## What is an Autoregressive model?

An autoregressive (AR) model is a type of statistical model that uses past values of a time series to predict future values. It is based on the assumption that the current value of the time series depends on its past values, with the relationship between the current and past values described by a set of coefficients.

**To forecast the outcome of the next time step, an autoregressive model uses the results of prior time steps as inputs into a regression model.**

An autoregressive model can be written as:

**y(t) = c + w_1y(t-1) + w_2y(t-2) + … + w_py(t-p) + e(t)**

where:

- y(t) is the current value of the time series;
- y(t-1) is the value of the time series at the previous time step;
- y(t-2) is the value of the time series two-time steps ago, and so on;
- The coefficients w_1, w_2, …, w_p are the autoregressive coefficients that determine the strength and direction of the relationship between the current value of the time series and its past values; and
- The constant c is a bias term, and e(t) is a random error term.

## Autoregressive language model

An autoregressive language model is a type of Machine Learning model that uses autoregressive techniques to predict the next word in a sequence of words based on the words that have come before it. This can be used for tasks such as natural language processing and machine translation.

In the programming language R, the arima() function can be used to fit an autoregressive model in R to a time series.

## VAR and CAR

A Vector Autoregressive Model (VAR) is a type of autoregressive model that is used to model the relationship between multiple time series.

A Conditional Autoregressive Model (CAR) is a type of autoregressive model that is used to model spatial data such as data collected from a grid of locations. It is based on the assumption that the value of a variable at a given location depends on the values of the variable at neighboring locations.

## Methods of an Autoregressive Model

To forecast the value of the next time step, autoregression models typically analyze the degree of correlation between data at earlier time steps (the lag variables).

With a positive correlation, the two variables move in the same direction, either up or down. It is said that there is a negative correlation between two variables if the values of these variables tend to fluctuate in opposing ways, such as one rising while the other falls. In any case, the relationship between the outcome and the input may be measured using some elementary statistical analysis.

It’s more probable that the past may foretell the future if this association is strong, whether positive or negative. Or, to put it another way, the greater significance of this value in the DL algorithm.

The term “autocorrelation” is used to describe the relationship between the variable and itself at earlier times.

In addition, the predictability of the time series dataset is diminished if all input variables have a weak or nonexistent relationship with the final output variable — which may be put to good use while teaching deep neural networks.