Introduction to Regression
Regression is one of the most powerful statistical tools that we have at our disposal. It allows us to understand the relationship between two or more variables and predict future observations. To put it simply, regression is like trying to find the most fitting line through a scatter plot of data points.
Let's imagine you're an insurance company that insures Swedish motor vehicles. You have data on how many claims were made by each policyholder, and the total amount of those claims. You wonder, "Is there a relationship between the number of claims and the total claim amount?" This is where regression comes into play!
Swedish Motor Insurance Data: We will use this data to understand regression. The dataset includes variables like the number of claims and total claim amount.
For example, a row in this data may look like this:
{"Number of Claims": 10, "Total Claim Amount": 2000}
Descriptive Statistics
Descriptive statistics is like the bedrock upon which regression analysis is built. It gives us insights into the basic features of the data and helps us understand the structure and distribution of our variables.
When trying to understand the relationship between two variables, one term that often comes up is "correlation." Correlation measures the degree to which two variables move in relation to each other. For example, in our Swedish motor insurance data, we could calculate the correlation between the number of claims and the total claim amount.
import pandas as pd
# assuming the data is stored in a pandas DataFrame named 'df'
correlation = df["Number of Claims"].corr(df["Total Claim Amount"])
print("Correlation: ", correlation)
The output might look like: "Correlation: 0.82". This indicates a strong positive relationship between the number of claims and the total claim amount.
Basics of Regression
In regression analysis, we work with two types of variables: response (or dependent) and explanatory (or independent) variables. Our response variable is what we want to predict or explain, while explanatory variables are those that we believe have an influence on our response variable.
Regression can be mainly divided into two types: Linear Regression and Logistic Regression. Linear Regression is used when our response variable is continuous (e.g., total claim amount), while Logistic Regression is used when our response variable is categorical (e.g., whether a claim was made or not).
Visualization Techniques for Regression
Data visualization is a great way to understand the relationship between variables. A scatter plot is a common way to visualize two numeric variables. In our insurance example, we can plot the number of claims on the x-axis and the total claim amount on the y-axis.
We can add a trend line, or "line of best fit", to our scatter plot. This line seeks to best summarize the trend in the data.
import matplotlib.pyplot as plt
# assuming 'df' is our DataFrame
plt.scatter(df["Number of Claims"], df["Total Claim Amount"])
plt.title('Scatter plot: Number of Claims vs Total Claim Amount')
plt.xlabel('Number of Claims')
plt.ylabel('Total Claim Amount')
plt.show()
Here, you'd see a scatter plot of your data points. Now let's add a trend line.
import numpy as np
# calculate trendline
z = np.polyfit(df["Number of Claims"], df["Total Claim Amount"], 1)
p = np.poly1d(z)
plt.scatter(df["Number of Claims"], df["Total Claim Amount"])
plt.plot(df["Number of Claims"], p(df["Number of Claims"]), "r--")
plt.title('Scatter plot: Number of Claims vs Total Claim Amount')
plt.xlabel('Number of Claims')
plt.ylabel('Total Claim Amount')
plt.show()
This code would add a red trendline to your scatter plot, helping you visualize the general trend of the relationship between number of claims and total claim amount.
Workflow and Tools for Regression
The general workflow in regression analysis includes data collection, data cleaning, exploratory data analysis, model building, and model evaluation. Python, being a versatile language, has several packages for regression analysis. Two widely used ones are statsmodels for its statistical insights and scikit-learn for its machine learning capabilities.
Which package to use often depends on your needs. If you are more interested in interpreting your model and less concerned with prediction, you might prefer statsmodels. If your main goal is prediction, you might prefer scikit-learn.
# importing the libraries
import statsmodels.api as sm
import sklearn
Working with Linear Regression
Linear Regression is based on the idea that the relationship between the response and explanatory variables can be explained by a straight line. This might seem like a simplified approach, and it is. But it's also a powerful and effective one.
The equation of a straight line is typically written as y = mx + c, where m represents the slope of the line, c is the intercept, and x is the explanatory variable. In regression, the slope tells us how much y changes for each unit change in x, while the intercept tells us the value of y when x is zero.
# Import the library
import statsmodels.api as sm
# We add a constant to our model to get an intercept
X = sm.add_constant(df["Number of Claims"])
Y = df["Total Claim Amount"]
# Fit the model
model = sm.OLS(Y, X)
results = model.fit()
# Print the results
print(results.summary())
Here, coef for Number of Claims represents the slope and coef for const is the intercept. The R-squared value tells us the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
Categorical Explanatory Variables in Regression
So far, we've been dealing with numeric explanatory variables. But what if our explanatory variable is categorical?
Let's consider a hypothetical Fish dataset where we want to predict a fish's weight based on its species. Here, "species" is a categorical variable with values like "Bream", "Roach", etc.
First, we can visualize our data using a boxplot to understand how the fish weight differs by species.
import seaborn as sns
sns.boxplot(x='species', y='weight', data=fish_data)
plt.title('Boxplot: Fish weight by Species')
plt.xlabel('Species')
plt.ylabel('Weight')
plt.show()
We can then calculate the mean weight by species.
mean_weights = fish_data.groupby('species')['weight'].mean()
print(mean_weights)
Next, let's run a linear regression with the categorical explanatory variable.
X = sm.add_constant(pd.get_dummies(fish_data['species'], drop_first=True))
Y = fish_data['weight']
model = sm.OLS(Y, X)
results = model.fit()
print(results.summary())
In the results, each species will have its own coefficient. This is because when dealing with categorical variables, we create dummy variables - binary variables for each category. The coefficients represent the difference in mean weight from the reference category (the one we dropped).
We've covered both numeric and categorical explanatory variables, linear regression model, and how to interpret their coefficients. These techniques should give you a solid start in regression analysis.
Conclusion
Congratulations! You've made it to the end of this tutorial. We've explored regression analysis from its basics to handling numeric and categorical variables. We've also seen how to visualize data and interpret the results. As with any skill, practice is key to mastering regression analysis. Keep exploring more datasets and trying out different types of regression models to enhance your data analysis skills. Happy analyzing!