What Are Autoregressive Models? Types, Usage, and Benefits

Autoregressive (AR) models are statistical tools used by data scientists in time series forecasting. These models use past data to make predictions.

Many artificial neural networks use autoregressive models to foresee outcomes using their training data. Say for instance a sales leader wants to look ahead to next month’s sales, an AR model studies the previous month's sales data to make the prediction. 

The AR model was developed for forecasting and data augmentation purposes. They serve as a reliable source of creating training data for AI models in economics, finance, or weather forecasting sectors.

For example, these models deliver accurate predictions of temperature changes using historical patterns. They’re also valuable in economics, where they predict inflation rates and gross domestic product (GDP) growth to help policymakers enact effective plans. 

This technique is used in time-series analysis, which assumes that the current time series value is a function of its past values. These models use mathematical techniques to assess the probabilistic correlation between elements in a sequence. Based on the knowledge acquired, they guess the unknown element. 

Let’s take an example to understand this further. An autoregressive model that processes different English language statements identifies that the word “is” always follows “here.” When it generates a new sequence of words, it will automatically write “here is” together.  

Types of autoregressive models

Here, we have some notable types of autoregressive models for different purposes and data types. 

• Autoregressive (AR) models predict values based on past data.  

• Autoregressive integrated moving average (ARIMA) models assess the differences between values in the series instead of using actual values. The differencing of raw observations makes the time series stationary. The moving average part smoothens out short-term fluctuations by averaging past errors. 

• The vector autoregressive (VAR) model helps predict outcomes for all time series based on their past values and those of the other series. 

• The seasonal autoregressive integrated moving average (SARIMA) model is an extension of the non-seasonal ARIMA model, but it can handle seasonal patterns. SARIMA considers short-term and long-term dependencies within data.

Understanding autoregressive models in detail

Multiple regression models predict values by using the predictor's linear combination. However, autoregressive models use a combination of past values to make predictions. An AR (1) process predicts the current value based on the value that immediately precedes it. In the AR (2) process, the current value is based on two preceding values. The AR (0) process is used for white noise. It doesn’t depend on the terms in between.

Autoregressive models are based on past data and assume that the factors affecting the past data stay the same. However, if these factors change, the data can become incorrect.

Despite this, experts improve these models to better account for errors, seasonality, trends, and changing data for making predictions. One such advanced model is an autoregressive integrated moving average (ARIMA) model that makes accurate predictions.

Overall, their prediction accuracy helps businesses make better decisions and improve planning for the future.

How does autoregressive modeling work?

An autoregressive model uses linear regression to predict the following values. 

Consider the equation for a line (y=c*x+ m), where y is the dependent variable, and x is the independent variable; c and m are constants from all possible values of x and . 

For example, if you have the input dataset (x,y) as (1,4), (2,6), (3, 12). The values of constants c and m are 2 and 2. The equation becomes y = 2x+2. Plotting these coordinates on a straight line and extrapolating them will give a value of y as 10 when x equals 4. 

This is how linear regression works. Autoregression models apply linear regression with lagged variables from previous inputs. They don’t use independent variables like linear regression. The autoregressive model assumes earlier variables conditionally influence the next variable’s outcome.  

The expression below represents autoregressive modeling. 

Source: AWS

Y is the prediction outcomes of multiple previous values multiplied with their coefficients, ϕ (phi). The formula considers weights or parameters that influence predictor variables and also takes into account the random noise. It indicates that there’s scope for further improvements. 

Professionals that use autoregressive modeling either add more lagged values or increase the number of steps in time series to improve data accuracy. 

How generative AI uses autoregressive models

Autoregressive modeling plays a crucial role in helping generative AI understand user input. The generative pre-trained transformer (GPT) model’s decoder uses autoregressive language modeling to understand natural language and generate it in a way humans can understand. 

Autoregressive models also support deep learning models in image generation after analysis. Different image-processing neural networks like PixelRNN and PixelCNN predict visual data using autoregressive modeling. They also have applications with regard to the likelihood of time-series events. 

In some situations, machine learning (ML) engineers have a shortage of training datasets. If that’s the case, they turn to autoregressive modeling to generate new and realistic deep learning training that helps AI models improve performance. 

Benefits and limitations of autoregressive models

Several benefits of autoregressive models make it a suitable choice for data scientists. 

Benefits

You can use the autocorrelation function to tell if there’s a lack of randomness in the datasets. Moreover, using a self-variable series lets you predict possible outcomes even when information is lacking. 

The self-variable series uses the lagged values of dependent variables as the independent variables in the model. The autoregressive model is also capable of forecasting recurring patterns in data.

Limitations

The autoregressive model must have an autocorrelation coefficient of at least 0.5 to make an accurate prediction. The autocorrelation coefficient measures how correlated a time series is with itself over time. 

Different regressive techniques for analyzing variables

Besides autoregressive models, data scientists employ different regressive techniques to analyze variables and their interdependencies. 

• Linear regression uses several independent variables to predict outcomes within the same timeframe. Autoregression predicts future outcomes using one variable type and expanding it over several points.
• Polynomial regression captures relationships between non-linear variables that can’t be represented in a straight line. 
• Logistic regression predicts an event’s likelihood in probabilistic terms. The outcome is expressed in percentages rather than a range of numbers.
• Ridge regression is similar to linear regression, but it restricts the coefficient of a model. This technique helps when the algorithm is prone to overfitting. Overfitting occurs when a model becomes over-trained on a data set and returns inaccurate outcomes for real-world data. 
• Lasso regression restricts variable coefficients with a penalty factor. It lets data scientists simplify complex models by ignoring non-critical parameters. 

Making predictions easier

AR models are easy to implement and interpret since they focus only on linear relationships between current and past data. However, the model is based on the assumption that past data sufficiently captures all relevant information needed to predict future values, which largely influences its effectiveness. 

These models are a fundamental part of time series analysis. Their ease of use and ability to model temporal dependencies make them suitable for plenty of applications in the real world. 

Learn more about the different types of regression analysis you can use to interpret business data.