metric, Dataset Coverage.

Runs in parallel with joblib.

Parameters:

• rules: List of rule dictionaries.

• transactions: One-hot encoded pandas DataFrame.

• max_workers: Number of parallel threads (via joblib).

Returns:

• A dictionary of average metrics (support, confidence, zhangs_metric, coverage)

• A list of updated rules

Citation

If you use pyaerial in your work, please cite the following paper:

@misc{karabulut2025neurosymbolic,
    title={Neurosymbolic Association Rule Mining from Tabular Data},
    author={Erkan Karabulut and Paul Groth and Victoria Degeler},
    year={2025},
    eprint={2504.19354},
    archivePrefix={arXiv},
    primaryClass={cs.AI}
}

Contact

For questions, suggestions, or collaborations, please contact:

Erkan Karabulut
📧 [e-mail]
📧 [e-mail]

Contributing

Contributions, feedback, and issue reports are very welcome!

Feel free to open a pull request or create an issue if you have ideas for improvements.