Time series data is a sequence of data points collected or recorded at specific time intervals. Modeling time series data is crucial in various fields such as finance, economics, environmental science, and many others. One of the simplest yet powerful methods to model time series data is using linear regression. This article will delve into the technical aspects of modeling time series data with linear regression, covering the fundamental concepts, steps involved, and practical applications.

Time Series Model With Linear Regression

Before diving deep into the modelling of time series data with linear regression, let us have some basic understanding on these two important concepts on time series data and linear regression.

• Linear Regression: In Regression Analysis, we try to find the best fit curve or line representing the relationship between these two variables. If the relation between dependent variable and independent variable is a straight line or establishing a linear relationship, then the Regression is called as Linear Regression. Hence this model makes predictions based on this straight line or linear relationship.
• Time Series Data: Time Series Data is basically a specific data points that are recorded over a regular intervals of time. This data is used to anayzae trends and patterns and make predictions. Examples of such data include weather measurements during weather forecasting, economic indicators, stock prices, and many more.

Building a Time Series Model With Linear Regression

In Modelling Time Series Data With Linear Regression, the dependent variable is the time series, and the independent variable will be either non-time series variables or other time series variables. In this section, you will go through the implementation of modelling time series data with linear regression with the help of several python libraries like numpy, pandas, matplotlib, and sklearn, in a step-by-step manner.

Problem Statement

• Objective: Develop a forecasting model for airline passenger numbers using time series data and linear regression.
• Data: Historical airline passenger data, collected monthly.

Step 1: Import Libraries and Load Data

Ensure that you have already installed the required libraries like numpy, pandas, matplotlib, and sklearn, before diving deep into the modelling of time data with linear regression.

# Importing the Required libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# Loading the dataset from the URL
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
data = pd.read_csv(url, parse_dates=['Month'], index_col='Month')
data.head()

Step 2: Visualize the Time Series Data

Now you can plot the time series to identify the patterns or trends, since understanding the data through visualization is very important.

# plotting the time series data to visualize the trends or patterns
plt.figure(figsize=(10, 6))
plt.plot(data, label='Monthly airline passengers')
plt.title('Airline Passengers Over Time')
plt.xlabel('Date')
plt.ylabel('Number of Passengers')
plt.legend()
# saving the plot as an image
plt.savefig('timeseries_plot.png')
plt.show()

Step 3: Data Preparation for Linear Regression

In this section, we will apply linear regression, for that we need to convert the time series problem into a supervised learning problem. we can create lagged features to do this task, and then you can split the final data into the training and testing sets in proportion of 80% and 20% respectively.

# Creating the lagged features and preparing the data for linear regression
data['Month'] = data.index
data['Month'] = data['Month'].map(pd.Timestamp.toordinal)

# Defining the feature (X) and target (y) variables
X = data[['Month']]
y = data['Passengers']

# Splitting the data into training and test sets in proportion of 80% and 20% respectively
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, shuffle=False)

Step 4: Fitting the Linear Regression Model

Now train the linear regression model using the training data available. Below is the code for fitting the linear regression model.

# Initializing the linear regression model
model = LinearRegression()

# Training the model on the training data available
model.fit(X_train, y_train)

Step 5: Make Predictions

Now predict the values for the testing set, using the trained model. Below is the code for making predictions.

# Making predictions on the testing data
y_pred = model.predict(X_test)

Step 6: Model Evaluation

In this step, you need to evaluate the model performance using evaluation metrics such as Mean Squared Error (MSE). Below is the code for printing the value of mean squared error of the model.

# Evaluating the models using MSE metric
mse = mean_squared_error(y_test, y_pred)
print(f'Mean Squared Error: {mse}')

Step 7: Visualize the Results

Finally plot the actual vs predicted values to visually understand the performance of the model.

# Plotting the actual vs predicted values inorder to visualize model performance
plt.figure(figsize=(10, 6))
plt.plot(data.index, data['Passengers'], label='Actual')
plt.plot(X_test.index, y_pred, label='Predicted', color='red')
plt.title('Actual vs Predicted Airline Passengers')
plt.xlabel('Date')
plt.ylabel('Number of Passengers')
plt.legend()
# Saving the plot as an image
plt.savefig('actual_vs_predicted.png')
plt.show()

Output:

Mean Squared Error: 5450.723647259961

Airline Passengers

Actual vs Predicted Passengers Over time

The actual passenger numbers are consistently higher than the predicted passenger numbers. This suggests that the forecasting model used to create the predicted passenger numbers was not very accurate. There could be a number of reasons for this, such as the model not taking into account all of the relevant factors that affect airline passenger numbers. Hence, there is not use of forecasting.

Advantages and Disadvantages of Linear Regression for Time Series Data

Advantages of Linear Regression for Time Series Data:

• Interpretability: Linear regression provides coefficients that can be interpreted to understand the impact of variables on the forecast.
• Simplicity: Easy to implement and understand, making it accessible for initial modeling efforts.
• Computational Efficiency: Training and prediction times are generally fast compared to more complex models.

Disadvantages of Linear Regression for Time Series Data:

• Assumption of Linearity: Linear regression assumes a linear relationship between variables, which may not always hold true for complex time series patterns.
• Limited Flexibility: It may struggle with capturing nonlinear relationships and intricate seasonal variations without feature engineering.
• Sensitivity to Outliers: Outliers can significantly affect the regression line and model predictions.

When to Choose Linear Regression for Time Series Data

• Stable Trends: When the time series data exhibits a stable trend that can be adequately captured by a linear relationship.
• Initial Exploration: Useful for quick initial analysis or as a baseline model before exploring more complex techniques.
• Interpretability: When understanding the impact of predictors on the outcome is crucial.

Conclusion

In conclusion, linear regression offers a straightforward approach to modeling time series data, especially when trends are linear and interpretability is important. However, its effectiveness can be limited by the assumptions of linearity and the complexity of the underlying patterns. It serves well in scenarios where simplicity and initial insights are prioritized, but more advanced methods may be necessary to handle nonlinearities and complex seasonal variations effectively.