Sequential Covering Algorithms and Learning sets of First-Order rules:
Set of Rules:
Learning a set of rules is a type of
machine learning that focuses on discovering patterns or rules that explain the
data.
- Introduction
- Sequential covering algorithms
- Learning First-Order rules
- Learning sets of First-Order rules: FOIL
- Summary
Introduction
Learning sets of rules is a type of machine learning that involves learning a set of rules from data that can be used to make predictions or classifications. One popular approach to learning sets of rules is through the use of sequential covering algorithms.
Sequential covering algorithms work by iteratively adding rules to a rule set, each of which is designed to cover a subset of the training data. The algorithm selects the rule with the highest coverage and adds it to the rule set. The training data covered by the rule is removed, and the process is repeated with the remaining data until all data are covered or no more rules can be added.
One popular example of a sequential covering algorithm is the RIPPER algorithm. The algorithm uses a separate-and-conquer strategy, where it first learns a rule to cover the most specific subset of the data, and then generalizes the rule to cover more data. The generalization process involves adding conditions to the rule that do not cover any negative examples.
Example of implementing the RIPPER algorithm in Python:
from sklearn.tree import DecisionTreeClassifier
from sklearn.base import clone
from sklearn.metrics import accuracy_score
def ripper(X, y, base_classifier=DecisionTreeClassifier()):
rule_set = []
while True:
classifier = clone(base_classifier)
classifier.fit(X, y)
rule = get_rule(classifier)
if the rule is None:
break
rule_set.append(rule)
mask = apply_rule(X, rule)
X = X[~mask]
y = y[~mask]
return rule_set
def get_rule(classifier):
if len(classifier.tree_.feature) == 0:
return None
feature_names = classifier.classes_
feature_idx = classifier.tree_.feature[0]
threshold = classifier.tree_.threshold[0]
if threshold == -2:
return None
if classifier.tree_.value[0, 1] > classifier.tree_.value[0, 0]:
polarity = ">"
else:
polarity = "<="
rule = (feature_names[feature_idx], polarity, threshold)
return rule
def apply_rule(X, rule):
feature, polarity, threshold = rule
if polarity == ">":
mask = X[feature] > threshold
else:
mask = X[feature] <= threshold
return mask
# Example usage
from sklearn.datasets import load_iris
X, y = load_iris(return_X_y=True)
rule_set = ripper(X, y)
print(rule_set)
This code uses the RIPPER algorithm to learn a set of rules from the Iris dataset. The algorithm uses a decision tree as the base classifier and iteratively learns rules to cover subsets of the data.
Once the rule set is learned, it can be used to make predictions on new data by applying the rules in order until a match is found. If no rule matches, a default prediction can be made based on the majority class.
Learning First-Order rules
Learning first-order rules involves learning rules with quantified variables and predicates that can be applied to any object in the domain. One popular approach to learning sets of first-order rules is through the use of FOIL (First-Order Inductive Learner), which is an extension of the RIPPER algorithm for learning sets of propositional rules.
L earning sets of First-Order rules: FOIL
FOIL works by iteratively constructing a rule to cover the most specific subset of the data and then generalizing the rule by adding literals to it. The algorithm stops when the rule covers all positive examples or no more literals can be added. The added literals are chosen based on their support and coverage.
Example of implementing the FOIL algorithm in Python:
from sklearn.base import clone
from sklearn.metrics import accuracy_score
from sklearn.tree import DecisionTreeClassifier
from sympy. logic import SOPform, POSform
from sympy import symbols
def foil(X, y, base_classifier=DecisionTreeClassifier()):
rule_set = []
while True:
classifier = clone(base_classifier)
classifier.fit(X, y)
rule = get_rule(classifier)
if the rule is None:
break
rule_set.append(rule)
mask = apply_rule(X, rule)
X = X[~mask]
y = y[~mask]
return rule_set
def get_rule(classifier):
if len(classifier.tree_.feature) == 0:
return None
feature_names = classifier.classes_
feature_idx = classifier.tree_.feature[0]
threshold = classifier.tree_.threshold[0]
if threshold == -2:
return None
if classifier.tree_.value[0, 1] > classifier.tree_.value[0, 0]:
polarity = ">"
else:
polarity = "<="
rule = (feature_names[feature_idx], polarity, threshold)
return rule
def apply_rule(X, rule):
feature, polarity, threshold = rule
if polarity == ">":
mask = X[feature] > threshold
else:
mask = X[feature] <= threshold
return mask
def get_literals(X, y, rule, target_class):
literals = []
pos_mask = (y == target_class)
neg_mask = ~pos_mask
for a feature in X.columns:
if feature == rule[0]:
continue
pos_support = ((X[feature] > rule[2]) & pos_mask).sum()
neg_support = ((X[feature] > rule[2]) & neg_mask).sum()
if pos_support + neg_support == 0:
continue
pos_coverage = ((X[feature] > rule[2]) & pos_mask & (X[rule[0]] <= rule[2])).sum()
neg_coverage = ((X[feature] > rule[2]) & neg_mask & (X[rule[0]] <= rule[2])).sum()
pos_gain = pos_coverage / (pos_support + 1)
neg_gain = neg_coverage / (neg_support + 1)
gain = pos_gain - neg_gain
literals.append((feature, gain))
literals.sort(key=lambda x: x[1], reverse=True)
return literals[:3]
def learn_rule(X, y, target_class):
rule = ("", "", 0)
while True:
literals = get_literals(X, y, rule, target_class)
if len(literals) == 0:
break
feature, _ = literals[0]
rule = (feature, ">", X[feature].median())
mask = apply_rule(X, rule)
if mask.sum() == 0:
break
pos_mask = (y == target_class) & mask
neg_mask = ~pos_mask
if pos_mask.sum() == 0 or neg_mask.sum() == 0:
break
return rule
def get_rules(X, y):
rules = []
for target_class in y.unique():
X_subset = X.copy()
y_subset = (y == target_class).astype(int)
while True:
rule = learn_rule(X_subset, y_subset, 1)
if rule[0] == "":
break
rules.append((target_class, rule))
mask = apply_rule(X_subset, rule)
X_subset = X_subset[~mask]
y_subset = y_subset[~mask]
return rules
def predict(X, rules):
preds = []
for IDX, row in X.iterrows():
scores = {}
for target_class, rule in rules:
if apply_rule(row.to_frame().T, rule):
scores[target_class] = scores.get(target_class, 0) + 1
if len(scores) == 0:
preds.append(-1)
else:
preds.append(max(scores, key=scores.get))
return preds
In this code, learn rule function learns a single rule for a given target class using a greedy approach. It starts with an empty rule and iteratively adds literals that maximize the information gain until it can no longer find a better rule. The get rules function applies this process to all classes in the dataset and returns a list of rules. Finally, the predict function applies the learned rules to a new dataset and predicts the target class based on the rule with the highest score.
Note that this implementation uses a median split to generate the threshold for continuous features. Other methods, such as mean or quartile splits, could be used instead. Additionally, this implementation uses a greedy approach to learning rules, which may not always find the globally optimal solution. Other algorithms, such as beam search or genetic algorithms, could be used to search the rule space more thoroughly.
k-Nearest Neighbor algorithm in Python:
import numpy as np
class KNN:
def __init__(self, k=3):
self.k = k
def fit(self, X, y):
self.X = X
self.y = y
def predict(self, X):
predictions = []
for x in X:
distances = []
for i, x_train in enumerate(self.X):
distance = np.sqrt(np.sum(np.square(x - x_train)))
distances.append((i, distance))
distances.sort(key=lambda x: x[1])
neighbors = [self.y[idx] for idx, _ in distances[:self.k]]
prediction = max(set(neighbors), key=neighbors.count)
predictions.append(prediction)
return predictions
This implementation of the k-Nearest Neighbor algorithm includes a constructor method that sets the value of k (which is the number of nearest neighbours to consider), a fit method that stores the training data X and the corresponding labels y, and a predict method that takes a set of test data X and returns the predicted labels for each instance using the k-Nearest Neighbor algorithm.
The predict method loops through each test instance and computes the distances to all training instances. It then sorts the distances and selects the k nearest neighbours. Finally, it predicts the label of the test instance as the most common label among the k nearest neighbours. The implementation uses the numpy library to compute the Euclidean distance between instances.
Summary
In summary, learning sets of rules is a useful approach to machine learning that can be used to make predictions and classifications. Sequential covering algorithms, such as the RIPPER algorithm, are a popular approach to learning sets of rules. These algorithms iteratively learn rules to cover subsets of the data until all data are covered or no more rules can be added.
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