Machine Learning and Deep Learning Algorithm Trading, Implementation of LDA using sklearn

1. Introduction

Automated trading in the stock market has become an attractive option for many investors, and machine learning and deep learning technologies are bringing innovations to such trading. This article will explain the basic principles of LDA and step by step how to apply it to real financial data.

2. What is LDA?

LDA is an algorithm primarily used for classification problems, which maximizes the separation between classes and minimizes the variance within classes. In stock trading, LDA is useful for predicting whether stock prices will rise or fall.

The basic mathematical concepts are as follows:

• Mean of Class
• Overall Mean
• Between-Class Scatter Matrix
• Within-Class Scatter Matrix

The goal of LDA is to find the optimal axis that separates the classes.

3. Mathematical Foundations of LDA

LDA operates based on specific mathematical formulas and performs maximum likelihood estimation (MLE) when the distribution follows a normal distribution. It assumes that the means of the two classes are the same and that the covariance matrices are identical.

3.1. Mathematical Formulas

To calculate the class-wise means and overall mean, the following formulas are used:

$$ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i $$

$$ S_W = \sum_{i=1}^{k} \sum_{x \in C_i}(x – \mu_i)(x – \mu_i)^T $$

$$ S_B = \sum_{i=1}^{k} N_i(\mu_i – \mu)(\mu_i – \mu)^T $$

4. Implementing LDA Using sklearn

Now let’s implement LDA using the sklearn library in Python. Here are the main steps:

1. Data collection
2. Data preprocessing
3. Feature selection and applying LDA
4. Model evaluation

4.1. Data Collection

Use Python’s pandas library to collect historical stock price datasets. The following code snippet shows how to download data from Yahoo Finance:

import pandas as pd
import yfinance as yf

# Download data
data = yf.download('AAPL', start='2020-01-01', end='2023-01-01')
data = data[['Open', 'High', 'Low', 'Close', 'Volume']]
data.head()

            

Specific features need to be created based on this data to perform LDA.

4.2. Data Preprocessing

Preprocess the data to create features and generate the target variable. The following code is an example of setting the target as price increase:

# Create target variable: whether price increases the next day
data['Target'] = (data['Close'].shift(-1) > data['Close']).astype(int)

# Remove missing values
data.dropna(inplace=True)

            

4.3. Feature Selection and Applying LDA

Now we are ready to apply LDA for feature selection. Prepare X and y to train the LDA model:

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis

# Separate features (X) and target (y)
X = data[['Open', 'High', 'Low', 'Close', 'Volume']]
y = data['Target']

# Initialize and train LDA model
lda = LinearDiscriminantAnalysis()
lda.fit(X, y)

            

4.4. Model Evaluation

Since the model has been trained, we can now evaluate its performance using test data:

from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score

# Train/Test split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Train the model
lda.fit(X_train, y_train)

# Prediction
y_pred = lda.predict(X_test)

# Evaluate accuracy
accuracy = accuracy_score(y_test, y_pred)
print(f'Accuracy: {accuracy:.2f}')

            

5. Results Interpretation

We will explain in detail how to analyze the class separation generated by the LDA model and interpret the results. Besides accuracy, model performance can also be evaluated using confusion matrices, ROC curves, etc.

6. Conclusion

This course covered the basic principles of algorithmic trading using LDA and detailed implementation methods through sklearn. There are various other machine learning techniques that can be utilized for stock price prediction, so continuing learning is encouraged.