Introduction
In Chapter 2, we are focusing on learning the key algorithms commonly used in machine learning. In this session, we will start with linear regression, a fundamental model for predicting numerical values based on data. Despite its simplicity, linear regression has numerous applications. Understanding its basic concepts and usage will help you deepen your understanding of machine learning.
What is Linear Regression?
Basic Concept
Linear regression assumes a linear relationship between the independent variables (explanatory variables) and the dependent variable (target variable) to make predictions. For example, when predicting the relationship between the price of a product and its sales volume, we might assume that as the price increases, sales decrease. This type of relationship is modeled by linear regression.
Mathematical Background
The linear regression model is represented by the following linear equation:
\[
y = \beta_0 + \beta_1x + \epsilon
\]
Where:
- \( y \) is the target variable we want to predict,
- \( x \) is the independent variable,
- \( \beta_0 \) is the intercept (the point where the regression line intersects the y-axis),
- \( \beta_1 \) is the slope or regression coefficient (indicating how much the independent variable influences the target variable),
- \( \epsilon \) is the error term.
For example, when predicting house prices based on area (independent variable), if the regression coefficient \( \beta_1 \) is positive, it means that as the area increases, the price also rises.
Estimating Coefficients Using the Least Squares Method
When building a linear regression model, we need to estimate the unknown coefficients \( \beta_0 \) and \( \beta_1 \). The most common way to do this is through the least squares method. This method minimizes the sum of the squared differences between the observed data and the predicted regression line.
For example, consider the following dataset:
| Area (㎡) | Price (in million yen) |
|---|---|
| 50 | 15 |
| 80 | 22 |
| 120 | 28 |
Using the least squares method, we can calculate the slope \( \beta_1 \) and intercept \( \beta_0 \) of the regression line that best predicts the price based on the area.
Scope and Limitations of Linear Regression
While linear regression is simple and intuitive, it is not suitable for all datasets. For example:
- Non-linear data: If the relationship between the independent and dependent variables is non-linear, linear regression may struggle to make accurate predictions.
- Influence of outliers: Outliers can significantly affect the model, leading to inaccurate predictions.
- Multicollinearity: When multiple independent variables are highly correlated with each other, it can cause instability in the model’s coefficients.
Applications of Linear Regression
Predicting Real Estate Prices
In the real estate industry, linear regression is widely used to predict property prices. Variables such as area, number of rooms, and age of the building are used as independent variables to model how they affect the price of a property. This model allows us to predict the price of new properties based on these features.
Sales Forecasting
In marketing, linear regression is used to forecast future sales based on past data. For example, by analyzing the relationship between advertising spend and sales, companies can predict how much sales will increase if advertising spend is raised.
Risk Assessment in Healthcare
In the healthcare field, linear regression is used to predict the risk of disease based on factors such as weight, age, and blood pressure. For example, to predict the risk of heart disease, these health indicators are used as independent variables in a predictive model.
Limitations of Linear Regression and How to Address Them
The Assumption of Linearity
Linear regression assumes a linear relationship between the independent and dependent variables, but real-world data doesn’t always follow this pattern. For instance, if demand for a product sharply drops after it reaches a certain price point, linear regression may not be able to accurately model this. In such cases, more complex models like polynomial regression or non-linear regression might be more appropriate.
Influence of Outliers
Linear regression models are sensitive to outliers. If outliers are present, they can skew the regression line, leading to reduced prediction accuracy. To address this, methods such as robust regression or outlier removal can be used.
The Problem of Multicollinearity
When independent variables are strongly correlated, multicollinearity can occur, making the coefficient estimates unstable and reducing the model’s reliability. To handle this, techniques such as Principal Component Analysis (PCA) or Ridge Regression can be used to reduce dimensionality or add regularization.
Conclusion
Linear regression is a fundamental model in machine learning and is widely used for solving prediction problems based on numerical data. Its simplicity and ease of understanding make it ideal for beginners and a perfect starting point for learning data science. However, it’s important to be aware of its limitations and select the appropriate model based on the characteristics of the data.
In the next session, we will explore logistic regression, a basic model for binary classification problems. Logistic regression is especially useful for Yes/No predictions. Stay tuned!
Notes
- Least squares method: A method used to estimate the coefficients of a linear regression model by minimizing the squared differences between observed data points and the predicted regression line.
- Outliers: Data points that deviate significantly from other points in the dataset, potentially affecting model accuracy.
- Multicollinearity: A condition in which independent variables are highly correlated, making the model’s coefficients less reliable.
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