Adventures in Machine Learning

Mastering Residual Plots: Evaluating Linear Regression Models in Python

Creating a Residual Plot in Python

Have you ever heard of a residual plot? It may not be a term that’s familiar to everyone, but it’s a tool that statisticians and data scientists often use to evaluate the performance of their linear regression models.

The concept appears intimidating, but once you get familiar with it, making a residual plot in Python is a straightforward process.

Simple Linear Regression

Simple linear regression enables us to model the relationships between two variables, typically to estimate the presumed cause-and-effect relationship between them. Simple linear regression’s main purpose is to foresee the impact on the dependent variable by altering the independent variable using a line of best fit.

However, sometimes it’s difficult to evaluate the precision of our predictions using a line graph. This is where a residual plot comes in handy.

A residual plot displays the differences between the predicted values of the dependent variable and the actual values by plotting the residuals (vertical axis) against the independent variable (horizontal axis). In other terms, we plot the distance between the predicted dependent variable and the real dependent variable’s value.

The residual plot also allows us to identify clear trends in the errors: if there’s a pattern, then our regression assumptions are violated, implying that our predictions are not precise enough. The residual plot for simple linear regression can be easily made using Python.

We’ll build a simple linear regression model using the Scikit-learn library and draw the residual plot using Seaborn.

Multiple Linear Regression

Multiple linear regression models are just as common as simple linear regression models and might be more helpful when modeling more complex real-world problems. Before starting, let’s recap: in multiple linear regression, we have numerous dependent variables and one independent variable that predict the response variable.

We’ll figure out how to generate a residual plot for the multiple linear regression model we created in the preceding part. Drawing a residual plot for a multiple linear regression model is similar to doing it for a simple linear regression model.

The difference is that instead of plotting the independent variable’s values on the x-axis, we’ll use the predicted response variable’s values.

Dataset Description

Before constructing the regression models, let’s first understand the dataset efficiently. Basketball is a sport that requires physical strength, agility, and endurance.

The dataset includes both physical characteristics, such as height, weight, and wingspan, as well as statistical data on the NBA player’s performances.

Data Attributes

The dataset contains several attributes, including age, years of experience, field goals, rebounds, points, weight, height, and more.

The data we have on basketball players can be used to create an algorithm that predicts their performance given the physical attributes and experience statistics available in the data.

Data Preparation

Before we make the residual plots, we’ll have to prepare the dataset using Pandas, a Python library used for data manipulation and analysis. To begin, we may load the dataset file and convert it to a pandas dataframe.

We could try removing missing values or using imputation methods to fill in those values. Before creating regression models, we may also scale and normalize data to ensure that all variables are in the same range, making it easier to compare them.

Conclusion

Residual plots and data preparation are essential components of linear regression model development. Residual plots are used to assess the performance of the model, while data preparation is used to clean and transform raw data into an appropriate format for machine learning models.

Python has numerous libraries that make it easy to create residual plots for both simple and complex linear regression models. Furthermore, Pandas is ideal for preparing data in a suitable format for modeling.

Therefore, these two basic concepts, combined with the Python packages shown above, provide a strong foundation for individuals interested in machine learning and predictive analytics. Residual Plot for

Simple Linear Regression

Linear regression is a statistical method that determines the relationship between two or more variables by fitting a linear equation to the data.

Simple linear regression is used when there is a single predictor variable, such as predicting the rating of a basketball player based on their points scored for a season.

Regression Model

Let’s say we have data on points scored and rating for a group of basketball players, and we want to determine how well points predict a player’s rating. We can create a linear regression model using Python’s scikit-learn library:

“`

from sklearn.linear_model import LinearRegression

import pandas as pd

# Load the dataset

data = pd.read_csv(‘basketball_data.csv’)

# Split the data into predictor and response variables

X = data[‘Points’].values.reshape(-1, 1)

y = data[‘Rating’].values.reshape(-1, 1)

# Create the linear regression model

regressor = LinearRegression()

regressor.fit(X, y)

“`

Our model is created, and now we need to evaluate its performance. One way to do this is by creating a residual plot.

Residual vs. Fitted Plot

A residual plot is a graph that displays the differences between the predicted values and the actual values of the response variable.

In other words, a residual plot shows the error of the regression model.

To create a residual plot in Python, we can use the Seaborn library:

“`

import seaborn as sns

# Create the residual plot

sns.residplot(X, y, lowess=True, color=”g”)

“`

In this plot, the residuals (vertical axis) are plotted against the fitted values (horizontal axis). The fitted values are the predicted values that the regression line passes through.

Ideally, the residual plot should show a random scatter of points centered around zero. If there is a pattern in the residual plot, such as an increasing or decreasing trend, it can indicate heteroscedasticity, which is a violation of the linear regression assumptions.

Heteroscedasticity is when the variance of the residuals is correlated with the predictors. This means that the size of the error terms differs across the range of the predictor variable.

When heteroscedasticity is present, it means that the model is not capturing all the variation in the data, and the point predictions may be too narrow.

To detect heteroscedasticity in a residual plot, look for a funnel-like shape where the scatter of points grows wider or narrower as the predictor variable increases.

In this case, you may need to add additional predictor variables to the regression model to correct for the heteroscedasticity. Residual Plots for

Multiple Linear Regression

Multiple Linear Regression is used when there are two or more predictor variables. For example, to predict a player’s rating based on points, assists, and rebounds.

Regression Model

We can create a multiple linear regression model using Python’s scikit-learn library:

“`

from sklearn.linear_model import LinearRegression

import pandas as pd

# Load the dataset

data = pd.read_csv(‘basketball_data.csv’)

# Split the data into predictor and response variables

X = data[[‘Points’, ‘Assists’, ‘Rebounds’]].values

y = data[‘Rating’].values.reshape(-1, 1)

# Create the linear regression model

regressor = LinearRegression()

regressor.fit(X, y)

“`

Residual vs. Predictor Plot for Assists

We can create a residual plot for each predictor variable to evaluate its impact on the model’s performance.

For example, we can create a residual vs. predictor plot for assists:

“`

import seaborn as sns

# Create a dataframe of the predictor and response variables

data_for_plot = pd.DataFrame(data={‘Assists’: X[:,1], ‘Rating’: y[:,0]})

# Create the residual plot

sns.residplot(x=’Assists’, y=’Rating’, data=data_for_plot, color=”g”)

“`

In this plot, the residuals (vertical axis) are plotted against the predictor variable (horizontal axis). The residual plot for assists helps us see how well assists are predicting the rating.

If we see a clear pattern in the residual plot, it may indicate a nonlinear relationship between the predictor and response variables, which would require using a nonlinear regression model instead.

Residual vs.

Predictor Plot for Rebounds

Similarly, we can create a residual plot vs. predictor plot for rebounds:

“`

# Create a dataframe of the predictor and response variables

data_for_plot = pd.DataFrame(data={‘Rebounds’: X[:,2], ‘Rating’: y[:,0]})

# Create the residual plot

sns.residplot(x=’Rebounds’, y=’Rating’, data=data_for_plot, color=”g”)

“`

By examining the residual plot for rebounds, we can assess how well rebounds are predicting the rating.

In conclusion, creating residual plots are essential in evaluating the regression model’s performance. Python offers numerous libraries to create residual plots for both simple and multiple linear regression models, making it easy to assess how well predictor variables are predicting the response variable.

In conclusion, residual plots are valuable tools in evaluating the performance of linear regression models. For simple linear regression models, residual plots help identify whether there is an increasing or decreasing trend, indicating heteroscedasticity.

On the other hand, multiple linear regression models require creating residual plots for each predictor variable to assess how well the predictor variables are predicting the response variable. Python offers several libraries, such as scikit-learn and Seaborn, to generate residual plots, enabling us to better predict the rating of basketball players based on their physical and statistical data.

Understanding residual plots provides a foundation for developing accurate regression models, which can be applied to various fields beyond basketball.

Popular Posts