Overview of classification problems using the Fisher computation method.
Fisher’s Linear Discriminant (Fisher’s Linear Discriminant) is a method for constructing a linear discriminant model to distinguish between two classes, which will aim to find a projection that maximizes the variance between classes and minimizes the variance within classes. Specifically, the following steps are used to construct the model
1. compute the within-class and between-class variance matrices:
For each data point within each class, we compute the within-class variance matrix, and we also compute the between-class variance matrix using the overall mean and the mean of each class.
2. maximizing the variance ratio:
The eigenvectors of the inverse of the within-class variance matrix multiplied by the between-class variance matrix are calculated, which defines the direction of projection of the data. In this case, we find the direction that maximizes the interclass variance.
3. selection of the projection vector:
After finding the direction that maximizes the variance ratio, a new feature space is constructed by projecting the data in this direction. This projection improves the separation between classes.
4. classification:
Using the data obtained in the new feature space, a discriminative model is constructed to classify the new samples into classes.
The Fisher computation method is particularly effective when the two classes are linearly separated, and although the method was originally designed for two-class problems, it can be extended to multi-class problems.
Fisher’s method, also called Linear Discriminant Analysis (LDA), is also approached from the statistical side, where LDA considers not only class separation, but also information about the shape of the data distribution for each class.
Algorithms related to classification problems using the Fisher computation method.
The algorithm for the classification problem based on Fisher’s Linear Discriminant follows these steps
1. computation of the within-class and between-class variance matrices:
For each data point within each class, we compute the Within-class scatter matrix, and we also compute the Between-class scatter matrix using the overall mean and the mean of each class.
\[
S_W = \sum_{i=1}^{C} \sum_{j=1}^{n_i} (\mathbf{x}_j^i – \mathbf{m}_i)(\mathbf{x}_j^i – \mathbf{m}_i)^T
\]
\[
S_B = \sum_{i=1}^{C} n_i (\mathbf{m}_i – \mathbf{m})(\mathbf{m}_i – \mathbf{m})^T
\]
where \(C\) is the number of classes, \(n_i\) is the number of samples in each class, \(\mathbf{x}_j^i\) is the \(j\)th sample in class \(i\), \(\mathbf{m}_i\) is the mean vector for class \(i\), and \(\mathbf{m}\) is the overall mean vector.
2. maximizing the variance ratio:
The eigenvectors of the inverse of the within-class variance matrix multiplied by the between-class variance matrix are obtained. This finds the direction that maximizes the between-class variance.
\[
S_W^{-1}S_B \mathbf{v} = \lambda \mathbf{v}
\]
where \(\lambda\) is the eigenvalue and \(\mathbf{v}\) is the eigenvector.
3. selection of the projection vectors:
The direction corresponding to the eigenvector that maximizes the variance ratio is the direction to project the data.
4. projection of data:
Construct a new feature space by projecting the data in the direction that maximizes the variance ratio.
5. classification:
Using the data obtained in the new feature space, a discriminative model is constructed and the new samples are classified into classes.
The Fisher computation method is effective as a linear discriminant model because it optimizes the separation between classes in terms of variance ratio maximization. However, the assumption is that the data in each class follows a multivariate normal distribution, and since it is designed for two-class problems, it needs to be extended for multi-class problems.
Application of the Fisher Computation Method to Classification Problems
Fisher’s Linear Discriminant is a method mainly applied to 2-class classification problems.
1. medical diagnosis:
It is used to distinguish between patients and healthy controls in biometric and image analysis. For example, to detect tumor grade and abnormalities.
2. quality control:
Used in manufacturing processes and product quality control to detect abnormal or defective products. Fisher’s method is useful in classifying abnormal products from normal products.
3. customer segmentation:
In the marketing field, it is used to classify different customer segments based on their purchase history and attributes. For example, targeting different products and services based on customer preferences and behavior patterns.
4. security:
Used in biometrics and security systems to classify individuals as legitimate users or unauthorized accessors. Examples include fingerprints and facial recognition.
5. financial transaction fraud detection:
Used to analyze financial transaction data to detect fraudulent activity. It helps to identify and properly classify fraudulent transaction patterns and anomalous behavior.
Example implementation of a classification problem using the Fisher computation method
The Fisher computation method is typically used to reduce the dimensionality of features. Here is a simple example of implementing the Fisher computation method on the Iris dataset using the scikit-learn library in Python.
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import load_iris
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
# Reading Iris Data Sets
iris = load_iris()
X = iris.data
y = iris.target
# Split data into training and test sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Application of the Fisher Calculation Method
lda = LinearDiscriminantAnalysis()
X_train_lda = lda.fit_transform(X_train, y_train)
X_test_lda = lda.transform(X_test)
# Construct a classifier (using logistic regression as an example)
from sklearn.linear_model import LogisticRegression
classifier = LogisticRegression()
classifier.fit(X_train_lda, y_train)
# Prediction on test data
y_pred = classifier.predict(X_test_lda)
# Classifier Evaluation
accuracy = accuracy_score(y_test, y_pred)
print(f"Accuracy: {accuracy:.2f}")
# Visualization of the projection obtained by the Fisher computation method
plt.figure(figsize=(8, 6))
for label, marker, color in zip(range(3), ('^', 's', 'o'), ('blue', 'red', 'green')):
plt.scatter(X_train_lda[y_train == label, 0], X_train_lda[y_train == label, 1], marker=marker, color=color, label=f'Class {label}')
plt.xlabel('LD1')
plt.ylabel('LD2')
plt.legend()
plt.title('LDA Projection of Iris Dataset')
plt.show()In this example, the Iris data set is loaded and split into a training set and a test set. The Fisher computation method is then applied using the LinearDiscriminantAnalysis class, the resulting projection is used to train the logistic regression model, and finally, predictions are made on the test data to evaluate the classifier’s performance.
In the above example, two Fisher projections are used to visualize the data, and it is expected that such projections will allow the data to be better separated by class.
Challenges and Remedies for Classification Problems Using the Fisher Computation Method
Although the Fisher computation method is powerful and effective, several challenges exist. The following is a description of the main challenges and how they are addressed.
1. handling the case of singular covariance matrices:
Challenge: When the within-class covariance matrix is singular, it is difficult to maximize the variance ratio because the inverse matrix cannot be computed.
Solution: To deal with singular covariance matrices, either use regularization methods or reduce the rank of the covariance matrix by applying a principal component analysis (PCA) or other methods as data preprocessing.
2. if the number of classes is greater than the number of samples:
Challenge: When the number of classes is larger than the number of samples, the within-class variance matrix tends to be singular and the inverse matrix cannot be computed.
Solution: If the number of classes is larger than the number of samples, problems are likely to occur, especially for high-dimensional data. The number of classes may be reduced or regularization or other techniques may be used to deal with the problem.
3. difficulty in applying the method to nonlinear class separation problems:
Challenge: Fisher’s method learns linear discriminative surfaces, so there are limitations when nonlinear class separation is required.
Solution: To deal with nonlinear class separation problems, nonlinear transformations and kernel methods are used. Kernel Fisher Discriminant Analysis (KFDA) is a typical method.
4. problems that depend on the distribution of the data:
Challenge: The Fisher method assumes that each class follows a multivariate normal distribution. This assumption may not be valid for real data.
Solution: If assumptions about the distribution of the data are not valid, consider nonparametric or model-free methods (e.g., support vector machines).
Reference Information and Reference Books
For more information on optimization in machine learning, see also “Optimization for the First Time Reading Notes” “Sequential Optimization for Machine Learning” “Statistical Learning Theory” “Stochastic Optimization” etc.
Reference books include Optimization for Machine Learning
“Machine Learning, Optimization, and Data Science“
“Linear Algebra and Optimization for Machine Learning: A Textbook“
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