? This tutorial will show you the most simple and straightforward way to implement linear regression in Python—by using scikit-learn’s linear regression functionality. I have written this tutorial as part of my book Python One-Liners where I present how expert coders accomplish a lot in a little bit of code.
It is really simple to implement linear regression with the sklearn (short for scikit-learn) library. Have a quick look at this code snippet—we’ll explain everything afterward!
from sklearn.linear_model import LinearRegression import numpy as np ## Data (Apple stock prices) apple = np.array([155, 156, 157]) n = len(apple) ## One-liner model = LinearRegression().fit(np.arange(n).reshape((n,1)), apple) ## Result & puzzle print(model.predict([,])) # What is the output of this code?
This one-liner uses two Python libraries: NumPy and scikit-learn. The former is the de-facto standard library for numerical computations (e.g. matrix operations). The latter is the most comprehensive library for machine learning which implements hundreds of machine learning algorithms and techniques.
So let’s explore the code snippet step by step.
We create a simple dataset of three values: three stock prices of the Apple stock in three consecutive days. The variable
apple holds this dataset as a one-dimensional NumPy array. We also store the length of the NumPy array in the variable
Let’s say the goal is to predict the stock value of the next two days. Such an algorithm could be useful as a benchmark for algorithmic trading applications (using larger datasets of course).
To achieve this goal, the one-liner uses linear regression and creates a model via the function
fit(). But what exactly is a model?
Background: What is a Model?
Think of a machine learning model as a black box. You put stuff into the box. We call the input “features” and denote them using the variable
x which can be a single value or a multi-dimensional vector of values. Then the box does its magic and processes your input. After a bit of time, you get back the result
Now, there are two separate phases: the training phase and the inference phase. During the training phase, you tell your model your “dream” output
y’. You change the model as long as it does not generate your dream output
As you keep telling the model your “dream” outputs for many different inputs, you “train” the model using your “training data”. Over time, the model will learn which output you would like to get for certain outputs.
That’s why data is so important in the 21st century: your model will only be as good as it’s training data. Without good training data, it is guaranteed to fail.
So why is machine learning such a big deal nowadays? The main reason is that models “generalize”, i.e., they can use their experience from the training data to predict outcomes for completely new inputs which they have never seen before. If the model generalizes well, these outputs can be surprisingly accurate compared to the “real” but unknown outputs.
Now, let’s deconstruct the one-liner which creates the model:
model = LinearRegression().fit(np.arange(n).reshape((n,1)), apple)
First, we create a new “empty” model by calling
LinearRegression(). How does this model look like?
Every linear regression model consists of certain parameters. For linear regression, the parameters are called “coefficients” because each parameter is the coefficient in a linear equation combining the different input features.
With this information, we can shed some light into our black box.
Given the input features
x_k. The linear regression model combines the input features with the coefficients
a_k to calculate the predicted output y using the formula:
In our example, we have only a single input feature
x so the formula becomes easier:
In other words, our linear regression model describes a line in the two-dimensional space. The first axis describes the input
x. The second axis describes the output
x. The line describes the (linear) relationship between input and output.
What is the training data in this space? In our case, the input of the model simply takes the indices of the days:
[0, 1, 2] – one day for each stock price [155, 156, 157]. To put it differently:
x=0should cause output
x=1should cause output
x=2should cause output
Now, which line fits best to our training data
[155, 156, 157]?
Here is what the linear regression model computes:
## Data (Apple stock prices) apple = np.array([155, 156, 157]) n = len(apple) ## One-liner model = LinearRegression().fit(np.arange(n).reshape((n,1)), apple) ## Result print(model.coef_) # [1.] print(model.intercept_) # 155.0
You can see that we have two coefficients: 1.0 and 155.0. Let’s put them in our formula for linear regression:
Let’s plot both the line and the training data in the same space:
A perfect fit! Using this model, we can predict the stock price for any value of
x. Of course, whether this prediction accurately reflects the real world is another story.
After having trained the model, we use it to predict the two next days. The Apple dataset consists of three values 155, 156, and 157. We want to know the fourth and fifth value in this series. Thus, we predict the values for indices 3 and 4.
Note that both the function
fit() and the function
predict() require an array with the following format:
Each training data value is a sequence of feature value:
<training_data> = [feature_1, feature_2, …, feature_k]
Again, here is our one-liner:
model = LinearRegression().fit(np.arange(n).reshape((n,1)), apple)
In our case, we only have a single feature
x. Therefore, we reshape the NumPy array to the strange looking matrix form:
fit() function takes two arguments: the input features of the training data (see the last paragraph) and the “dream outputs” of these inputs. Of course, our dream outputs are the real stock prices of the Apple stock. The function then repeats testing and tweaking different model parameters (i.e., lines) so that the difference between the predicted model values and the “dream outputs” is minimal. This is called “error minimization”. (To be more precise, the function minimizes the squared difference from the predicted model values and the “dream outputs” so that outliers have a larger impact on the error.)
In our case, the model perfectly fits the training data, so the error is zero. But often it is not possible to find such a linear model. Here is an example of training data that cannot be fit by a single straight line:
from sklearn.linear_model import LinearRegression import numpy as np import matplotlib.pyplot as plt ## Data (Apple stock prices) apple = np.array([157, 156, 159]) n = len(apple) ## One-liner model = LinearRegression().fit(np.arange(n).reshape((n,1)), apple) ## Result print(model.predict([,])) # [158. 159.] x = np.arange(5) plt.plot(x[:len(apple)], apple, "o", label="apple stock price") plt.plot(x, model.intercept_ + model.coef_*x, ":", label="prediction") plt.ylabel("y") plt.xlabel("x") plt.ylim((154,164)) plt.legend() plt.show()
In this case, the
fit() function finds the line that minimizes the squared error between the training data and the predictions as described above.
Where to Go from Here?
Do you feel like you need to brush up your coding skills? Then join my free “Coffee Break Python Email Course”. I’ll send you cheat sheets, daily Python lessons, and code contests. It’s fun!
While working as a researcher in distributed systems, Dr. Christian Mayer found his love for teaching computer science students.
To help students reach higher levels of Python success, he founded the programming education website Finxter.com. He’s author of the popular programming book Python One-Liners (NoStarch 2020), coauthor of the Coffee Break Python series of self-published books, computer science enthusiast, freelancer, and owner of one of the top 10 largest Python blogs worldwide.
His passions are writing, reading, and coding. But his greatest passion is to serve aspiring coders through Finxter and help them to boost their skills. You can join his free email academy here.