Mastering Linear Regression: A Step-by-Step Guide

Author(s): Anushka sonawane

Originally published on Towards AI.

From Basics to Advanced: A Comprehensive Guide to Linear Regression

Have you ever wondered how we could use machine learning to predict future outcomes based on past data? One of the most fundamental techniques used in machine learning is linear regression. In this article, we will explore the basics of linear regression and how it can be applied to solve real-world problems.

Let’s say you are the owner of a ropeway system that transports tourists up a mountain for sightseeing. You want to predict the number of daily visitors based on the weather conditions and the price of the ropeway tickets.

You collect data on the number of visitors, the daily temperature, the amount of precipitation, and the ticket price for each day over the past year. You can then use linear regression to build a model to predict the number of visitors based on the weather conditions and the ticket price.

First, you would identify the independent variables, which in this case are the daily temperature, the amount of precipitation, and the ticket price. The dependent variable is the number of visitors.

Linear regression is a commonly used statistical method that is a type of predictive analysis. It’s a supervised algorithm and the central idea of regression is to study the mathematical relationship between variables. Linear regression helps to identify how strongly the dependent variable(outcome) is associated with the independent variable(input) and the nature of that relationship, whether it is positive or negative. We find the relationship between them with the help of the best-fit line, which is also known as the Regression line.

The equation of a line is, y = mx + c

In this equation:

y represents the dependent variable, which in this case is the number of visitors to the ropeway.

x represents the independent variable, which in this case can be the temperature, precipitation, or ticket price.

m represents the slope of the line, which determines the relationship between the independent variable and the dependent variable. In other words, it represents how much the dependent variable changes for each unit increase in the independent variable.

c represents the y-intercept, which is the point where the line intersects the y-axis.

In the context of the ropeway example, the equation can be written as:

Visitors = m1 x Temperature + m2 x Precipitation + m3 x Ticket Price + c

Let's understand the brute-force code of linear regression in Python.

import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression


# Load data into a Pandas DataFrame
df = pd.read_csv('your dataset location')
# Split data into independent and dependent variables
X = 'Your independent features'
y = 'Your dependent features'

# Create a linear regression model and fit it to the data
model = LinearRegression()
model.fit(X, y)


# Use the model to make predictions
new_data ='Your transformed data'
predicted_visitors = model.predict(new_data)

# Create a scatter plot of actual visitors versus predicted visitors
plt.scatter(y, model.predict(X))
plt.scatter(predicted_visitors, predicted_visitors, color='red')
plt.xlabel('your actual labels ')
plt.ylabel('your Predicted labels')
plt.show()

Key assumptions of a linear regression model :

1. Linearity: The independent and dependent variables should be in a linear relationship, such that a change in the independent variable will result in a proportional change in the dependent variable.

2. No multicollinearity: The difference between the actual and predicted values should be normally distributed for all levels of the independent variables(the data should follow a normal distribution in multiple dimensions).

3. Normality: The errors of the regression model should be normally distributed around zero.

4. Homoscedasticity: The variability of the dependent variable is consistent across all values of the independent variable.

Techniques for model selection:

Stepwise regression: There are two types of stepwise regression: forward selection and backward elimination.
a. Forward selection: This method starts with a model that includes no variables and adds variables one at a time based on their significance. At each step, the variable that provides the largest improvement in the fit of the model is added to the model, until there are no more variables that improve the fit.
b. Backward elimination: This method starts with a model that includes all variables and removes variables one at a time based on their significance. At each step, the variable that provides the smallest improvement in the fit of the model is removed from the model, until there are no more variables that can be removed without significantly reducing the fit.

Information criteria: The most commonly used information criteria are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which provide a way to compare different models based on their goodness-of-fit as well as the number of parameters used in the model.

3. Regularization: Regularization methods, such as ridge regression and lasso regression, add a penalty term to the regression equation to shrink the coefficients of less important variables toward zero. This helps to reduce the effects of multicollinearity and improve the generalizability of the model.

4. Cross-validation: Cross-validation involves splitting the data into training and testing sets, and then evaluating the performance of the model on the testing set. This helps to prevent overfitting and ensures that the model is generalizable to new data.

Techniques to Interpret Results of a Machine Learning Model –

R-squared: R-squared is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variables. It ranges from 0 to 1, with higher values indicating a better fit of the model to the data.
In the ropeway example, the R-squared value of the linear regression model tells us how much of the variation in the number of visitors can be explained by the independent variables (temperature, precipitation, and ticket price). For example, if the R-squared value is 0.8, it means that 80% of the variation in the number of visitors can be explained by the independent variables, while the remaining 20% is due to other factors that are not included in the model.

Mean Absolute Error (MAE): Mean Absolute Error (MAE) is a metric that measures the average absolute difference between the actual and predicted values of a model. In the context of the ropeway example, MAE can be used to evaluate how well the linear regression model predicts the number of visitors to the ropeway based on temperature, precipitation, and ticket price.

Root Mean Squared Error (RMSE): Root Mean Squared Error (RMSE) is another popular metric used for evaluating the performance of a regression model. It measures the square root of the average of the squared differences between the predicted values and the actual values.

Mean Squared Error (MSE): This technique calculates the average of the squared differences between the actual and predicted values. It gives an idea of how much variance there is in the error term.

Real-life examples-

Traffic flow analysis: Linear regression can be used to model the relationship between traffic flow and various factors such as time of day, weather conditions, etc.

Sports analysis: Linear regression can be a useful tool in sports analysis to gain insights into player performance and make informed decisions around team management and player selection.

Asset pricing: Linear regression can be used to analyze the relationship between various financial variables and asset prices. This can help investors make informed decisions about which assets to buy or sell based on their expected future returns

Price optimization: Linear regression can be used to analyze the relationship between product pricing and sales volume. This can help companies optimize their pricing strategy to maximize sales and profits.

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Published via Towards AI